When does a Composition Table Provide a Complete and TractableProof Procedure for a Relational Constraint Language ?

Originating in Allen's analysis of temporal relations , the notion of a composition table (CT) has become a key technique in providing an ef-cient inference mechanism for a wide class of theories. In this paper we challenge researchers working with CTs to give a general character-isation of the class of theories and relational constraint languages for which a complete proof procedure can be speciied by a CT. Several related problems and conjectures will be discussed. A secondary aim is to clarify the terminology used to describe CTs and to establish a general conceptual framework applicable to any CT whatever relations are involved. One of the main advantages of using CTs is that they can lead to tractable computation of sig-niicant classes of inference. An important aspect of our proposed research programme is to separate computational from purely logical issues and then to systematically investigate relationships between logical and computational complexity. In representing and reasoning about many domains useful information can often be stated in terms of a limited vocabulary of binary relations holding among objects. Typically such families of relations will be logically constrained in that certain combinations of relations are possible whilst others are impossible. The logical dependencies between relations may be stated in many ways. If such representations are to be useful for some computational application, we must have a practical method of determining consequences of sets of relational facts. The traditional approach to this problem is to formulate dependencies between relations as a formal theory stated in some general-purpose logical language (e.g. 1st-order logic). If this theory is conjoined with a set of relational facts, consequences of these facts can be determined using any proof procedure which is complete for that language. From a computational point of view, this approach is unsatisfactory for all but the simplest sets of relations. Reasoning with a suuciently expressive general-purpose logic is in most cases undecidable and at best an NP-complete problem. Another method widely employed is to formulate the task as a constraint satisfaction problem (CSP) Tsang, 1993]: each relation is interpreted as a constraint restricting possible values of its arguments. Although testing satissability of CSPs is in general intractable, for many restricted classes of CSP eeective algorithms exist. We are concerned with investigating relational constraint languages consisting of a nite vocabulary of basic relational expressions (a basis) and an enumerable set of constants. In the current work we restrict …