Sequentiality, Second Order Monadic 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. L evy deene in 9] the sequentiality of a predicate P on partially evaluated terms. We show here that the sequentiality of P is deenable in SkS, the second-order monadic logic with k successors, provided P is deenable in SkS. We derive several known an new consequences of this remark: 1{strong sequentiality, as deened in 9], of a left linear (possibly overlapping) rewrite system is decidable, 2{ NV-sequentiality, as deened in 17] 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 se-quentiality 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.