Branching time temporal logic and amorphous tree automata
An automata-theoretic framework for branching-time temporal logics is presented. We introduce a new type of finite automata on infinite trees, the amorphous automata, and use them as a formalism to represent efficiently CTL formulas. In addition, we introduce simultaneous trees, and associate with every model for CTL, a simultaneous tree that enables a tree automaton to visit different nodes on the same path of the tree simultaneously. With every formula ψ, we associate an amorphous automaton Uψ, that accepts exactly those simultaneous trees (of any branching degree) that originate from models that satisfy ψ. This enables to use the automaton both for model checking which is reduced to the membership problem, and for satisfiability decision, which is reduced to testing the nonemptiness of an extension of Uψ that does not assume simultaneous input trees.
The amorphous automata for CTL use the Büchi acceptance condition. The size of Uψ is linear in ¦ψ¦ and the extension required for satisfiability is exponential. Based on that, we get a polynomial model checking procedure and an exponential decision procedure for CTL, both match the known lower bounds. This is the first time that a model checking algorithm for a branching-time temporal logic is placed in the automata-theoretic framework.
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- [CE81]Clarke, E.M. and Emerson, E.A., Design and synthesis of synchronization skeletons using branching time temporal logic. Proc. Workshop on Logic of Programs, LNCS 131 (1981) 52–71Google Scholar
- [Em90]Emerson, E.A., Temporal and modal logic. Handbook of theoretical computer science. (1990) 997–1072 North-Holland.Google Scholar
- [Ha82]Harel, D., Effective Transformations on Infinite Trees, with Applications to High Undecidability, Dominoes and Fairness. J. Assoc. Comput. Mach. 33 (1986) 224–248.Google Scholar
- [Pn77]Pnueli A., The temporal logic of programs. Proc. 18th Symp. on Foundation of Computer Science. (1977) 46–57.Google Scholar
- [Ra69]Rabin, M.O., Decidability of second order theories and automata on infinite trees. Trans. AMS, 141 (1969) 1–35.Google Scholar
- [Th90]Thomas W., Automata on Infinite Objects. Handbook of theoretical computer science.(1990) 165–191, North-Holland.Google Scholar
- [Va89]Vardi M.Y., Automata theory for database theoreticians. Proc. 8th ACM Symp. on Principles of Data Systems. (1989) 83–92.Google Scholar
- [VW86b]Vardi M.Y. and Wolper P., An Automata theoretic approach to automatic program verification. Proc. Symp. on Logic in Computer Science, (1986) 322–331Google Scholar
- [Wo87]Wolper, P., On the relations of programs and computations to models of temporal logic, Temporal Logic in Specification, LNCS 398, (1989) 75–123Google Scholar
- [WVS83]Wolper P., Vardi M.Y., and Sistla P., Reasoning about infinite computation paths, Proc. 24th Symp. on Foundations of Computer Science, (1983) 185–194.Google Scholar