The Logic of Exact Covers: Completeness and Uniform Interpolation

Abstract

Abstract-We show that all (not necessarily normal or monotone) modal logics that can be axiomatised in rank-1 have the interpolation property, and that in fact interpolation is uniform if the logics just have finitely many modal operators. As immediate applications, we obtain previously unknown interpolation theorems for a range of modal logics, containing probabilistic and graded modal logic, alternating temporal logic and some variants of conditional logic.Technically, this is achieved by translating to and from a new (coalgebraic) logic introduced in this paper, the logic of exact covers. It is interpreted over coalgebras for an endofunctor on the category of sets that also directly determines the syntax. Apart from closure under bisimulation quantifiers (and hence interpolation), we also provide a complete tableaux calculus and establish both the Hennessy-Milner and the small model property for this logic.

Countermodel Construction via Optimal Hypersequent Calculi for Non-normal Modal Logics

“…In future research, we intend to extend the calculi to further non-normal modal logics obtained by adding standard modal axioms, possibly including also regular logics which have a non standard relational semantics. Moreover, we intend to use the calculi also for metalogical investigation, e.g., for obtaining proof-theoretic constructive proofs of interpolation complementing the general result in [17] and completing [15]. Finally we wish to study the formal relation with other recent calculi in the literature such as [4,13] in the form of mutual simulation.…”

Lattice-theoretic progress measures and coalgebraic model checking

“…They are on: expressivity (i.e. that bisimilarity is captured) [36,45]; sound and complete axiomatizations [45,52]; satisfiability complexity [52]; cut elimination and interpolation [46,47]; and so on. Fixed-point operators in coalgebraic modal logics have been actively studied too.…”

Sequent Calculi and Interpolation for Non-Normal Modal and Deontic Logics

“…For a thorough study of interpolation in modal logics we refer the reader to [10]. A model-theoretic proof of interpolation for E is given in [15], and a coalgebraic proof of (uniform) interpolation for all the logics considered here, as well as all other rank-1 modal logics (see below), is given in [34]. As it is explained in Example 5.5, we have not been able to prove interpolation for calculi containing the non-standard rule LR-C (see Figure 6) and, as far as we know, it is still an open problem whether it is possible to give a constructive proof of interpolation for these logics.…”

“…The modal rules of inference presented in Figure 6 are obtained from the rules presented in [21] by adding weakening contexts to the conclusion of the rules. This minor modification, used also in [20,34,35] for several modal rules, allows us to shift from set-based sequents to multiset-based ones and to prove not only that cut is admissible, as it is done in [19,20,21], but also that weakening and contraction are heightpreserving admissible. Given that implicit contraction is not eliminable from set-based sequents, the decision procedure for non-normal logics given in [21] is based on a model-theoretic inversion technique so that it is possible to define a procedure that outputs a derivation for all valid sequents and a finite countermodel for all invalid ones.…”

“…Finally, it is worth noticing that all the non-normal logics we consider here are rank-1 logics in the sense of [34,35,37]  i.e., logics whose modal axioms are propositional combinations of formulas of the form φ, where φ is purely propositional  and the calculi we give for the modal logics E, M, K and KD are explicitly considered in [34,37]. Thus, they are part of the family of modal coalgebraic logics [34,35,37] and most of the results in this paper can be seen as instances of general results that hold for rank-1 (coalgebraic) logics. If, in particular, we consider cut-elimination for coalgebraic logics [35] then all our calculi absorb congruence and Theorem 3.5 and case 3 of Theorem 3.6 show that they absorb contraction and cut.…”