So, I've been through a period of time where, somehow, everybody talking about me says, 'But, he's just decidable like anybody else.
|
|
In some ways, it is the only argument worth having, since the specific cases are not decidable in advance in one way or another.
|
|
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.
|
|
The deadlock problem consists in deciding whether there is a reachable configuration without successor. This problem is decidable over lossy channel system and trivially decidable over machine capable of insertion of errors. It is also decidable over counter machine.
|
|
The Bernays–Schönfinkel class of first-order formulas is also decidable. Decidable subsets of first-order logic are also studied in the framework of description logics.
|
|
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.
|
|
An undecidable problem is a problem that is not decidable.
|
|
This fact allowed Tarski to prove that Euclidean geometry is decidable.
|
|
It is decidable whether a predicate defined in Presburger arithmetic is regular.
|
|
In model theory, Tarski's exponential function problem asks whether the theory of the real numbers together with the exponential function is decidable. Alfred Tarski had previously shown that the theory of the real numbers (without the exponential function) is decidable.
|
|
MISRA C:2012 classifies the rules (but not the directives) as Decidable or Undecidable.
|
|
For example one may speak of languages decidable on a non-deterministic Turing machine. Therefore, whenever an ambiguity is possible, the synonym for "recursive language" used is Turing-decidable language, rather than simply decidable. The class of all recursive languages is often called R, although this name is also used for the class RP. This type of language was not defined in the Chomsky hierarchy of . All recursive languages are also recursively enumerable.
|
|
The boundedness problem consists in deciding whether the set of reachable configuration is finite. I.e. the length of the content of each channel is bounded. This problem is trivially decidable over machine capable of insertion of errors. It is also decidable over counter machine.
|
|
The recursively enumerable sets, although not decidable in general, have been studied in detail in recursion theory.
|
|
Whereas the graph isomorphism problem is not known to be decidable in polynomial time and not known to be NP-complete, the fractional graph isomorphism problem is decidable in polynomial time because it is a special case of the linear programming problem, for which there is an efficient solution.
|
|
Whether a particular theory is decidable or not depends whether the theory is variable-free or on other conditions.
|
|
If satisfiability were also a semi- decidable problem, then the problem of the existence of counter-models would be too (a formula has counter-models iff its negation is satisfiable). So the problem of logical validity would be decidable, which contradicts the Church- Turing theorem, a result stating the negative answer for the Entscheidungsproblem.
|
|
This generalization is commonly called satisfiability modulo theories. The question whether a sentence in propositional logic is satisfiable is a decidable problem. In general, the question whether sentences in first-order logic are satisfiable is not decidable. In universal algebra and equational theory, the methods of term rewriting, congruence closure and unification are used to attempt to decide satisfiability.
|
|
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 .
|
|
Every computable real function is continuous (Weihrauch 2000, p. 6). The arithmetic operations on real numbers are computable. There is a subset of the real numbers called the computable numbers, which by the results above is a real closed field. While the equality relation is not decidable, the greater-than predicate on unequal real numbers is decidable.
|
|
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.
|
|
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\.
|
|
In early model theory, quantifier elimination was used to demonstrate that various theories possess properties like decidability and completeness. A common technique was to show first that a theory admits elimination of quantifiers and thereafter prove decidability or completeness by considering only the quantifier-free formulas. This technique can be used to show that Presburger arithmetic is decidable. Theories could be decidable yet not admit quantifier elimination.
|
|
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.
|
|
He won the 2002 Gödel Prize "for proving that equivalence of deterministic pushdown automata is decidable". In 2003 he was awarded with the Gay-Lussac Humboldt Prize.
|
|
A finite- valued logic is decidable (sure to determine outcomes of the logic when it is applied to propositions) if and only if it has a computational semantics.
|
|
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.
|
|
By contrast, the Tarski–Seidenberg theorem says that the first-order theory of the real field is decidable, so it is not possible to remove the sine function entirely.
|
|
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.
|
|
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.
|
|
Term algebras can be shown decidable using quantifier elimination. The complexity of the decision problem is in NONELEMENTARY.Jeanne Ferrante; Charles W. Rackoff (1979). The Computational Complexity of Logical Theories. Springer.
|
|
Multiple variants of channel systems have been introduced. The two variants introduced below does not allow to simulate a Turing machine and thus allows multiple problem of interest to be decidable.
|
|
A formal system is said to be recursive (i.e. effective) or recursively enumerable if the set of axioms and the set of inference rules are decidable sets or semidecidable sets, respectively.
|
|
Static type checking for Turing-complete languages is inherently conservative. That is, if a type system is both sound (meaning that it rejects all incorrect programs) and decidable (meaning that it is possible to write an algorithm that determines whether a program is well-typed), then it must be incomplete (meaning there are correct programs, which are also rejected, even though they do not encounter runtime errors)."... any sound, decidable type system must be incomplete" —D. Remy (2017). p.
|
|
Later, this result was refined showing exponential space completeness of rank 2 intersection type inhabitation and undecidability of rank 3 intersection type inhabitation. Remarkably, principal type inhabitation is decidable in polynomial time.
|
|
Real polynomial arithmetic, also known as the theory of real closed fields, is decidable; this is the Tarski–Seidenberg theorem, which has been implemented in computers by using the cylindrical algebraic decomposition.
|
|
As a qualitative property, safety of a FD-DEVS network is decidable by (1) generating RG of the given network and (2) checking whether some bad states are reachable or not [HZ06b].
|
|
Around 1945, Alfred Tarski asked whether the free groups on two or more generators have the same first order theory, and whether this theory is decidable. answered the first question by showing that any two nonabelian free groups have the same first order theory, and answered both questions, showing that this theory is decidable. A similar unsolved (as of 2011) question in free probability theory asks whether the von Neumann group algebras of any two non-abelian finitely generated free groups are isomorphic.
|
|
In this sense, the word problem on a lattice has a solution, namely, the set of all equivalent words is the free lattice. One of the most deeply studied cases of the word problem is in the theory of semigroups and groups. There are many groups for which the word problem is not decidable, in that there is no Turing machine that can determine the equivalence of two arbitrary words in a finite time. The word problem on ground terms is not decidable.
|
|
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, .
|
|
In the case of classical propositional logic, satisfiability is decidable for propositional formulae. In particular, satisfiability is an NP-complete problem, and is one of the most intensively studied problems in computational complexity theory.
|
|
Given a system S, there is no algorithm which computes a finite state machine representing R(S) for the class of lossy channel system. This problem is decidable over machine capable of insertion of error .
|
|
If L1, L2 both are regular tree languages, then the tree sets L1 ∩ L2, L1 ∪ L2, and L1 \ L2 are also regular tree languages, and it is decidable whether L1 ⊆ L2, and whether L1 = L2.
|
|
Their proofs demonstrate a connection between the unsolvability of the decision problem for first-order logic and the unsolvability of the halting problem. There are systems weaker than full first-order logic for which the logical consequence relation is decidable. These include propositional logic and monadic predicate logic, which is first-order logic restricted to unary predicate symbols and no function symbols. Other logics with no function symbols which are decidable are the guarded fragment of first-order logic, as well as two-variable logic.
|
|
As a partial converse, proved that, whenever a family of graphs has a decidable MSO2 satisfiability problem, the family must have bounded treewidth. The proof is based on a theorem of Robertson and Seymour that the families of graphs with unbounded treewidth have arbitrarily large grid minors. Seese also conjectured that every family of graphs with a decidable MSO1 satisfiability problem must have bounded clique-width; this has not been proven, but a weakening of the conjecture that extends MSO1 with modular counting predicates is true.
