User:IssaRice/Computability and logic/Entscheidungsproblem

Entscheidungsproblem, also called Hilbert's decision problem is a problem in mathematical logic.

Equivalent formulations

Some things to note:

• A relation $R$ is "decidable" means that there is some algorithm such that if $R(x)$ is true, then the algorithm outputs "yes", and if $R(x)$ is false, then the algorithm outputs "no".

Label	Syntactic or semantic	Statement	Notes
In terms of validity	Semantic	Take as input a sentence of first-order logic, and decide whether it is valid (a.k.a. universally valid, true-in-every-interpretation).
In terms of provability	Syntactic	Take as input a sentence of first-order logic, and decide whether it is provable (using only logical axioms).	By Godel's completeness theorem, validity and provability are equivalent.
In terms of satisfiability	Semantic	Take as input a sentence of first-order logic, and decide whether it is satisfiable.
In terms of semantic implication	Semantic	Take as input a set of sentences $\Gamma$ and a sentence $\phi$ (both of first-order logic), and decide whether $\Gamma$ semantically implies (a.k.a. logically implies) $\phi$.

Decidability for first-order logic versus decidability for a particular theory

The question of decidability (i.e. is this thing decidable or undecidable?) can be asked of both logics/formal systems and of theories. This is similar to how both soundness and completeness can be asked of logics and theories (and why we have both the completeness theorem and the incompleteness theorems!).

Even though first-order logic is undecidable, a particular first-order theory (i.e. a theory specified in first-order logic via some non-logical axioms) may still be decidable.

See also

• wikipedia:Entscheidungsproblem