Bertram Felgenhauer

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.

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.

Reachability Analysis with State-Compatible Automata

Regular tree languages are a popular device for reachability analysis over term rewrite systems, with many applications like analysis of cryptographic protocols, or confluence and termination analysis. At the heart of this approach lies tree automata completion, first introduced by Genet for left-linear rewrite systems. Korp and Middeldorp introduced so-called quasi-deterministic automata to extend the technique to non-left-linear systems. In this paper, we introduce the simpler notion of quasi-compatible automata, which are slightly more general than quasi-deterministic, compatible automata. This notion also allows us to decide whether a regular tree language is closed under rewriting, a problem which was not known to be decidable before. Several of our results have been formalized in the theorem prover Isabelle/HOL. This allows to certify automatically generated non-confluence and termination proofs that are using tree automata techniques.

CSI: New Evidence – A Progress Report

CSI is a strong automated confluence prover for rewrite systems which has been in development since 2010. In this paper we report on recent extensions that make CSI more powerful, secure, and useful. These extensions include improved confluence criteria but also support for uniqueness of normal forms. Most of the implemented techniques produce machine-readable proof output that can be independently verified by an external tool, thus increasing the trust in CSI. We also report on CSI\(\mathbf {\hat{~}}\)oho, a tool built on the same framework and similar ideas as CSI that automatically checks confluence of higher-order rewrite systems.

Deciding Confluence of Ground Term Rewrite Systems in Cubic Time.

It is well known that the confluence property of ground term rewrite systems (ground TRSs) is decidable in polynomial time. For an efficient implementation, the degree of this polynomial is of great interest. The best complexity bound in the literature is given by Comon, Godoy and Nieuwenhuis (2001), who describe an O(n^5) algorithm, where n is the size of the ground TRS. In this paper we improve this bound to O(n^3). The algorithm has been implemented in the confluence tool CSI.

Reachability, confluence, and termination analysis with state-compatible automata ☆

Abstract Regular tree languages are a popular device for reachability analysis over term rewrite systems, with many applications like analysis of cryptographic protocols, or confluence and termination analysis. At the heart of this approach lies tree automata completion, first introduced by Genet for left-linear rewrite systems. Korp and Middeldorp introduced so-called quasi-deterministic automata to extend the technique to non-left-linear systems. In this paper, we introduce the simpler notion of state-compatible automata, which are slightly more general than quasi-deterministic, compatible automata. This notion also allows us to decide whether a regular tree language is closed under rewriting, a problem which was not known to be decidable before. The improved precision has a positive impact in applications which are based on reachability analysis, namely termination and confluence analysis. Our results have been formalized in the theorem prover Isabelle/HOL. This allows to certify automatically generated proofs that are using tree automata techniques.

Improving Automatic Confluence Analysis of Rewrite Systems by Redundant Rules.

We describe how to utilize redundant rewrite rules, i.e., rules that can be simulated by other rules, when (dis)proving confluence of term rewrite systems. We demonstrate how automatic confluence provers benefit from the addition as well as the removal of redundant rules. Due to their simplicity, our transformations were easy to formalize in a proof assistant and are thus amenable to certification. Experimental results show the surprising gain in power.

Proof Orders for Decreasing Diagrams

We present and compare some well-founded proof orders for decreasing diagrams. These proof orders order a conversion above another conversion if the latter is obtained by filling any peak in the former by a (locally) decreasing diagram. Therefore each such proof order entails the decreasing diagrams technique for proving confluence. The proof orders dier with respect to monotonicity and complexity. Our results are developed in the setting of involutive monoids. We extend these results to obtain a decreasing diagrams technique for confluence modulo. 1998 ACM Subject Classification F.4 Mathematical Logic and Formal Languages

Layer Systems for Proving Confluence

We introduce layer systems for proving generalizations of the modularity of confluence for first-order rewrite systems. Layer systems specify how terms can be divided into layers. We establish structural conditions on those systems that imply confluence. Our abstract framework covers known results like many-sorted persistence, layer-preservation and currying. We present a counterexample to an extension of the former to order-sorted rewriting and derive new sufficient conditions for the extension to hold.

Constructing Cycles in the Simplex Method for DPLL(T)

Modern SMT solvers use a special DPLL(T) variant of the simplex algorithm to solve satisfiability problems in linear real arithmetic. Termination is guaranteed by Bland’s pivot selection rule, but it is not immediately obvious that such a rule is required. For the traditional simplex method non-termination is well-understood, but the cycling examples from the literature do not immediately carry over to the DPLL(T) variant. We present two SMT encodings of the problem of finding cycles, using linear and nonlinear real arithmetic.