Which means that calculating the approximate maximum-winning probability for nonlocal games is undecidable, just like the halting problem.
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In technical terms, Turing proved that this halting problem is undecidable — even the most powerful computer imaginable couldn't solve it.
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Due to The Jackpot, some dark black undecidable Event way beyond Badiou, as of CE 2500, 80 percent of humanity will have been disappeared.
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Computational irreducibility implies that at least after an infinite time it's actually formally undecidable (in the sense of Gödel's Theorem or the Halting Problem) what can happen.
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The halting problem is an important undecidable decision problem; for more examples, see list of undecidable problems.
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Many, if not most all, undecidable problems in mathematics can be posed as word problems; see the list of undecidable problems for many examples.
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While the existence of undecidable statements had been known since Gödel's incompleteness theorem of 1931, previous examples of undecidable statements (such as the continuum hypothesis) had all been in pure set theory. The Whitehead problem was the first purely algebraic problem to be proved undecidable. later showed that the Whitehead problem remains undecidable even if one assumes the continuum hypothesis. The Whitehead conjecture is true if all sets are constructible.
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An undecidable problem is a problem that is not decidable.
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Thus G_w has property P if and only if w=_G 1. Since it is undecidable whether w=_G 1, it follows that it is undecidable whether a finitely presented group has property P.
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Peano arithmetic is also incomplete by Gödel's incompleteness theorem. In his 1953 Undecidable theories, Tarski et al. showed that many mathematical systems, including lattice theory, abstract projective geometry, and closure algebras, are all undecidable.
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Kaplansky's conjecture is thus an example of a statement undecidable in ZFC.
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The simplest example of an undecidable word problem occurs in combinatory logic: when are two strings of combinators equivalent? Because combinators encode all possible Turing machines, and the equivalence of two Turing machines is undecidable, it follows that the equivalence of two strings of combinators is undecidable. Likewise, one has essentially the same problem in (untyped) lambda calculus: given two distinct lambda expressions, there is no algorithm which can discern whether they are equivalent or not; equivalence is undecidable. For several typed variants of the lambda calculus, equivalence is decidable by comparison of normal forms.
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Limits can be difficult to compute. There exist limit expressions whose modulus of convergence is undecidable. In recursion theory, the limit lemma proves that it is possible to encode undecidable problems using limits.Recursively enumerable sets and degrees, Soare, Robert I.
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MISRA C:2012 classifies the rules (but not the directives) as Decidable or Undecidable.
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In computability theory, an undecidable problem is a type of computational problem that requires a yes/no answer, but where there cannot possibly be any computer program that always gives the correct answer; that is, any possible program would sometimes give the wrong answer or run forever without giving any answer. More formally, an undecidable problem is a problem whose language is not a recursive set; see the article Decidable language. There are uncountably many undecidable problems, so the list below is necessarily incomplete. Though undecidable languages are not recursive languages, they may be subsets of Turing recognizable languages: i.e.
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These are natural mathematical equivalents of the Gödel "true but undecidable" sentence. They can be proved in a larger system which is generally accepted as a valid form of reasoning, but are undecidable in a more limited system such as Peano Arithmetic. In 1977, Paris and Harrington proved that the Paris-Harrington principle, a version of the infinite Ramsey theorem, is undecidable in (first- order) Peano arithmetic, but can be proved in the stronger system of second- order arithmetic. Kirby and Paris later showed that Goodstein's theorem, a statement about sequences of natural numbers somewhat simpler than the Paris- Harrington principle, is also undecidable in Peano arithmetic.
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For example, the properties of being nontrivial, infinite, nonabelian, etc., for finitely presentable groups are undecidable. However, there do exist examples of interesting undecidable properties such that neither these properties nor their complements are Markov. Thus Collins (1969) Donald J. Collins, On recognizing Hopf groups, Archiv der Mathematik, vol.
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In 1973, Saharon Shelah showed the Whitehead problem in group theory is undecidable, in the first sense of the term, in standard set theory. In 1977, Paris and Harrington proved that the Paris- Harrington principle, a version of the Ramsey theorem, is undecidable in the axiomatization of arithmetic given by the Peano axioms but can be proven to be true in the larger system of second-order arithmetic. Kruskal's tree theorem, which has applications in computer science, is also undecidable from the Peano axioms but provable in set theory. In fact Kruskal's tree theorem (or its finite form) is undecidable in a much stronger system codifying the principles acceptable on basis of a philosophy of mathematics called predicativism.
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Gödel's theorems do not hold when any one of the seven axioms above is dropped. These fragments of Q remain undecidable, but they are no longer essentially undecidable: they have consistent decidable extensions, as well as uninteresting models (i.e., models which are not end-extensions of the standard natural numbers).
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We say that these structures are interpretable. A key fact is that one can translate sentences from the language of the interpreted structures to the language of the original structure. Thus one can show that if a structure M interprets another whose theory is undecidable, then M itself is undecidable.
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Grzegorczyk proved the elementary theory of closure algebras undecidable.Andrzej Grzegorczyk (1951), "Undecidability of some topological theories," Fundamenta Mathematicae 38: 137–52.According to footnote 19 in McKinsey and Tarski, 1944, the result had been proved earlier by S. Jaskowski in 1939, but remained unpublished and not accessible in view of the present [at the time] war conditions. Naturman demonstrated that the theory is hereditarily undecidable (all its subtheories are undecidable) and demonstrated an infinite chain of elementary classes of interior algebras with hereditarily undecidable theories.
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Like automatic groups, automatic semigroups have word problem solvable in quadratic time. Kambites & Otto (2006) showed that it is undecidable whether an element of an automatic monoid possesses a right inverse. Cain (2006) proved that both cancellativity and left-cancellativity are undecidable for automatic semigroups. On the other hand, right- cancellativity is decidable for automatic semigroups (Silva & Steinberg 2004).
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Beyond section Termination and convergence, additional subtleties are to be considered for term rewriting systems. Termination even of a system consisting of one rule with a linear left-hand side is undecidable. Termination is also undecidable for systems using only unary function symbols; however, it is decidable for finite ground systems. The following term rewrite system is normalizing,i.e.
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The halting problem is historically important because it was one of the first problems to be proved undecidable. (Turing's proof went to press in May 1936, whereas Alonzo Church's proof of the undecidability of a problem in the lambda calculus had already been published in April 1936 [Church, 1936].) Subsequently, many other undecidable problems have been described.
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A statement that is neither provable nor disprovable from a set of axioms is called undecidable (from those axioms). One example is the parallel postulate, which is neither provable nor refutable from the remaining axioms of Euclidean geometry. Mathematicians have shown there are many statements that are neither provable nor disprovable in Zermelo–Fraenkel set theory with the axiom of choice (ZFC), the standard system of set theory in mathematics (assuming that ZFC is consistent); see list of statements undecidable in ZFC. Gödel's (first) incompleteness theorem shows that many axiom systems of mathematical interest will have undecidable statements.
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Higman's embedding theorem also implies the Novikov-Boone theorem (originally proved in the 1950s by other methods) about the existence of a finitely presented group with algorithmically undecidable word problem. Indeed, it is fairly easy to construct a finitely generated recursively presented group with undecidable word problem. Then any finitely presented group that contains this group as a subgroup will have undecidable word problem as well. The usual proof of the theorem uses a sequence of HNN extensions starting with R and ending with a group G which can be shown to have a finite presentation.
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This question asks if a group is finite if the group has a definite number of generators and meets the criteria xn=1, for x in the group. Many word problems are undecidable based on the Post correspondence problem. Any two homomorphisms g,h with a common domain and a common codomain form an instance of the Post correspondence problem, which asks whether there exists a word w in the domain such that g(w)=h(w). Post proved that this problem is undecidable; consequently, any word problem that can be reduced to this basic problem is likewise undecidable.
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The connection between these two is that if a decision problem is undecidable (in the recursion theoretical sense) then there is no consistent, effective formal system which proves for every question A in the problem either "the answer to A is yes" or "the answer to A is no". Because of the two meanings of the word undecidable, the term independent is sometimes used instead of undecidable for the "neither provable nor refutable" sense. The usage of "independent" is also ambiguous, however. It can mean just "not provable", leaving open whether an independent statement might be refuted.
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With this encoding of action tables as strings, it becomes possible, in principle, for Turing machines to answer questions about the behaviour of other Turing machines. Most of these questions, however, are undecidable, meaning that the function in question cannot be calculated mechanically. For instance, the problem of determining whether an arbitrary Turing machine will halt on a particular input, or on all inputs, known as the Halting problem, was shown to be, in general, undecidable in Turing's original paper. Rice's theorem shows that any non-trivial question about the output of a Turing machine is undecidable.
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The language consisting of all Turing machine descriptions paired with all possible input streams on which those Turing machines will eventually halt, is not recursive. The halting problem is therefore called non-computable or undecidable. An extension of the halting problem is called Rice's theorem, which states that it is undecidable (in general) whether a given language possesses any specific nontrivial property.
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Kruskal's tree theorem, which has applications in computer science, is also undecidable from Peano arithmetic but provable in set theory. In fact Kruskal's tree theorem (or its finite form) is undecidable in a much stronger system codifying the principles acceptable based on a philosophy of mathematics called predicativism. The related but more general graph minor theorem (2003) has consequences for computational complexity theory.
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The unexpected existence of aperiodic tilings, although not Berger's explicit construction of them, follows from another result proved by Berger: that the so-called domino problem is undecidable, disproving a conjecture of Hao Wang, Berger's advisor. The result is analogous to a 1962 construction used by Kahr, Moore, and Wang, to show that a more constrained version of the domino problem was undecidable.
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148) Finally, in only 64 words and symbols Turing proves by reductio ad absurdum that "the Hilbert Entscheidungsproblem can have no solution" (Undecidable, p. 145).
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Hayward's "undecidable monument" In December 1968 he wrote "Blivets: Research and Development" to The Worm Runner's Digest in which he presented interpretations of impossible objects.
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One problem considered in the study of combinatorics on words in group theory is the following: for two elements x,y of a semigroup, does x=y modulo the defining relations of x and y. Post and Markov studied this problem and determined it undecidable. Undecidable means the theory cannot be proved. The Burnside question was proved using the existence of an infinite cube-free word.
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Subsequent authors have greatly extended Dehn's algorithm and applied it to a wide range of group theoretic decision problems. It was shown by Pyotr Novikov in 1955 that there exists a finitely presented group G such that the word problem for G is undecidable. It follows immediately that the uniform word problem is also undecidable. A different proof was obtained by William Boone in 1958.
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There are two distinct senses of the word "undecidable" in mathematics and computer science. The first of these is the proof-theoretic sense used in relation to Gödel's theorems, that of a statement being neither provable nor refutable in a specified deductive system. The second sense, which will not be discussed here, is used in relation to computability theory and applies not to statements but to decision problems, which are countably infinite sets of questions each requiring a yes or no answer. Such a problem is said to be undecidable if there is no computable function that correctly answers every question in the problem set (see undecidable problem).
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Because of the two meanings of the word undecidable, the term independent is sometimes used instead of undecidable for the "neither provable nor refutable" sense. Undecidability of a statement in a particular deductive system does not, in and of itself, address the question of whether the truth value of the statement is well-defined, or whether it can be determined by other means. Undecidability only implies that the particular deductive system being considered does not prove the truth or falsity of the statement. Whether there exist so-called "absolutely undecidable" statements, whose truth value can never be known or is ill-specified, is a controversial point in the philosophy of mathematics.
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The emptiness problem, the universality problem and the containability problem for OCATA is decidable but is a nonelementary problem. Those three problems are undecidable over ATAs.
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20, 1969, pp. 235–240. proved that the property of being Hopfian is undecidable for finitely presentable groups, while neither being Hopfian nor being non- Hopfian are Markov.
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A Source Book in Mathematical Logic, 1879-1931. Harvard Univ. Press: 228-51. Adding a single binary relation symbol to monadic logic, however, results in an undecidable logic.
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The emptiness problem is undecidable for context-sensitive grammars, a fact that follows from the undecidability of the halting problem. It is, however, decidable for context- free grammars.
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Third proof: "Corresponding to each computing machine M we construct a formula Un(M) and we show that, if there is a general method for determining whether Un(M) is provable, then there is a general method for determining whether M ever prints 0" (Undecidable, p. 145). The third proof requires the use of formal logic to prove a first lemma, followed by a brief word-proof of the second: :"Lemma 1: If S1 [symbol "0"] appears on the tape in some complete configuration of M, then Un(M) is provable" (Undecidable, p. 147) :"Lemma 2: [The converse] If Un(M) is provable then S1 [symbol "0"] appears on the tape in some complete configuration of M" (Undecidable, p.
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Undecidable, p. 300) Like Turing he defined erasure as printing a symbol "S0". And so his model admitted quadruplets of only three types (cf. Undecidable, p. 294): : qi Sj L ql, : qi Sj R ql, : qi Sj Sk ql At this time he was still retaining the Turing state- machine convention – he had not formalized the notion of an assumed sequential execution of steps until a specific test of a symbol "branched" the execution elsewhere.
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The structural termination problem consists in deciding, given a channel system S if the termination problem holds for S for every initial configuration. This problem is undecidable even over counter machine.
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He further argues that the future of deconstruction faces a perhaps undecidable choice between a theological approach and a technological approach, represented first of all by the work of Bernard Stiegler.
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In 2000, Schmidhuber explicitly constructed limit-computable, deterministic universes whose pseudo- randomness based on undecidable, Gödel-like halting problems is extremely hard to detect but does not at all prevent formal TOEs describable by very few bits of information. Related critique was offered by Solomon Feferman, among others. Douglas S. Robertson offers Conway's game of life as an example: The underlying rules are simple and complete, but there are formally undecidable questions about the game's behaviors. Analogously, it may (or may not) be possible to completely state the underlying rules of physics with a finite number of well-defined laws, but there is little doubt that there are questions about the behavior of physical systems which are formally undecidable on the basis of those underlying laws.
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The following example shows how to use reduction from the halting problem to prove that a language is undecidable. Suppose H(M, w) is the problem of determining whether a given Turing machine M halts (by accepting or rejecting) on input string w. This language is known to be undecidable. Suppose E(M) is the problem of determining whether the language a given Turing machine M accepts is empty (in other words, whether M accepts any strings at all).
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The 1-halting problem is the problem of deciding of any algorithm whether it defines a function with this property, i.e., whether the algorithm halts on input 1. By Rice's theorem, the 1-halting problem is undecidable. Similarly the question of whether a Turing machine T terminates on an initially empty tape (rather than with an initial word w given as second argument in addition to a description of T, as in the full halting problem) is still undecidable.
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56--70, 1995, Elsevier or they may be incompatible, in which case either the applications can sometimes be executed sequentially, or one can even preclude the other. It can be used as a language for software design and programming (usually a variant working on richer structures than graphs is chosen). Termination for DPO graph rewriting is undecidable because the Post correspondence problem can be reduced to it., "Termination of graph rewriting is undecidable", Detlef Plump, Fundamenta Informaticae, vol.
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It is modeled by an infinite ray, but violates Euler's handshaking lemma for finite graphs. However, it follows from the negative solution to the Entscheidungsproblem (by Alonzo Church and Alan Turing in the 1930s) that satisfiability of first-order sentences for graphs that are not constrained to be finite remains undecidable. It is also undecidable to distinguish between the first-order sentences that are true for all graphs and the ones that are true of finite graphs but false for some infinite graphs.
