Local Variations on a Loose Theme: Modal Logic and Decidability

7.1 Introduction This chapter is about decidability and complexity issues in modal logic; more specifically, we confine ourselves to satisfiability (and the complementary validity) problems. The satisfiability problem is the following: for a fixed class of models, to determine whether a given formula ϕ is satisfiable in some model of that class (a more precise definition will follow). The general picture is that modal logic behaves quite well in this respect. In fact, many authors follow Vardi [58] in calling modal logic robustly decidable on the ground that most of the nice computational properties of modal logic are preserved if one considers extensions or variants of the basic system. The main aim of this chapter is to refine and analyze this picture. To start with, we should clarify what we are talking about when using the term " modal logic ". Traditionally, propositional modal logic would be described as an extension of propositional logic with operators 2 and 3 for talking about the necessity and possibility of a formula being true. However , nowadays the term " modal logic " is used for a plethora of formalisms, with applications in various disciplines ranging from linguistics to economics, see [11, 17, 39, 56] for a sample of applications in computer science. And while (propositional) modal logics will usually still be an extension of classical propositional logic with a number of modal operators, the intended meanings of these operators differ enormously. For instance, the formula 2 a ϕ could mean " player a knows that ϕ is the case " in a formalization of game theory, or " after the execution of program a, ϕ will be the case " in a formal language for program verification. Fortunately, on a technical level, all these formalisms still have a lot in common. That is why this chapter first introduces the notion of a modal system as a triple consisting of a (propositional) modal language, a class of models and a truth function. This definition covers most of the systems that appear in the literature under the name " modal logic " ; in particular, the familiar system of basic modal logic, to be discussed in section 7.3.