Complexity of Simple Dependent Bimodal Logics

Abstract

We characterize the computational complexity of simple dependent bimodal logics. We define an operator ⊕ ⊆ between logics that almost behaves as the standard joint operator ⊕ except that the inclusion axiom [2]p ⇒ [1]p is added. Many multimodal logics from the literature are of this form or contain such fragments. For the standard modal logics K,T ,B,S4 and S5 we study the complexity of the satisfiability problem of the joint in the sense of ⊕ ⊆. We mainly establish the PSPACE upper bounds by designing tableaux-based algorithms in which a particular attention is given to the formalization of termination and to the design of a uniform framework. Reductions into the packed guarded fragment with only two variables introduced by M. Marx are also used. E. Spaan proved that K ⊕ ⊆ S5 is EXPTIME-hard. We show that for L1, L2 ∈ {K, T, B} × {S4, S5}, L1 ⊕ ⊆ L2 is also EXPTIME-hard.

Modal Logics with Hard Diamond-Free Fragments

“…In [5], Demri established that D ⊕ ⊆ K4-satisfiability (and because of the following section's results also D4 2 ⊕ ⊆ K4-satisfiability) is EXP-complete. In this paper, though, we establish that the complexity of these two logics' diamond-free (and one-variable) fragments are PSPACE-complete (in this section we establish the PSPACE upper bounds, while in the next one the lower bounds), which is a drop in complexity (assuming PSPACE = EXP), but not one that makes the problem tractable (assuming P = PSPACE).…”

Modal logics with hard diamond-free fragments

“…In [5], Demri shows that satisfiability for L 1 ⊕ ⊆ L 2 ⊕ ⊆ · · · ⊕ ⊆ L n is EXP-complete, as long as there are i < j ≤ n for which L i ⊕ ⊆ L j is EXPhard. On the other hand, Corollary 1 shows that for all these logics, their diamond-free fragment is in NP, as long as L 1 has frames with transitive (or reflexive) accessibility relations.…”

Modularisation of Sequent Calculi for Normal and Non-normal Modalities

“…In a Hilbert-style presentation of these logics, the axioms are given by extensions of the axioms of modal logic K for every modality i together with interaction axioms of the form i A → j A. A simple example of such a logic is the simply dependent bimodal logic KT ⊕ ⊆ S4 from [Demri 2000], whose language contains the two modalities 1 and 2 . Its Hilbert-style axiomatisation is given by the axioms and rules of classical propositional logic together with the axioms and rules of modal logic KT for the modality 1 , i.e., axioms K and T and the rule nec, the S4 axioms for the modality 2 , i.e., axioms K, T, 4 and rule nec, and the single interaction axiom 2 A → 1 A.…”