The unification problem for term rewriting systems (TRSs) is the problem of deciding, for a given TRS R and two terms M and N, whether there exists a substitution θ such that Mθ and Nθ are congruent modulo R (i.e., Mθ↔R*Nθ). In this paper, the unification problem for confluent right-ground TRSs is shown to be decidable. To show this, the notion of minimal terms is introduced and a new unification algorithm for obtaining a substitution whose range consists of minimal terms is proposed. Our result extends the decidability of unification for canonical (i.e., terminating and confluent) right-ground TRSs given by Hullot [Proceedings of the 5th Conference on Automated Deduction, LNCS, vol. 87, 1980, p. 318] in the sense that the termination condition can be omitted.

In 1979 Huet and Levy introduced the class of sequential term-rewriting systems in which call-by-need computations are possible (without look-ahead) and defined the subclass called strongly sequential systems for which needed redexes in a given term are effectively found [chapter in Computational Logic: Essays in Honor of Alan Robinson, J.-L. Lassez and G. Plotkin, eds., MIT Press, Cambridge, MA, 1991]. This paper introduces a larger subclass that is a natural extension of strong sequentiality and is based on the analysis of both the left-hand sides and part of the right-hand sides (i.e., the nonvariable parts) of systems, whereas strong sequentiality is based on the analysis of left-hand sides alone. This new sequentiality is called NV-sequentiality. It is shown that (i) the class of NV-sequential systems properly includes the class of strongly sequential systems, (ii) there exists an algorithm for finding needed redexes for a given term when a system is NV-sequential, and (iii) it is decidable whether an arbitrary left-linear system is NV-sequential.

The equivalence problem for deterministic real-time pushdown automata is shown to be decidable. This result is obtained by showing that Valiant's parallel stacking technique using a replacement function introduced in this paper succeeds for deterministic real-time pushdown automata. Equivalence is also decidable for two deterministic pushdown automata, one of which is real-time.