Local Variations on a Loose Theme: Modal Logic and Decidability
Maarten Marx, Yde Venema
Abstract:
This chapter is about the satisfiability problem for modal logics and
related formalisms. We discuss and explain the good computational
behaviour of many modal systems. We show how finite and other simple
structures are used in the proofs of various decidability and
complexity results. In doing so, we introduce some of the most
important proof methods that are used in establishing these results.
Concerning the first theme, we single out some principles of the basic
modal logic that determine its computational properties. The most
important of these is the {\em looseness principle} which says that
any satisfiable modal formula is satisfiable on a loose model; but we
will also discuss two {\em locality principles} which are two
formalizations of the intuition that the truth of a modal formula at a
state of a model only depends on a small, local part of the model.
Concerning the second theme, we give detailed expositions of the
selection of points method, the method of filtration and the mosaic
method.
The chapter starts with a general introduction to modal logic, and
then focuses on a detailed proof of the decidability of the basic
modal system. After that, decidability is discussed for some
variations of this basic modal system, and the picture is refined by
taking complexity issues into account. Finally, the observation that
modal logic is nothing but a nice, decidable fragment of first-order
logic, is employed to generalize this to larger parts of first-order
logic, such as the guarded and packed fragments.
Keyword(s): modal logic, decidability, complexity, guarded fragments