We present a new method for proving termination of term rewriting systems automatically. It is a generalization of the match bound method for string rewriting. To prove that a term rewriting system terminates on a given regular language of terms, we first construct an enriched system over a new signature that simulates the original derivations. The enriched system is an infinite system over an infinite signature, but it is locally terminating: every restriction of the enriched system to a finite signature is terminating. We then construct iteratively a finite tree automaton that accepts the enriched given regular language and is closed under rewriting modulo the enriched system. If this procedure stops, then the enriched system is compact: every enriched derivation involves only a finite signature. Therefore, the original system terminates. We present three methods to construct the enrichment: top heights, roof heights, and match heights. Top and roof heights work for left-linear systems, while match heights give a powerful method for linear systems. For linear systems, the method is strengthened further by a forward closure construction. Using these methods, we give examples for automated termination proofs that cannot be obtained by standard methods.

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Abstract. Davies and Wakerly show that Byzantine fault tolerance can be achieved by a cascade of broadcasts and middle value select functions. We present an extension of the Davies and Wakerly protocol, the unified protocol, and its proof of correctness. We prove that it satisfies validity and agreement properties for communication of exact values. We then introduce bounded communication error into the model. Inexact communication is inherent for clock synchronization protocols. We prove that validity and agreement properties hold for inexact communication, and that exact communication is a special case. As a running example, we illustrate the unified protocol using the SPIDER family of fault-tolerant architectures. In particular we demonstrate that the SPIDER interactive consistency, distributed diagnosis, and clock synchronization protocols are instances of the unified protocol.

List homomorphisms are functions that are parallelizable using the divide-and-conquer paradigm. We study the problem of finding homomorphic representations of functions in the Bird–Meertens constructive theory of lists, by means of term rewriting and theorem proving techniques. A previous work proved that to each pair of leftward and rightward sequential representations of a function, based on cons- and snoc-lists, respectively, there is also a representation as a homomorphism. Our contribution is a mechanizable method to extract the homomorphism representation from a pair of sequential representations. The method is decomposed to a generalization problem and an inductive claim, both solvable by term rewriting techniques. To solve the former we present a sound generalization procedure which yields the required representation, and terminates under reasonable assumptions. The inductive claim is provable automatically. We illustrate the method and the procedure by the systematic parallelization of the scan-function (parallel prefix) and of the maximum segment sum problem.