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It is also easy to see that the halting problem is not in NP since all problems in NP are decidable in a finite number of operations, but the halting problem, in general, is undecidable. There are also NP-hard problems that are neither NP-complete nor Undecidable. For instance, the language of true quantified Boolean formulas is decidable in polynomial space, but not in non- deterministic polynomial time (unless NP = PSPACE).More precisely, this language is PSPACE-complete; see, for example, .
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A property that is undecidable already for context-free languages or finite intersections of them, must be undecidable also for conjunctive grammars; these include: emptiness, finiteness, regularity, context-freeness,Given a conjunctive grammar, is its generated language empty / finite / regular / context-free? inclusion and equivalence.Given two conjunctive grammars, is the first's generated language a subset of / equal to the second's? The family of conjunctive languages is closed under union, intersection, concatenation and Kleene star, but not under string homomorphism, prefix, suffix, and substring.
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Later work has shown that the question of solvability of a Diophantine equation is undecidable even if the equation only has 9 natural number variables (Matiyasevich, 1977) or 11 integer variables (Zhi Wei Sun, 1992).
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Turing's proof is complicated by a large number of definitions, and confounded with what Martin Davis called "petty technical details" and "...technical details [that] are incorrect as given" (Davis's commentary in Undecidable, p. 115). Turing himself published "A correction" in 1937: "The author is indebted to P. Bernays for pointing out these errors" (Undecidable, p. 152). Specifically, in its original form the third proof is badly marred by technical errors. And even after Bernays' suggestions and Turing's corrections, errors remained in the description of the universal machine.
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Any such foundation would have to include axioms powerful enough to describe the arithmetic of the natural numbers (a subset of all mathematics). Yet Gödel proved that, for any consistent recursively enumerable axiomatic system powerful enough to describe the arithmetic of the natural numbers, there are (model-theoretically) true propositions about the natural numbers that cannot be proved from the axioms. Such propositions are known as formally undecidable propositions. For example, the continuum hypothesis is undecidable in the Zermelo-Fraenkel set theory as shown by Cohen.
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The Post correspondence problem is an undecidable decision problem that was introduced by Emil Post in 1946. Because it is simpler than the halting problem and the Entscheidungsproblem it is often used in proofs of undecidability.
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Given a CFG, is it ambiguous? The undecidability of this problem follows from the fact that if an algorithm to determine ambiguity existed, the Post correspondence problem could be decided, which is known to be undecidable.
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In model checking, the Metric Interval Temporal Logic (MITL) is a fragment of Metric Temporal Logic (MTL). This fragment is often preferred to MTL because some problems that are undecidable for MTL become decidable for MITL.
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Verifying sequential consistency through model checking is undecidable in general, even for finite-state cache coherence protocols. Consistency models define rules for the apparent order and visibility of updates, and are on a continuum with tradeoffs.
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The following properties of finitely presented groups are Markov and therefore are algorithmically undecidable by the Adian–Rabin theorem: #Being the trivial group. #Being a finite group. #Being an abelian group. #Being a finitely generated free group.
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There are two distinct senses of the word "undecidable" in contemporary use. The first of these is the sense used in relation to Gödel's theorems, that of a statement being neither provable nor refutable in a specified deductive system. The second sense is used in relation to computability theory and applies not to statements but to decision problems, which are countably infinite sets of questions each requiring a yes or no answer. Such a problem is said to be undecidable if there is no computable function that correctly answers every question in the problem set.
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Goodstein's theorem is a statement about the Ramsey theory of the natural numbers that Kirby and Paris showed is undecidable in Peano arithmetic. Gregory Chaitin produced undecidable statements in algorithmic information theory and proved another incompleteness theorem in that setting. Chaitin's theorem states that for any theory that can represent enough arithmetic, there is an upper bound c such that no specific number can be proven in that theory to have Kolmogorov complexity greater than c. While Gödel's theorem is related to the liar paradox, Chaitin's result is related to Berry's paradox.
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Anatoly Maltsev also made important contributions to group theory during this time; his early work was in logic in the 1930s, but in the 1940s he proved important embedding properties of semigroups into groups, studied the isomorphism problem of group rings, established the Malçev correspondence for polycyclic groups, and in the 1960s return to logic proving various theories within the study of groups to be undecidable. Earlier, Alfred Tarski proved elementary group theory undecidable.Tarski, Alfred (1953) "Undecidability of the elementary theory of groups" in Tarski, Mostowski, and Raphael Robinson Undecidable Theories. North-Holland: 77-87.
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Problems which are undecidable using classical computers remain undecidable using quantum computers. What makes quantum algorithms interesting is that they might be able to solve some problems faster than classical algorithms because the quantum superposition and quantum entanglement that quantum algorithms exploit probably can't be efficiently simulated on classical computers (see Quantum supremacy). The best-known algorithms are Shor's algorithm for factoring and Grover's algorithm for searching an unstructured database or an unordered list. Shor's algorithms runs much (almost exponentially) faster than the best-known classical algorithm for factoring, the general number field sieve.
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In 1973, Saharon Shelah showed that the Whitehead problem in group theory is undecidable, in the first sense of the term, in standard set theory. Gregory Chaitin produced undecidable statements in algorithmic information theory and proved another incompleteness theorem in that setting. Chaitin's incompleteness theorem states that for any system that can represent enough arithmetic, there is an upper bound c such that no specific number can be proved in that system to have Kolmogorov complexity greater than c. While Gödel's theorem is related to the liar paradox, Chaitin's result is related to Berry's paradox.
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The incompleteness theorem is closely related to several results about undecidable sets in recursion theory. Stephen Cole Kleene (1943) presented a proof of Gödel's incompleteness theorem using basic results of computability theory. One such result shows that the halting problem is undecidable: there is no computer program that can correctly determine, given any program P as input, whether P eventually halts when run with a particular given input. Kleene showed that the existence of a complete effective system of arithmetic with certain consistency properties would force the halting problem to be decidable, a contradiction.
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An algorithm known as the chase takes as input an instance that may or may not satisfy a set of EDs, and, if it terminates (which is a priori undecidable), output an instance that does satisfy the EDs.
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For most typed calculi, the type inhabitation problem is very hard. Richard Statman proved that for simply typed lambda calculus the type inhabitation problem is PSPACE-complete. For other calculi, like System F, the problem is even undecidable.
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We have previously shown, however, that the halting problem is undecidable. We have a contradiction, and we have thus shown that our assumption that M exists is incorrect. The complement of the halting language is therefore not recursively enumerable.
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Determining whether an arbitrary Turing machine is a busy beaver is undecidable. This has implications in computability theory, the halting problem, and complexity theory. The concept was first introduced by Tibor Radó in his 1962 paper, "On Non-Computable Functions".
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The concept borrows directly from and is an example of the broader notion of running a program on itself as input, used also in various proofs in theoretical computer science, such as the proof that the halting problem is undecidable.
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It is known to be undecidable when 9 pairs are used (however, Stephen Wolfram (2002) suggested that it is also undecidable with just 3 pairs). The undecidability of his Post correspondence problem turned out to be exactly what was needed to obtain undecidability results in the theory of formal languages. In an influential address to the American Mathematical Society in 1944, he raised the question of the existence of an uncomputable recursively enumerable set whose Turing degree is less than that of the halting problem. This question, which became known as Post's problem, stimulated much research.
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Géraud Sénizergues (1997) proved that the equivalence problem for deterministic PDA (i.e. given two deterministic PDA A and B, is L(A)=L(B)?) is decidable, -- Full version: a proof that earned him the 2002 Gödel Prize. For nondeterministic PDA, equivalence is undecidable.
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An algorithm known as the chase takes as input an instance that may or may not satisfy a set of TGDs (or more generally EDs), and, if it terminates (which is a priori undecidable), outputs an instance that does satisfy the TGDs.
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The problem remains undecidable even if the language is produced by a "linear" context-free grammar (i.e., with at most one nonterminal in each rule's right-hand side, cf. Exercise 4.20, p. 105). nor whether it is an LL(k) language for a given k.
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A fundamental distinction is extensional vs intensional type theory. In extensional type theory definitional (i.e., computational) equality is not distinguished from propositional equality, which requires proof. As a consequence type checking becomes undecidable in extensional type theory because programs in the theory might not terminate.
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In certain cases algorithms or other methods exist for proving that a given expression is non-zero, or of showing that the problem is undecidable. For example, if x1, ..., xn are real numbers, then there is an algorithm for deciding whether there are integers a1, ..., an such that : a_1 x_1 + \cdots + a_n x_n = 0\,. If the expression we are interested in contains an oscillating function, such as the sine or cosine function, then it has been shown that the problem is undecidable, a result known as Richardson's theorem. In general, methods specific to the expression being studied are required to prove that it cannot be zero.
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For instance, in this model it is undecidable to determine whether a given point belongs to the Mandelbrot set. She published a book on the subject, and in 1990 she gave an address at the International Congress of Mathematicians on computational complexity theory and real computation.
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"Difficult", in this sense, is described in terms of the computational resources needed by the most efficient algorithm for a certain problem. The field of recursion theory, meanwhile, categorizes undecidable decision problems by Turing degree, which is a measure of the noncomputability inherent in any solution.
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The problem of finding truth value assignments to make a conjunction of propositional Horn clauses true is a P-complete problem, solvable in linear time, and sometimes called HORNSAT. (The unrestricted Boolean satisfiability problem is an NP-complete problem however.) Satisfiability of first-order Horn clauses is undecidable.
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Gödel in Undecidable, p. 9 See more below about Gödel's proof. Alan Turing constructed this paradox with a machine and proved that this machine could not answer a simple question: will this machine be able to determine if any machine (including itself) will become trapped in an unproductive ‘infinite loop’ (i.e.
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The recurrent state problem consists in deciding, given a channel system S and an initial configuration \gamma and a state s whether there exists a run of S, starting at \gamma, going infinitely often through state s. This problem is undecidable over lossy channel system, even with a single channel.
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Two examples of the latter can be found in Hilbert's problems. Work on Hilbert's 10th problem led in the late twentieth century to the construction of specific Diophantine equations for which it is undecidable whether they have a solution,M. Davis. "Hilbert's Tenth Problem is Unsolvable." American Mathematical Monthly 80, pp.
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Conjunctive queries are one of the great success stories of database theory in that many interesting problems that are computationally hard or undecidable for larger classes of queries are feasible for conjunctive queries.Serge Abiteboul, Richard B. Hull, Victor Vianu: Foundations of Databases. Addison-Wesley, 1995. For example, consider the query containment problem.
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The eventuality property, or inevitability property problem consists in deciding, given a channel system S and a set \Gamma of configurations whether all run of S starting at \gamma goes through a configuration of \Gamma. This problem is undecidable for lossy channel system with impartiality and with the two other fairness constraints.
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So-called Oracle machines have access to various "oracles" which provide the solution to specific undecidable problems. For example, the Turing machine may have a "halting oracle" which answers immediately whether a given Turing machine will ever halt on a given input. These machines are a central topic of study in recursion theory.
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It was shown by that it is an undecidable problem to determine, given a finite presentation of a group, whether the group is Hopfian. Unlike the undecidability of many properties of groups this is not a consequence of the Adian–Rabin theorem, because Hopficity is not a Markov property, as was shown by .
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In mathematics, Richardson's theorem establishes a limit on the extent to which an algorithm can decide whether certain mathematical expressions are equal. It states that for a certain fairly natural class of expressions, it is undecidable whether a particular expression E satisfies the equation E = 0, and similarly undecidable whether the functions defined by expressions E and F are everywhere equal (in fact, E = F if and only if E − F = 0). It was proved in 1968 by computer scientist Daniel Richardson of the University of Bath. Specifically, the class of expressions for which the theorem holds is that generated by rational numbers, the number π, the number ln 2, the variable x, the operations of addition, subtraction, multiplication, composition, and the sin, exp, and abs functions.
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The above applies to first order theories, such as Peano arithmetic. However, for a specific model that may be described by a first order theory, some statements may be true but undecidable in the theory used to describe the model. For example, by Gödel's incompleteness theorem, we know that any theory whose proper axioms are true for the natural numbers cannot prove all first order statements true for the natural numbers, even if the list of proper axioms is allowed to be infinite enumerable. It follows that an automated theorem prover will fail to terminate while searching for a proof precisely when the statement being investigated is undecidable in the theory being used, even if it is true in the model of interest.
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In abstract algebra, the group isomorphism problem is the decision problem of determining whether two given finite group presentations present isomorphic groups. The isomorphism problem was identified by Max Dehn in 1911 as one of three fundamental decision problems in group theory; the other two being the word problem and the conjugacy problem. All three problems are undecidable: there does not exist a computer algorithm that correctly solves every instance of the isomorphism problem, or of the other two problems, regardless of how much time is allowed for the algorithm to run. In fact the problem of deciding whether a group is trivial is undecidable, a consequence of the Adian-Rabin theorem due to Sergei Adian and Michael O. Rabin.
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In the mathematical subject of group theory, the Adian–Rabin theorem is a result which states that most "reasonable" properties of finitely presentable groups are algorithmically undecidable. The theorem is due to Sergei Adian (1955)S. I. Adian, Algorithmic unsolvability of problems of recognition of certain properties of groups. Doklady Akademii Nauk SSSR vol.
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An algorithm known as the chase takes as input an instance that may or may not satisfy a set of EGDs (or more generally a set of EDs), and, if it terminates (which is a priori undecidable), output an instance that does satisfy the EGDs. An important subclass of equality-generating dependencies are functional dependencies.
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A decision problem A is decidable or effectively solvable if A is a recursive set. A problem is partially decidable, semidecidable, solvable, or provable if A is a recursively enumerable set. Problems that are not decidable are undecidable. For those it is not possible to create an algorithm, efficient or otherwise, that solves them.
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Hilbert's problem is not concerned with finding the solutions. It only asks whether, in general, we can decide whether one or more solutions exist. The answer to this question is negative, in the sense that no "process can be devised" for answering that question. In modern terms, Hilbert's 10th problem is an undecidable problem.
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All the systems mentioned so far, with the exception of the untyped lambda calculus, are strongly normalizing: all computations terminate. Therefore, they cannot describe all Turing-computable functions.since the halting problem for the latter class was proven to be undecidable As another consequence they are consistent as a logic, i.e. there are uninhabited types.
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The reachability problem consists in deciding, given a channel system S and two initial configurations \gamma and \gamma' whether there is a run of S from \gamma to \gamma'. This problem is undecidable over perfect channel systems and decidable but nonprimitive recursive over lossy channel system. This problem is decidable over machine capable of insertion of errors .
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In quantum many-body systems, ground states of gapped Hamiltonians have exponential decay of correlations. In 2015 it was shown that the problem of determining the existence of a spectral gap is undecidable. The authors used an aperiodic tiling of quantum Turing machines and showed that this hypothetical material becomes gapped if and only if it halts.
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A real number is called computable if there exists an algorithm that yields its digits. Because there are only countably many algorithms, but an uncountable number of reals, almost all real numbers fail to be computable. Moreover, the equality of two computable numbers is an undecidable problem. Some constructivists accept the existence of only those reals that are computable.
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In other words, quantum computers provide no additional power over classical computers in terms of computability. This means that quantum computers cannot solve undecidable problems like the halting problem and the existence of quantum computers does not disprove the Church–Turing thesis.Nielsen, p. 126 As of yet, quantum computers do not satisfy the strong Church thesis.
