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 di3culty is to do this computation e3ciently. 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 6rst 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 de6ned 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 satis6ability problem for some extensions of LTL characterizing solvable and regular languages is PSPACE-complete. c © 2003 Elsevier B.V. All rights reserved.