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

Abstract

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.

Reachability Analysis with State-Compatible Automata

“…In this paper we are largely concerned with over-approximations of the terms reachable from a regular tree language L 0 by rewriting using a term rewrite system R, that is, we are interested in regular tree languages L such that R * (L 0 ) ⊆ L. Such over-approximations have been used, among other things, in the analysis of cryptographic protocols [6], for termination analysis [7,10] and for establishing non-confluence of term rewrite systems [15].…”

Proving Termination of Rewrite Systems Using Bounds

“…General knowledge of the match-bound technique [7,10] will be helpful. Below we recall important definitions and results from the latter paper.…”

“…The general idea [6,10] is to look for violations of the compatibility requirement: lσ → * ∆ d q and not rσ → * ∆ q for some rewrite rule l → r, state substitution σ : Var(l) → Q d , and state q. Then we add new states and transitions to the current automaton to ensure rσ → * ∆ q.…”

“…The following concept has been introduced in [10]. It enables the simulation of narrowing in the definition of right-hand sides of forward closures by rewriting.…”

“…Initially proposed for string rewriting by Geser, Hofbauer, and Waldmann [7], the method has recently been extended to left-linear term rewrite systems [10]. Variations and improvements are discussed in [5,8,9].…”