Vectorial languages and linear temporal logic

Determining for a given deterministic complete automaton the sequence of visited states while reading a given word is the core of important problems with automata-based solutions, such as approximate string matching. The main difficulty is to do this computation efficiently. Considering words as vectors and working on them using vectorial operations allows to solve the problem faster than using local operations. In this paper, we show first that the set of vectorial operations needed by an algorithm representing a given automaton depends on the language accepted by the automaton. We give precise characterizations for star-free, solvable and regular languages using vectorial algorithms. We also study classes of languages associated with restricted sets of vectorial operations and relate them with languages defined by fragments of linear temporal logic. Finally, we consider the converse problem of constructing an automaton from a given vectorial algorithm. As a byproduct, we show that the satisfiability problem for some extensions of LTL characterizing solvable and regular languages is PSPACE-complete.