Elementary Modal Logics over Transitive Structures

Elementary Modal Logics over Transitive Structures∗ Jakub Michaliszyn1,2 and Jan Otop1,3 1 2 3 University Of Wrocław Imperial College London IST Austria Abstract We show that modal logic over universally first-order definable classes of transitive frames is decidable. More precisely, let K be an arbitrary class of transitive Kripke frames definable by a universal first-order sentence. We show that the global and finite global satisfiability problems of modal logic over K are decidable in NP, regardless of choice of K. We also show that the local satisfiability and the finite local satisfiability problems of modal logic over K are decidable in NExpTime. 1998 ACM Subject Classification F.4.1 Mathematical Logic Keywords and phrases Modal logic, Transitive frames, Elementary modal logics, Decidability Digital Object Identifier 10.4230/LIPIcs.CSL.2013.563 1 Introduction Modal logic was first introduced by philosophers as the study of the deductive behaviour of the expressions ‘it is necessary that’ and ‘it is possible that’. Nowadays, it is widely used in several areas of computer science, including formal verification and artificial intelligence. Syntactically, modal logic extends propositional logic by two unary operators: ♦ and . The formal semantics is usually given in terms of Kripke structures. Basically, a Kripke structure is a directed graph, called a frame, together with a valuation of propositional variables. Vertices of this graph are called worlds. For each world truth values of all propositional variables can be defined independently. In this semantics, ♦ϕ means the current world is connected to some world in which ϕ is true; and ϕ, equivalent to ¬♦¬ϕ, means ϕ is true in all worlds to which the current world is connected. “Classical” modal logic, defined as above, is very simple, and therefore it has limited applications. For that reason, many modifications of modal logic are studied. One way to enrich modal logic is to add more modalities and obtain so called multimodal logic. Another popular modification is to add some constraints on the interpretation of operators, e.g. by requiring that the modal operator represents a relation that is reflexive and transitive (S4). Finally, by combining these two techniques, we may obtain multimodal logics with nonuniform modal operators, like Linear Temporal Logic (LTL), Computation Tree Logic (CTL) or Halpern–Shoham logic (HS). ∗ The first author was generously supported by Polish National Science Center based on the decision number DEC-2011/03/N/ST6/00415. The second author was supported in part by the Austrian Science Fund NFN RiSE (Rigorous Systems Engineering) and by the ERC Advanced Grant QUAREM (Quantitative Reactive Modeling). © Jakub Michaliszyn and Jan Otop; licensed under Creative Commons License CC-BY Computer Science Logic 2013 (CSL’13). Editor: Simona Ronchi Della Rocca; pp. 563–577 Leibniz International Proceedings in Informatics Schloss Dagstuhl – Leibniz-Zentrum für Informatik, Dagstuhl Publishing, Germany 564 Elementary Modal Logics over Transitive Structures Variants of modal logic vary in the complexity and the decidability of the satisfiability problem. While there are some logics, like S5, that are NP-complete, most of them are PSpace-hard. Modal logic itself is PSpace-complete, and so is the temporal logic LTL. Logics CTL and CTL∗ are even harder, ExpTime-complete and 2-ExpTime-complete, resp. Finally, the Halpern–Shoham logic is a simple example of a temporal logic that is undecidable, even if we consider some unimodal fragments [3, 16]. There are several ways of adding constraints on the interpretation of operators. The syntactic way goes by adding some additional axioms and considering only modal logics that satisfy these axioms. Another way is by restricting the class of the admissible frames. It also can be done in many ways, but one of the simplest is to define the class of frames by a first order logic sentence that uses a single binary relation R, which is interpreted as the transition relation. For example, the sentence ∀xyz.xRy ∧ yRz ⇒ xRz defines the class of all transitive frames. Modal logic over a class of frames definable by a first order logic sentence is called an elementary modal logic. The main goal of our work is to classify all elementary modal logics with respect to decidability of their satisfiability problems. In [9], it was shown that there is an universal first-order formula such that the global satisfiability problem over the class of frames that satisfy this formula is undecidable. A slight modification of that formula yields an analogous result for the local satisfiability problem. In [12] it was shown that even a very simple formula with three variables without equality leads to undecidability. Many modal logics used in automatic verification contain operators that are interpreted as transitive relations. For example, Linear Temporal Logic (LTL) contains transitive operators F and G [2]. In epistemic modal logics, the knowledge operators Ki are interpreted as relations that are not only transitive, but also reflexive and symmetric [4]. Another example is the logic of subintervals [16], which is a fragment of Halpern–Shoham logic with a single modality that can be read as “in some subinterval”. Main results. In this paper, an logic is called a subframe logic if it is an unimodal logic defined by restricting the class of the admissible frames to a class that is closed under subframes. Fine[5] showed that there are undecidable transitive subframe logics. An open question suggested in [9] is whether there is an undecidable transitive subframe logic that is an elementary modal logic. Due to [5], such a logic would not be finitely axiomatizable. We show that such a logic does not exist. The Łoś–Tarski preservation theorem (see, e.g., [10]) states that a first order definable class of frames is closed under subframes if and only if it is universally first order definable. Therefore, to answer the question discussed above, we may restrict our attention to universal formulae. We prove the following theorem. I Theorem 1. Let K be a class of frames defined by a universal first order formula that implies transitivity. The local and the global satisfiability problems for unimodal logic over K are decidable. The finiteness constraint can make the satisfiability problem easier or harder. There are decidable modal logics that are finitely undecidable, and there are undecidable modal logics that are finitely decidable [6, 19]. However, this is not the case here — the finite satisfiability problems are decidable as well. I Theorem 2. Let K be a class of frames defined by a universal first order formula that implies transitivity. The local and the global finite satisfiability problems for unimodal logic over K are decidable. J. Michaliszyn and J. Otop 565 We focus on the case where a first-order formula, that defines the class of frames, is a parameter of the problem, an (...truncated)