Detecting Useless Critical Pairs

The critical pair/completion (CPC) technique was initiated in the mid sixties in three separated areas: theorem proving [12], polynomial ideal theory [3] and term rewriting [9]. For theoretical and practical reasons, improvements of CPC algorithms were developed in two main directions. The rst one concerns strategies for selecting critical pairs. In [10], strategies consisting in adding the new critical pairs in a stack or in queue are investigated: the rst one can fail even if another strategy succeeds whereas the second one always succeeds in the same situation. Other strategies consist in reducing critical pairs in parallel [1] and have shown to be e cient since the previously intractable cyclic 9 problem is solved using such a strategy in [6]. The second direction of improvement consists in nding criteria for detecting useless critical pairs. Buchberger introduced such a criterion in the context of polynomial ideal theory in [4] which was adapted to term rewriting systems in [13]. The presented work concerns the detection of useless critical pairs of rewriting systems whose underlying set of terms is a vector space. We are studying such rewriting systems since the theory of Gröbner bases concerns rewriting in a large class of algebraic structures (polynomial, tensor or Lie algebras, operads, invariant rings...) and we want that our framework generalises these various structures. For these structures, several criteria based on the so-called syzygies were introduced [7, 8, 11] for avoiding useless critical pairs during completion. As shown in [2], the computation of syzygies does not only enable us to reject critical pairs but to reject useless reductions. By useless reductions, we mean that all the critical pairs created from these reductions are useless. The downside of this approach is that useless reductions cannot be used to reduce terms into normal forms. In this work, we introduce a lattice criterion for reducing the number of examinations of critical pairs. For that, we consider rewrite relations −→ on a vector space V which admit decompositions