Deciding Confluence and Normal Form Properties of Ground Term Rewrite Systems Efficiently

It is known that the first-order theory of rewriting is decidable for ground term rewrite systems, but the general technique uses tree automata and often takes exponential time. For many properties, including confluence (CR), uniqueness of normal forms with respect to reductions (UNR) and with respect to conversions (UNC), polynomial time decision procedures are known for ground term rewrite systems. However, this is not the case for the normal form property (NFP). In this work, we present a cubic time algorithm for NFP, an almost cubic time algorithm for UNR, and an almost linear time algorithm for UNC, improving previous bounds. We also present a cubic time algorithm for CR

Labelings for Decreasing Diagrams

This article 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 article have been implemented and experimental results demonstrate how, e.g., the rule labeling benefits from our contributions

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 differ 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

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

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

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(n5) algorithm, where n is the size of the ground TRS. In this paper we improve this bound to O(n3). The algorithm has been implemented in the confluence tool CSI

Confluence of term rewriting : theory and automation

Thema dieser Dissertation ist die Theorie der Konfluenz von Termersetzungssystemen und deren Automatisierung. Termersetzungssysteme sind ein Berechnungsmodell bei dem Terme sukzessiv abgeändert werden, indem man Instanzen der linken Seite von Gleichungen durch entsprechende Instanzen der rechten Seite ersetzt. Konfluenz ist eine wichtige Eigenschaft von Ersetzungssystemen welche eng mit der Eindeutigkeit von Normalformen verbunden ist,und daher mit der Eindeutigkeit von Funktionsdefinitionen. Für den Fall dass Termination eines Systems nicht gegeben ist, drückt Konfluenz eine Art deterministisches Verhalten aus: wenn zwei Berechnungen vom gleichen Ausgangszustand begonnen werden, so ist es auch möglich, diese wieder zu einem gemeinsamen Zustand zusammenzuführen. Wie die meisten interessanten Eigenschaften in der Informatik ist auch Konfluenz eine unentscheidbare Eigenschaft. Festzustellen ob ein Ersetzungssystem konfluent ist, ist deshalb oft eine Herausforderung. Dennoch gibt es automatische Beweiser für Konfluenz, die versuchen, diese Aufgabe selbständig zu bewältigen. Das Ziel dieser Arbeit ist es, den Stand der Technik auf dem Gebiet des automatischen Beweisens von Konfluenz voranzubringen. Zu diesem Zweck erweitern wir die Theorie der Konfluenz in mehreren Bereichen um Konfluenzkriterien abzuleiten, die sowohl mächtig als auch programmierbar sind.The topic of this thesis is confluence of first-order term rewriting systems, both its theory and its automation. Term rewrite systems are a model of computation in which terms are successively modified by replacing instances of left-hand sides of equations by the corresponding instance of the right-hand side. Confluence is an important property of rewrite systems which is intimately connected to uniqueness of normal forms, and therefore well-definedness of functions. In the absence of termination, confluence expresses a kind of deterministic behavior: for any two computations starting at the same initial state, it is always possible to continue both of them until they reach a common state again. Like most interesting properties in computer science, confluence is an undecidable property of term rewrite systems. Therefore, determining whether a system is confluent is often challenging. Nevertheless, there are automated confluence provers that attempt to solve this task automatically. The goal of this thesis is to advance the state of the art of the field of automated confluence proving. To this end, we extend the theory of confluence in several areas to obtain confluence criteria that are both powerful and implementable.by Bertram FelgenhauerInnsbruck, Univ., Diss., 2015OeBB(VLID)36200