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The mathematical statements discussed below are independent of ZFC (the canonical axiomatic set theory of contemporary mathematics, consisting of the Zermelo–Fraenkel axioms plus the axiom of choice), assuming that ZFC is consistent. A statement is independent of ZFC (sometimes phrased "undecidable in ZFC") if it can neither be proven nor disproven from the axioms of ZFC.
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The class of real closed rings is first-order axiomatizable and undecidable. The class of all real closed valuation rings is decidable (by Cherlin-Dickmann) and the class of all real closed fields is decidable (by Tarski). After naming a definable radical relation, real closed rings have a model companion, namely von Neumann regular real closed rings.
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Turing's proof is a proof by Alan Turing, first published in January 1937 with the title "On Computable Numbers, with an Application to the Entscheidungsproblem." It was the second proof (after Church's theorem) of the conjecture that some purely mathematical yes-no questions can never be answered by computation; more technically, that some decision problems are "undecidable" in the sense that there is no single algorithm that infallibly gives a correct "yes" or "no" answer to each instance of the problem. In Turing's own words: "...what I shall prove is quite different from the well- known results of Gödel ... I shall now show that there is no general method which tells whether a given formula U is provable in K [Principia Mathematica]..." (Undecidable, p. 145). Turing followed this proof with two others.
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Undecidability of a statement in a particular deductive system does not, in and of itself, address the question of whether the truth value of the statement is well- defined, or whether it can be determined by other means. Undecidability only implies that the particular deductive system being considered does not prove the truth or falsity of the statement. Whether there exist so-called "absolutely undecidable" statements, whose truth value can never be known or is ill-specified, is a controversial point among various philosophical schools. One of the first problems suspected to be undecidable, in the second sense of the term, was the word problem for groups, first posed by Max Dehn in 1911, which asks if there is a finitely presented group for which no algorithm exists to determine whether two words are equivalent.
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The halting problem for Turing-complete computational models states that the decision problem of whether a program will halt on a particular input, or on all inputs, is undecidable. Therefore, a general algorithm for proving any program to halt does not exist. Size- change termination is decidable because it only asks whether termination is provable from given size-change graphs.
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In some OWL flavors like OWL1-DL, entities can be either classes or instances, but cannot be both. This limitations forbids metaclasses and metamodeling. This is not the case in the OWL1 full flavor, but this allows the model to be computationally undecidable. In OWL2, metaclasses can implemented with punning, that is a way to treat classes as if they were individuals.
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And confusingly, since Turing was unable to correct his original paper, some text within the body harks to Turing's flawed first effort. Bernays' corrections may be found in Undecidable, pp. 152–154; the original is to be found as: :"On Computable Numbers, with an Application to the Entscheidungsproblem. A Correction," Proceedings of the London Mathematical Society (2), 43 (1936–37), 544-546.
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Languages generated by context-free grammars are known as context-free languages (CFL). Different context-free grammars can generate the same context-free language. It is important to distinguish the properties of the language (intrinsic properties) from the properties of a particular grammar (extrinsic properties). The language equality question (do two given context-free grammars generate the same language?) is undecidable.
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It is decidable whether a given grammar is a regular grammar,This is easy to see from the grammar definitions. as well as whether it is an LL(k) grammar for a given k≥0. If k is not given, the latter problem is undecidable. Given a context-free language, it is neither decidable whether it is regular,, Exercise 8.10a, p. 214\.
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Satisfiability is undecidable and indeed it isn't even a semidecidable property of formulae in first-order logic (FOL). This fact has to do with the undecidability of the validity problem for FOL. The question of the status of the validity problem was posed firstly by David Hilbert, as the so-called Entscheidungsproblem. The universal validity of a formula is a semi-decidable problem.
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In computability theory and computational complexity theory, an undecidable problem is a decision problem for which it is proved to be impossible to construct an algorithm that always leads to a correct yes-or-no answer. The halting problem is an example: it can be proven that there is no algorithm that correctly determines whether arbitrary programs eventually halt when run.
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Formal languages allow formalizing the concept of well-formed expressions. In the 1930s, a new type of expressions, called lambda expressions, were introduced by Alonzo Church and Stephen Kleene for formalizing functions and their evaluation. They form the basis for lambda calculus, a formal system used in mathematical logic and the theory of programming languages. The equivalence of two lambda expressions is undecidable.
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J. Katoen and P. Stevens, Eds. Lecture Notes In Computer Science, vol. 2280. Springer-Verlag, London, 357-370. He has also contributed to research on Message Sequence Charts(MSC), where it was shown that weak realizability is undecidable for bounded MSC- graphs and that safe-realizability is in EXPSPACE, along with other interesting results related to the verification of MSC-graphs.
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246 in The Undecidable, plus footnote 13 with regards to the need for an additional operator, boldface added). But the need for the mu-operator is a rarity. As indicated above by Kleene's list of common calculations, a person goes about their life happily computing primitive recursive functions without fear of encountering the monster numbers created by Ackermann's function (e.g. super- exponentiation ).
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Description numbers play a key role in many undecidability proofs, such as the proof that the halting problem is undecidable. In the first place, the existence of this direct correspondence between natural numbers and Turing machines shows that the set of all Turing machines is denumerable, and since the set of all partial functions is uncountably infinite, there must certainly be many functions that cannot be computed by Turing machines. By making use of a technique similar to Cantor's diagonal argument, it is possible exhibit such an uncomputable function, for example, that the halting problem in particular is undecidable. First, let us denote by U(e, x) the action of the universal Turing machine given a description number e and input x, returning 0 if e is not the description number of a valid Turing machine.
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In 1930 Gödel attended the Second Conference on the Epistemology of the Exact Sciences, held in Königsberg, 5–7 September. Here he delivered his incompleteness theorems. Gödel published his incompleteness theorems in (called in English "On Formally Undecidable Propositions of and Related Systems"). In that article, he proved for any computable axiomatic system that is powerful enough to describe the arithmetic of the natural numbers (e.g.
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Language equations with concatenation and Boolean operations were first studied by Parikh, Chandra, Halpern and Meyer who proved that the satisfiability problem for a given equation is undecidable, and that if a system of language equations has a unique solution, then that solution is recursive. Later, Okhotin proved that the unsatisfiability problem is RE-complete and that every recursive language is a unique solution of some equation.
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Most problem related to perfect channel system are undecidable. This is due to the fact that such a machine may simulates the run of a Turing machine. This simulation is now sketched. Given a Turing machine M, there exists a perfect channel system S such that any run of M of length n can be simulated by a run of S of length O(n^2).
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In computer science, polymorphic recursion (also referred to as Milner-Mycroft typability or the Milner-Mycroft calculus) refers to a recursive parametrically polymorphic function where the type parameter changes with each recursive invocation made, instead of staying constant. Type inference for polymorphic recursion is equivalent to semi-unification and therefore undecidable and requires the use of a semi-algorithm or programmer supplied type annotations.
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Examples include Paul Erdős and Kurt Gödel. Gödel believed in an objective mathematical reality that could be perceived in a manner analogous to sense perception. Certain principles (e.g., for any two objects, there is a collection of objects consisting of precisely those two objects) could be directly seen to be true, but the continuum hypothesis conjecture might prove undecidable just on the basis of such principles.
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Cayenne is a dependently typed functional programming language created by Lennart Augustsson in 1998,Augustsson, Lennart (1998). "Cayenne -- a language with dependent types". making it one of the earliest dependently typed programming languages (as opposed to proof assistants or logical frameworks). A notable design decision is that the language allows unbounded recursive functions to be used on the type level, making type checking undecidable.
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" > "Whoever wants to live well (eudaimonia) must consider these three > questions: First, how are pragmata (ethical matters, affairs, topics) by > nature? Secondly, what attitude should we adopt towards them? Thirdly, what > will be the outcome for those who have this attitude?" Pyrrho's answer is > that "As for pragmata they are all adiaphora (undifferentiated by a logical > differentia), astathmēta (unstable, unbalanced, not measurable), and > anepikrita (unjudged, unfixed, undecidable).
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In complexity theory and computability theory, an oracle machine is an abstract machine used to study decision problems. It can be visualized as a Turing machine with a black box, called an oracle, which is able to solve certain decision problems in a single operation. The problem can be of any complexity class. Even undecidable problems, such as the halting problem, can be used.
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" > "Whoever wants to live well (eudaimonia) must consider these three > questions: First, how are pragmata (ethical matters, affairs, topics) by > nature? Secondly, what attitude should we adopt towards them? Thirdly, what > will be the outcome for those who have this attitude?" Pyrrho's answer is > that "As for pragmata they are all adiaphora (undifferentiated by a logical > differentia), astathmēta (unstable, unbalanced, not measurable), and > anepikrita (unjudged, unfixed, undecidable).
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In 2000, he published, in the style of Dr. Seuss, a proof of Turing's theorem that the Halting Problem is recursively unsolvable.Pullum, Geoffrey K. (2000) "Scooping the loop snooper: An elementary proof of the undecidability of the halting problem". Mathematics Magazine 73.4 (October 2000), 319–320. A corrected version appears on the author's website as "Scooping the loop snooper: A proof that the Halting Problem is undecidable".
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Given a CFG, does it generate the language of all strings over the alphabet of terminal symbols used in its rules?, Theorem 5.10, p. 181. A reduction can be demonstrated to this problem from the well-known undecidable problem of determining whether a Turing machine accepts a particular input (the halting problem). The reduction uses the concept of a computation history, a string describing an entire computation of a Turing machine.
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It developed into a study of abstract computability, which became known as recursion theory.Many of the foundational papers are collected in The Undecidable (1965) edited by Martin Davis The priority method, discovered independently by Albert Muchnik and Richard Friedberg in the 1950s, led to major advances in the understanding of the degrees of unsolvability and related structures. Research into higher-order computability theory demonstrated its connections to set theory.
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William Werner Boone (16 January 1920 in Cincinnati – 14 September 1983 in Urbana, Illinois) was an American mathematician. Alonzo Church was his Ph.D. advisor at Princeton, and Kurt Gödel was his friend at the Institute for Advanced Study. Pyotr Novikov showed in 1955 that there exists a finitely presented group G such that the word problem for G is undecidable. A different proof was obtained by Boone in 1958.
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There are approaches to automatically proving properties of hybrid systems (e.g., some of the tools mentioned below). Common techniques for proving safety of hybrid systems are computation of reachable sets, abstraction refinement, and barrier certificates. Most verification tasks are undecidable,Thomas A. Henzinger, Peter W. Kopke, Anuj Puri, and Pravin Varaiya: What's Decidable about Hybrid Automata, Journal of Computer and System Sciences, 1998 making general verification algorithms impossible.
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The version of System F used in this article is as an explicitly typed, or Church-style, calculus. The typing information contained in λ-terms makes type-checking straightforward. Joe Wells (1994) settled an "embarrassing open problem" by proving that type checking is undecidable for a Curry-style variant of System F, that is, one that lacks explicit typing annotations. Wells's result implies that type inference for System F is impossible.
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Post devised a method of 'auxiliary symbols' by which he could canonically represent any Post-generative language, and indeed any computable function or set at all. Correspondence systems were introduced by Post in 1946 to give simple examples of undecidability. He showed that the Post Correspondence Problem (PCP) of satisfying their constraints is, in general, undecidable. With 2 string pairs, PCP was shown to be decidable in 1981.
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Semantic garbage is data that will not be accessed, either because it is unreachable (hence also syntactic garbage), or is reachable but will not be accessed; this latter requires analysis of the code, and is in general an undecidable problem. Syntactic garbage is a (usually strict) subset of semantic garbage, as it is entirely possible for an object to hold a reference to another object without ever using that object.
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An implementation of a Turing machine As Turing wrote in The Undecidable, p. 128 (italics added): This finding is now taken for granted, but at the time (1936) it was considered astonishing. The model of computation that Turing called his "universal machine"—"U" for short—is considered by some (cf. Davis (2000)) to have been the fundamental theoretical breakthrough that led to the notion of the stored-program computer.
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Since the 2005 Economic Journal article on Markets As Complex Adaptive Systems, Markose has underscored the relevance of Gödel logic for what has been held as the sine qua non of Complex Adaptive Systems, viz their capacity to produce novelty and surprises as in a Red Queen style arms race. Such innovation based structure changing arms races seen in the immune system, technology and regulator-regulatee arms races in economic systems are shown to correspond to undecidable Type 4 dynamicsSelf-referential basis of undecidable dynamics: from The Liar Paradox and The Halting Problem to The Edge of Chaos, Mikhail Prokopenko, Michael Harré, Joseph Lizier, Fabio Boschetti, Pavlos Peppas, Stuart Kauffman,2017 of the Wolfram-Chomsky schema. In Gödel logic, the Liar which represents a negation or a contrarian position is key to novelty production and heterogeneity. Markose has shown that in a Nash equilibrium, the only agent who needs to be ‘surprised’ is the Liar who will negate rules with predictable outcomes.
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What Turing called "the state formula" includes both the current instruction and all the symbols on the tape: Earlier in his paper Turing carried this even further: he gives an example where he placed a symbol of the current "m-configuration"—the instruction's label—beneath the scanned square, together with all the symbols on the tape (The Undecidable, p. 121); this he calls "the complete configuration" (The Undecidable, p. 118). To print the "complete configuration" on one line, he places the state-label/m-configuration to the left of the scanned symbol. A variant of this is seen in Kleene (1952) where Kleene shows how to write the Gödel number of a machine's "situation": he places the "m-configuration" symbol q4 over the scanned square in roughly the center of the 6 non-blank squares on the tape (see the Turing-tape figure in this article) and puts it to the right of the scanned square.
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Due to many forms of static analysis being computationally undecidable, the mechanisms for doing it will not always terminate with the right answer either because they sometimes return a false negative ("no problems found" when the code does in fact have problems) or a false positive, or because they never return the wrong answer but sometimes never terminate. Despite their limitations, the first type of mechanism might reduce the number of vulnerabilities, while the second can sometimes give strong assurance of the lack of a certain class of vulnerabilities. Incorrect optimizations are highly undesirable. So, in the context of program optimization, there are two main strategies to handle computationally undecidable analysis: # An optimizer that is expected to complete in a relatively short amount of time, such as the optimizer in an optimizing compiler, may use a truncated version of an analysis that is guaranteed to complete in a finite amount of time, and guaranteed to only find correct optimizations.
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As in the case of the next-bit test, the predicting collection used in the above definition can be replaced by a probabilistic Turing machine, working in polynomial time. This also yields a strictly stronger definition of Yao's test (see Adleman's theorem). Indeed, One could decide undecidable properties of the pseudo-random sequence with the non-uniform circuits described above, whereas BPP machines can always be simulated by exponential-time deterministic Turing machines.
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However, frequently it remains undecidable whether or not a particular equivalence holds (for the reason that the proofs can become very long). During and after World War II, Birkhoff's interests gravitated towards what he called "engineering" mathematics. During the war, he worked on radar aiming and ballistics, including the bazooka. In the development of weapons, mathematical questions arose, some of which had not yet been addressed by the literature on fluid dynamics.
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The problem of deciding the truth of a statement in Presburger arithmetic requires even more time. Fischer and Rabin proved in 1974 that every algorithm that decides the truth of Presburger statements of length n has a runtime of at least 2^{2^{cn}} for some constant c. Hence, the problem is known to need more than exponential run time. Even more difficult are the undecidable problems, such as the halting problem.
