Classes of Term Rewrite Systems with PolynomialCon uence

The connuence property of ground (i.e., variable-free) term rewrite systems (TRS) is well-known to be decidable. This was proved independently in Dauchet et al. 1987; Dauchet et al. 1990] and in Oyamaguchi 1987] using tree automata techniques and ground tree transducer techniques (originated from this problem), yielding EXPTIME decision procedures (PSPACE for strings). Since then, and until last year, the optimality of this bound had been a well-known longstanding open question (see, e.g., RTA-LOOP 2001]). In Comon et al. 2001] we gave the rst polynomial-time algorithm for deciding the connuence of ground TRS. Later in Tiwari 2002] this result was extended, using abstract congruent closure techniques, to linear shallow TRS, i.e., TRS where no variable occurs twice in the same rule nor at depth greater than one. Here, we give a new and much simpler proof of the latter result.