Editing Decidability (logic)

{{Short description|Whether a decision problem has an effective method to derive the answer}}
In [[logic]], a true/false [[decision problem]] is '''decidable''' if there exists an [[effective method]] for deriving the correct answer. [[Zeroth-order logic]] (propositional logic) is decidable, whereas [[first-order logic|first-order]] and [[higher-order logic|higher-order]] logic are not. [[Formal system|Logical system]]s are decidable if membership in their set of [[Validity (logic)|logically valid]] formulas (or theorems) can be effectively determined. A [[Theory (mathematical logic)|theory]] (set of sentences [[Deductive closure|closed]] under [[logical consequence]]) in a fixed logical system is decidable if there is an effective method for determining whether arbitrary formulas are included in the theory. Many important problems are [[undecidable problem|undecidable]], that is, it has been proven that no effective method for determining membership (returning a correct answer after finite, though possibly very long, time in all cases) can exist for them.

==Decidability of a logical system==

Each [[logical system]] comes with both a [[Syntax (logic)|syntactic component]], which among other things determines the notion of [[formal proof|provability]], and a [[Formal semantics (logic)|semantic component]], which determines the notion of [[logical validity]]. The logically valid formulas of a system are sometimes called the '''theorems''' of the system, especially in the context of first-order logic where [[Gödel's completeness theorem]] establishes the equivalence of semantic and syntactic consequence. In other settings, such as [[linear logic]], the syntactic consequence (provability) relation may be used to define the theorems of a system.

A logical system is decidable if there is an effective method for determining whether arbitrary formulas are theorems of the logical system. For example, [[propositional logic]] is decidable, because the [[truth table|truth-table]] method can be used to determine whether an arbitrary [[propositional formula]] is logically valid.

[[First-order logic]] is not decidable in general; in particular, the set of logical validities in any [[signature (logic)|signature]] that includes equality and at least one other predicate with two or more arguments is not decidable.<ref>{{cite journal|journal=Doklady AN SSSR |volume=88|year=1953|pages=935–956|title=On recursive separability |language=russian|author=[[Boris Trakhtenbrot]]}}</ref> Logical systems extending first-order logic, such as [[second-order logic]] and [[type theory]], are also undecidable.

The validities of [[monadic predicate calculus]] with identity are decidable, however. This system is first-order logic restricted to those signatures that have no function symbols and whose relation symbols other than equality never take more than one argument.

Some logical systems are not adequately represented by the set of theorems alone. (For example, [[ternary logic|Kleene's logic]] has no theorems at all.) In such cases, alternative definitions of decidability of a logical system are often used, which ask for an effective method for determining something more general than just validity of formulas; for instance, validity of [[sequent]]s, or the [[logical consequence|consequence relation]] {(Г, ''A'') | Г ⊧ ''A''} of the logic.

==Decidability of a theory==

A [[theory (mathematical logic)|theory]] is a set of formulas, often assumed to be [[deductively closed|closed]] under [[logical consequence]]. Decidability for a theory concerns whether there is an effective procedure that decides whether the formula is a member of the theory or not, given an arbitrary formula in the signature of the theory. The problem of decidability arises naturally when a theory is defined as the set of logical consequences of a fixed set of axioms.

There are several basic results about decidability of theories. Every (non-[[Paraconsistent logic|paraconsistent]]) inconsistent theory is decidable, as every formula in the signature of the theory will be a logical consequence of, and thus a member of, the theory. Every [[complete theory|complete]] [[recursively enumerable]] first-order theory is decidable. An extension of a decidable theory may not be decidable. For example, there are undecidable theories in propositional logic, although the set of validities (the smallest theory) is decidable.

A consistent theory that has the property that every consistent extension is undecidable is said to be '''essentially undecidable'''. In fact, every consistent extension will be essentially undecidable. The theory of fields is undecidable but not essentially undecidable. [[Robinson arithmetic]] is known to be essentially undecidable, and thus every consistent theory that includes or interprets Robinson arithmetic is also (essentially) undecidable.

Examples of decidable first-order theories include the theory of [[real closed field]]s, and [[Presburger arithmetic]], while the theory of [[group (mathematics)|groups]] and [[Robinson arithmetic]] are examples of undecidable theories.