|
|
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.
|
|
As a partial converse, proved that, whenever a family of graphs has a decidable MSO2 satisfiability problem, the family must have bounded treewidth. The proof is based on a theorem of Robertson and Seymour that the families of graphs with unbounded treewidth have arbitrarily large grid minors.. Seese also conjectured that every family of graphs with a decidable MSO1 satisfiability problem must have bounded clique-width; this has not been proven, but a weakening of the conjecture that replaces MSO1 by CMSO1 is true..
|
|
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.
|
|
In predicate logic, a predicate P over some domain is called decidable if for every x in the domain, either P(x) is true, or P(x) is not true. This is not trivially true constructively. For a decidable predicate P over the natural numbers, Markov's principle then reads: :(\forall n (P(n) \vee eg P(n)) \wedge eg \forall n\, eg P(n)) \rightarrow \exists n\, P(n) That is, if P cannot be false for all natural numbers n, then it is true for some n.
|
|
Strictly speaking, the theory of the additive natural numbers did not admit quantifier elimination, but it was an expansion of the additive natural numbers that was shown to be decidable. Whenever a theory is decidable, and the language of its valid formulas is countable, it is possible to extend the theory with countably many relations to have quantifier elimination (for example, one can introduce, for each formula of the theory, a relation symbol that relates the free variables of the formula). Example: Nullstellensatz for algebraically closed fields and for differentially closed fields.
|
|
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.
|
|
As a consequence of the complementation it is decidable whether a deterministic PDA accepts all words over its input alphabet, by testing its complement for emptiness. This is not possible for context-free grammars (hence not for general PDA).
|
|
Antwerp: ICALP, 1984 Boundedness, deadlocks, and unspecified reception state are all decidable in polynomial time (which means that a particular problem can be solved in tractable, not infinite, amount of time) since the decision problems regarding them are nondeterministic logspace complete.
|
|
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.
|
|
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.
|
|
BAN logic, and logics in the same family, are decidable: there exists an algorithm taking BAN hypotheses and a purported conclusion, and that answers whether or not the conclusion is derivable from the hypotheses. The proposed algorithms use a variant of magic sets.
|
|
Some first-order theories are algorithmically decidable; examples of this include Presburger arithmetic, real closed fields and static type systems of many programming languages. The general first-order theory of the natural numbers expressed in Peano's axioms cannot be decided with an algorithm, however.
|
|
The remaining 4 are too loosely formulated to be stated as solved or not. In 1929 and 1930, it was proved the truth or falsity of all statements formulated about the natural numbers plus one of addition and multiplication, was decidable, i.e. could be determined by some algorithm.
|
|
Sakarovitch (2009) p.171 Regular languages of star-height 0 are also known as star-free languages. The theorem of Schützenberger provides an algebraic characterization of star-free languages by means of aperiodic syntactic monoids. In particular star-free languages are a proper decidable subclass of regular languages.
|
|
Methods for checking both safety and liveness properties have been developed and intensively studied over the last 20 years. It has been shown that the state reachability problem for timed automata is decidable,Rajeev Alur , David L. Dill. 1994 A Theory of Timed Automata. In Theoretical Computer Science, vol.
|
|
In 1930, he proved this theory decidable because it can be mapped into another theory he had already proved decidable, namely his first-order theory of the real numbers. In 1929 he showed that much of Euclidean solid geometry could be recast as a first- order theory whose individuals are spheres (a primitive notion), a single primitive binary relation "is contained in", and two axioms that, among other things, imply that containment partially orders the spheres. Relaxing the requirement that all individuals be spheres yields a formalization of mereology far easier to exposit than Lesniewski's variant. Near the end of his life, Tarski wrote a very long letter, published as Tarski and Givant (1999), summarizing his work on geometry.
|
|
Computable model theory is a branch of model theory which deals with questions of computability as they apply to model-theoretical structures. Computable model theory introduces the ideas of computable and decidable models and theories and one of the basic problems is discovering whether or not computable or decidable models fulfilling certain model-theoretic conditions can be shown to exist. Computable model theory was developed almost simultaneously by mathematicians in the West, primarily located in the United States and Australia, and Soviet Russia during the middle of the 20th century. Because of the Cold War there was little communication between these two groups and so a number of important results were discovered independently.
|
|
For example, whether a machine runs for more than 100 steps on a particular input is a decidable property, even though it is non-trivial. Implementing exactly the same language, two different machines might require a different number of steps to recognize the same input. Similarly, whether a machine has more than 5 states is a decidable property of the machine, as the number of states can simply be counted. Where a property is of the kind that either of the two machines may or may not have it, while still implementing exactly the same language, the property is of the machines and not of the language, and Rice's Theorem does not apply.
|
|
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).
|
|
For some fixed value k, rank-k polymorphism is a system in which a quantifier may not appear to the left of k or more arrows (when the type is drawn as a tree). Type inference for rank-2 polymorphism is decidable, but reconstruction for rank-3 and above is not.
|
|
In the absence of replication/recursion, the -calculus ceases to be Turing- powerful. This can be seen by the fact that bisimulation equivalence becomes decidable for the recursion-free calculus and even for the finite-control -calculus where the number of parallel components in any process is bounded by a constant.
|
|
In the language of computability theory, Markov's principle is a formal expression of the claim that if it is impossible that an algorithm does not terminate, then it does terminate. This is equivalent to the claim that if a set and its complement are both computably enumerable, then the set is decidable.
|
|
Computing the output of a given Boolean circuit on a specific input is P-complete problem. If the input is an integer circuit, however, it is unknown whether this problem is decidable. Circuit complexity attempts to classify Boolean functions with respect to the size or depth of circuits that can compute them.
|
|
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.
|
|
The stricter requirement of DLOGTIME-uniformity is of particular interest in the study of shallow-depth circuit-classes such as AC0 or TC0. When no resource bounds are specified, a language is recursive (i.e., decidable by a Turing machine) if and only if the language is decided by a uniform family of Boolean circuits.
|
|
Given two CFGs, can the first one generate all strings that the second one can generate? If this problem was decidable, then language equality could be decided too: two CFGs G1 and G2 generate the same language if L(G1) is a subset of L(G2) and L(G2) is a subset of L(G1).
|
|
G. Japaridze, "Decidable and enumerable predicate logics of provability". Studia Logica 49 (1990), pages 7-21. In the same paper he showed that, on the condition of the 1-completeness of the underlying arithmetical theory, predicate provability logic with non-iterated modalities is recursively enumerable. InG. Japaridze, "Predicate provability logic with non-modalized quantifiers".
|
|
As a weak counterexample, suppose θ(x) is some decidable predicate of a natural number such that it is not known whether any x satisfies θ. For example, θ may say that x is a formal proof of some mathematical conjecture whose provability is not known. Let φ the formula . Then is trivially provable.
|
|
A classic example of a decidable decision problem is the set of prime numbers. It is possible to effectively decide whether a given natural number is prime by testing every possible nontrivial factor. Although much more efficient methods of primality testing are known, the existence of any effective method is enough to establish decidability.
|
|
Sub-linear time algorithms are typically randomized, and provide only approximate solutions. In fact, the property of a binary string having only zeros (and no ones) can be easily proved not to be decidable by a (non- approximate) sub-linear time algorithm. Sub-linear time algorithms arise naturally in the investigation of property testing.
