Sequentiality, monadic second-order logic and tree automata

Abstract

Given a term rewriting system R and a normalizable term t, a redex is needed if in any reduction sequence of t to a normal form, this redex will be contracted. Roughly, R is sequential if there is an optimal reduction strategy in which only needed redexes are contracted. More generally, G. Huet and J.-J. Lévy have defined the sequentiality of a predicate P on partially evaluated terms (1991, "Computational Logic: Essays in Honor of Alan Robinson", MIT Press, Cambridge, MA, pp. 415-443). We show here that the sequentiality of P is definable in SkS, the monadic second-order logic with k successors, provided P is definable in SkS. We derive several known and new consequences of this remark: (1) strong sequentiality, as defined by Huet and Lévy of a left linear (possibly overlapping) rewrite system is decidable, (2) NV-sequentiality, as defined in (M. Oyamaguchi, 1993, SIAM J. Comput. 19, 424-437), is decidable, even in the case of overlapping rewrite systems (3) sequentiality of any linear shallow rewrite system is decidable. Then we describe a direct construction of a tree automaton recognizing the set of terms that do have needed redexes, which again, yields immediate consequences: (1) Strong sequentiality of possibly overlapping linear rewrite systems is decidable in EXPTIME, (2) For strongly sequential rewrite systems, needed redexes can be read directly on the automaton. © 2000 Academic Press.