MATHEMATICAL MODAL LOGIC: A VIEW OF ITS EVOLUTION

70 Robert GoldblattThe logic CTL* was defined semantically, and a sound and complete axiomatisationof it was hard to find. Eventually one was provided by Mark Reynolds[2001].A property of paths not expressible in linear time logic, or even in CTL*, isthat a formula be true at every even state along the path (and possibly at others).Sets of sequences that have this property can be generated by formal grammars, orcharacterised by finite-state automata that process infinite strings. Pierre Wolper[1983] showed that any regular grammar gives rise to a temporal connective creatingformulas that are true just of paths generated by that grammar in a certainway. He also showed that the linear time connectives G, F , X and U can eachbe expressed by such a grammar, and dubbed this formalism ETL for “ExtendedTemporal Logic”. The idea can be applied to branching time systems, and leadsto a logic ECTL* into which CTL* can be translated (see [Thomas, 1989]).Surveys of computational temporal logic, and its various applications to reasoningabout programs, are given in [Emerson, 1990] and [Stirling, 1992].A different kind of use of modalities of the branching-time type was made byGlynn Winskel [1985] in constructing powerdomains. These structures arise in thedenotational semantics of programs, and are intended to provide domain-theoreticanalogues of powersets. In dynamic logic a non-deterministic program is modelledas a binary transition relation R on a set S of possible program states. Alternativelythis can be viewed as a function from S to its powerset P(S), taking eachstate x ∈ S to the set {y : xRy} of states that can be reached by different possibleexecutions of the program. Analogously, given a domain D, a non-deterministicprogram may be modelled as a function from D to its powerdomain.There are several different powerdomain constructions, and Winskel shows howto build them out of formulas of some modal languages associated with D. Thisinvolves tree-like models of the languages that represent certain computations.For the “Smyth” powerdomain a modality ✷ is used that it read “inevitably”. ✷αhas the same meaning in these models as the CTL-modality ∀F α, i.e. along everyfuture path there is a state at which α holds. The construction of the “Hoare”powerdomain uses ✸, for “possibly”, with ✸α meaning that there is a future pathwith α true somewhere, i.e. ∃F α. For the “Plotkin” powerdomain, both of thesemodalities are involved.7.4 The Modal µ-CalculusMathematics and computer science abound with concepts and objects that aredefined recursively, or self-referentially. Many of these have an elegant formulationas special fixed points of certain operations. The µ-calculus L µ of Kozen [1982;1983] admits formulas that are interpreted as fixed points, and is expressively morepowerful than any of the modal program logics considered above.Let Θ : P(S) → P(S) be an operation on the powerset of a set S. Tarski appliedthe term “fixpoint” to any subset T of S such that Θ(T ) = T . If Θ is monotonic