Proving Termination by Invariance Relations

Abstract

We propose a new constraint-based approach to termination analysis, applicable to Logic Programming (LP) and Constraint Handling Rules (CHR). Our approach further extends the existing constraintbased approaches for LP based on polynomial interpretations and introduces a whole new level of expressivity. We can handle problems such as bounded increase and integer arithmetic, elegantly. Furthermore, we are able to prove termination of programs that only terminate for subsets of the considered queries. Examples are algorithms that manipulate graphs and that only terminate if the graph in the input is cycle-free. This information cannot be represented, using the existing techniques in termination analysis. Therefore, we introduce invariance relations, representing relations among terms that hold on atoms during calls to the program. These relations can also be derived in a constraint-based manner and they can be used as a basis for a more expressive interpretation of the atoms of the program. We discuss our technique in the context of CHR, solving an important class of open problems containing transitivity rules. We also demonstrate the technique in an LP context and show that it is more powerful than existing constraint-based approaches. The following CHR program [3, 7], computes the transitive closure of a graph. transitivity @ arc(X, Y), arc(Y, Z) ⇒ arc(X, Z).

CLP projection for constraint handling rules

“…It goes without saying there exist today ad-hoc CHR analyzers that provide better results than Pilozzi et al's transformation-based approach. For instance, Pilozzi's CHRisTA system [17] can prove the termination of convert and weight. Nevertheless, to the best of our knowledge, the CLP projection together with AProVE pro vides the flrst automatic termination proof for the CHR implementation of ackerman.…”

A complete and terminating execution model for Constraint Handling Rules

“…. The program's termination behavior in ω t and ω p has been investigated in Pilozzi and De Schreye (2009), where it is shown to terminate for acyclic graphs. However, states containing cyclic graphs entail non-terminating behavior (cf.…”

Research Summary: Termination of CHR

“…To be able to handle programs with bounded increase or that terminate only for subsets of the considered queries, we have developed a new approach. Preliminary work has been accepted for ICLP '09 [7]. Using this approach, we are developing an automated analysis for CHR on top of LP, to obtain as such the most powerful integrated termination analysis tool for both CHR and LP.…”