==Some decidable theories==

Some decidable theories include (Monk 1976, p.&nbsp;234):<ref name=Monk1976 />
* The set of first-order logical validities in the signature with only equality, established by [[Leopold Löwenheim]] in 1915.
* The  set of first-order logical validities in a signature with equality and one unary function, established by Ehrenfeucht in 1959.
* The first-order theory of the natural numbers in the signature with equality and addition, also called [[Presburger arithmetic]]. The completeness was established by [[Mojżesz Presburger]] in 1929.
* The first-order theory of the natural numbers in the signature with equality and multiplication, also called [[Skolem arithmetic]].
* The first-order theory of [[Boolean algebras canonically defined|Boolean algebras]], established by [[Alfred Tarski]] in 1940 (found in 1940 but announced in 1949).
* The first-order theory of [[algebraically closed field]]s of a given [[characteristic (algebra)|characteristic]], established by Tarski in 1949.
* The [[Decidability of First-order Theory of Real Numbers|first-order theory of real-closed ordered fields]], [[Tarski–Seidenberg theorem|established by Tarski in 1949]] (see also [[Tarski's exponential function problem]]).
* The first-order theory of [[Euclidean geometry]], established by Tarski in 1949.
* The first-order theory of [[Abelian group]]s, established by Szmielew in 1955.
* The first-order theory of [[hyperbolic geometry]], established by Schwabhäuser in 1959.
* Specific [[decidable sublanguages of set theory]] investigated in the 1980s through today.(Cantone ''et al.'', 2001)
* The [[monadic second-order logic|monadic second-order]] theory of [[tree (graph theory)|trees]] (see [[S2S (mathematics)|S2S]]).

Methods used to establish decidability include [[quantifier elimination]], [[model completeness]], and the [[Łoś-Vaught test]].

==Some undecidable theories==
Some undecidable theories include:<ref name="Monk1976">{{cite book |first=Donald |last=Monk |year =1976 |title =Mathematical Logic |publisher =Springer |page=279 |isbn =9780387901701 |url-access =registration |url =https://archive.org/details/mathematicallogi00jdon }}</ref>

* The set of logical validities in any first-order signature with equality and either: a relation symbol of [[arity]] no less than 2, or two unary function symbols, or one function symbol of arity no less than 2, established by [[Boris Trakhtenbrot|Trakhtenbrot]] in 1953.
* The first-order theory of the natural numbers with addition, multiplication, and equality, established by Tarski and [[Andrzej Mostowski]] in 1949.
* The first-order theory of the rational numbers with addition, multiplication, and equality, established by [[Julia Robinson]] in 1949.
* The first-order theory of [[Group (mathematics)|groups]], established by [[Alfred Tarski]] in 1953.<ref>{{Citation| last1=Tarski| first1=A. | last2=Mostovski| first2=A. | last3=Robinson| first3=R. |  title=Undecidable Theories | publisher=North-Holland, Amsterdam | series=Studies in Logic and the Foundation of Mathematics | year=1953 |url=https://books.google.com/books?id=XtLbjZjB1B8C&pg=PP1 |isbn=9780444533784}}</ref>  Remarkably, not only the general theory of groups is undecidable, but also several more specific theories, for example (as established by Mal'cev 1961) the theory of finite groups. Mal'cev also established that the theory of semigroups and the theory of [[Ring (mathematics)|rings]] are undecidable. Robinson established in 1949 that the theory of [[Field (mathematics)|fields]] is undecidable.
*[[Robinson arithmetic]] (and therefore any consistent extension, such as [[Peano arithmetic]]) is essentially undecidable, as established by [[Raphael Robinson]] in 1950.
* The first-order theory with equality and two function symbols.<ref>{{cite journal |last=Gurevich |first=Yuri |date=1976 |title= The Decision Problem for Standard Classes|url=http://dblp.uni-trier.de/rec/bib/journals/jsyml/Gurevich76 |journal= J. Symb. Log.|volume=41 |issue=2 |pages=460–464 |doi= 10.1017/S0022481200051513|access-date=5 August 2014|citeseerx=10.1.1.360.1517 |s2cid=798307 }}</ref>

The [[interpretability]] method is often used to establish undecidability of theories. If an essentially undecidable theory ''T'' is interpretable in a consistent theory ''S'', then ''S'' is also essentially undecidable. This is closely related to the concept of a [[many-one reduction]] in [[computability theory]].

==Semidecidability==

A property of a theory or logical system weaker than decidability is '''semidecidability'''.  A theory is semidecidable if there is a well-defined method whose result, given an arbitrary formula, arrives as positive, if the formula is in the theory; otherwise, may never arrive at all; otherwise, arrives as negative. A logical system is semidecidable if there is a well-defined method for generating a sequence of theorems such that each theorem will eventually be generated. This is different from decidability because in a semidecidable system there may be no effective procedure for checking that a formula is ''not'' a theorem.

Every decidable theory or logical system is semidecidable, but in general the converse is not true; a theory is decidable if and only if both it and its complement are semi-decidable. For example, the set of logical validities ''V'' of first-order logic is semi-decidable, but not decidable. In this case, it is because there is no effective method for determining for an arbitrary formula ''A'' whether ''A'' is not in ''V''. Similarly, the set of logical consequences of any [[recursively enumerable set]] of first-order axioms is semidecidable. Many of the examples of undecidable first-order theories given above are of this form.

== Relationship with completeness ==

Decidability should not be confused with [[complete theory|completeness]]. For example, the theory of [[algebraically closed field]]s is decidable but incomplete, whereas the set of all true first-order statements about nonnegative integers in the language with + and × is complete but undecidable. 
Unfortunately, as a terminological ambiguity, the term "undecidable statement" is sometimes used as a synonym for [[independence (mathematical logic)|independent statement]].

