Deriving constraints among argument sizes in logic programs (extended abstract)

• Authors: Allen van Gelder
• Venue: PODS '90 Proceedings of the ninth ACM SIGACT-SIGMOD-SIGART symposium on Principles of database systems
• Year: 1990

In a logic program the feasible argument sizes of derivable facts involving an n-ary predicate are viewed as a set of points in the positive orthant of Rn. We investigate a method of deriving constraints on the feasible set in the form of a polyhedral convex set in the positive orthant, which we call a polycone. Faces of this polycone represent inequalities proven to hold among the argument sizes. These inequalities are often useful for selecting an evaluation method that is guaranteed to terminate for a given logic procedure. The methods may be applicable to other languages in which the sizes of data structures can be determined syntactically.We introduce a generalized Tucker representation for systems of linear equations and show how needed operations on polycones are performed in this representation. We prove that every polycone has a unique normal form in this representation, and give an algorithm to produce it. This in turn gives a decision procedure for the question of whether two set of linear equations define the same polycone.When a predicate has several rules, the union of the individual rule's polycones gives the set of feasible argument size vectors for the predicate. Because this set is not necessarily convex, we instead operate with the smallest enclosing polycone, which is the closure of the convex hull of the union. Retaining convexity is one of the key features of our technique.Recursion is handled by finding a polycone that is a fixpoint of a transformation that is derived from both the recursive and nonrecursive rules. Some methods for finding a fixpoint are presented, but there are many unresolved problems in this area.