Practical use of polynomials over the reals in proofs of termination

• Authors: Salvador Lucas
• Venue: Proceedings of the 9th ACM SIGPLAN international conference on Principles and practice of declarative programming
• Year: 2007

Nowadays, polynomial interpretations are an essential ingredient in the development of tools for proving termination. We have recently proven that polynomial interpretations over the reals are strictly better for proving polynomial termination of rewriting than those which only use integer coefficients. Some essential aspects of their practical use, though, remain unexplored or underdeveloped. In this paper, we compare the two current frameworks for using polynomial intepretations over the reals and show that one of them is strictly better than the other, thus making a suitable choice for implementations. We also prove that the use of algebraic real co-efficients in the interpretations suffice for termination proofs. We also discuss the use of algorithms and techniques from Tarski's first-order logic of the real closed fields for implementing their use in proofs of termination. We argue that more standard constraint-solving techniques are better suited for this. We propose an algorithm to solve the polynomial constraints which arise when specific finite subsets of rational (or even algebraic real) numbers are considered for giving value to the coefficients. We provide a preliminary experimental evaluation of the algorithm which has been implemented as part of the termination tool MU-TERM.