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While the P versus NP problem is generally considered unsolved, many amateur and some professional researchers have claimed solutions. Gerhard J. Woeginger maintains a list that, as of 2018, contains 62 purported proofs of P = NP, 50 proofs of P ≠ NP, 2 proofs the problem is unprovable, and one proof that it is undecidable. Some attempts at resolving P versus NP have received brief media attention, though these attempts have since been refuted.
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An example of an unsound technique is one that searches only a subset of the possibilities, for instance only integers up to a certain number, and give a "good-enough" result. Techniques can also be decidable, meaning that their algorithmic implementations are guaranteed to terminate with an answer, or undecidable, meaning that they may never terminate. Because they are bounded, unsound techniques are often more likely to be decidable than sound ones.
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Wang tiles can be generalized in various ways, all of which are also undecidable in the above sense. For example, Wang cubes are equal-sized cubes with colored faces and side colors can be matched on any polygonal tessellation. Culik and Kari have demonstrated aperiodic sets of Wang cubes.. Winfree et al. have demonstrated the feasibility of creating molecular "tiles" made from DNA (deoxyribonucleic acid) that can act as Wang tiles.. Mittal et al.
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The formal system described above is sometimes called the pure monadic predicate calculus, where "pure" signifies the absence of function letters. Allowing monadic function letters changes the logic only superficially, whereas admitting even a single binary function letter results in an undecidable logic. Monadic second-order logic allows predicates of higher arity in formulas, but restricts second-order quantification to unary predicates, i.e. the only second-order variables allowed are subset variables.
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The termination problem consists in deciding, given a channel system S and an initial configuration \gamma whether all runs of S starting at \gamma are finite. This problem is undecidable over perfect channel systems, even when the system is a counter machine or when it is a one-channel machine. This problem is decidable but nonprimitive recursive over lossy channel system. This problem is trivially decidable over machine capable of insertion of errors.
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A decision problem A is called decidable or effectively solvable if A is a recursive set. A problem is called partially decidable, semi-decidable, solvable, or provable if A is a recursively enumerable set. This means that there exists an algorithm that halts eventually when the answer is yes but may run for ever if the answer is no. Partially decidable problems and any other problems that are not decidable are called undecidable.
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In computability theory, the halting problem is a decision problem which can be stated as follows: :Given the description of an arbitrary program and a finite input, decide whether the program finishes running or will run forever. Alan Turing proved in 1936 that a general algorithm running on a Turing machine that solves the halting problem for all possible program-input pairs necessarily cannot exist. Hence, the halting problem is undecidable for Turing machines.
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Church's paper An Unsolvable Problem of Elementary Number Theory (1936) proved that the Entscheidungsproblem was undecidable within the λ-calculus and Gödel-Herbrand's general recursion; moreover Church cites two theorems of Kleene's that proved that the functions defined in the λ-calculus are identical to the functions defined by general recursion: :"Theorem XVI. Every recursive function of positive integers is λ-definable.16 :"Theorem XVII. Every λ-definable function of positive integers is recursive.
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The boundedness problem for Datalog asks, given a Datalog program, whether it is bounded, i.e., the maximal recursion depth reached when evaluating the program on an input database can be bounded by some constant. In other words, this question asks whether the Datalog program could be rewritten as a nonrecursive Datalog program. Solving the boundedness problem on arbitrary Datalog programs is undecidable, but it can be made decidable by restricting to some fragments of Datalog.
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An impossible object (also known as an impossible figure or an undecidable figure) is a type of optical illusion. It consists of a two-dimensional figure which is instantly and subconsciously interpreted by the visual system as representing a projection of a three-dimensional object. In most cases the impossibility becomes apparent after viewing the figure for a few seconds. However, the initial impression of a 3D object remains even after it has been contradicted.
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If R rejects N, then the language accepted by N is nonempty, so M does halt on input w, so S can accept. Thus, if we had a decider R for E, we would be able to produce a decider S for the halting problem H(M, w) for any machine M and input w. Since we know that such an S cannot exist, it follows that the language E is also undecidable.
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For some grammars, it can do this by peeking on the unread input (without reading). In our example, if the parser knows that the next unread symbol is ( , the only correct rule that can be used is 2. Generally, an LL(k) parser can look ahead at k symbols. However, given a grammar, the problem of determining if there exists a LL(k) parser for some k that recognizes it is undecidable.
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Fred Cohen experimented with computer viruses and confirmed Neumann's postulate and investigated other properties of malware such as detectability and self-obfuscation using rudimentary encryption. His 1988 Doctoral dissertation was on the subject of computer viruses.Fred Cohen, "Computer Viruses", PhD Thesis, University of Southern California, ASP Press, 1988. Cohen's faculty advisor, Leonard Adleman, presented a rigorous proof that, in the general case, algorithmic determination of the presence of a virus is undecidable.
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The full classification of n-manifolds for n greater than three is known to be impossible; it is at least as hard as the word problem in group theory, which is known to be algorithmically undecidable. In fact, there is no algorithm for deciding whether a given manifold is simply connected. There is, however, a classification of simply connected manifolds of dimension ≥ 5.Žubr A.V. (1988) Classification of simply-connected topological 6-manifolds.
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The computational complexity of some problems related to timed automata are now given. The emptiness problem for timed automaton can be solved by constructing a region automaton and checking whether it accepts the empty language. This problem is PSPACE-complete. The universality problem of non-deterministic timed automaton is undecidable, and more precisely Π. However, when the automaton contains a single clock, the property is decidable, however it is not primitive recursive.
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In mathematics, Robinson arithmetic is a finitely axiomatized fragment of first-order Peano arithmetic (PA), first set out by R. M. Robinson in 1950. It is usually denoted Q. Q is almost PA without the axiom schema of mathematical induction. Q is weaker than PA but it has the same language, and both theories are incomplete. Q is important and interesting because it is a finitely axiomatized fragment of PA that is recursively incompletable and essentially undecidable.
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Through various reasonings, he determines the pharmakon of writing to be a bad thing for the Egyptian people. The pharmakon, the undecidable, has been returned decided. The problem, as Derrida reasons, is this: since the word pharmakon, in the original Greek, means both a remedy and a poison, it cannot be determined as fully remedy or fully poison. Amon rejected writing as fully poison in Socrates' retelling of the tale, thus shutting out the other possibilities.
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Yuri Matijasevich, (1967) "Simple examples of undecidable associative calculi", Soviet Mathematics - Doklady 8(2) pp 555–557. The word problem on free Heyting algebras is difficult.Peter T. Johnstone, Stone Spaces, (1982) Cambridge University Press, Cambridge, . (See chapter 1, paragraph 4.11) The only known results are that the free Heyting algebra on one generator is infinite, and that the free complete Heyting algebra on one generator exists (and has one more element than the free Heyting algebra).
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Hamkins introduced with Jeff Kidder and Andy Lewis the theory of infinite-time Turing machines, a part of the subject of hypercomputation, with connections to descriptive set theory. In other computability work, Hamkins and Miasnikov proved that the classical halting problem for Turing machines, although undecidable, is nevertheless decidable on a set of asymptotic probability one, one of several results in generic-case complexity showing that a difficult or unsolvable problem can be easy on average.
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Gödel's paper was published in the Monatshefte in 1931 under the title "Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I" ("On Formally Undecidable Propositions in Principia Mathematica and Related Systems I"). As the title implies, Gödel originally planned to publish a second part of the paper in the next volume of the Monatshefte; the prompt acceptance of the first paper was one reason he changed his plans (van Heijenoort 1967:328, footnote 68a).
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With this model, Turing was able to answer two questions in the negative: (1) does a machine exist that can determine whether any arbitrary machine on its tape is "circular" (e.g., freezes, or fails to continue its computational task); similarly, (2) does a machine exist that can determine whether any arbitrary machine on its tape ever prints a given symbol.Turing 1936 in The Undecidable 1965:132-134; Turing's definition of "circular" is found on page 119.
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Once they had improved the compiler to the point where it could compile its own source code, it was self-hosting. This technique is only possible when an interpreter already exists for the very same language that is to be compiled. It borrows directly from the notion of running a program on itself as input, which is also used in various proofs in theoretical computer science, such as the proof that the halting problem is undecidable.
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T2 aims to find whether a program can run infinitely (called a termination analysis). It supports nested loops and recursive functions, pointers and side-effects, and function-pointers as well as concurrent programs. Like all programs for termination analysis it tries to solve the halting problem for particular cases, since the general problem is undecidable. It provides a solution which is sound, meaning that when it states that a program does always terminate, the result is dependable.
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The level of abstraction included in a programming language can influence its overall usability. The Cognitive dimensions framework includes the concept of abstraction gradient in a formalism. This framework allows the designer of a programming language to study the trade-offs between abstraction and other characteristics of the design, and how changes in abstraction influence the language usability. Abstractions can prove useful when dealing with computer programs, because non-trivial properties of computer programs are essentially undecidable (see Rice's theorem).
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Stephen M. Omohundro, "Modelling Cellular Automata with Partial Differential Equations", Physica D, 10D (1984) 128-134. The asymptotic behavior of these PDEs is therefore logically undecidable. With John David Crawford he showed that the orbits of three- dimensional period doubling systems can form an infinite number of topologically distinct torus knots and described the structure of their stable and unstable manifolds.John David Crawford and Stephen M. Omohundro, "On the Global Structure of Period Doubling Flows", Physica D, 12D (1984), pp. 161-180.
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In theoretical computer science and formal language theory, the equivalence problem is the question of determining, given two representations of formal languages, whether they denote the same formal language. The complexity and decidability of this decision problem depends upon the type of representation under consideration. For instance, in the case of finite-state automata, equivalence is decidable, and the problem is PSPACE-complete, whereas it is undecidable for pushdown automata, context-free grammars, etc.J. E. Hopcroft and J. D. Ullman.
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The exact static call graph is an undecidable problem, so static call graph algorithms are generally overapproximations. That is, every call relationship that occurs is represented in the graph, and possibly also some call relationships that would never occur in actual runs of the program. Call graphs can be defined to represent varying degrees of precision. A more precise call graph more precisely approximates the behavior of the real program, at the cost of taking longer to compute and more memory to store.
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In general it is undecidable whether a Java program using generics is well-typed or not, so any type checker will have to go into an infinite loop or time out for some programs. For the programmer, it leads to complicated type error messages. Java type checks wildcard types by replacing the wildcards with fresh type variables (so-called capture conversion). This can make error messages harder to read, because they refer to type variables that the programmer did not directly write.
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Using the universal type \omega allowed for a fine-grained analysis of head normalization, normalization, and strong normalization. In collaboration with Henk Barendregt, a filter λ-model for an intersection type system was given, tying intersection types ever more closely to λ-calculus semantics. Due to the correspondence with normalization, typability in prominent intersection type systems (excluding the universal type) is undecidable. Complementarily, undecidability of the dual problem of type inhabitation in prominent intersection type systems was proven by Paweł Urzyczyn.
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In logic, Richard's paradox is a semantical antinomy of set theory and natural language first described by the French mathematician Jules Richard in 1905. The paradox is ordinarily used to motivate the importance of distinguishing carefully between mathematics and metamathematics. Kurt Gödel specifically cites Richard's antinomy as a semantical analogue to his syntactical incompleteness result in the introductory section of "On Formally Undecidable Propositions in Principia Mathematica and Related Systems I". The paradox was also a motivation of the development of predicative mathematics.
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A type error is an unintended condition which might manifest in multiple stages of a program's development. Thus a facility for detection of the error is needed in the type system. In some languages, such as Haskell, for which type inference is automated, lint might be available to its compiler to aid in the detection of error. Type safety contributes to program correctness, but might only guarantee correctness at the cost of making the type checking itself an undecidable problem.
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Meister 2009, p. 163. William James in his essay "The Will to Believe" argues for a pragmatic conception of religious belief. For James, religious belief is justified if one is presented with a question which is rationally undecidable and if one is presented with genuine and live options which are relevant for the individual.Rowe 2007, pp 98 For James, religious belief is defensible because of the pragmatic value it can bring to one's life, even if there is no rational evidence for it.
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Menger conjectured that in ZFC every Menger metric space is σ-compact. Fremlin and Miller proved that Menger's conjecture is false, by showing that there is, in ZFC, a set of real numbers that is Menger but not σ-compact. The Fremlin- Miller proof was dichotomic, and the set witnessing the failure of the conjecture heavily depends on whether a certain (undecidable) axiom holds or not. Bartoszyński and Tsaban gave a uniform ZFC example of a Menger subset of the real line that is not σ-compact.
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The proof by Pythagoras (or more likely one of his students) about 500 BCE has had a profound effect on mathematics. It shows that the square root of 2 cannot be expressed as the ratio of two integers (counting numbers). The proof bifurcated "the numbers" into two non-overlapping collections—the rational numbers and the irrational numbers. This bifurcation was used by Cantor in his diagonal method, which in turn was used by Turing in his proof that the Entscheidungsproblem, the decision problem of Hilbert, is undecidable.
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In 1953 he became a member of the Communist Party. In 1960, Markov obtained fundamental results showing that the classification of four- dimensional manifolds is undecidable: no general algorithm exists for distinguishing two arbitrary manifolds with four or more dimensions. This is because four-dimensional manifolds have sufficient flexibility to allow us to embed any algorithm within their structure, so that classification of all four-manifolds would imply a solution to Turing's halting problem. This result has profound implications for the limitations of mathematical analysis.
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This definition makes the concept of proof amenable to study. Indeed, the field of proof theory studies formal proofs and their properties, the most famous and surprising being that almost all axiomatic systems can generate certain undecidable statements not provable within the system. The definition of a formal proof is intended to capture the concept of proofs as written in the practice of mathematics. The soundness of this definition amounts to the belief that a published proof can, in principle, be converted into a formal proof.
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Harrison, Ruzzo and Ullman discussed whether there is an algorithm that takes an arbitrary initial configuration and answers the following question: is there an arbitrary sequence of commands that adds a generic right into a cell of the access matrix where it has not been in the initial configuration? They showed that there is no such algorithm, thus the problem is undecidable in the general case. They also showed a limitation of the model to commands with only one primitive operation to render the problem decidable.
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The frame syntax of the Rule Interchange Format Basic Logic Dialect (RIF BLD) standardized by the World Wide Web Consortium is based on F-logic; RIF BLD however does not include non-monotonic reasoning features of F-logic. In contrast to description logic based ontology formalism the semantics of F-logic are normally that of a closed world assumption as opposed to DL's open world assumption. Also, F-logic is generally undecidable, whereas the SHOIN description logic that OWL DL is based on is decidable.
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The French philosopher Gilles Deleuze attempted to recoup the novelty of Bergson's idea in his book Bergsonism, though the term itself underwent substantial changes by Deleuze. No longer considered a mystical, elusive force acting on brute matter, as it was in the vitalist debates of the late 19th century, élan vital in Deleuze's hands denotes an internal force,K. Ansell-Pearson, Germinal Life (2012) p. 21 a substance in which the distinction between organic and inorganic matter is indiscernible, and the emergence of life undecidable.
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Derrida would say that the difference is "undecidable", in that it cannot be discerned in everyday experiences. Deconstruction perceives that language, especially ideal concepts such as truth and justice, is irreducibly complex, unstable, or impossible to determine. Many debates in continental philosophy surrounding ontology, epistemology, ethics, aesthetics, hermeneutics, and philosophy of language refer to Derrida's observations. Since the 1980s, these observations have inspired a range of theoretical enterprises in the humanities, including the disciplines of law, anthropology, historiography, linguistics, sociolinguistics, psychoanalysis, LGBT studies, and feminism.
