MATHEMATICAL MODAL LOGIC: A VIEW OF ITS EVOLUTION

Mathematical Modal Logic: A View of its Evolution 49by complete and atomic algebras at all.6.2 Decidability and ComplexityThe finite model property does not give a universal method for proving the decidabilityof modal logics. Although every finitely axiomatisable logic with the finitemodel property is decidable, the converse is not true. This was shown by DovGabbay, building on some earlier work of Makinson [1969] which had exhibitedthe first example of a normal logic that lacked the finite model property. Makinson’sexample is a proper sublogic of S4, but all of its finite algebras satisfy S4 aswell.Gabbay’s paper [1972] extended Makinson’s idea to produce finitely axiomatisablemodal and tense logics that lacked the finite model property, but could still beshown to be decidable by appealing to a powerful result of Michael Rabin [1969].This concerns the decidability of monadic second-order theories of successor functions,and has many applications. For each ordinal n with 2 ≤ n ≤ ω, considerthe structureS n = (T n , {s m : m < n}, ≤ , ),where T n is the n-ary branching tree of all finite sequences of elements of the set[n) = {m ∈ ω : m < n}, s m is the successor function x ↦→ xm on the tree, ≤ isthe “initial segment” ordering of sequences, and is their lexicographical orderinginduced by the natural ordering < on [n). Rabin proved that the monadic secondordertheory SnS of the structure S n is decidable. To do this he developed atheory of finite-state automata that process infinite labelled trees, and establishedthe decidability of the emptiness problem of whether any given automaton acceptsat least one tree. The decidability of SnS was then reduced to this emptynessproblem. It was later shown that the decision problem for SnS is intractable:Albert Meyer [1975] proved that no algorithm for deciding if a sentence is inSnS can run in elementary time, i.e. time bounded by some fixed number ofcompositions of exponential functions.Gabbay developed a method of coding Kripke models into the structure S ω andthereby reducing the decidability problem for certain logics to Rabin’s decidabilityresults for SωS. The technique is explained in Part 5 of the book [Gabbay, 1976],where it is used to establish decidability results for many modal systems.Gabbay’s method was later used by Cresswell [1984] in adapting an incompletelogic from [van Benthem, 1979] to construct a decidable modal logic that is finitelyaxiomatisable but incomplete with respect to Kripke frames (and hence lacks thefinite model property). Cresswell’s example is a proper sublogic of the logic characterisedby the class of finite strict linear orderings, but the two logics are validatedby exactly the same frames.For any logic Λ, the problem of deciding if a given formula is Λ-provable is thesame as the Λ-validity problem of deciding if a given formula is true in all modelsM such that M |= Λ. The Λ-satisfiability problem of whether a given formula is