|
|
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).
|
|
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.
|
|
A major open problem in finite semigroup theory is the decidability of complexity: is there an algorithm that will compute the Krohn–Rhodes complexity of a finite semigroup, given its multiplication table? Upper bounds and ever more precise lower bounds on complexity have been obtained (see, e.g. Rhodes & Steinberg, 2009). Rhodes has conjectured that the problem is decidable.
|
|
Unlike Peano arithmetic, Presburger arithmetic is a decidable theory. This means it is possible to algorithmically determine, for any sentence in the language of Presburger arithmetic, whether that sentence is provable from the axioms of Presburger arithmetic. The asymptotic running-time computational complexity of this decision problem is at least doubly exponential, however, as shown by .
|
|
However, there are many interesting special cases that are decidable. In particular, it is possible to reason about the behaviour of a network of finite-state machines. One example is telling whether a given network of interacting (asynchronous and non-deterministic) finite-state machines can reach a deadlock. This problem is PSPACE-complete,, Section 19.3. i.e.
|
|
As a qualitative property, liveness of a FD- DEVS network is decidable by (1) generating RG of the given network, (2) from RG, generating kernel directed acyclic graph (KDAG) in which a vertex is strongly connected component, and (3) checking if a vertex of KDAG contains a state transition cycle which contains a set of liveness states[HZ06b].
|
|
When extending the type inference for the simply-typed lambda calculus towards polymorphism, one has to define when deriving an instance of a value is admissible. Ideally, this would be allowed with any use of a bound variable, as in: (λ id . ... (id 3) ... (id "text") ... ) (λ x . x) Unfortunately, type inference in polymorphic lambda calculus is not decidable.
|
|
It might seem that the potentially infinite memory capacity is an unrealizable attribute, but any decidable problem solved by a Turing machine will always require only a finite amount of memory. So in principle, any problem that can be solved (decided) by a Turing machine can be solved by a computer that has a finite amount of memory.
|
|
In 1924, he and Stefan Banach proved that, if one accepts the Axiom of Choice, a ball can be cut into a finite number of pieces, and then reassembled into a ball of larger size, or alternatively it can be reassembled into two balls whose sizes each equal that of the original one. This result is now called the Banach–Tarski paradox. In A decision method for elementary algebra and geometry, Tarski showed, by the method of quantifier elimination, that the first-order theory of the real numbers under addition and multiplication is decidable. (While this result appeared only in 1948, it dates back to 1930 and was mentioned in Tarski (1931).) This is a very curious result, because Alonzo Church proved in 1936 that Peano arithmetic (the theory of natural numbers) is not decidable.
|
|
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.
|
|
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.
|
|
Presburger arithmetic can be extended to include multiplication by constants, since multiplication is repeated addition. Most array subscript calculations then fall within the region of decidable problems. This approach is the basis of at least five proof-of-correctness systems for computer programs, beginning with the Stanford Pascal Verifier in the late 1970s and continuing through to Microsoft's Spec# system of 2005.
|
|
David Hilbert supervised the preparation of his doctoral thesis, Die Antinomie der transfiniten Zahl und ihre Auflösung durch die Theorie von Russell und Whitehead. In 1922 Behmann proved that the monadic predicate calculus is decidable. In 1938 he obtained a professorial chair in mathematics at Halle (Saale). In 1945 he was dismissed for having been a member of the NSDAP.
|
|
As an alternative analysis method against the sampling-based simulation method, an exhaustive generating behavior approach, generally called verification has been applied for analysis of DEVS models. It is proven that infinite states of a given DEVS model (especially a coupled DEVS model ) can be abstracted by behaviorally isomorphic finite structure, called a reachability graph when the given DEVS model is a sub-class of DEVS such as Schedule-Preserving DEVS (SP-DEVS), Finite & Deterministic DEVS (FD- DEVS) [HZ09], and Finite & Real-time DEVS (FRT-DEVS) [Hwang12]. As a result, based on the rechability graph, (1) dead-lock and live-lock freeness as qualitative properties are decidable with SP-DEVS [Hwang05], FD-DEVS [HZ06], and FRT-DEVS [Hwang12]; and (2) min/max processing time bounds as a quantitative property are decidable with SP-DEVS so far by 2012.
|
|
Size-change graphs express both the possible presence of a function call as well as whether parameters within function call decrease or do not increase. In order to derive termination from size-change graphs, Lee at al. formulate a sufficient condition in terms of the graphs (with no reference to the underlying program). This condition is decidable by an algorithm that operates solely on the graphs.
|
|
A proof of x=y is simply the trivial algorithm if x evaluates to the same number that y does (which is always decidable for natural numbers), otherwise there is no proof. These are then built up by induction into more complex algorithms. If one takes lambda calculus as defining the notion of a function, then the BHK interpretation describes the correspondence between natural deduction and functions.
|
|
In type-based program analysis polymorphic recursion is often essential in gaining high precision of the analysis. Notable examples of systems employing polymorphic recursion include Dussart, Henglein and Mossin's binding-time analysis and the Tofte-Talpin region-based memory management system. As these systems assume the expressions have already been typed in an underlying type system (not necessary employing polymorphic recursion), inference can be made decidable again.
|
|
In mathematical logic the theory of pure equality is a first-order theory. It has a signature consisting of only the equality relation symbol, and includes no non-logical axioms at all (Monk 1976:240-242). This theory is consistent, as any set with the usual equality relation provides an interpretation. The theory of pure equality was proven to be decidable by Löwenheim in 1915.
|
|
We say that s is decidable if both s and its complement –s are recursively enumerable. An extension of such a theory to the general case of the L-subsets is possible (see Gerla 2006). The proposed definitions are well related with fuzzy logic. Indeed, the following theorem holds true (provided that the deduction apparatus of the considered fuzzy logic satisfies some obvious effectiveness property).
|
|
Nominal unification is efficiently decidable. This fact led to the development of alphaProlog, a Prolog-like logic programming language with facilities for binding names in terms, where Prolog's standard first-order unification algorithm is replaced with nominal unification. Nominal term embeddings may be seen as alternatives to de Bruijn encodings and higher-order abstract syntax, where the latter uses the simply typed lambda calculus as a metalanguage.
|
|
It is decidable in polynomial-time whether an UFA's language is a subset of another UFA's language. Let A and B be two UFAs. Let L(A) and L(B) be the languages accepted by those automata. Then L(A)⊆L(B) if and only if L(A∩B)=L(A), where A∩B denotes the Cartesian product automaton, which can be proven to be also unambiguous.
|
|
This gap was closed in 1978 by Vaughan Pratt who showed that PDL was decidable in deterministic exponential time. In 1977, Krister Segerberg proposed a complete axiomatization of PDL, namely any complete axiomatization of modal logic K together with axioms A1-A6 as given above. Completeness proofs for Segerberg's axioms were found by Gabbay (unpublished note), Parikh (1978), Pratt (1979), and Kozen and Parikh (1981).
|
|
If Heath-Brown's conjecture is true, the problem is decidable. In this case, an algorithm could correctly solve the problem by computing n modulo 9, returning false when this is 4 or 5, and otherwise returning true. Heath-Brown's research also includes more precise conjectures on how far an algorithm would have to search to find an explicit representation rather than merely determining whether one exists.