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If the remainder is zero the answer is 'yes', otherwise it is 'no'. A decision problem which can be solved by an algorithm is called decidable. Decision problems typically appear in mathematical questions of decidability, that is, the question of the existence of an effective method to determine the existence of some object or its membership in a set; some of the most important problems in mathematics are undecidable. The field of computational complexity categorizes decidable decision problems by how difficult they are to solve.
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A computer with access to an infinite tape of data may be more powerful than a Turing machine: for instance, the tape might contain the solution to the halting problem or some other Turing-undecidable problem. Such an infinite tape of data is called a Turing oracle. Even a Turing oracle with random data is not computable (with probability 1), since there are only countably many computations but uncountably many oracles. So a computer with a random Turing oracle can compute things that a Turing machine cannot.
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A key part of the proof is a mathematical definition of a computer and program, which is known as a Turing machine; the halting problem is undecidable over Turing machines. It is one of the first cases of decision problems proven to be unsolvable. This proof is significant to practical computing efforts, defining a class of applications which no programming invention can possibly perform perfectly. Jack Copeland (2004) attributes the introduction of the term halting problem to the work of Martin Davis in the 1950s.
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Gödel used the completeness theorem to prove the compactness theorem, demonstrating the finitary nature of first- order logical consequence. These results helped establish first-order logic as the dominant logic used by mathematicians. In 1931, Gödel published On Formally Undecidable Propositions of Principia Mathematica and Related Systems, which proved the incompleteness (in a different meaning of the word) of all sufficiently strong, effective first-order theories. This result, known as Gödel's incompleteness theorem, establishes severe limitations on axiomatic foundations for mathematics, striking a strong blow to Hilbert's program.
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An important subfield of recursion theory studies algorithmic unsolvability; a decision problem or function problem is algorithmically unsolvable if there is no possible computable algorithm that returns the correct answer for all legal inputs to the problem. The first results about unsolvability, obtained independently by Church and Turing in 1936, showed that the Entscheidungsproblem is algorithmically unsolvable. Turing proved this by establishing the unsolvability of the halting problem, a result with far- ranging implications in both recursion theory and computer science. There are many known examples of undecidable problems from ordinary mathematics.
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Implementing an assertion quantified for an infinite set by definition results in an undecidable non-terminating program. However, the problem is deeper than not being able to implement infinite sets. As Levesque demonstrated, the closer a knowledge representation mechanism comes to FOL, the more likely it is to result in expressions that require infinite or unacceptably large resources to compute. As a result of this trade-off, a great deal of early work on knowledge representation for artificial intelligence involved experimenting with various compromises that provide a subset of FOL with acceptable computation speeds.
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For example, in the expression `(f(x)-1)/(f(x)+1)`, the function `f` must be called twice, because the two calls may return different results. Moreover, the value of `x` must be fetched again before the second call, since the first call may have changed it. Determining whether a subprogram may have a side effect is very difficult (indeed, undecidable by virtue of Rice's theorem). So, while those optimizations are safe in purely functional programming languages, compilers of typical imperative programming usually have to assume the worst.
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An impossible trident with backgrounds, to enhance the illusion Roger Hayward's Undecidable Monument An impossible trident,Andrew M. Colman, A Dictionary of Psychology, Oxford University Press, 2009, , p. 369 also known as an impossible fork,Article "Impossible Fork" at MathWorld blivet,The Hacker's Dictionary, article "Blivet"; It lists the impossible fork among numerous meanings of the term poiuyt, or devil's tuning fork,Brooks Masterton, John M. Kennedy, "Building the Devil's Tuning Fork", Perception, 1975, vol. 4, pp. 107-109 is a drawing of an impossible object (undecipherable figure), a kind of an optical illusion.
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In fact, only single rules over extensional predicate symbols can be easily rewritten as an equivalent conjunctive query. The problem of deciding whether for a given Datalog program there is an equivalent nonrecursive program (corresponding to a positive relational algebra query, or, equivalently, a formula of positive existential first-order logic, or, as a special case, a conjunctive query) is known as the Datalog boundedness problem and is undecidable.Gerd G. Hillebrand, Paris C. Kanellakis, Harry G. Mairson, Moshe Y. Vardi: Undecidable Boundedness Problems for Datalog Programs. J. Log. Program.
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Since \lambda lies in p's past, the Turing machine can signal (a solution) to p at any stage of this never-ending task. Meanwhile, the observer takes a quick trip (finite proper time) through spacetime to p, to pick up the solution. The set-up can be used to decide the halting problem, which is known to be undecidable by an ordinary Turing machine. All the observer needs to do is to prime the Turing machine to signal to p if and only if the Turing machine halts.
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During this year, Gödel also developed the ideas of computability and recursive functions to the point where he was able to present a lecture on general recursive functions and the concept of truth. This work was developed in number theory, using Gödel numbering. In 1934, Gödel gave a series of lectures at the Institute for Advanced Study (IAS) in Princeton, New Jersey, entitled On undecidable propositions of formal mathematical systems. Stephen Kleene, who had just completed his PhD at Princeton, took notes of these lectures that have been subsequently published.
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Church's system for computation developed into the modern λ-calculus, while the Turing machine became a standard model for a general-purpose computing device. It was soon shown that many other proposed models of computation were equivalent in power to those proposed by Church and Turing. These results led to the Church–Turing thesis that any deterministic algorithm that can be carried out by a human can be carried out by a Turing machine. Church proved additional undecidability results, showing that both Peano arithmetic and first-order logic are undecidable.
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Since δ is built up from what we have assumed are Turing machines as well then it too must have a description number, call it e. So, we can feed the description number e to the UTM again, and by definition, δ(k) = U(e, k), so δ(e) = U(e, e). But since TEST(e) is 1, by our other definition, δ(e) = U(e, e) + 1, leading to a contradiction. Thus, TEST(e) cannot exist, and in this way we have settled the halting problem as undecidable.
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The first incompleteness theorem applies only to axiomatic systems defining sufficient arithmetic to carry out the necessary coding constructions (of which Gödel numbering forms a part). The axioms of Q were chosen specifically to ensure they are strong enough for this purpose. Thus the usual proof of the first incompleteness theorem can be used to show that Q is incomplete and undecidable. This indicates that the incompleteness and undecidability of PA cannot be blamed on the only aspect of PA differentiating it from Q, namely the axiom schema of induction.
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It is intuitive to assume that inheritance creates a semantic "is a" relationship, and thus to infer that objects instantiated from subclasses can always be safely used instead of those instantiated from the superclass. This intuition is unfortunately false in most OOP languages, in particular in all those that allow mutable objects. Subtype polymorphism as enforced by the type checker in OOP languages (with mutable objects) cannot guarantee behavioral subtyping in any context. Behavioral subtyping is undecidable in general, so it cannot be implemented by a program (compiler).
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"Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I" ("On Formally Undecidable Propositions of Principia Mathematica and Related Systems I") is a paper in mathematical logic by Kurt Gödel. Dated November 17, 1930, it was originally published in German in the 1931 volume of Monatshefte für Mathematik. Several English translations have appeared in print, and the paper has been included in two collections of classic mathematical logic papers. The paper contains Gödel's incompleteness theorems, now fundamental results in logic that have many implications for consistency proofs in mathematics.
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It is undecidable whether a given first-order sentence can be realized by a finite undirected graph. writes that this undecidability result is well known, and attributes it to on the undecidability of first order satisfiability for more general classes of finite structures. There exist first-order sentences that are modeled by infinite graphs but not by any finite graph. For instance, the property of having exactly one vertex of degree one, with all other vertices having degree exactly two, can be expressed by a first order sentence.
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The Entscheidungsproblem (decision problem) was originally posed by German mathematician David Hilbert in 1928. Turing proved that his "universal computing machine" would be capable of performing any conceivable mathematical computation if it were representable as an algorithm. He went on to prove that there was no solution to the decision problem by first showing that the halting problem for Turing machines is undecidable: it is not possible to decide algorithmically whether a Turing machine will ever halt. This paper has been called "easily the most influential math paper in history".
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Pyrrho claimed that all pragmata (matters, affairs, questions, topics) are adiaphora (not differentiable, not clearly definable, negating Aristotle's use of "diaphora"), astathmēta (unstable, unbalanced, unmeasurable), and anepikrita (unjudgeable, undecidable). Therefore, neither our senses nor our beliefs and theories are able to identify truth or falsehood. Philologist Christopher Beckwith has demonstrated that Pyrrho's use of adiaphora reflects his effort to translate the Buddhist three marks of existence into Greek, and that adiaphora reflects Pyrrho's understanding of the Buddhist concept of anatta. Likewise he suggests that astathmēta and anepikrita may be compared to dukkha and anicca respectively.
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Unlike BPP, PP is a syntactic, rather than semantic class. Any polynomial-time probabilistic machine recognizes some language in PP. In contrast, given a description of a polynomial-time probabilistic machine, it is undecidable in general to determine if it recognizes a language in BPP. PP has natural complete problems, for example, MAJSAT. MAJSAT is a decision problem in which one is given a Boolean formula F. The answer must be YES if more than half of all assignments x1, x2, ..., xn make F true and NO otherwise.
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So far the focus has been on designing a distributed system that solves a given problem. A complementary research problem is studying the properties of a given distributed system. The halting problem is an analogous example from the field of centralised computation: we are given a computer program and the task is to decide whether it halts or runs forever. The halting problem is undecidable in the general case, and naturally understanding the behaviour of a computer network is at least as hard as understanding the behaviour of one computer.
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While the essay was under consideration, Gödel's "On formally undecidable sentences of Principia Mathematica and related systems I" announced the impossibility of formalizing within a theory that theory's consistency proof. Herbrand studied Gödel's essay and wrote an appendix to his own study explaining why Gödel's result did not contradict his own. In July of that year he was mountain-climbing in the French Alps with two friends when he fell to his death in the granite mountains of Massif des Écrins. "On the consistency of arithmetic" was published posthumously.
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Although it is unlikely that Hilbert had conceived of such a possibility, before going on to list the problems, he did presciently remark: > Occasionally it happens that we seek the solution under insufficient > hypotheses or in an incorrect sense, and for this reason do not succeed. The > problem then arises: to show the impossibility of the solution under the > given hypotheses or in the sense contemplated. Proving the 10th problem undecidable is then a valid answer even in Hilbert's terms, since it is a proof about "the impossibility of the solution".
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Unlike von Neumann–Bernays–Gödel set theory (NBG) and Morse–Kelley set theory (MK), ZFC does not admit the existence of proper classes. A further comparative weakness of ZFC is that the axiom of choice included in ZFC is weaker than the axiom of global choice included in NBG and MK. There are numerous mathematical statements undecidable in ZFC. These include the continuum hypothesis, the Whitehead problem, and the normal Moore space conjecture. Some of these conjectures are provable with the addition of axioms such as Martin's axiom or large cardinal axioms to ZFC.
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For one-dimensional cellular automata there are known algorithms for deciding whether a rule is reversible or irreversible. However, for cellular automata of two or more dimensions reversibility is undecidable; that is, there is no algorithm that takes as input an automaton rule and is guaranteed to determine correctly whether the automaton is reversible. The proof by Jarkko Kari is related to the tiling problem by Wang tiles. Reversible cellular automata are often used to simulate such physical phenomena as gas and fluid dynamics, since they obey the laws of thermodynamics.
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Moreover, for each consistent effectively generated system T, it is possible to effectively generate a multivariate polynomial p over the integers such that the equation p = 0 has no solutions over the integers, but the lack of solutions cannot be proved in T (Davis 2006:416, Jones 1980). Smorynski (1977, p. 842) shows how the existence of recursively inseparable sets can be used to prove the first incompleteness theorem. This proof is often extended to show that systems such as Peano arithmetic are essentially undecidable (see Kleene 1967, p. 274).
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In logic, finite model theory, and computability theory, Trakhtenbrot's theorem (due to Boris Trakhtenbrot) states that the problem of validity in first-order logic on the class of all finite models is undecidable. In fact, the class of valid sentences over finite models is not recursively enumerable (though it is co-recursively enumerable). Trakhtenbrot's theorem implies that Gödel's completeness theorem (that is fundamental to first-order logic) does not hold in the finite case. Also it seems counter-intuitive that being valid over all structures is 'easier' than over just the finite ones.
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Additionally, assuming a type of all types that includes itself as type leads into a paradox, as in the set of all sets, so one must proceed in steps of levels of abstraction. Research in second order lambda calculus, one step upwards, showed, that type inference is undecidable in this generality. Parts of one extra level has been introduced into Haskell named kind, where it is used helping to type monads. Kinds are left implicit, working behind the scenes in the inner mechanics of the extended type system.
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138) This is indeed the technique by which a deterministic (i.e., a-) Turing machine can be used to mimic the action of a nondeterministic Turing machine; Turing solved the matter in a footnote and appears to dismiss it from further consideration. An oracle machine or o-machine is a Turing a-machine that pauses its computation at state "o" while, to complete its calculation, it "awaits the decision" of "the oracle"—an unspecified entity "apart from saying that it cannot be a machine" (Turing (1939), The Undecidable, p. 166–168).
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Given the expressiveness of hybrid automata it is not surprising that simple reachability questions are undecidable for general hybrid automata. In fact, a straightforward reduction from Counter machine to three variables hybrid automata (two variables for storing counter values and one to restrict spending a unit-time per location) proves the undecidablity of the reachability problem for hybrid automata. A sub-class of hybrid automata are timed automata Alur, R. and Dill, D. L. "A Theory of Timed Automata". Theoretical Computer Science (TCS), 126(2), pages 183-235, 1995.
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OWL Full is based on a different semantics from OWL Lite or OWL DL, and was designed to preserve some compatibility with RDF Schema. For example, in OWL Full a class can be treated simultaneously as a collection of individuals and as an individual in its own right; this is not permitted in OWL DL. OWL Full allows an ontology to augment the meaning of the pre-defined (RDF or OWL) vocabulary. OWL Full is undecidable, so no reasoning software is able to perform complete reasoning for it.
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To describe such recognizers, formal language theory uses separate formalisms, known as automata theory. One of the interesting results of automata theory is that it is not possible to design a recognizer for certain formal languages.. For more on this subject, see undecidable problem. Parsing is the process of recognizing an utterance (a string in natural languages) by breaking it down to a set of symbols and analyzing each one against the grammar of the language. Most languages have the meanings of their utterances structured according to their syntax--a practice known as compositional semantics.
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The satisfiability problem for a formula of monadic second-order logic is the problem of determining whether there exists at least one graph (possibly within a restricted family of graphs) for which the formula is true. For arbitrary graph families, and arbitrary formulas, this problem is undecidable. However, satisfiability of MSO2 formulas is decidable for the graphs of bounded treewidth, and satisfiability of MSO1 formulas is decidable for graphs of bounded clique-width. The proof involves building a tree automaton for the formula and then testing whether the automaton has an accepting path.
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In computing, an optimizing compiler is a compiler that tries to minimize or maximize some attributes of an executable computer program. Common requirements are to minimize a program's execution time, memory requirement, and power consumption (the last two being popular for portable computers). Compiler optimization is generally implemented using a sequence of optimizing transformations, algorithms which take a program and transform it to produce a semantically equivalent output program that uses fewer resources and/or executes faster. It has been shown that some code optimization problems are NP-complete, or even undecidable.
