Essentially undecidable

In logic, the term decidable refers to the decision problem, the question of the existence of an effective method for determining membership in a set of formulas, or, more precisely, an algorithm that can and will return a Boolean true or false value (instead of looping indefinitely). Logical systems such as propositional logic are decidable if membership in their set of logically valid formulas (or theorems) can be effectively determined. A theory (set of sentences 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.

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 methods or in terms of computable functions. 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. 206ff.).

Decidability of a logical system

Each logical system comes with both a syntactic component, which among other things determines the notion of provability, and a 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 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 that includes equality and at least one other predicate with two or more arguments is not decidable.[1] 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 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, 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 sequents, or the consequence relation {(Г, A) | Г ⊧ A} of the logic.

Decidability of a theory

A theory is a set of formulas, which here is assumed to be closed under logical consequence. The question of decidability for a theory is whether there is an effective procedure that, given an arbitrary formula in the signature of the theory, decides whether the formula is a member of the theory or not. This problem arises naturally when a theory is defined as the set of logical consequences of a fixed set of axioms. Examples of decidable first-order theories include the theory of real closed fields, and Presburger arithmetic, while the theory of groups and Robinson arithmetic are examples of undecidable theories.

There are several basic results about decidability of theories. Every inconsistent theory is decidable, as every formula in the signature of the theory will be a logical consequence of, and thus member of, the theory. Every 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 which 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 which includes or interprets Robinson arithmetic is also (essentially) undecidable.

Some decidable theories

Some decidable theories include (Monk 1976, p. 234):

  • 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 integers 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 Boolean algebras, established by Alfred Tarski in 1949.
  • The first-order theory of algebraically closed fields of a given characteristic, established by Tarski in 1949.
  • The first-order theory of real-closed ordered fields, established by Tarski in 1949.
  • The first-order theory of Euclidean geometry, established by Tarski in 1949.
  • 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)

Methods used to establish decidability include quantifier elimination, model completeness, and Vaught's test.

Some undecidable theories

Some undecidable theories include (Monk 1976, p. 279):

  • 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 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 groups, established by Mal'cev in 1961. Mal'cev also established that the theory of semigroups and the theory of rings are undecidable. Robinson established in 1949 that the theory of 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 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 an effective method which, given an arbitrary formula, will always tell correctly when the formula is in the theory, but may give either a negative answer or no answer at all when the formula is not in the theory. A logical system is semidecidable if there is an effective method for generating theorems (and only theorems) such that every 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 completeness. For example, the theory of algebraically closed fields 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 independent statement.

See also

References

Notes

Bibliography

  • Cantone, D., E. G. Omodeo and A. Policriti, "Set Theory for Computing. From Decision Procedures to Logic Programming with Sets," Monographs in Computer Science, Springer, 2001.
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