Logic Programming with Sets by Membership-constraints 1.1 Sets as New Data Structures 1.3 Overview of the Rest

1 Motivation Implementations of logic programming languages have terms as their main data type. But often it may be useful and more natural to represent objects by sets instead of simple terms. General agreement exists about the usefulness of sets as very high-level representations of complex data structures. In particular, sets can be used conveniently in rapid software prototyp-ing. Therefore, it is worthwhile to introduce sets into logic programming. Sets shall be used on argument positions in (logic) programs. If we query the system e.g. fx; yg = fa; bg the system will answer with two variable bindings, namely fx a; y bg and fx b; y ag which represent the solutions of the corresponding set uniication problem. It is also useful to introduce more operators in the programming language, namely 2 and = 2. Furthermore set union and intersection \ may be desirable. There are constraint canonization algorithms like the ones stated in 2, 9]. Both need an algorithm for set uni-cation but they do not develop algorithms that are reasonable eecient in the average case. In this paper, we will concentrate on the implementation of an algorithm for set uniication which is needed in any logic programming language embedding ((nite) sets. Before we address this problem we have to give a deeni-tion of set uniication. A substitution is called a uni-er of two sets A = fx 1 ; : : : ; x m g and B = fy 1 ; : : : ; y n g ii for every x in A there exists some y in B (and vice versa too) such that xx = y 5] or (stated diierently) AA B and B AA. In the sequel, we will also consider sets with tails s written as fx 1 ; : : : ; x k jtg which has to be understood as the union fx 1 ; : : : ; x k g t; here t is a variable for a set, called the tail of s. In the following, we will present an algorithm that behaves eeciently in the average case. Firstly, the case where the set tails are empty can be treated by so-called membership-constraints which can be seen as a tricky variant of the predicate member/2 of Prolog using non-uniiability constraints 12]. Constraint techniques such as delayed execution and the rst-fail principle lead to even more improvements. Secondly, …