Yoshihito Toyama

Counterexamples to termination for the direct sum of term rewriting systems

Abstract The direct sum of two term rewriting systems is the union of systems having disjoint sets of function symbols. It is shown that the direct sum of two term rewriting systems is not terminating, even if these systems are both terminating.

Decidability for Left-Linaer Growing Term Rewriting Systems

A. term rewriting system is called growing if each variable occurring both the left-hand side and the right-hand side of a rewrite rule occurs at depth zero or one in the left-hand side. Jacquemard showed that the reachability and the sequentiality of linear (i.e., left-right-linear) growing term rewriting systems are decidable. In this paper we show that Jacquemards result can be extended to left-linear growing rewriting systems that may have right-non-linear rewrite rules. This implies that the reachability and the joinability of some class of right-linear term rewriting systems are decidable, which improves the results for rightground term rewriting systems by Oyamaguchi. Our result extends the class of left-linear term rewriting systems having a decidable call-by-need normalizing strategy. Moreover, we prove that the termination property is decidable for almost orthogonal growing term rewriting systems.

Termination for the direct sum of left-linear term rewriting systems

The direct sum of two term rewriting systems is the union of systems having disjoint sets of function symbols. It is shown that two term rewriting systems both are left-linear and complete if and only if the direct sum of these systems is so.

Proving Confluence of Term Rewriting Systems Automatically

We have developed an automated confluence prover for term rewriting systems (TRSs). This paper presents theoretical and technical ingredients that have been used in our prover. A distinctive feature of our prover is incorporation of several divide---and---conquer criteria such as those for commutative (Toyama, 1988), layer-preserving (Ohlebusch, 1994) and persistent (Aoto & Toyama, 1997) combinations. For a TRS to which direct confluence criteria do not apply, the prover decomposes it into components and tries to apply direct confluence criteria to each component. Then the prover combines these results to infer the (non-)confluence of the whole system. To the best of our knowledge, an automated confluence prover based on such an approach has been unknown.

Modularity of confluence: a simplified proof

Abstract In this note we present a simple proof of a result of Toyama which states that the disjoint union of confluent term rewriting is confluent.

Termination of S-Expression Rewriting Systems: Lexicographic Path Ordering for Higher-Order Terms

This paper expands the termination proof techniques based on the lexicographic path ordering to term rewriting systems over varyadic terms, in which each function symbol may have more than one arity. By removing the deletion property from the usual notion of the embedding relation, we adapt Kruskal’s tree theorem to the lexicographic comparison over varyadic terms. The result presented is that finite term rewriting systems over varyadic terms are terminating whenever they are compatible with the lexicographic path order. The ordering is simple, but powerful enough to handle most of higher-order rewriting systems without λ-abstraction, expressed as S-expression rewriting systems.

On Composable Properties of Term Rewriting Systems

A property of term rewriting system (TRS, for short) is said to be composable if it is preserved under unions. We present composable properties of TRSs on the base of modularity results for direct sums of TRSs. We propose a decomposition by a naive sort attachment, and show that modular properties for direct sums of TRSs are τ-composable for a naive sort attachment τ. Here, a decomposition of a TRS R is a pair (R1,R2) of (not necessary disjoint) subsets of R such that R = R1 U R2; and for a naive sort attachment T a property o of TRSs is said to be τ-composable if for any TRS R such that τ is consistent with R, o(R1) Λ φ(R2) implies φ(R) where (R1, R2) is the decomposition of R by τ.

Termination Proof of S-Expression Rewriting Systems with Recursive Path Relations

S-expression rewriting systems were proposed by the author (RTA 2004) for termination analysis of Lisp-like untyped higher-order functional programs. This paper presents a short and direct proof for the fact that every finite S-expression rewriting system is terminating if it is compatible with a recursive path relation with status. By considering well-founded binary relations instead of well-founded orders, we give a much simpler proof than the one depending on Kruskals tree theorem.

Fast Knuth-Bendix completion with a term rewriting system compiler

Abstract A term rewriting system compiler can greatly improve the execution speed of reductions by transforming rewriting rules into target code. In this report, we present a new application of the term rewriting system compiler: the Knuth-Bendix completion algorithm. The compiling technique proposed in this algorithm, is dynamic in the sense that rewriting rules are repeatedly compiled in the completion process. The execution time of the completion with dynamic compiling is ten or more times as fast as that obtained with a traditional term rewriting system interpreter.

Parallel Closure Theorem for Left-Linear Nominal Rewriting Systems

Nominal rewriting has been introduced as an extension of first-order term rewriting by a binding mechanism based on the nominal approach. In this paper, we extend Huet’s parallel closure theorem and its generalisation on confluence of left-linear term rewriting systems to the case of nominal rewriting. The proof of the theorem follows a previous inductive confluence proof for orthogonal uniform nominal rewriting systems, but the presence of critical pairs requires a much more delicate argument. The results include confluence of left-linear uniform nominal rewriting systems that are not \(\alpha \)-stable and thus are not represented by any systems in traditional higher-order rewriting frameworks.