|
|
In computational complexity theory, the parallel computation thesis is a hypothesis which states that the time used by a (reasonable) parallel machine is polynomially related to the space used by a sequential machine. The parallel computation thesis was set forth by Chandra and Stockmeyer in 1976. In other words, for a computational model which allows computations to branch and run in parallel without bound, a formal language which is decidable under the model using no more than t(n) steps for inputs of length n is decidable by a non-branching machine using no more than t(n)^k units of storage for some constant k. Similarly, if a machine in the unbranching model decides a language using no more than s(n) storage, a machine in the parallel model can decide the language in no more than s(n)^k steps for some constant k.
|
|
The first specific occurrence of aperiodic tilings arose in 1961, when logician Hao Wang tried to determine whether the Domino Problem is decidable — that is, whether there exists an algorithm for deciding if a given finite set of prototiles admits a tiling of the plane. Wang found algorithms to enumerate the tilesets that cannot tile the plane, and the tilesets that tile it periodically; by this he showed that such a decision algorithm exists if every finite set of prototiles that admits a tiling of the plane also admits a periodic tiling. In 1964 Robert Berger found an aperiodic set of prototiles from which he demonstrated that the tiling problem is in fact not decidable.. This first such set, used by Berger in his proof of undecidability, required 20,426 Wang tiles. Berger later reduced his set to 104, and Hans Läuchli subsequently found an aperiodic set requiring only 40 Wang tiles.
|
|
The theory of Abelian groups is decidable, but that of non-Abelian groups is not. In the 1920s and 30s, Tarski often taught high school geometry. Using some ideas of Mario Pieri, in 1926 Tarski devised an original axiomatization for plane Euclidean geometry, one considerably more concise than Hilbert's. Tarski's axioms form a first-order theory devoid of set theory, whose individuals are points, and having only two primitive relations.
|
|
Markov's rule is the formulation of Markov's principle as a rule. It states that \exists n\;P(n) is derivable as soon as eg eg \exists n\;P(n) is, for P decidable. Formally, :\forall n (P(n)\lor eg P(n)),\ eg eg \exists n\;P(n)\ \vdash\ \exists n\;P(n) Anne S. TroelstraAnne S. Troelstra. Metamathematical Investigation of Intuitionistic Arithmetic and Analysis, Springer Verlag (1973), Theorem 4.2.
|
|
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.
|
|
Recently there are attempts at handling the theory of the real numbers with functions such as exp, sin by relaxing decidability to the weaker notion of quasi-decidability. A theory is quasi-decidable if there is a procedure that decides satisfiability but that may run forever for inputs that are not robust in a certain, well-defined sense. Such a procedure exists for systems of n equations in n variables ().
|
|
In computer programming, predicate dispatch is a generalisation of multiple dispatch ("multimethods") that allows the method to call to be selected at runtime based on arbitrary decidable logical predicates and/or pattern matching attached to a method declaration. Raku supports predicate dispatch using "where" clauses that can execute arbitrary code against any function or method parameter. Julia has a package for it with PatternDispatch.jl but otherwise natively supports multiple dispatch.
|
|
On the topic of infinite chess, Hamkins, Brumleve and Schlicht proved that the mate-in-n problem of infinite chess is decidable. Hamkins and Evans investigated transfinite game values in infinite chess, proving that every countable ordinal arises as the game value of a position in infinite three-dimensional chess.C. D. A. Evans and J. D. Hamkins, "Transfinite game values in infinite chess," Integers, volume 14, Paper No. G2, 36, 2014.
|
|
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.
|
|
In computer science, specifically in the field of formal verification, well- structured transition systems (WSTSs) are a general class of infinite state systems for which many verification problems are decidable, owing to the existence of a kind of order between the states of the system which is compatible with the transitions of the system. WSTS decidability results can be applied to Petri nets, lossy channel systems, and more.
|
|
The strong reductions listed above restrict the manner in which oracle information can be accessed by a decision procedure but do not otherwise limit the computational resources available. Thus if a set A is decidable then A is reducible to any set B under any of the strong reducibility relations listed above, even if A is not polynomial-time or exponential-time decidable. This is acceptable in the study of recursion theory, which is interested in theoretical computability, but it is not reasonable for computational complexity theory, which studies which sets can be decided under certain asymptotical resource bounds. The most common reducibility in computational complexity theory is polynomial-time reducibility; a set A is polynomial-time reducible to a set B if there is a polynomial-time function f such that for every n, n is in A if and only if f(n) is in B. This reducibility is, essentially, a resource-bounded version of many-one reducibility.
|
|
This seminar, in essence, started a new and extremely fruitful school in model theory and decidability of elementary theories. During the early 1960s, Maltsev worked on problems of decidability of elementary theories of various algebraic structures. He showed the undecidability of the elementary theory of finite groups, of free nilpotent groups, of free soluble groups and many others. He also proved that the class of locally free algebras has a decidable theory.
|
|
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.
|
|
Any "axiomatizable" fuzzy theory is recursively enumerable. In particular, the fuzzy set of logically true formulas is recursively enumerable in spite of the fact that the crisp set of valid formulas is not recursively enumerable, in general. Moreover, any axiomatizable and complete theory is decidable. It is an open question to give supports for a "Church thesis" for fuzzy mathematics, the proposed notion of recursive enumerability for fuzzy subsets is the adequate one.
|
|
These models are based on game semantics (Hyland and Ong, 2000; Abramsky, Jagadeesan, and Malacaria, 2000) and Kripke logical relations (O'Hearn and Riecke, 1995). For a time it was felt that neither of these models was completely satisfactory, since they were not effectively presentable. However, Ralph Loader demonstrated that no effectively presentable fully abstract model could exist, since the question of program equivalence in the finitary fragment of PCF is not decidable.
|
|
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.
|
|
These type systems do not have decidable type inference and are difficult to understand and program with. But dependent types can express arbitrary propositions in predicate logic. Through the Curry–Howard isomorphism, then, well-typed programs in these languages become a means of writing formal mathematical proofs from which a compiler can generate certified code. While these languages are mainly of interest in academic research (including in formalized mathematics), they have begun to be used in engineering as well.
|
|
Then the theory at hand, which is assumed to prove certain facts about numbers, also proves facts about its own statements. Questions about the provability of statements are represented as questions about the properties of numbers, which would be decidable by the theory if it were complete. In these terms, the Gödel sentence states that no natural number exists with a certain, strange property. A number with this property would encode a proof of the inconsistency of the theory.
|
|
The most widely used subclass of the ω-languages is the set of ω-regular languages, which enjoy the useful property of being recognizable by Büchi automata. Thus the decision problem of ω-regular language membership is decidable using a Büchi automaton, and fairly straightforward to compute. If the language Σ is the power set of a set (called the "atomic propositions") then the ω-language is a linear time property, which are studied in model checking.
|
|
The notions of a "decidable subset" and "recursively enumerable subset" are basic ones for classical mathematics and classical logic. Thus the question of a suitable extension of them to fuzzy set theory is a crucial one. A first proposal in such a direction was made by E.S. Santos by the notions of fuzzy Turing machine, Markov normal fuzzy algorithm and fuzzy program (see Santos 1970). Successively, L. Biacino and G. Gerla argued that the proposed definitions are rather questionable.
|
|
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.
|
|
It can be CRCW, CREW, or EREW. See PRAM for descriptions of those models. Equivalently, NC can be defined as those decision problems decidable by a uniform Boolean circuit (which can be calculated from the length of the input, for NC, we suppose we can compute the Boolean circuit of size n in logarithmic space in n) with polylogarithmic depth and a polynomial number of gates. RNC is a class extending NC with access to randomness.
