Computing with conditional rewrite rules

Abstract

We present a method for validating abstract data type specifications. The method takes as input a set of ground terms L 0 and a set of conditional equations A 0 over L 0. The object of this method is to find a normal form function, Norm, for the pair 〈 L 0, A 0 〉. The function Norm is computed as a sequence of step functions S 1, S 2, ..., S n.Each step function S i, 0 ≤ i ≤ n, takes as input a pair 〈 L i−1, A i−1 〉, where L i−1 is a set of ground terms and A i is a set of conditional equations over the set of terms L i−1. At each step i, a set of equations E i is selected from the set of theorems of the pair 〈 L i−1, A i−1 〉. The set of equations E i is transformed into a set of reductions R i. The step function S i is defined as the top-down reduction extention of R i to L i−1. The output of S i is the pair 〈 L i, Ai 〉, where L i is the set of normal forms of L i−1 under the set of reductions R i and A i is the set of normal forms of the equations in A i−1 under the same set of reductions. This way, a theorem in the system 〈 L i−1, A i−1 〉 becomes a theorem in the system 〈 L i, Ai 〉. The last step, S n, has as output the pair 〈 L n, φ 〉. The only theorems in 〈 L n, φ 〉 are the identities. This way the sequence\(< L_0 ,A_0 > \mathop \to \limits^{S_1 } < L_1 ,A_1 > \mathop \to \limits^{S_2 } ... < L_{n - 1} ,A_{n - 1} > \mathop \to \limits^{S_n } < A_n ,\phi >\)gives us a procedure to compute the normal form of the terms in 〈 L 0, A 0 〉.In this paper we present criteria for choosing the sets of equations E i which simplify the pair 〈 L i−1, A i−1 〉. We also present results that characterize the output set 〈 L i, Ai 〉 of S i as a function of the set 〈 L i−1, A i−1 〉 and of the set of reductions R i. If the sets of reductions R i are confluent and terminating, then they can be combined, by using a priority system similar to the one developed by Baeten, Bergstra and Klop, to form a confluent and terminating set of reductions on the set 〈 L 0, A 0 〉.KeywordsNormal FormFree AlgebraTransfer PropertyComputation SequenceGround TermThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.