Harald Zankl

SAT solving for termination analysis with polynomial interpretations

Polynomial interpretations are one of the most popular techniques for automated termination analysis and the search for such interpretations is a main bottleneck in most termination provers. We show that one can obtain speedups in orders of magnitude by encoding this task as a SAT problem and by applying modern SAT solvers.

Satisfiability of non-linear (Ir)rational arithmetic

We present a novel way for reasoning about (possibly ir)rational quantifier-free non-linear arithmetic by a reduction to SAT/SMT. The approach is incomplete and dedicated to satisfiable instances only but is able to producemodels for satisfiable problems quickly. These characteristics suffice for applications such as termination analysis of rewrite systems. Our prototype implementation, called MiniSmt, is made freely available. Extensive experiments show that it outperforms current SMT solvers especially on rational and irrational domains.

Revisiting matrix interpretations for polynomial derivational complexity of term rewriting

Matrix interpretations can be used to bound the derivational complexity of term rewrite systems. In particular, triangular matrix interpretations over the natural numbers are known to induce polynomial upper bounds on the derivational complexity of (compatible) rewrite systems. Using techniques from linear algebra, we show how one can generalize the method to matrices that are not necessarily triangular but nevertheless polynomially bounded. Moreover, we show that our approach also applies to matrix interpretations over the real (algebraic) numbers. In particular, it allows triangular matrix interpretations to infer tighter bounds than the original approach.

KBO Orientability

This article presents three new approaches to prove termination of rewrite systems with the Knuth–Bendix order efficiently. The constraints for the weight function and for the precedence are encoded in (pseudo-)propositional logic or linear arithmetic and the resulting formula is tested for satisfiability using dedicated solvers. Any satisfying assignment represents a weight function and a precedence such that the induced Knuth–Bendix order orients the rules of the encoded rewrite system from left to right. This means that in contrast to the dedicated methods our approach does not directly solve the problem but transforms it to equivalent formulations for which sophisticated back-ends exist. In order to make all approaches complete we present a method to compute upper bounds on the weights. Furthermore, our encodings take dependency pairs into account to increase the applicability of the order.

Uncurrying for Termination

First-order applicative term rewrite systems provide a natural framework for modeling higher-order aspects. In this paper we present a transformation from untyped applicative term rewrite systems to functional term rewrite systems that preserves and reflects termination. Our transformation is less restrictive than other approaches. In particular, head variables in right-hand sides of rewrite rules can be handled. To further increase the applicability of our transformation, we present a version for dependency pairs.

CSI: a confluence tool

This paper describes a new confluence tool for term rewrite systems. Due to its modular design, the few techniques implemented so far can be combined flexibly. Methods developed for termination analysis are adapted to prove and disprove confluence. Preliminary experimental results show the potential of our tool.

Joint spectral radius theory for automated complexity analysis of rewrite systems

Matrix interpretations can be used to bound the derivational complexity of term rewrite systems. In particular, triangular matrix interpretations over the natural numbers are known to induce polynomial upper bounds on the derivational complexity of (compatible) rewrite systems. Recently two different improvements were proposed, based on the theory of weighted automata and linear algebra. In this paper we strengthen and unify these improvements by using joint spectral radius theory.

Satisfying KBO constraints

This paper presents two new approaches to prove termination of rewrite systems with the Knuth-Bendix order efficiently. The constraints for the weight function and for the precedence are encoded in (pseudo-)propositional logic and the resulting formula is tested for satisfiability. Any satisfying assignment represents a weight function and a precedence such that the induced Knuth-Bendix order orients the rules of the encoded rewrite system from left to right.

Modular Complexity Analysis for Term Rewriting

All current investigations to analyze the derivational complexity of term rewrite systems are based on a single termination method, possibly preceded by transformations. However, the exclusive use of direct criteria is problematic due to their restricted power. To overcome this limitation the article introduces a modular framework which allows to infer (polynomial) upper bounds on the complexity of term rewrite systems by combining different criteria. Since the fundamental idea is based on relative rewriting, we study how matrix interpretations and match-bounds can be used and extended to measure complexity for relative rewriting, respectively. The modular framework is proved strictly more powerful than the conventional setting. Furthermore, the results have been implemented and experiments show significant gains in power.

Labelings for Decreasing Diagrams

This paper is concerned with automating the decreasing diagrams technique of van Oostrom for establishing confluence of term rewrite systems. We study abstract criteria that allow to lexicographically combine labelings to show local diagrams decreasing. This approach has two immediate benefits. First, it allows to use labelings for linear rewrite systems also for left-linear ones, provided some mild conditions are satisfied. Second, it admits an incremental method for proving confluence which subsumes recent developments in automating decreasing diagrams. The techniques proposed in the paper have been implemented and experimental results demonstrate how, e.g., the rule labeling benefits from our contributions.