Alfons Geser

Match-Bounded String Rewriting Systems

Abstract.We introduce a new class of automated proof methods for the termination of rewriting systems on strings. The basis of all these methods is to show that rewriting preserves regular languages. To this end, letters are annotated with natural numbers, called match heights. If the minimal height of all positions in a redex is h then every position in the reduct will get height h+1. In a match-bounded system, match heights are globally bounded. Using recent results on deleting systems, we prove that rewriting by a match-bounded system preserves regular languages. Hence it is decidable whether a given rewriting system has a given match bound. We also provide a criterion for the absence of a match-bound. It is still open whether match-boundedness is decidable. Match-boundedness for all strings can be used as an automated criterion for termination, for match-bounded systems are terminating. This criterion can be strengthened by requiring match-boundedness only for a restricted set of strings, namely the set of right hand sides of forward closures.

A Unified Fault-Tolerance Protocol

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.

Abstractions for Fault-Tolerant Distributed System Verification

Four kinds of abstraction for the design and analysis of fault–tolerant distributed systems are discussed. These abstractions concern system messages, faults, fault–masking voting, and communication. The abstractions are formalized in higher–order logic, and are intended to facilitate specifying and verifying such systems in higher–order theorem–provers.

Parallelizing functional programs by generalization

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.

Termination Proofs for String Rewriting Systems via Inverse Match-Bounds

Annotating a letter by a number, one can record information about its history during a rewrite derivation. In each rewrite step, numbers in the reduct are updated depending on the redex numbering. A string rewriting system is called match-bounded if there is a global upper bound to these numbers. Match-boundedness is known to be a strong sufficient criterion for both termination and preservation of regular languages. We show that the string rewriting systems whose inverse (left and right hand sides exchanged) is match-bounded, also have exceptional properties, but slightly different ones. Inverse match-bounded systems need not terminate; they effectively preserve context-free languages; their sets of normalizable strings and their sets of immortal strings are effectively regular. These languages can be used to decide the normalization, the uniform normalization, the termination and the uniform termination problem for inverse match-bounded systems. We also prove that the termination problem is decidable in linear time, and that a certain strong reachability problem is decidable, thereby solving two open problems of McNaughton’s. Like match-bounds, inverse match-bounds entail linear derivational complexity on the set of terminating strings.

On tree automata that certify termination of left-linear term rewriting systems

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.

A Complete Characterization of Termination of 0 p 1 q → 1 r 0 s

Abstract. We characterize termination of one-rule string rewriting systems of the form 0p 1q→ 1r 0s for every choice of positive integers p, q, r, and s. In doing so we introduce a termination proof method that applies to some hard examples. For the simply terminating cases, i.e. string rewriting systems that can be ordered by a division order, we give the precise complexity of derivation lengths.

An Improved General Path Order

We define a strong and versatile termination order for term rewriting systems, called theImproved General Path Order, which simplifies and strengthens Dershowitz/Hoots General Path Order. We demonstrate the power of the Improved General Path Order by proofs of termination of non-trivial examples, among them a medium-scale term rewriting system that models a lift control.

Relative Undecidability in the Termination Hierarchy of Single Rewrite Rules

For a hierarchy of properties of term rewriting systems, related to termination, we prove relative undecidability even in the case of single rewrite rules: for implications X ⇒ Y in the hierarchy the property X is undecidable for rewrite rules satisfying Y.

Omega-Termination is Undecidable for Totally Terminating Term Rewriting Systems

We give a complete proof of the fact that the following problem is undecidable:Given:A term rewriting system, where the termination of its rewrite relation is provable by a total reduction order on ground terms,Wanted:Does there exist a strictly monotonic interpretation in the positive integers that proves termination?