|
|
Parametric polymorphism implies that types themselves are passed as parameters as if they were proper values. Passed as arguments to a proper functions, but also into "type functions" as in the "parametric" type constants, leads to the question how to more properly type types themselves. Meta types are used to create an even more expressive type system. Unfortunately, only unification is not longer decidable in the presence of meta types, rendering type inference impossible in this extend of generality.
|
|
The class of languages described by operator-precedence grammars, i.e., operator-precedence languages, is strictly contained in the class of deterministic context-free languages, and strictly contains visibly pushdown languages. Operator-precedence languages enjoy many closure properties: union, intersection, complementation, concatenation, and they are the largest known class closed under all these operations and for which the emptiness problem is decidable. Another peculiar feature of operator-precedence languages is their local parsability, that enables efficient parallel parsing.
|
|
In particular Moscow was trying to develop close relations with the new nationalist government of Turkey. In the end, the Soviet Union agreed to a division under which Zangezur would fall under the control of Armenia, while Karabakh and Nakhchivan would be under the control of Azerbaijan. "If the matter had been decidable without reference to the situation in the rest of Asia," says historian Robert Service, "Stalin would probably have left Karabagh inside Armenia despite Azerbaijani protests."Service, Robert.
|
|
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.
|
|
An important case is where X denotes some subclass of a bigger class of functions as studied in computability theory. Consider the subclass of total functions and note that being total is not a decidable property, i.e. there cannot be a constructive bijection between the total functions and the natural numbers. However, via enumeration of the codes of all possible partial functions (which also includes non-terminating functions), subsets of those, such as the total functions, are seen to be a subcountable sets.
|
|
The formula interpretation is such that whenever A is provable in Heyting arithmetic then there exists a sequence of closed terms t such that A_D(t; y) is provable in the system T. The sequence of terms t and the proof of A_D(t; y) are constructed from the given proof of A in Heyting arithmetic. The construction of t is quite straightforward, except for the contraction axiom A \rightarrow A \wedge A which requires the assumption that quantifier-free formulas are decidable.
|
|
In computational complexity theory, the class NC (for "Nick's Class") is the set of decision problems decidable in polylogarithmic time on a parallel computer with a polynomial number of processors. In other words, a problem is in NC if there exist constants c and k such that it can be solved in time using parallel processors. Stephen Cook coined the name "Nick's class" after Nick Pippenger, who had done extensive research on circuits with polylogarithmic depth and polynomial size.Arora & Barak (2009) p.
|
|
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.
|
|
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.
|
|
These cannot be replaced by any finite number of axioms, that is, Presburger arithmetic is not finitely axiomatizable in first-order logic. Presburger arithmetic can be viewed as first-order theory with equality containing precisely all consequences of the above axioms. Alternatively, it can be defined as the set of those sentences that are true in the intended interpretation: the structure of non-negative integers with constants 0, 1, and the addition of non-negative integers. Presburger arithmetic is designed to be complete and decidable.
|
|
In mathematics, logic and computer science, a formal language is called recursively enumerable (also recognizable, partially decidable, semidecidable, Turing-acceptable or Turing-recognizable) if it is a recursively enumerable subset in the set of all possible words over the alphabet of the language, i.e., if there exists a Turing machine which will enumerate all valid strings of the language. Recursively enumerable languages are known as type-0 languages in the Chomsky hierarchy of formal languages. All regular, context- free, context-sensitive and recursive languages are recursively enumerable.
|
|
With regard to Hilbert's problems posed by the famous mathematician David Hilbert in 1900, an aspect of problem #10 had been floating about for almost 30 years before it was framed precisely. Hilbert's original expression for #10 is as follows: By 1922, this notion of "Entscheidungsproblem" had developed a bit, and H. Behmann stated that By the 1928 international congress of mathematicians, Hilbert "made his questions quite precise. First, was mathematics complete ... Second, was mathematics consistent ... And thirdly, was mathematics decidable?" (Hodges p. 91, Hawking p. 1121).
|
|
Some of the most important methods and algorithms of the 20th century are: the simplex algorithm, the fast Fourier transform, error- correcting codes, the Kalman filter from control theory and the RSA algorithm of public-key cryptography. At the same time, deep insights were made about the limitations to mathematics. In 1929 and 1930, it was proved the truth or falsity of all statements formulated about the natural numbers plus one of addition and multiplication, was decidable, i.e. could be determined by some algorithm.
|
|
The question "Does any arbitrary "Diophantine equation" have an integer solution?" is undecidable.That is, it is impossible to answer the question for all cases. Franzén introduces Hilbert's tenth problem and the MRDP theorem (Matiyasevich- Robinson-Davis-Putnam theorem) which states that "no algorithm exists which can decide whether or not a Diophantine equation has any solution at all". MRDP uses the undecidability proof of Turing: "... the set of solvable Diophantine equations is an example of a computably enumerable but not decidable set, and the set of unsolvable Diophantine equations is not computably enumerable" (p. 71).
|
|
For some other operations, the abstraction may lose precision: for instance, it is impossible to know the sign of a sum whose operands are respectively positive and negative. Sometimes a loss of precision is necessary to make the semantics decidable (see Rice's theorem, halting problem). In general, there is a compromise to be made between the precision of the analysis and its decidability (computability), or tractability (computational cost). In practice the abstractions that are defined are tailored to both the program properties one desires to analyze, and to the set of target programs.
|
|
The absence of polyadic relation symbols severely restricts what can be expressed in the monadic predicate calculus. It is so weak that, unlike the full predicate calculus, it is decidable—there is a decision procedure that determines whether a given formula of monadic predicate calculus is logically valid (true for all nonempty domains).Heinrich Behmann, Beiträge zur Algebra der Logik, insbesondere zum Entscheidungsproblem, in Mathematische Annalen (1922)Löwenheim, L. (1915) "Über Möglichkeiten im Relativkalkül," Mathematische Annalen 76: 447-470. Translated as "On possibilities in the calculus of relatives" in Jean van Heijenoort, 1967.
|
|
Turing machines can decide any context-free language, in addition to languages not decidable by a push-down automaton, such as the language consisting of prime numbers. It is therefore a strictly more powerful model of computation. Because Turing machines have the ability to "back up" in their input tape, it is possible for a Turing machine to run for a long time in a way that is not possible with the other computation models previously described. It is possible to construct a Turing machine that will never finish running (halt) on some inputs.
|
|
The ordered real field R is a structure over the language of ordered rings Lor = (+,·,−,<,0,1), with the usual interpretation given to each symbol. It was proved by Tarski that the theory of the real field, Th(R), is decidable. That is, given any Lor-sentence φ there is an effective procedure for determining whether :\R\models\varphi. He then asked whether this was still the case if one added a unary function exp to the language that was interpreted as the exponential function on R, to get the structure Rexp.
|
|
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.
|
|
This was the first automated deduction system to demonstrate an ability to solve mathematical problems that were announced in the Notices of the American Mathematical Society before solutions were formally published. First-order theorem proving is one of the most mature subfields of automated theorem proving. The logic is expressive enough to allow the specification of arbitrary problems, often in a reasonably natural and intuitive way. On the other hand, it is still semi-decidable, and a number of sound and complete calculi have been developed, enabling fully automated systems.
