Abstract
Sequent calculi are proof systems that are exceptionally suitable for proving the decidability of a logic. Several relevance logics were proved decidable using a technique attributable to Curry and Kripke. Further enhancements led to a proof of the decidability of implicational ticket entailment by Bimbó and Dunn in [12, 13]. This paper uses a different adaptation of the same core proof technique to prove a group of positive modal logics (with disjunction but no conjunction) decidable.
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Notes
- 1.
- 2.
We may use other letters than \(\varphi \), from the latter part of the Greek alphabet, as variables for formulas.
- 3.
In this paper, we only have use for finite multisets; thus, we use the term in a narrower sense than it is used elsewhere in the literature.
- 4.
A quick approximation suggests that there are 89 logics that can be expected to be distinct.
- 5.
\(\times \) excludes a pair of rules;
shows that the rules are easily derivable, hence, it is better to omit them—for the sake of economy in proofs. - 6.
- 7.
This notion is an adaptation of a similar notion from Curry [17].
- 8.
More details of a triple-inductive proof of the admissibility of the cut rule for a logic with no lattice operators may be found in [8]. Various enhancements of a more usual double-inductive proof of the cut theorem were introduced in [6, 7], where a goal was to accommodate constants like \(\textsf {Y},\textsf {y}\) and \(\varvec{t}\).
- 9.
- 10.
See Meyer [31] for a discussion of conceptual links that can be created between Dickson’s lemma and Kripke’s lemma.
- 11.
- 12.
The so-called mix rule in [23] and the multicut rule explicitly stated, for example, in Dunn [18] are versions of the cut that were introduced specifically to facilitate the inductive proof of the cut theorem for the single cut. An early publication that exhibits a suitable version of cut in connection to a decidability proof using the Curry–Kripke method is [4], which is a precursor of the more readily available [5].
- 13.
We defined heap numbers in a very liberal manner in order to make sure that all the necessary contractions are permitted. However, even if
, for example, it may happen that in the \(L\mathfrak {X}_{3}^{\text {*}}\) logic no contraction will be applied to the formula, because it occurs on the left-hand side of the 
. (Similarly, but dually for 
.) This does not cause any problem in the proof search, because the heap number (like the 
notation) does not force contractions, rather, it places a limit on the number of potential applications of the contraction rules.
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Acknowledgments
I am grateful to the organizers of the TABLEAUX, FroCoS and ITP conferences for their invitation for me to speak at those conferences, which triggered the writing of this paper.
I would also like to thank the program committee for helpful comments on the first version of this paper.
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Bimbó, K. (2017). On the Decidability of Certain Semi-Lattice Based Modal Logics. In: Schmidt, R., Nalon, C. (eds) Automated Reasoning with Analytic Tableaux and Related Methods. TABLEAUX 2017. Lecture Notes in Computer Science(), vol 10501. Springer, Cham. https://doi.org/10.1007/978-3-319-66902-1_3
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