MATHEMATICAL MODAL LOGIC: A VIEW OF ITS EVOLUTION

Mathematical Modal Logic: A View of its Evolution 69Since there are many different execution sequences with a given starting stateany particular sequence is just one “branch” or “path” of the “tree” of all possiblefuture states. Considering the tree as a whole gives rise to some interesting newmodalities that can formalise reasoning about future behaviour. This line waspursued by Ben-Ari, Pnueli and Manna [Ben-Ari et al., 1983], defining a systemUB (the unified system of branching time), which combined G and X with thesymbols ∀, ∃ for quantification over paths to produce the following modal forms:∀Gα :∃Gα :∀Xα :along all future paths, α is true at all states.along some path, α is true at all states.along all paths, α is true at the next state.Dual modalities were defined by writing ∃F for ¬∀G¬, ∀F for ¬∃G¬, and ∃X for¬∀X¬. The logic UB was shown to be finitely axiomatisable and have the finitemodel property, using semantic tableaux methods. It was also stated that, incontrast to PLTL, no temporal language for branching time with a finite number ofmodalities could be expressively complete, this theorem being credited to Gabbay.The until connective U was added to UB by Edmund Clarke and Allen Emerson[1981] to define the system CTL of Computation Tree Logic, which was axiomatisedand shown to have the finite model property by Emerson and Joseph Halpern[1982; 1985]. CTL has the limitation that the path quantifiers ∀ and ∃ are tiedto a single linear-time state quantifier (modality) as in the forms ∀G, ∃F , or asingle instance of U as in ∃(α Uβ) etc. It does not allow a combination like ∃GF α,expressing “there is a path along which α is true infinitely often”, a property ofrelevance to fair scheduling conditions. Emerson and Halpern [1983; 1986] deviseda new system CTL* that allows such formations. It distinguishes between stateformulas, which are true or false at each state, and path formulas, which are trueor false of each path. The path formulas include the state formulas and both categoriesare closed under the truth-functional connectives. If α, β are path formulasthen αUβ, Gα and Xα are path formulas, while ∀α and ∃α are state formulas.∀α (respectively ∃α) is true at state s iff α is true of all (respectively some) pathsthat start at s.In addition to being more expressive than CTL, CTL* is more complex. WhereasCTL and PDL are decidable by algorithms that run in deterministic exponentialtime, the complexity of CTL* is that of deterministic doubly exponential time. Thelower bound here was established by Moshe Vardi and Larry Stockmeyer [1985],and the upper bound by Emerson and Charanjit Jutla [1988; 1999]. Methods fromtree automata theory are used to prove decidability results in this context. Modelscan be viewed as infinite branching trees, or at least can be “unravelled” into suchtree structures. Associated with each formula α is an automaton A α that acceptsa tree model iff it it satisfies α at its root. Thus the satisfiability problem for manylogics can be reduced to the emptiness problem for automata on infinite trees thatwas shown to be decidable in [Rabin, 1969] (see section 6.2). This technique wasfirst developed in the 1980 Masters thesis of Robert Streett (see [1982]) who usedit to prove the decidability of PDL with the repeat construct.