|
|
Ethereum's smart contracts are based on different computer languages, which developers use to program their own functionalities. Smart contracts are high-level programming abstractions that are compiled down to EVM bytecode and deployed to the Ethereum blockchain for execution. They can be written in Solidity (a language library with similarities to C and JavaScript), Serpent (similar to Python, but deprecated), LLL (a low-level Lisp-like language), and Mutan (Go-based, but deprecated). There is also a research-oriented language under development called Vyper (a strongly-typed Python-derived decidable language).
|
|
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.
|
|
The method of quantifier elimination can be used to show that definable sets in particular theories cannot be too complicated. Tarski (1948) established quantifier elimination for real-closed fields, a result which also shows the theory of the field of real numbers is decidable. (He also noted that his methods were equally applicable to algebraically closed fields of arbitrary characteristic.) A modern subfield developing from this is concerned with o-minimal structures. Morley's categoricity theorem, proved by Michael D. Morley (1965), states that if a first-order theory in a countable language is categorical in some uncountable cardinality, i.e.
|
|
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.
|
|
A simpler, but related, problem is proof verification, where an existing proof for a theorem is certified valid. For this, it is generally required that each individual proof step can be verified by a primitive recursive function or program, and hence the problem is always decidable. Since the proofs generated by automated theorem provers are typically very large, the problem of proof compression is crucial and various techniques aiming at making the prover's output smaller, and consequently more easily understandable and checkable, have been developed. Proof assistants require a human user to give hints to the system.
|
|
Every decision problem can be converted into the function problem of computing the characteristic function of the set associated to the decision problem. If this function is computable then the associated decision problem is decidable. However, this reduction is more liberal than the standard reduction used in computational complexity (sometimes called polynomial-time many-one reduction); for example, the complexity of the characteristic functions of an NP-complete problem and its co-NP-complete complement is exactly the same even though the underlying decision problems may not be considered equivalent in some typical models of computation.
|
|
However, this doesn't prevent extensional type theory from being a basis for a practical tool, for example, NuPRL is based on extensional type theory. In contrast in intensional type theory type checking is decidable, but the representation of standard mathematical concepts is somewhat more cumbersome, since intensional reasoning requires using setoids or similar constructions. There are many common mathematical objects, which are hard to work with or can't be represented without this, for example, integer numbers, rational numbers, and real numbers. Integers and rational numbers can be represented without setoids, but this representation isn't easy to work with.
|
|
Maimonides himself held that neither creation nor Aristotle's infinite time were provable, or at least that no proof was available. (According to scholars of his work, he didn't make a formal distinction between unprovability and the simple absence of proof.) Thomas Aquinas was influenced by this belief, and held in his Summa Theologica that neither hypothesis was demonstrable. Some of Maimonides' Jewish successors, including Gersonides and Crescas, conversely held that the question was decidable, philosophically. John Philoponus was probably the first to use the argument that infinite time is impossible in order to establish temporal finitism.
|
|
Büchi arithmetic of base k is the first-order theory of the natural numbers with addition and the function V_k(x) which is defined as the largest power of k dividing x, named in honor of the Swiss mathematician Julius Richard Büchi. The signature of Büchi arithmetic contains only the addition operation, V_k and equality, omitting the multiplication operation entirely. Unlike Peano arithmetic, Büchi arithmetic is a decidable theory. This means it is possible to effectively determine, for any sentence in the language of Büchi arithmetic, whether that sentence is provable from the axioms of Büchi arithmetic.
|
|
Here he developed the arithmetic of the natural numbers by first defining objects by primitive recursion, then devising another system to prove properties of the objects defined by the first system. These two systems enabled him to define prime numbers and to set out a considerable amount of number theory. If the first of these systems can be considered as a programming language for defining objects, and the second as a programming logic for proving properties about the objects, Skolem can be seen as an unwitting pioneer of theoretical computer science. In 1929, Presburger proved that Peano arithmetic without multiplication was consistent, complete, and decidable.
|
|
The conjecture of Kontsevich and Zagier would imply that equality of periods is also decidable: inequality of computable reals is known recursively enumerable; and conversely if two integrals agree, then an algorithm could confirm so by trying all possible ways to transform one of them into the other one. It is not expected that Euler's number e and Euler-Mascheroni constant γ are periods. The periods can be extended to exponential periods by permitting the product of an algebraic function and the exponential function of an algebraic function as an integrand. This extension includes all algebraic powers of e, the gamma function of rational arguments, and values of Bessel functions.
|
|
In mathematics, logic and computer science, a formal language (a set of finite sequences of symbols taken from a fixed alphabet) is called recursive if it is a recursive subset of the set of all possible finite sequences over the alphabet of the language. Equivalently, a formal language is recursive if there exists a total Turing machine (a Turing machine that halts for every given input) that, when given a finite sequence of symbols as input, accepts it if it belongs to the language and rejects it otherwise. Recursive languages are also called decidable. The concept of decidability may be extended to other models of computation.
|
|
Since there is no general solution to the halting problem, total correctness is not decidable. A termination proof is a type of mathematical proof that plays a critical role in formal verification because total correctness of an algorithm depends on termination. For example, successively searching through integers 1, 2, 3, … to see if we can find an example of some phenomenon--say an odd perfect number--it is quite easy to write a partially correct program (using long division by two to check n as perfect or not). But to say this program is totally correct would be to assert something currently not known in number theory.
|
|
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.
|
|
Upon completing the Principia, three volumes of extraordinarily abstract and complex reasoning, Russell was exhausted, and he felt his intellectual faculties never fully recovered from the effort.The Autobiography of Bertrand Russell, the Early Years, p. 202. Although the Principia did not fall prey to the paradoxes in Frege's approach, it was later proven by Kurt Gödel that neither Principia Mathematica, nor any other consistent system of primitive recursive arithmetic, could, within that system, determine that every proposition that could be formulated within that system was decidable, i.e. could decide whether that proposition or its negation was provable within the system (See: Gödel's incompleteness theorem).
|
|
For a nondeterministic finite automaton A with n states and an m letter alphabet, it is decidable in time O(n2m) whether A is unambiguous. It suffices to use a fixpoint algorithm to compute the set of pairs of states q and q' such that there exists a word w which leads both to q and to q' . The automaton is unambiguous if and only if there is no such a pair such that both states are accepting. There are at most O(n2) state pairs, and for each pair there are m letters to consider to resume the fixpoint algorithm, hence the computation time.
|
|
Depending on the underlying logic, the problem of deciding the validity of a formula varies from trivial to impossible. For the frequent case of propositional logic, the problem is decidable but co-NP- complete, and hence only exponential-time algorithms are believed to exist for general proof tasks. For a first order predicate calculus, Gödel's completeness theorem states that the theorems (provable statements) are exactly the logically valid well-formed formulas, so identifying valid formulas is recursively enumerable: given unbounded resources, any valid formula can eventually be proven. However, invalid formulas (those that are not entailed by a given theory), cannot always be recognized.
|
|
A language is called computable (synonyms: recursive, decidable) if there is a computable function such that for each word over the alphabet, if the word is in the language and if the word is not in the language. Thus a language is computable just in case there is a procedure that is able to correctly tell whether arbitrary words are in the language. A language is computably enumerable (synonyms: recursively enumerable, semidecidable) if there is a computable function such that is defined if and only if the word is in the language. The term enumerable has the same etymology as in computably enumerable sets of natural numbers.
