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, A_i 〉, 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, A_i 〉. 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, A_i 〉 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 〉.