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Russell and Whitehead thought they could derive all mathematical truth using axioms and inference rules of formal logic, in principle opening up the process to automatisation. In 1920, Thoralf Skolem simplified a previous result by Leopold Löwenheim, leading to the Löwenheim–Skolem theorem and, in 1930, to the notion of a Herbrand universe and a Herbrand interpretation that allowed (un)satisfiability of first-order formulas (and hence the validity of a theorem) to be reduced to (potentially infinitely many) propositional satisfiability problems. In 1929, Mojżesz Presburger showed that the theory of natural numbers with addition and equality (now called Presburger arithmetic in his honor) is decidable and gave an algorithm that could determine if a given sentence in the language was true or false.) However, shortly after this positive result, Kurt Gödel published On Formally Undecidable Propositions of Principia Mathematica and Related Systems (1931), showing that in any sufficiently strong axiomatic system there are true statements which cannot be proved in the system. This topic was further developed in the 1930s by Alonzo Church and Alan Turing, who on the one hand gave two independent but equivalent definitions of computability, and on the other gave concrete examples for undecidable questions.
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The decision problem of whether a given string s can be generated by a given unrestricted grammar is equivalent to the problem of whether it can be accepted by the Turing machine equivalent to the grammar. The latter problem is called the Halting problem and is undecidable. Recursively enumerable languages are closed under Kleene star, concatenation, union, and intersection, but not under set difference; see Recursively enumerable language#Closure properties. The equivalence of unrestricted grammars to Turing machines implies the existence of a universal unrestricted grammar, a grammar capable of accepting any other unrestricted grammar's language given a description of the language.
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An alt= The subject of aperiodic tilings received new interest in the 1960s when logician Hao Wang noted connections between decision problems and tilings. In particular, he introduced tilings by square plates with colored edges, now known as Wang dominoes or tiles, and posed the "Domino Problem": to determine whether a given set of Wang dominoes could tile the plane with matching colors on adjacent domino edges. He observed that if this problem were undecidable, then there would have to exist an aperiodic set of Wang dominoes. At the time, this seemed implausible, so Wang conjectured no such set could exist.
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Badiou's ultimate ethical maxim is therefore one of: 'decide upon the undecidable'. It is to name the indiscernible, the generic set, and thus name the event that re-casts ontology in a new light. He identifies four domains in which a subject (who, it is important to note, becomes a subject through this process) can potentially witness an event: love, science, politics and art. By enacting fidelity to the event within these four domains one performs a 'generic procedure', which in its undecidability is necessarily experimental, and one potentially recasts the situation in which being takes place.
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Since the halting problem is undecidable, Ω cannot be computed. The algorithm proceeds as follows. Given the first n digits of Ω and a k ≤ n, the algorithm enumerates the domain of F until enough elements of the domain have been found so that the probability they represent is within 2−(k+1) of Ω. After this point, no additional program of length k can be in the domain, because each of these would add 2−k to the measure, which is impossible. Thus the set of strings of length k in the domain is exactly the set of such strings already enumerated.
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The LALR(j) parsers are incomparable with LL(k) parsers: for any j and k both greater than 0, there are LALR(j) grammars that are not LL(k) grammars and conversely. In fact, it is undecidable whether a given LL(1) grammar is LALR(k) for any k > 0. Depending on the presence of empty derivations, a LL(1) grammar can be equal to a SLR(1) or a LALR(1) grammar. If the LL(1) grammar has no empty derivations it is SLR(1) and if all symbols with empty derivations have non-empty derivations it is LALR(1).
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This was shown to be the case in 1952. The combined work of Gödel and Paul Cohen has given two concrete examples of undecidable statements (in the first sense of the term): The continuum hypothesis can neither be proved nor refuted in ZFC (the standard axiomatization of set theory), and the axiom of choice can neither be proved nor refuted in ZF (which is all the ZFC axioms except the axiom of choice). These results do not require the incompleteness theorem. Gödel proved in 1940 that neither of these statements could be disproved in ZF or ZFC set theory.
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Since we have only one equation but n variables, infinitely many solutions exist (and are easy to find) in the complex plane; however, the problem becomes impossible if solutions are constrained to integer values only. Matiyasevich showed this problem to be unsolvable by mapping a Diophantine equation to a recursively enumerable set and invoking Gödel's Incompleteness Theorem. In 1936, Alan Turing proved that the halting problem—the question of whether or not a Turing machine halts on a given program—is undecidable, in the second sense of the term. This result was later generalized by Rice's theorem.
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HNN-extensions play a key role in Higman's proof of the Higman embedding theorem which states that every finitely generated recursively presented group can be homomorphically embedded in a finitely presented group. Most modern proofs of the Novikov–Boone theorem about the existence of a finitely presented group with algorithmically undecidable word problem also substantially use HNN-extensions. Both HNN- extensions and amalgamated free products are basic building blocks in the Bass–Serre theory of groups acting on trees. The idea of HNN extension has been extended to other parts of abstract algebra, including Lie algebra theory.
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Since the class V may be considered to be the arena for most of mathematics, it is important to establish that it "exists" in some sense. Since existence is a difficult concept, one typically replaces the existence question with the consistency question, that is, whether the concept is free of contradictions. A major obstacle is posed by Gödel's incompleteness theorems, which effectively imply the impossibility of proving the consistency of ZF set theory in ZF set theory itself, provided that it is in fact consistent.See article On Formally Undecidable Propositions of Principia Mathematica and Related Systems and .
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The satisfiability problem for a formula of monadic second-order logic is the problem of determining whether there exists at least one graph (possibly within a restricted family of graphs) for which the formula is true. For arbitrary graph families, and arbitrary formulas, this problem is undecidable. However, satisfiability of MSO2 formulas is decidable for the graphs of bounded treewidth, and satisfiability of MSO1 formulas is decidable for graphs of bounded clique-width. The proof involves using Courcelle's theorem to build an automaton that can test the property, and then examining the automaton to determine whether there is any graph it can accept.
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It is often said that "Only perl can parse Perl," meaning that only the Perl interpreter (`perl`) can parse the Perl language (Perl), but even this is not, in general, true. Because the Perl interpreter can simulate a Turing machine during its compile phase, it would need to decide the halting problem in order to complete parsing in every case. It is a long-standing result that the halting problem is undecidable, and therefore not even perl can always parse Perl. Perl makes the unusual choice of giving the user access to its full programming power in its own compile phase.
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' Drawing upon Georg Simmel's sociology and the philosophy of Jacques Derrida, Bauman came to write of the stranger as the person who is present yet unfamiliar, society's undecidable. In Modernity and Ambivalence Bauman attempted to give an account of the different approaches modern society adopts toward the stranger. He argued that, on the one hand, in a consumer- oriented economy the strange and the unfamiliar is always enticing; in different styles of food, different fashions and in tourism it is possible to experience the allure of what is unfamiliar. Yet this strange-ness also has a more negative side.
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The main difficulty in proving the second incompleteness theorem is to show that various facts about provability used in the proof of the first incompleteness theorem can be formalized within the system using a formal predicate for provability. Once this is done, the second incompleteness theorem follows by formalizing the entire proof of the first incompleteness theorem within the system itself. Let p stand for the undecidable sentence constructed above, and assume that the consistency of the system can be proved from within the system itself. The demonstration above shows that if the system is consistent, then p is not provable.
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The argument is usually attributed to the Pyrrhonist philosopher Agrippa the Skeptic as part of what has become known as "Agrippa's trilemma". The argument can be seen as a response to the suggestion in Plato's Theaetetus that knowledge is justified true belief. The Pyrrhonist philosopher Sextus Empiricus described Agrippa's trope as follows: > According to the mode deriving from dispute, we find that undecidable > dissension about the matter proposed has come about both in ordinary life > and among philosophers. Because of this we are not able to choose or to rule > out anything, and we end up with suspension of judgement.
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Many patterns in the Game of Life eventually become a combination of still lifes, oscillators, and spaceships; other patterns may be called chaotic. A pattern may stay chaotic for a very long time until it eventually settles to such a combination. The Game of Life is undecidable, which means that given an initial pattern and a later pattern, no algorithm exists that can tell whether the later pattern is ever going to appear. This is a corollary of the halting problem: the problem of determining whether a given program will finish running or continue to run forever from an initial input.
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Also provably unsolvable are so- called undecidable problems, such as the halting problem for Turing machines. Many abstract problems can be solved routinely, others have been solved with great effort, for some significant inroads have been made without having led yet to a full solution, and yet others have withstood all attempts, such as Goldbach's conjecture and the Collatz conjecture. Some well-known difficult abstract problems that have been solved relatively recently are the four- colour theorem, Fermat's Last Theorem, and the Poincaré conjecture. The all of mathematical new ideas which develop a new horizon on our imagination not correspond to the real world.
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Thus first-order logical consequence is semidecidable: it is possible to make an effective enumeration of all pairs of sentences (φ,ψ) such that ψ is a logical consequence of φ. Unlike propositional logic, first-order logic is undecidable (although semidecidable), provided that the language has at least one predicate of arity at least 2 (other than equality). This means that there is no decision procedure that determines whether arbitrary formulas are logically valid. This result was established independently by Alonzo Church and Alan Turing in 1936 and 1937, respectively, giving a negative answer to the Entscheidungsproblem posed by David Hilbert and Wilhelm Ackermann in 1928.
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'' The on-line version of Turing's paper has these corrections in an addendum; however, corrections to the Universal Machine must be found in an analysis provided by Emil Post. At first, the only mathematician to pay close attention to the details of the proof was Post (cf. Hodges p. 125) — mainly because he had arrived simultaneously at a similar reduction of "algorithm" to primitive machine-like actions, so he took a personal interest in the proof. Strangely (perhaps World War II intervened) it took Post some ten years to dissect it in the Appendix to his paper Recursive Unsolvability of a Problem of Thue, 1947 (reprinted in Undecidable, p. 293).
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The decision problem that asks whether a certain string s belongs to the language of a given context-sensitive grammar G, is PSPACE-complete. Moreover, there are context-sensitive grammars whose languages are PSPACE-complete. In other words, there is a context-sensitive grammar G such that deciding whether a certain string s belongs to the language of G is PSPACE-complete (so G is fixed and only s is part of the input of the problem).An example of such a grammar, designed to solve the QSAT problem, is given in The emptiness problem for context-sensitive grammars (given a context-sensitive grammar G, is L(G)=∅ ?) is undecidable.
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After graduating, Robinson continued in graduate studies at Berkeley. As a graduate student, Robinson was employed as a teaching assistant with the Department of Mathematics and later as a statistics lab assistant by Jerzy Neyman in the Berkeley Statistical Laboratory, where her work resulted in her first published paper, titled "A Note on Exact Sequential Analysis". Robinson received her Ph.D. degree in 1948 under Alfred Tarski with a dissertation on "Definability and Decision Problems in Arithmetic". Her dissertation showed that the theory of the rational numbers was an undecidable problem, by demonstrating that elementary number theory could be defined in terms of the rationals.
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Puzzles commonly ask for tiling a given region with a given set of polyominoes, such as the 12 pentominoes. Golomb's and Gardner's books have many examples. A typical puzzle is to tile a 6×10 rectangle with the twelve pentominoes; the 2339 solutions to this were found in 1960. Where multiple copies of the polyominoes in the set are allowed, Golomb defines a hierarchy of different regions that a set may be able to tile, such as rectangles, strips, and the whole plane, and shows that whether polyominoes from a given set can tile the plane is undecidable, by mapping sets of Wang tiles to sets of polyominoes.
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However, this does not achieve much, because even though we can solve the new problem, performing the reduction is just as hard as solving the old problem. Likewise, a reduction computing a noncomputable function can reduce an undecidable problem to a decidable one. As Michael Sipser points out in Introduction to the Theory of Computation: "The reduction must be easy, relative to the complexity of typical problems in the class [...] If the reduction itself were difficult to compute, an easy solution to the complete problem wouldn't necessarily yield an easy solution to the problems reducing to it." Therefore, the appropriate notion of reduction depends on the complexity class being studied.
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In mathematical logic, independence is the unprovability of a sentence from other sentences. A sentence σ is independent of a given first-order theory T if T neither proves nor refutes σ; that is, it is impossible to prove σ from T, and it is also impossible to prove from T that σ is false. Sometimes, σ is said (synonymously) to be undecidable from T; this is not the same meaning of "decidability" as in a decision problem. A theory T is independent if each axiom in T is not provable from the remaining axioms in T. A theory for which there is an independent set of axioms is independently axiomatizable.
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In languages that allow side effects, like most object-oriented languages, subtyping is generally not sufficient to guarantee that a function can be safely used in the context of another. Liskov's work in this area focused on behavioral subtyping, which besides the type system safety discussed in this article also requires that subtypes preserve all invariants guaranteed by the supertypes in some contract.Barbara Liskov, Jeannette Wing, A behavioral notion of subtyping, ACM Transactions on Programming Languages and Systems, Volume 16, Issue 6 (November 1994), pp. 1811–1841. An updated version appeared as CMU technical report: This definition of subtyping is generally undecidable, so it cannot be verified by a type checker.
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A fundamental trade-off identified with knowledge representation in artificial intelligence is between expressive power and computability. As Levesque demonstrated in his classic paper on the topic, the more powerful a knowledge-representation formalism one designs, the closer the formalism will come to the expressive power of first order logic. As Levesque also demonstrated, the closer a language is to First Order Logic, the more probable that it will allow expressions that are undecidable or require exponential processing power to complete. In the implementation of KBE systems, this trade off is reflected in the choice to use powerful knowledge- based environments or more conventional procedural and object-oriented programming environments.
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The combined work of Gödel and Paul Cohen has given two concrete examples of undecidable statements (in the first sense of the term): The continuum hypothesis can neither be proved nor refuted in ZFC (the standard axiomatization of set theory), and the axiom of choice can neither be proved nor refuted in ZF (which is all the ZFC axioms except the axiom of choice). These results do not require the incompleteness theorem. Gödel proved in 1940 that neither of these statements could be disproved in ZF or ZFC set theory. In the 1960s, Cohen proved that neither is provable from ZF, and the continuum hypothesis cannot be proved from ZFC.
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In computability theory, Rice's theorem states that all non-trivial, semantic properties of programs are undecidable. A semantic property is one about the program's behavior (for instance, does the program terminate for all inputs), unlike a syntactic property (for instance, does the program contain an if- then-else statement). A property is non-trivial if it is neither true for every computable function, nor false for every computable function. Rice's theorem can also be put in terms of functions: for any non-trivial property of partial functions, no general and effective method can decide whether an algorithm computes a partial function with that property.
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Computability theory deals primarily with the question of the extent to which a problem is solvable on a computer. The statement that the halting problem cannot be solved by a Turing machine is one of the most important results in computability theory, as it is an example of a concrete problem that is both easy to formulate and impossible to solve using a Turing machine. Much of computability theory builds on the halting problem result. Another important step in computability theory was Rice's theorem, which states that for all non-trivial properties of partial functions, it is undecidable whether a Turing machine computes a partial function with that property.
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Church's paper (published 15 April 1936) showed that the Entscheidungsproblem was indeed "undecidable" and beat Turing to the punch by almost a year (Turing's paper submitted 28 May 1936, published January 1937). In the meantime, Emil Post submitted a brief paper in the fall of 1936, so Turing at least had priority over Post. While Church refereed Turing's paper, Turing had time to study Church's paper and add an Appendix where he sketched a proof that Church's lambda-calculus and his machines would compute the same functions. And Post had only proposed a definition of calculability and criticized Church's "definition", but had proved nothing.
