User:IssaRice/Computability and logic/Entscheidungsproblem: Difference between revisions
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| In terms of satisfiability || Semantic || Take as input a sentence of first-order logic, and decide whether it is satisfiable. | | In terms of satisfiability || Semantic || Take as input a sentence of first-order logic, and decide whether it is satisfiable. | ||
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| In terms of semantic implication of a finite set of sentences || Semantic || Take as input a finite set of sentences <math>\Gamma</math> and a sentence <math>\phi</math> (both of first-order logic), and decide whether <math>\Gamma</math> semantically implies (a.k.a. logically implies) <math>\phi</math>. || This is version can be used (where <math>\Gamma</math> is the set of axioms of Robinson arithmetic or the group axioms)<ref>https://math.stackexchange.com/a/696106/35525</ref> to show that first-order logic is undecidable. | | In terms of semantic implication of a finite set of sentences || Semantic || Take as input a finite set of sentences <math>\Gamma</math> and a sentence <math>\phi</math> (both of first-order logic), and decide whether <math>\Gamma</math> semantically implies (a.k.a. logically implies) <math>\phi</math>. || This is version can be used (where <math>\Gamma</math> is the set of axioms of Robinson arithmetic<ref>https://math.stackexchange.com/a/2190149/35525</ref> or the group axioms)<ref>https://math.stackexchange.com/a/696106/35525</ref> to show that first-order logic is undecidable. | ||
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Revision as of 03:27, 9 February 2019
Entscheidungsproblem, also called Hilbert's decision problem is a problem in mathematical logic.
Equivalent formulations
Some things to note:
- A relation is "decidable" means that there is some algorithm such that if is true, then the algorithm outputs "yes", and if 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 and a sentence (both of first-order logic), and decide whether semantically implies (a.k.a. logically implies) . | This is version can be used (where is the set of axioms of Robinson arithmetic[1] or the group axioms)[2] 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.