A many-sorted polyadic modal logic

Abstract

We propose a general system that combines the powerful features of modal logic and many-sorted reasoning. Its algebraic semantics leads to a many-sorted generalization of boolean algebras with operators, for which we prove the analogue of the Jónsson-Tarski theorem. Our goal was to deepen the connections between modal logic and program verification, and we test the expressivity of our system by defining a small imperative language and its operational semantics.

From Hybrid Modal Logic to Matching Logic and Back

“…In this paper, we continue our work from [9,10], where we defined a (hybrid) many-sorted polyadic modal logic, for which we proved soundness and completeness, generalizing well-known results from the mono-sorted setting [5].…”

“…For (S, Σ) a many-sorted signature, the many-sorted polyadic modal logic H Σ defined in [9] is recalled in Figure 1. The system H Σ (∀), defined in Section 2, is a fragment of the system introduced by [10] which enriches H Σ with nominals, state variables and the forall binder.…”

“…Let (S, Σ) be a many-sorted signature. In this section we perform hybridization on top of H Σ , the manysorted polyadic modal logic defined in [9].…”

“…for any s ∈ S. The model M = (F ,V ) will be simply denoted as M = (W, (R σ ) σ ∈Σ ,V ). For s ∈ S, w ∈ W s and φ a formula of sort s, the many-sorted satisfaction relation M , w | s = φ is defined by structural induction of formulas (see [9] for details). Moreover, let s ∈ S and assume φ is a formula of sort s. Then φ is satisfiable if M , w | s = φ for some model M and some w ∈ W s .…”

“…The formula φ is valid in a model M if M , w | s = φ for any w ∈ W s ; in this case we write M | s = φ . The deductive system of H Σ is recalled in 1 and the completeness theorem is proved in [9]. The hybridization of our many-sorted modal logic is developed using a combination of ideas and techniques from [1,2,3,5,7,8], but for this section we drew our inspiration mainly form [3].…”