|
|
A type system defines how a programming language classifies values and expressions into types, how it can manipulate those types and how they interact. The goal of a type system is to verify and usually enforce a certain level of correctness in programs written in that language by detecting certain incorrect operations. Any decidable type system involves a trade-off: while it rejects many incorrect programs, it can also prohibit some correct, albeit unusual programs. In order to bypass this downside, a number of languages have type loopholes, usually unchecked casts that may be used by the programmer to explicitly allow a normally disallowed operation between different types.
|
|
The terminology for recursive functions and sets is not completely standardized. The definition in terms of μ-recursive functions as well as a different definition of rekursiv functions by Gödel led to the traditional name recursive for sets and functions computable by a Turing machine. The word decidable stems from the German word Entscheidungsproblem which was used in the original papers of Turing and others. In contemporary use, the term "computable function" has various definitions: according to Cutland (1980), it is a partial recursive function (which can be undefined for some inputs), while according to Soare (1987) it is a total recursive (equivalently, general recursive) function.
|
|
This subset of higher-order unification is decidable and solvable unification problems have most-general unifiers. Many computer systems that contain higher-order unification, such as the higher-order logic programming languages λProlog and Twelf, often implement only the pattern fragment and not full higher-order unification. In computational linguistics, one of the most influential theories of ellipsis is that ellipses are represented by free variables whose values are then determined using Higher-Order Unification (HOU). For instance, the semantic representation of "Jon likes Mary and Peter does too" is and the value of R (the semantic representation of the ellipsis) is determined by the equation .
|
|
Therefore, the system, which can prove certain facts about numbers, can also indirectly prove facts about its own statements, provided that it is effectively generated. Questions about the provability of statements within the system are represented as questions about the arithmetical properties of numbers themselves, which would be decidable by the system if it were complete. Thus, although the Gödel sentence refers indirectly to sentences of the system F, when read as an arithmetical statement the Gödel sentence directly refers only to natural numbers. It asserts that no natural number has a particular property, where that property is given by a primitive recursive relation (Smith 2007, p. 141).
|
|
Gödel's second incompleteness theorem also implies that a system F1 satisfying the technical conditions outlined above cannot prove the consistency of any system F2 that proves the consistency of F1. This is because such a system F1 can prove that if F2 proves the consistency of F1, then F1 is in fact consistent. For the claim that F1 is consistent has form "for all numbers n, n has the decidable property of not being a code for a proof of contradiction in F1". If F1 were in fact inconsistent, then F2 would prove for some n that n is the code of a contradiction in F1.
|
|
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.
|
|
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.
|
|
In the interest of precise language, a direct answer has been defined:Nuel Belnap & T.B. Steel Jr. (1976) The Logic of Questions and Answers, pages 3, 12 & 13, Yale University Press "A direct answer to a given question is a piece of language that completely, but just completely, answers the question...What is crucial is that it be effectively decidable whether a piece of language is a direct answer to a specific question." and "To each clear question there corresponds a set of statements which are directly responsive. ... A direct answer must provide an unarguably final resolution of the question." More information on these issues can be found in the articles yes–no question, yes and no, and answer ellipsis.
|
|
There are two equivalent major definitions for the concept of a recursive language: # A recursive formal language is a recursive subset in the set of all possible words over the alphabet of the language. # A recursive language is a formal language for which there exists a Turing machine that, when presented with any finite input string, halts and accepts if the string is in the language, and halts and rejects otherwise. The Turing machine always halts: it is known as a decider and is said to decide the recursive language. By the second definition, any decision problem can be shown to be decidable by exhibiting an algorithm for it that terminates on all inputs.
|
|
In computability theory, the Turing jump or Turing jump operator, named for Alan Turing, is an operation that assigns to each decision problem a successively harder decision problem with the property that is not decidable by an oracle machine with an oracle for . The operator is called a jump operator because it increases the Turing degree of the problem . That is, the problem is not Turing-reducible to . Post's theorem establishes a relationship between the Turing jump operator and the arithmetical hierarchy of sets of natural numbers.. Informally, given a problem, the Turing jump returns the set of Turing machines which halt when given access to an oracle that solves that problem.
|
|
This is a method for producing complete theories through the semantic route, with examples including the set of true sentences under the structure (N, +, ×, 0, 1, =), where N is the set of natural numbers, and the set of true sentences under the structure (R, +, ×, 0, 1, =), where R is the set of real numbers. The first of these, called the theory of true arithmetic, cannot be written as the set of logical consequences of any enumerable set of axioms. The theory of (R, +, ×, 0, 1, =) was shown by Tarski to be decidable; it is the theory of real closed fields (see Decidability of first-order theories of the real numbers for more).
|
|
Within this theory, it is possible to prove interesting statements such as "The complement of the Mandelbrot set is only partially decidable." These hypothetical computing machines can be viewed as idealised analog computers which operate on real numbers, whereas digital computers are limited to computable numbers. They may be further subdivided into differential and algebraic models (digital computers, in this context, should be thought of as topological, at least insofar as their operation on computable reals is concerned). Depending on the model chosen, this may enable real computers to solve problems that are inextricable on digital computers (For example, Hava Siegelmann's neural nets can have noncomputable real weights, making them able to compute nonrecursive languages.) or vice versa.
|
|
Because Presburger arithmetic is decidable, automatic theorem provers for Presburger arithmetic exist. For example, the Coq proof assistant system features the tactic omega for Presburger arithmetic and the Isabelle proof assistant contains a verified quantifier elimination procedure by . The double exponential complexity of the theory makes it infeasible to use the theorem provers on complicated formulas, but this behavior occurs only in the presence of nested quantifiers: Oppen and Nelson (1980) describe an automatic theorem prover which uses the simplex algorithm on an extended Presburger arithmetic without nested quantifiers to prove some of the instances of quantifier-free Presburger arithmetic formulas. More recent satisfiability modulo theories solvers use complete integer programming techniques to handle quantifier-free fragment of Presburger arithmetic theory ().
|
|
According to Maimonides, to argue that "because I have never observed something coming into existence without coming from a substratum it cannot occur" is equivalent to arguing that "because I cannot empirically observe eternity it does not exist." Maimonides himself held that neither creation nor Aristotle's infinite time were provable, or at least that no proof was available. (According to scholars of his work, he didn't make a formal distinction between unprovability and the simple absence of proof.) However, some of Maimonides' Jewish successors, including Gersonides and Crescas, conversely held that the question was decidable, philosophically. In the West, the 'Latin Averroists' were a group of philosophers writing in Paris in the middle of the thirteenth century, who included Siger of Brabant, Boethius of Dacia.
|
|
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.
|
|
Unification in Boolean rings is decidable, that is, algorithms exist to solve arbitrary equations over Boolean rings. Both unification and matching in finitely generated free Boolean rings are NP-complete, and both are NP-hard in finitely presented Boolean rings. (In fact, as any unification problem f(X) = g(X) in a Boolean ring can be rewritten as the matching problem f(X) + g(X) = 0, the problems are equivalent.) Unification in Boolean rings is unitary if all the uninterpreted function symbols are nullary and finitary otherwise (i.e. if the function symbols not occurring in the signature of Boolean rings are all constants then there exists a most general unifier, and otherwise the minimal complete set of unifiers is finite).
|
|
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.
