Branching time temporal logic and amorphous tree automata

Abstract

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 U _ψ, 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 U _ψ that does not assume simultaneous input trees.

The amorphous automata for CTL use the Büchi acceptance condition. The size of U _ψ 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.