The complexity of tree automata and logics of programs

The computational complexity of testing nonemptiness of finite-state automata on infinite trees is investigated. It is shown that for tree automata with m states and n pairs nonemptiness can be tested in time O((mn)³ⁿ), even though the problem is in general NP-complete. The nonemptiness algorithm is used to obtain exponentially improved, essentially tight upper bounds for numerous important modal logics of programs, interpreted with the usual semantics over structures generated by binary relations. For example, it is shown that satisfiability for the full branching time logic CTL* can be tested in deterministic double exponential time. It also follows that satisfiability for propositional dynamic logic with a repetition construct (PDL-delta) and for the propositional mu-calculus ( L μ) can be tested in deterministic single exponential time