The Completeness Problem for Modal Logic
We introduce the completeness problem for Modal Logic and examine its complexity. For a definition of completeness for formulas, given a formula of a modal logic, the completeness problem asks whether the formula is complete for that logic. We discover that completeness and validity have the same complexity — with certain exceptions for which there are, in general, no complete formulas. To prove upper bounds, we present a non-deterministic polynomial-time procedure with an oracle from PSPACE that combines tableaux and a test for bisimulation, and determines whether a formula is complete.
KeywordsModal logic Completeness Computational complexity Bisimulation
The author is grateful to Luca Aceto for valuable comments that helped improve the quality of this paper.
- 4.Artemov, S.: Syntactic epistemic logic. In: Book of Abstracts, 15th Congress of Logic, Methodology and Philosophy of Science CLMPS 2015, pp. 109–110 (2015)Google Scholar
- 5.Artemov, S.: Syntactic epistemic logic and games (2016)Google Scholar
- 13.Achilleos, A.: The completeness problem for modal logic. CoRR abs/1605.01004 (2016)Google Scholar