== Relationship to computability ==

As with the concept of a [[decidable set]], the definition of a decidable theory or logical system can be given either in terms of ''[[effective method]]s'' or in terms of ''[[computable function]]s''. These are generally considered equivalent per [[Church's thesis]]. Indeed, the proof that a logical system or theory is undecidable will use the formal definition of computability to show that an appropriate set is not a decidable set, and then invoke Church's thesis to show that the theory or logical system is not decidable by any effective method (Enderton 2001, pp.&nbsp;206''ff.'').

==In context of games==
Some [[games]] have been classified as to their decidability:
* Mate in ''n'' in [[infinite chess]] (with limitations on rules and gamepieces) is decidable.<ref name="Decidability">[https://mathoverflow.net/q/27967 Mathoverflow.net/Decidability-of-chess-on-an-infinite-board Decidability-of-chess-on-an-infinite-board]</ref><ref name="Mate in N">{{cite book |first1=Dan |last1=Brumleve |first2=Joel David |last2=Hamkins |first3=Philipp |last3=Schlicht |chapter=The Mate-in-n Problem of Infinite Chess Is Decidable |chapter-url=https://link.springer.com/chapter/10.1007%2F978-3-642-30870-3_9 |title=Conference on Computability in Europe |publisher=Springer |date=2012 |isbn=978-3-642-30870-3 |pages=78–88 |doi=10.1007/978-3-642-30870-3_9 |series=Lecture Notes in Computer Science |volume=7318 |arxiv=1201.5597|s2cid=8998263 }}</ref>  However, there are positions (with finitely many pieces) that are forced wins, but not mate in ''n'' for any finite ''n''.<ref>{{Cite web|url=https://mathoverflow.net/q/63423 |title=Lo.logic – Checkmate in $\omega$ moves?}}</ref>
* Some team games with imperfect information on a finite board (but with unlimited time) are undecidable.<ref>{{cite book |first=Bjorn |last=Poonen |chapter=10. Undecidable Problems: A Sampler: §14.1 Abstract Games |chapter-url=https://books.google.com/books?id=ulw3BAAAQBAJ&pg=PA211 |editor-first=Juliette |editor-last=Kennedy |title=Interpreting Gödel: Critical Essays |publisher=Cambridge University Press |date=2014 |isbn=9781107002661 |pages=211–241 See p. 239 |arxiv=1204.0299 |citeseerx=10.1.1.679.3322}}</ref>

==See also==
{{Portal|Philosophy}}
*[[Entscheidungsproblem]]
*[[Existential quantification]]<!--
Full citations are needed:
*[[László Kalmár]] (1936)
*[[Alonzo Church]] (1956)
*[[W. V. O. Quine]] (1953)
*Meyer and [[Karel Lambert|Lambert]] (1967)-->

==References==

===Notes===
{{Reflist}}

===Bibliography===
* {{Citation | first1=Jon | last1= Barwise | authorlink = Jon Barwise | editor1-last=Barwise | editor1-first=Jon | chapter=Introduction to first-order logic | title=Handbook of Mathematical Logic | publisher=North-Holland | location=Amsterdam | series=Studies in Logic and the Foundations of Mathematics | isbn=978-0-444-86388-1 | year=1982}}
* {{Citation |last1=Cantone |first1=D. |first2=E. G. |last2=Omodeo |first3=A. |last3=Policriti |title=Set Theory for Computing. From Decision Procedures to Logic Programming with Sets |publisher=Springer |series=Monographs in Computer Science |date=2013 |orig-year=2001 |isbn=9781475734522 |pages= |url=https://books.google.com/books?id=KWvlBwAAQBAJ&pg=PR2}}
* {{Citation | last1=Chagrov | first1=Alexander | last2=Zakharyaschev | first2=Michael | title=Modal logic | publisher=Oxford University Press | series=Oxford Logic Guides | isbn=978-0-19-853779-3 | mr=1464942 | year=1997 | volume=35}}
* {{Citation | last1=Davis | first1=Martin | author1-link=Martin Davis (mathematician) | title=Computability and Unsolvability | publisher=Dover |isbn=9780486151069 |date=2013 |orig-year=1958 |url=https://books.google.com/books?id=nbOqAAAAQBAJ&pg=PP1}}
* {{Citation | last1=Enderton | first1=Herbert | title=A mathematical introduction to logic | publisher=[[Academic Press]] | edition=2nd | isbn=978-0-12-238452-3 | year=2001}}
* {{Citation | first1= H. J. | last1= Keisler | authorlink = H. J. Keisler | editor1-last=Barwise | editor1-first=Jon | chapter=Fundamentals of model theory | title=Handbook of Mathematical Logic | publisher=North-Holland | location=Amsterdam | series=Studies in Logic and the Foundations of Mathematics | isbn=978-0-444-86388-1 | year=1982}}
* {{Citation | last1=Monk | first1=J. Donald | title=Mathematical Logic | publisher=[[Springer-Verlag]] |date=2012 |isbn=9781468494525 |orig-year=1976}}

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