MATHEMATICAL MODAL LOGIC: A VIEW OF ITS EVOLUTION

Mathematical Modal Logic: A View of its Evolution 71in the sense that T ⊆ T ′ implies Θ(T ) ⊆ Θ(T ′ ), then Θ has a least fixpoint µΘand a greatest fixpoint νΘ, given byµΘ = ⋂ {T ⊆ S : Θ(T ) ⊆ T },νΘ = ⋃ {T ⊆ S : T ⊆ Θ(T )}.The fact that Θ has a fixpoint was first shown by Tarski and B. Knaster in 1927.In 1939 Tarski generalised this to any monotonic function on a complete lattice,showing that its fixpoints also form a complete lattice, with greatest and leastelements specified by the lattice versions of the definitions just given (see [Tarski,1955b] for this historical background).Pratt [1981] introduced the idea of using a “minimisation” operator in a PDLlikecontext, but interpreted µ as a least root operator rather than a least fixpointone. He developed a language of terms intended to denote elements of a Booleanalgebra, with a term of the form µQ.τ(Q) interpreted as the least solution of theequation “τ(Q) = 0”. A syntactic restriction was imposed on τ to ensure thatat least one solution exists. A translation of PDL into the resulting calculus wasgiven, and the system was shown to have the finite model property by a refinementof the McKinsey method. A deterministic exponential time algorithm was givenfor the problem of deciding satsfiability terms.Pratt’s work provided the inspiration for Kozen’s development of the calculusLµ, whose language is generated from some collection Π of atomic programs (oraction labels) π. Lµ-formulas are constructed from propositional variables usingthe truth-functional connectives, the modalities [π] and 〈π〉 for π ∈ Π, and theconstructions µp.α and νp.α, where p is a propositional variable and α is a formula.The operations µp and νp function like quantifiers, binding occurrences of p in α.µp.α and νp.α are only allowed to be formed when α is positive in the sense that allfree occurrences of p in α are within the scope of an even number of negations ¬.This condition is satisfied for instance by any formula constructed from variablesusing only ⊤, ⊥, ∧, ∨, [π], 〈π〉, µp and νp. The “binder” ν is definable in terms ofµ by taking νp.α as ¬µp.¬α(¬p/p). Vice versa, µ could be defined in terms of ν.An Lµ model M = (S, { π −→: π ∈ Π}, Φ) is just like a Kripke model for dynamiclogic, or a labelled transition system for Hennessy–Milner logic augmented by avaluation Φ to interpret the variables p. M gives each formula α the interpretationM(α) = {x ∈ S : M |= x α}. If α contains the variable p, then varying theinterpretation of p causes the interpretation of α to vary, and in this way α inducesan operation on P(S). To make this precise, for T ⊆ S let M p:=T be the modelthat is identical to M except in interpreting p as T , i.e. M p:=T (p) = T . Then theoperation induced by α on P(S) relative to M is the functionΘ M α: T ↦−→ M p:=T (α).If α is positive, then Θ α is monotonic. Assuming inductively that Θ α has beenspecified, M(µp.α) and M(νp.α) are defined to be the least and greatest fixpointsµΘ M α and νΘ M α given by the Tarski–Knaster Theorem.