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RA can express any (and up to logical equivalence, exactly the) first-order logic (FOL) formulas containing no more than three variables. (A given variable can be quantified multiple times and hence quantifiers can be nested arbitrarily deeply by "reusing" variables.) Surprisingly, this fragment of FOL suffices to express Peano arithmetic and almost all axiomatic set theories ever proposed. Hence RA is, in effect, a way of algebraizing nearly all mathematics, while dispensing with FOL and its connectives, quantifiers, turnstiles, and modus ponens. Because RA can express Peano arithmetic and set theory, Gödel's incompleteness theorems apply to it; RA is incomplete, incompletable, and undecidable. (N.
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In 1992, Roger Heath-Brown conjectured that every n unequal to 4 or 5 modulo 9 has infinitely many representations as sums of three cubes. The case n=33 of this problem was used by Bjorn Poonen as the opening example in a survey on undecidable problems in number theory, of which Hilbert's tenth problem is the most famous example. Although this particular case has since been resolved, it is unknown whether representing numbers as sums of cubes is decidable. That is, it is not known whether an algorithm can, for every input, test in finite time whether a given number has such a representation.
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Common patterns of recursion can be abstracted away using higher-order functions, with catamorphisms and anamorphisms (or "folds" and "unfolds") being the most obvious examples. Such recursion schemes play a role analogous to built-in control structures such as loops in imperative languages. Most general purpose functional programming languages allow unrestricted recursion and are Turing complete, which makes the halting problem undecidable, can cause unsoundness of equational reasoning, and generally requires the introduction of inconsistency into the logic expressed by the language's type system. Some special purpose languages such as Coq allow only well-founded recursion and are strongly normalizing (nonterminating computations can be expressed only with infinite streams of values called codata).
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Thirdly, what > will be the outcome for those who have this attitude?" Pyrrho's answer is > that "As for pragmata they are all adiaphora (undifferentiated by a logical > differentia), astathmēta (unstable, unbalanced, not measurable), and > anepikrita (unjudged, unfixed, undecidable). Therefore, neither our sense- > perceptions nor our doxai (views, theories, beliefs) tell us the truth or > lie; so we certainly should not rely on them. Rather, we should be adoxastoi > (without views), aklineis (uninclined toward this side or that), and > akradantoi (unwavering in our refusal to choose), saying about every single > one that it no more is than it is not or it both is and is not or it neither > is nor is not.
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GTS is mutually interpretable with Peano arithmetic (thus it has the same proof-theoretic strength as PA); The most remarkable fact about ST (and hence GST), is that these tiny fragments of set theory give rise to such rich metamathematics. While ST is a small fragment of the well-known canonical set theories ZFC and NBG, ST interprets Robinson arithmetic (Q), so that ST inherits the nontrivial metamathematics of Q. For example, ST is essentially undecidable because Q is, and every consistent theory whose theorems include the ST axioms is also essentially undecidable.Burgess (2005), 2.2, p. 91. This includes GST and every axiomatic set theory worth thinking about, assuming these are consistent.
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The idea of constraining adjacent tiles to match each other occurs in the game of dominoes, so Wang tiles are also known as Wang dominoes.. The algorithmic problem of determining whether a tile set can tile the plane became known as the domino problem. According to Wang's student, Robert Berger, > The Domino Problem deals with the class of all domino sets. It consists of > deciding, for each domino set, whether or not it is solvable. We say that > the Domino Problem is decidable or undecidable according to whether there > exists or does not exist an algorithm which, given the specifications of an > arbitrary domino set, will decide whether or not the set is solvable.
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3 [3 Afterward he would regret his compliance, > for the published volume was marred throughout by sloppy typography and > numerous misprints.] The translation by Elliott Mendelson appears in the collection The Undecidable (Davis 1965:5ff). This translation also received a harsh review by Bauer- Mengelberg (1966), who in addition to giving a detailed list of the typographical errors also described what he believed to be serious errors in the translation. A translation by Jean van Heijenoort appears in the collection From Frege to Gödel: A Source Book in Mathematical Logic (van Heijenoort 1967). A review by Alonzo Church (1972) described this as "the most careful translation that has been made" but also gave some specific criticisms of it.
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Thirdly, what > will be the outcome for those who have this attitude?" Pyrrho's answer is > that "As for pragmata they are all adiaphora (undifferentiated by a logical > differentia), astathmēta (unstable, unbalanced, not measurable), and > anepikrita (unjudged, unfixed, undecidable). Therefore, neither our sense- > perceptions nor our doxai (views, theories, beliefs) tell us the truth or > lie; so we certainly should not rely on them. Rather, we should be adoxastoi > (without views), aklineis (uninclined toward this side or that), and > akradantoi (unwavering in our refusal to choose), saying about every single > one that it no more is than it is not or it both is and is not or it neither > is nor is not.
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Two incomparable families examined at length are WRB (languages generated by normal regular- based W-grammars) and WS (languages generated by simple W-grammars). Both properly contain the context-free languages and are properly contained in the family of quasirealtime languages. In addition, WRB is closed under nested iterate ... "An Infinite Hierarchy of Context-Free Languages," Journal of the ACM, Volume 16 Issue 1, January 1969 "A New Normal-Form Theorem for Context- Free Phrase Structure Grammars," JACM, Volume 12 Issue 1, January 1965 "The Unsolvability of the Recognition of Linear Context-Free Languages," JACM, Volume 13 Issue 4, October 1966 :The problem of whether a given context-free language is linear is shown to be recursively undecidable.
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Suslin's problem asks whether a specific short list of properties characterizes the ordered set of real numbers R. This is undecidable in ZFC. A Suslin line is an ordered set which satisfies this specific list of properties but is not order-isomorphic to R. The diamond principle ◊ proves the existence of a Suslin line, while MA + ¬CH implies EATS (every Aronszajn tree is special),Baumgartner, J., J. Malitz, and W. Reiehart, Embedding trees in the rationals, Proc. Natl. Acad. Sci. U.S.A., 67, pp. 1746 – 1753, 1970 which in turn implies (but is not equivalent to)Shelah, S., Free limits of forcing and more on Aronszajn trees, Israel Journal of Mathematics, 40, pp.
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In the late 1970s Pour-El began working on computable analysis. Her "most famous and surprising result", co-authored with Minnesota colleague J. Ian Richards, was that for certain computable initial conditions, determining the behavior of the wave equation is an undecidable problem. Their result was later taken up by Roger Penrose in his book The Emperor's New Mind; Penrose used this result as a test case for the Church–Turing thesis, but concluded that the non-smoothness of the initial conditions makes it implausible that a computational device could use this phenomenon to exceed the limits of conventional computing. Freeman Dyson used the same result to argue for the evolutionary superiority of analog to digital forms of life.
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And the two theorems of Turing's in question are really the following. There is no Turing ... machine which, when supplied with an arbitrary positive integer n, will determine whether n is the D.N of a Turing computing ... machine that is circle-free. [Secondly], There is no Turing convention-machine which, when supplied with an arbitrary positive integer n, will determine whether n is the D.N of a Turing computing ... machine that ever prints a given symbol (0 say)" (Post in Undecidable, p. 300) Anyone who has ever tried to read the paper will understand Hodges' complaint: :"The paper started attractively, but soon plunged (in typical Turing manner) into a thicket of obscure German Gothic type in order to develop his instruction table for the universal machine.
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One does not necessarily need to refer to constructible language to conceive of a 'set of dominations', which he refers to as the indiscernible set, or the generic set. It is therefore, he continues, possible to think beyond the strictures of the relativistic constructible universe of language, by a process Cohen calls forcing. And he concludes in following that while ontology can mark out a space for an inhabitant of the constructible situation to decide upon the indiscernible, it falls to the subject – about which the ontological situation cannot comment – to nominate this indiscernible, this generic point; and thus nominate, and give name to, the undecidable event. Badiou thereby marks out a philosophy by which to refute the apparent relativism or apoliticism in post-structuralist thought.
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While higher-order unification is undecidable,Claudio Lucchesi: The Undecidability of the Unification Problem for Third Order Languages (Research Report CSRR 2059; Department of Computer Science, University of Waterloo, 1972) Gérard Huet gave a semi-decidable (pre-)unification algorithmGérard Huet: A Unification Algorithm for typed Lambda-Calculus [] that allows a systematic search of the space of unifiers (generalizing the unification algorithm of Martelli- Montanari with rules for terms containing higher-order variables) that seems to work sufficiently well in practice. HuetGérard Huet: Higher Order Unification 30 Years Later and Gilles DowekGilles Dowek: Higher-Order Unification and Matching. Handbook of Automated Reasoning 2001: 1009–1062 have written articles surveying this topic. Dale Miller has described what is now called higher-order pattern unification.
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In computability theory, one of the basic undecidable problems is the halting problem: deciding whether a deterministic Turing machine (DTM) halts. One of the most fundamental EXPTIME- complete problems is a simpler version of this, which asks if a DTM halts in at most k steps. It is in EXPTIME because a trivial simulation requires O(k) time, and the input k is encoded using O(log k) bits which causes exponential number of simulations. It is EXPTIME-complete because, roughly speaking, we can use it to determine if a machine solving an EXPTIME problem accepts in an exponential number of steps; it will not use more.. The same problem with the number of steps written in unary is P-complete.
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In computability theory, a set of natural numbers is called recursive, computable or decidable if there is an algorithm which takes a number as input, terminates after a finite amount of time (possibly depending on the given number) and correctly decides whether the number belongs to the set or not. A set which is not computable is called noncomputable or undecidable. A more general class of sets than the decidable ones consists of the recursively enumerable sets, also called semidecidable sets. For these sets, it is only required that there is an algorithm that correctly decides when a number is in the set; the algorithm may give no answer (but not the wrong answer) for numbers not in the set.
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We write R \subseteq S for two database relations R, S of the same schema if and only if each tuple occurring in R also occurs in S. Given a query Q and a relational database instance I, we write the result relation of evaluating the query on the instance simply as Q(I). Given two queries Q_1 and Q_2 and a database schema, the query containment problem is the problem of deciding whether for all possible database instances I over the input database schema, Q_1(I) \subseteq Q_2(I). The main application of query containment is in query optimization: Deciding whether two queries are equivalent is possible by simply checking mutual containment. The query containment problem is undecidable for relational algebra and SQL but is decidable and NP-complete for conjunctive queries.
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For subtraction games with a fixed (but possibly infinite) subtraction set, such as subtract a square, the partition into P-positions and N-positions of the numbers up to a given value n may be computed in time O(n\log^2 n). The nim-values of all numbers up to n may be computed in time O(\min(ns,nm\log^2 n)) where s denotes the size of the subtraction set (up to n) and m denotes the largest nim-value occurring in this computation. For generalizations of subtraction games, played on vectors of natural numbers with a subtraction set whose vectors can have positive as well as negative coefficients, it is an undecidable problem to determine whether two such games have the same P-positions and N-positions.
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The most promising ideas about program-development parallels seem to us to be ones that point to an apparently close analogy between processes within cells, and the low-level operation of modern computers. Thus, biological systems are like computational machines that process input information to compute next states, such that biological systems are closer to a computation than classical dynamical system. Furthermore, following concepts from computational theory, micro processes in biological organisms are fundamentally incomplete and undecidable (completeness (logic)), implying that “there is more than a crude metaphor behind the analogy between cells and computers. The analogy to computation extends also to the relationship between inheritance systems and biological structure, which is often thought to reveal one of the most pressing problems in explaining the origins of life.
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An inclusion dependency over two (possibly identical) predicates R and S from a schema is written R[A_1, ..., A_n] \subseteq S[B_1, ..., B_n], where the A_i, B_i are distinct attributes (column names) of R and S. It implies that the tuples of values appearing in columns A_1, ..., A_n for facts of R must also appear as a tuple of values in columns B_1, ..., B_n for some fact of S. Logical implication between inclusion dependencies can be axiomatized by inference rules and can be decided by a PSPACE algorithm. The problem can be shown to be PSPACE-complete by reduction from the acceptance problem for a linear bounded automaton. However, logical implication between dependencies that can be inclusion dependencies or functional dependencies is undecidable by reduction from the word problem for monoids.
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We show that E is undecidable by a reduction from H. To obtain a contradiction, suppose R is a decider for E. We will use this to produce a decider S for H (which we know does not exist). Given input M and w (a Turing machine and some input string), define S(M, w) with the following behavior: S creates a Turing machine N that accepts only if the input string to N is w and M halts on input w, and does not halt otherwise. The decider S can now evaluate R(N) to check whether the language accepted by N is empty. If R accepts N, then the language accepted by N is empty, so in particular M does not halt on input w, so S can reject.
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He has done work on the theory of generic multiverses and the related concept of Ω-logic, which suggested an argument that the continuum hypothesis is either undecidable or false in the sense of mathematical platonism. Woodin criticizes this view arguing that it leads to a counterintuitive reduction in which all truths in the set theoretical universe can be decided from a small part of it. He claims that these and related mathematical results lead (intuitively) to the conclusion that Continuum Hypothesis has a truth value and the Platonistic approach is reasonable. Woodin now predicts that there should be a way of constructing an inner model for almost all known large cardinals, which he calls the Ultimate L and which would have similar properties as Gödel's constructible universe.
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Cohen is noted for developing a mathematical technique called forcing, which he used to prove that neither the continuum hypothesis (CH) nor the axiom of choice can be proved from the standard Zermelo–Fraenkel axioms (ZF) of set theory. In conjunction with the earlier work of Gödel, this showed that both of these statements are logically independent of the ZF axioms: these statements can be neither proved nor disproved from these axioms. In this sense, the continuum hypothesis is undecidable, and it is the most widely known example of a natural statement that is independent from the standard ZF axioms of set theory. For his result on the continuum hypothesis, Cohen won the Fields Medal in mathematics in 1966, and also the National Medal of Science in 1967.
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Early in his paper (1936) Turing makes a distinction between an "automatic machine"—its "motion ... completely determined by the configuration" and a "choice machine": Turing (1936) does not elaborate further except in a footnote in which he describes how to use an a-machine to "find all the provable formulae of the [Hilbert] calculus" rather than use a choice machine. He "suppose[s] that the choices are always between two possibilities 0 and 1. Each proof will then be determined by a sequence of choices i1, i2, ..., in (i1 = 0 or 1, i2 = 0 or 1, ..., in = 0 or 1), and hence the number 2n \+ i12n-1 \+ i22n-2 \+ ... +in completely determines the proof. The automatic machine carries out successively proof 1, proof 2, proof 3, ..." (Footnote ‡, The Undecidable, p.
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He proved that some such machine would be capable of performing any conceivable mathematical computation if it were representable as an algorithm. He went on to prove that there was no solution to the Entscheidungsproblem by first showing that the halting problem for Turing machines is undecidable: in general, it is not possible to decide algorithmically whether a given Turing machine will ever halt. He also introduced the notion of a "universal machine" (now known as a universal Turing machine), with the idea that such a machine could perform the tasks of any other machine, or in other words, it is provably capable of computing anything that is computable by executing a program stored on tape, allowing the machine to be programmable. Von Neumann acknowledged that the central concept of the modern computer was due to this paper.
