Difference between revisions of "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".
• Deciding is different from semi-deciding (a.k.a. recognizing).
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 of a finite set of sentences Semantic Take as input a finite set of sentences $\Gamma$ and a sentence $\phi$ (both of first-order logic), and decide whether $\Gamma$ semantically implies (a.k.a. logically implies) $\phi$. This is version can be used (where $\Gamma$ is the set of axioms of Robinson arithmetic or the group axioms) to show that first-order logic is undecidable.

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.