Sequentiality, Monadic Second-Order Logic and Tree Automata

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.