|
|
A. N. Prior created modern temporal logic, closely related to modal logic, in 1957 by adding modal operators [F] and [P] meaning "eventually" and "previously". Vaughan Pratt introduced dynamic logic in 1976. In 1977, Amir Pnueli proposed using temporal logic to formalise the behaviour of continually operating concurrent programs. Flavors of temporal logic include propositional dynamic logic (PDL), propositional linear temporal logic (PLTL), linear temporal logic (LTL), computation tree logic (CTL), Hennessy–Milner logic, and T. The mathematical structure of modal logic, namely Boolean algebras augmented with unary operations (often called modal algebras), began to emerge with J. C. C. McKinsey's 1941 proof that S2 and S4 are decidable, and reached full flower in the work of Alfred Tarski and his student Bjarni Jónsson (Jónsson and Tarski 1951–52).
|
|
However, better complexities are known for the decision problem: Ben-Or, Kozen, and Reif (1986) proved that the theory of real closed fields is decidable in exponential space, and therefore in double exponential time. Moreover, the parameter that appears in the second exponent is not the size of the formula, nor the number of variables (as with cylindrical algebraic decomposition), but the number of the quantifier changes (from \exists to \forall and vice versa) in the prenex normal form of the input formula. For purely existential formulas, that is for formulas of the form : where stands for either or , the complexity is lower. Basu and Roy (1996) provided a well-behaved algorithm to decide the truth of such an exstential formula with complexity of arithmetic operations and polynomial space.
|
|
The main form of computability studied in recursion theory was introduced by Turing (1936). A set of natural numbers is said to be a computable set (also called a decidable, recursive, or Turing computable set) if there is a Turing machine that, given a number n, halts with output 1 if n is in the set and halts with output 0 if n is not in the set. A function f from the natural numbers to themselves is a recursive or (Turing) computable function if there is a Turing machine that, on input n, halts and returns output f(n). The use of Turing machines here is not necessary; there are many other models of computation that have the same computing power as Turing machines; for example the μ-recursive functions obtained from primitive recursion and the μ operator.
|
|
The problem of constructing practical algorithms to determine whether sentences with large numbers of propositional variables are tautologies is an area of contemporary research in the area of automated theorem proving. The method of truth tables illustrated above is provably correct – the truth table for a tautology will end in a column with only T, while the truth table for a sentence that is not a tautology will contain a row whose final column is F, and the valuation corresponding to that row is a valuation that does not satisfy the sentence being tested. This method for verifying tautologies is an effective procedure, which means that given unlimited computational resources it can always be used to mechanistically determine whether a sentence is a tautology. This means, in particular, the set of tautologies over a fixed finite or countable alphabet is a decidable set.
|
|
An alternative characterization of PSPACE is the set of problems decidable by an alternating Turing machine in polynomial time, sometimes called APTIME or just AP.Arora & Barak (2009) p.100 A logical characterization of PSPACE from descriptive complexity theory is that it is the set of problems expressible in second- order logic with the addition of a transitive closure operator. A full transitive closure is not needed; a commutative transitive closure and even weaker forms suffice. It is the addition of this operator that (possibly) distinguishes PSPACE from PH. A major result of complexity theory is that PSPACE can be characterized as all the languages recognizable by a particular interactive proof system, the one defining the class IP. In this system, there is an all-powerful prover trying to convince a randomized polynomial-time verifier that a string is in the language.
|
|
Gödel's second incompleteness theorem (see Gödel's incompleteness theorems), another celebrated result, shows that there are inherent limitations in what can be achieved with formal proofs in mathematics. The name for the incompleteness theorem refers to another meaning of complete (see model theory – Using the compactness and completeness theorems): A theory T is complete (or decidable) if for every formula f in the language of T either T\vdash f or T\vdash eg f. Gödel's second incompleteness theorem states that in any consistent effective theory T containing Peano arithmetic (PA), a formula CT like CT = eg (0 = 1) expressing the consistency of T cannot be proven within T. The completeness theorem implies the existence of a model of T in which the formula CT is false. Such a model (precisely, the set of "natural numbers" it contains) is necessarily a non-standard model, as it contains the code number of a proof of a contradiction of T. But T is consistent when viewed from the outside.
|
|
The halting problem is theoretically decidable for linear bounded automata (LBAs) or deterministic machines with finite memory. A machine with finite memory has a finite number of configurations, and thus any deterministic program on it must eventually either halt or repeat a previous configuration: :...any finite-state machine, if left completely to itself, will fall eventually into a perfectly periodic repetitive pattern. The duration of this repeating pattern cannot exceed the number of internal states of the machine... (italics in original, Minsky 1967, p. 24) Minsky notes, however, that a computer with a million small parts, each with two states, would have at least 21,000,000 possible states: :This is a 1 followed by about three hundred thousand zeroes ... Even if such a machine were to operate at the frequencies of cosmic rays, the aeons of galactic evolution would be as nothing compared to the time of a journey through such a cycle (Minsky 1967 p.
|
|
A well-structured systemParosh Aziz Abdulla, Kārlis Čerāns, Bengt Jonsson, Yih-Kuen Tsay: Algorithmic Analysis of Programs with Well Quasi-ordered Domains (2000), Information and Computation, Vol. 160 issues 1-2, pp. 109--127 is a transition system (S,\to) with state set S = Q \times D made up from a finite control state set Q, a data values set D, furnished with a decidable pre-order \leq \subseteq D \times D which is extended to states by (q,d)\le(q',d') \Leftrightarrow q=q' \wedge d\le d', which is well-structured as defined above (\to is monotonic, i.e. upward compatible, with respect to \le) and in addition has a computable set of minima for the set of predecessors of any upward closed subset of S. Well- structured systems adapt the theory of well-structured transition systems for modelling certain classes of systems encountered in computer science and provide the basis for decision procedures to analyse such systems, hence the supplementary requirements: the definition of a WSTS itself says nothing about the computability of the relations \le, \to.
|
|
Cobham's thesis, also known as Cobham–Edmonds thesis (named after Alan Cobham and Jack Edmonds), asserts that computational problems can be feasibly computed on some computational device only if they can be computed in polynomial time; that is, if they lie in the complexity class P. In modern terms, it identifies tractable problems with the complexity class P. Formally, to say that a problem can be solved in polynomial time is to say that there exists an algorithm that, given an n-bit instance of the problem as input, can produce a solution in time O(nc), the letter O is big-O notation and c is a constant that depends on the problem but not the particular instance of the problem. Alan Cobham's 1965 paper entitled "The intrinsic computational difficulty of functions" is one of the earliest mentions of the concept of the complexity class P, consisting of problems decidable in polynomial time. Cobham theorized that this complexity class was a good way to describe the set of feasibly computable problems. Jack Edmonds's 1965 paper "Paths, trees, and flowers" is also credited with identifying P with tractable problems.
|
|
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.
|
|
Because of the many important problems in this class, there have been extensive efforts to find polynomial- time algorithms for problems in NP. However, there remain a large number of problems in NP that defy such attempts, seeming to require super-polynomial time. Whether these problems are not decidable in polynomial time is one of the greatest open questions in computer science (see P versus NP ("P=NP") problem for an in-depth discussion). An important notion in this context is the set of NP-complete decision problems, which is a subset of NP and might be informally described as the "hardest" problems in NP. If there is a polynomial-time algorithm for even one of them, then there is a polynomial- time algorithm for all the problems in NP. Because of this, and because dedicated research has failed to find a polynomial algorithm for any NP- complete problem, once a problem has been proven to be NP-complete this is widely regarded as a sign that a polynomial algorithm for this problem is unlikely to exist. However, in practical uses, instead of spending computational resources looking for an optimal solution, a good enough (but potentially suboptimal) solution may often be found in polynomial time.
|
|