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At the International Congress of Mathematicians (ICM) in 1900 in Paris the famous mathematician David Hilbert posed a set of problems – now known as Hilbert's problems – his beacon illuminating the way for mathematicians of the twentieth century. Hilbert's 2nd and 10th problems introduced the Entscheidungsproblem (the "decision problem"). In his 2nd problem he asked for a proof that "arithmetic" is "consistent". Kurt Gödel would prove in 1931 that, within what he called "P" (nowadays called Peano Arithmetic), "there exist undecidable sentences [propositions]".Gödel 1931a in (Davis 1965:6), 1930 in (van Heijenoort 1967:596) Because of this, "the consistency of P is unprovable in P, provided P is consistent".Gödel’s theorem IX, Gödel 1931a in (Davis 1965:36) While Gödel’s proof would display the tools necessary for Alonzo Church and Alan Turing to resolve the Entscheidungsproblem, he himself would not answer it.
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Higher order logics include the offshoots of Church's Simple theory of typesAlonzo Church, A formulation of the simple theory of types, The Journal of Symbolic Logic 5(2):56-68 (1940) and the various forms of intuitionistic type theory. Gérard Huet has shown that unifiability is undecidable in a type theoretic flavor of third-order logic, that is, there can be no algorithm to decide whether an arbitrary equation between third-order (let alone arbitrary higher-order) terms has a solution. Up to a certain notion of isomorphism, the powerset operation is definable in second-order logic. Using this observation, Jaakko Hintikka established in 1955 that second-order logic can simulate higher-order logics in the sense that for every formula of a higher order-logic one can find an equisatisfiable formula for it in second-order logic.
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In computability theory, the mortality problem is a decision problem which can be stated as follows: :Given a Turing machine, decide whether it halts when run on any configuration (not necessarily a starting one) In the statement above, the configuration is a pair , where q is one of the machine's states (not necessarily its initial state) and w is an infinite sequence of symbols representing the initial content of the tape. Note that while we usually assume that in the starting configuration all but finitely many cells on the tape are blanks, in the mortality problem the tape can have arbitrary content, including infinitely many non-blank symbols written on it. Philip K. Hooper proved in 1966 that the mortality problem is undecidable. However, it can be shown that the set of Turing machines which are mortal (i.e.
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Gu et al. presented a class of physical systems that exhibits non-computable macroscopic properties. More precisely, if one could compute certain macroscopic properties of these systems from the microscopic description of these systems, then one would be able to solve computational problems known to be undecidable in computer science. Gu et al. concluded that > Although macroscopic concepts are essential for understanding our world, > much of fundamental physics has been devoted to the search for a 'theory of > everything', a set of equations that perfectly describe the behavior of all > fundamental particles. The view that this is the goal of science rests in > part on the rationale that such a theory would allow us to derive the > behavior of all macroscopic concepts, at least in principle. The evidence we > have presented suggests that this view may be overly optimistic.
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His most prolific period spawned from his collaboration with Newton da Costa, a Brazilian logician and one of the founders of paraconsistent logic, which began in 1985. He is currently Professor of Communications, Emeritus, at UFRJ and a member of the Brazilian Academy of Philosophy. His main achievement (with Brazilian logician and philosopher Newton da Costa) is the proof that chaos theory is undecidable (published in 1991), and when properly axiomatized within classical set theory, is incomplete in the sense of Gödel. The decision problem for chaotic dynamical systems had been formulated by mathematician Morris Hirsch. More recently da Costa and Dória introduced a formalization for the P = NP hypothesis which they called the “exotic formalization,” and showed in a series of papers that axiomatic set theory together with exotic P = NP is consistent if set theory is consistent.
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The time- reversed dynamics of a reversible cellular automaton can always be described by another cellular automaton rule, possibly on a much larger neighborhood. Several methods are known for defining cellular automata rules that are reversible; these include the block cellular automaton method, in which each update partitions the cells into blocks and applies an invertible function separately to each block, and the second-order cellular automaton method, in which the update rule combines states from two previous steps of the automaton. When an automaton is not defined by one of these methods, but is instead given as a rule table, the problem of testing whether it is reversible is solvable for block cellular automata and for one-dimensional cellular automata, but is undecidable for other types of cellular automata. Reversible cellular automata form a natural model of reversible computing, a technology that could lead to ultra-low-power computing devices.
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In mathematics and computer science, a word problem for a set S with respect to a system of finite encodings of its elements is the algorithmic problem of deciding whether two given representatives represent the same element of the set. The problem is commonly encountered in abstract algebra, where given a presentation of an algebraic structure by generators and relators, the problem is to determine if two expressions represent the same element; a prototypical example is the word problem for groups. Less formally, the word problem in an algebra is: given a set of identities E, and two expressions x and y, is it possible to transform x into y using the identities in E as rewriting rules in both directions? While answering this question may not seem hard, the remarkable (and deep) result that emerges, in many important cases, is that the problem is undecidable.
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His PhD thesis, titled "Systems of Logic Based on Ordinals", contains the following definition of "a computable function": When Turing returned to the UK he ultimately became jointly responsible for breaking the German secret codes created by encryption machines called "The Enigma"; he also became involved in the design of the ACE (Automatic Computing Engine), "[Turing's] ACE proposal was effectively self- contained, and its roots lay not in the EDVAC [the USA's initiative], but in his own universal machine" (Hodges p. 318). Arguments still continue concerning the origin and nature of what has been named by Kleene (1952) Turing's Thesis. But what Turing did prove with his computational-machine model appears in his paper "On Computable Numbers, with an Application to the Entscheidungsproblem" (1937): Turing's example (his second proof): If one is to ask for a general procedure to tell us: "Does this machine ever print 0", the question is "undecidable".
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Suppose that X is the first uncountable ordinal, with the finite measure where the measurable sets are either countable (with measure 0) or the sets of countable complement (with measure 1). The (non-measurable) subset E of X×X given by pairs (x,y) with x The stronger versions of Fubini's theorem on a product of two unit intervals with Lebesgue measure, where the function is no longer assumed to be measurable but merely that the two iterated integrals are well defined and exist, are independent of the standard Zermelo–Fraenkel axioms of set theory. The continuum hypothesis and Martin's axiom both imply that there exists a function on the unit square whose iterated integrals are not equal, while showed that it is consistent with ZFC that a strong Fubini-type theorem for [0, 1] does hold, and whenever the two iterated integrals exist they are equal. See List of statements undecidable in ZFC.
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Grzegorczyk's undecidability of Alfred Tarski's concatenation theory is based on the philosophical motivation claiming that investigation of formal systems should be done with a help of operations on visually comprehensible objects, and the most natural element of this approach is the notion of text. On his research, Tarski's simple theory is undecidable although seems to be weaker than the weak arithmetic, whereas, instead of computability, he applies more epistemological notion of the effective recognizability of properties of a text and relationships between different texts. In 2011, Grzegorczyk introduced yet one more logical system, which today is known as the Grzegorczyk non-Fregean logic or the logic of descriptions (LD), to cover the basic features of descriptive equivalence of sentences, wherein he assumed that a human language is applied primarily to form descriptions of reality represented formally by logical connectives. According to this system, the logical language is equipped in at least four logical connectives negation (¬), conjunction (∧), disjunction (∨), and equivalence (≡).
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During his lifetime three English translations of Gödel's paper were printed, but the process was not without difficulty. The first English translation was by Bernard Meltzer; it was published in 1963 as a standalone work by Basic Books and has since been reprinted by Dover and reprinted by Hawking (God Created the Integers, Running Press, 2005:1097ff). The Meltzer version—described by Raymond Smullyan as a 'nice translation'—was adversely reviewed by Stefan Bauer-Mengelberg (1966). According to Dawson's biography of Gödel (Dawson 1997:216), > Fortunately, the Meltzer translation was soon supplanted by a better one > prepared by Elliott Mendelson for Martin Davis's anthology The Undecidable; > but it too was not brought to Gödel's attention until almost the last > minute, and the new translation was still not wholly to his liking ... when > informed that there was not time enough to consider substituting another > text, he declared that Mendelson's translation was 'on the whole very good' > and agreed to its publication.
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In work with Aaron Clauset, David Kempe, and Dimitris Achlioptas, Moore showed that the appearance of power laws in the degree distribution of networks can be illusory: network models such as the Erdős–Rényi model, whose degree distribution does not obey a power law, may nevertheless appear to exhibit one when measured using traceroute-like tools.. In work with Clauset and Mark Newman, Moore developed a probabilistic model of hierarchical clustering for complex networks, and showed that their model predicts clustering robustly in the face of changes to the link structure of the network... Other topics in Moore's research include modeling undecidable problems by physical systems,. phase transitions in random instances of the Boolean satisfiability problem, the unlikelihood of success in the search for extraterrestrial intelligence due to the indistinguishability of advanced signaling technologies from random noise,.. the inability of certain types of quantum algorithm to solve graph isomorphism, and attack-resistant quantum cryptography..
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In the following simple stack implementation in Java, each element popped from the stack becomes semantic garbage once there are no outside references to it: public class Stack { private Object[] elements; private int size; public Stack(int capacity) { elements = new Object[capacity]; } public void push(Object e) { elements[size++] = e; } public Object pop() { return elements[--size]; } } This is because `elements[]` still contains a reference to the object, but the object will never be accessed again through this reference, because `elements[]` is private to the class and the `pop` method only returns references to elements it has not already popped. (After it decrements `size`, this class will never access that element again.) However, knowing this requires analysis of the code of the class, which is undecidable in general. If a later `push` call re-grows the stack to the previous size, overwriting this last reference, then the object will become syntactic garbage, because it can never be accessed again, and will be eligible for garbage collection.
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Gödel specifically cites Richard's paradox and the liar paradox as semantical analogues to his syntactical incompleteness result in the introductory section of "On Formally Undecidable Propositions in Principia Mathematica and Related Systems I". The liar paradox is the sentence "This sentence is false." An analysis of the liar sentence shows that it cannot be true (for then, as it asserts, it is false), nor can it be false (for then, it is true). A Gödel sentence G for a system F makes a similar assertion to the liar sentence, but with truth replaced by provability: G says "G is not provable in the system F." The analysis of the truth and provability of G is a formalized version of the analysis of the truth of the liar sentence. It is not possible to replace "not provable" with "false" in a Gödel sentence because the predicate "Q is the Gödel number of a false formula" cannot be represented as a formula of arithmetic.
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Another way of stating Rice's theorem that is more useful in computability theory follows. Let S be a set of languages that is nontrivial, meaning # there exists a Turing machine that recognizes a language in S, # there exists a Turing machine that recognizes a language not in S. Then it is undecidable to determine whether the language recognized by an arbitrary Turing machine lies in S. In practice, this means that there is no machine that can always decide whether the language of a given Turing machine has a particular nontrivial property. Special cases include the undecidability of whether a Turing machine accepts a particular string, whether a Turing machine recognizes a particular recognizable language, and whether the language recognized by a Turing machine could be recognized by a nontrivial simpler machine, such as a finite automaton. It is important to note that Rice's theorem does not say anything about those properties of machines or programs that are not also properties of functions and languages.
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See general set theory for more details. Q is fascinating because it is a finitely axiomatized first-order theory that is considerably weaker than Peano arithmetic (PA), and whose axioms contain only one existential quantifier, yet like PA is incomplete and incompletable in the sense of Gödel's incompleteness theorems, and essentially undecidable. Robinson (1950) derived the Q axioms (1)–(7) above by noting just what PA axioms are required to prove (Mendelson 1997: Th. 3.24) that every computable function is representable in PA. The only use this proof makes of the PA axiom schema of induction is to prove a statement that is axiom (3) above, and so, all computable functions are representable in Q (Mendelson 1997: Th. 3.33, Rautenberg 2010: 246). The conclusion of Gödel's second incompleteness theorem also holds for Q: no consistent recursively axiomatized extension of Q can prove its own consistency, even if we additionally restrict Gödel numbers of proofs to a definable cut (Bezboruah and Shepherdson 1976; Pudlák 1985; Hájek & Pudlák 1993:387).
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Throughout her work, Johnson emphasizes both the difficulty of applying deconstruction to political action and of separating linguistic contradictions, complexities, and polysemy from political questions. In A World of Difference, she makes a turn to a “real world,” but one which is always left in quotation marks—"real," but nonetheless inseparable from its textual, written aspect. In a chapter of the book entitled, “Is Writerliness Conservative?” Johnson examines the political implications of undecidablility in writing, as well as the consequences of labeling the poetic and the undecidable as politically inert. She writes that, if “poetry makes nothing happen,” poetry also “makes nothing happen”—the limits of the political are themselves fraught with political implications (p. 30). Harold Schweizer writes in his introduction to The Wake of Deconstruction that “[i]f interpretive closure always violates textual indeterminacy, if authority is perhaps fundamentally non-textual, reducing to identity what should remain different, Johnson’s work could best be summarized as an attempt to delay the inevitable reductionist desire for meaning” (p. 8).
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Gödel's first incompleteness theorem first appeared as "Theorem VI" in Gödel's 1931 paper "On Formally Undecidable Propositions of Principia Mathematica and Related Systems I". The hypotheses of the theorem were improved shortly thereafter by J. Barkley Rosser (1936) using Rosser's trick. The resulting theorem (incorporating Rosser's improvement) may be paraphrased in English as follows, where "formal system" includes the assumption that the system is effectively generated. > First Incompleteness Theorem: "Any consistent formal system F within which a > certain amount of elementary arithmetic can be carried out is incomplete; > i.e., there are statements of the language of F which can neither be proved > nor disproved in F." (Raatikainen 2015) The unprovable statement GF referred to by the theorem is often referred to as "the Gödel sentence" for the system F. The proof constructs a particular Gödel sentence for the system F, but there are infinitely many statements in the language of the system that share the same properties, such as the conjunction of the Gödel sentence and any logically valid sentence.
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Radó's 1962 paper proved that if f: ℕ → ℕ is any computable function, then Σ(n) > f(n) for all sufficiently large n, and hence that Σ is not a computable function. Moreover, this implies that it is undecidable by a general algorithm whether an arbitrary Turing machine is a busy beaver. (Such an algorithm cannot exist, because its existence would allow Σ to be computed, which is a proven impossibility. In particular, such an algorithm could be used to construct another algorithm that would compute Σ as follows: for any given n, each of the finitely many n-state 2-symbol Turing machines would be tested until an n-state busy beaver is found; this busy beaver machine would then be simulated to determine its score, which is by definition Σ(n).) Even though Σ(n) is an uncomputable function, there are some small n for which it is possible to obtain its values and prove that they are correct. It is not hard to show that Σ(0) = 0, Σ(1) = 1, Σ(2) = 4, and with progressively more difficulty it can be shown that Σ(3) = 6 and Σ(4) = 13 .
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The ease of designing reversible block cellular automata, and of testing block cellular automata for reversibility, is in strong contrast to cellular automata with other non-block neighborhood structures, for which it is undecidable whether the automaton is reversible and for which the reverse dynamics may require much larger neighborhoods than the forward dynamics. Any reversible cellular automaton may be simulated by a reversible block cellular automaton with a larger number of states; however, because of the undecidability of reversibility for non-block cellular automata, there is no computable bound on the radius of the regions in the non-block automaton that correspond to blocks in the simulation, and the translation from a non-block rule to a block rule is also not computable.; Block cellular automata are also a convenient formalism in which to design rules that, in addition to reversibility, implement conservation laws such as the conservation of particle number, conservation of momentum, etc.. For instance, if the rule within each block preserves the number of live cells in the block, then the global evolution of the automaton will also preserve the same number. This property is useful in the applications of cellular automata to physical simulation.
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