Title:

Using automata to characterise fixed point temporal logics

This work examines propositional fixed point temporal and modal logics called mucalculi and their relationship to automata on infinite strings and trees. We use correspondences between formulae and automata to explore definability in mucalculi and their fragments, to provide normal forms for formulae, and to prove completeness of axiomatisations. The study of such methods for describing infinitary languages is of fundamental importance to the areas of computer science dealing with nonterminating computations, in particular to the specification and verification of concurrent and reactive systems. To emphasise the close relationship between formulae of mucalculi and alternating automata, we introduce a new first recurrence acceptance condition for automata, checking intuitively whether the first infinitely often occurring state in a run is accepting. Alternating first recurrence automata can be identified with mucalculus formulae, and ordinary, nonalternating first recurrence automata with formulae in a particular normal form, the strongly aconjunctive form. Automata with more traditional Büchi and Rabin acceptance conditions can be easily unwound to first recurrence automata, i.e. to mucalculus formulae. In the other direction, we describe a powerset operation for automata that corresponds to fixpoints, allowing us to translate formulae inductively to ordinary Büchi and Rabinautomata. These translations give easy proofs of the facts that Rabinautomata, the full mucalculus, its strongly aconjunctive fragment and the monadic secondorder calculus of n successors SnS are all equiexpressive, that Büchiautomata, the fixpoint alternation class Pi_2 and the strongly aconjunctive fragment of Pi_2 are similarly related, and that the weak SnS and the fixpointalternationfree fragment of mucalculus also coincide. As corollaries we obtain Rabin's complementation lemma and the powerful decidability result of SnS. We then describe a direct tableau decision method for modal and lineartime mucalculi, based on the notion of definition trees. The tableaux can be interpreted as first recurrence automata, so the construction can also be viewed as a transformation to the strongly aconjunctive normal form. Finally, we present solutions to two open axiomatisation problems, for the lineartime mucalculus and its extension with path quantifiers. Both completeness proofs are based on transforming formulae to normal forms inspired by automata. In extending the completeness result of the lineartime mucalculus to the version with path quantifiers, the essential problem is capturing the limit closure property of paths in an axiomatisation. To this purpose, we introduce a new \exists\nuinduction inference rule.
