arXiv subjects

Nao Hirokawa

Publications and source records attributed to Nao Hirokawa.

13 recordsLinked to original sources

arXiv2026

Lexicographic Combination of Reduction Pairs (Extended Version)

We present a simple criterion for combining reduction pairs lexicographically. The criterion is applicable to arbitrary classes of reduction pairs, such as the polynomial interpretation, the matrix interpretation, and the Knuth-Bendix order. In addition, we investigate a variant of the matrix interpretation where the lexicographic order is employed instead of the usual component-wise order. Effectiveness is demonstrated by experiments and examples, including Touzet's Hydra Battle.

cs.LO↗

arXiv2026

Unifying Semantic Path Order and Weighted Path Order

Monotonic semantic path orders and weighted path orders are powerful reduction orders for proving termination of term rewrite systems. In this paper we present their simple unification as reduction orders and reduction pairs. We also discuss the use of it as ground total reduction orders.

cs.LO↗

arXiv2025

Hydra Battles and AC Termination

We present a new encoding of the Battle of Hercules and Hydra as a rewrite system with AC symbols. Unlike earlier term rewriting encodings, it faithfully models any strategy of Hercules to beat Hydra. To prove the termination of our encoding, we employ type introduction in connection with many-sorted semantic labeling for AC rewriting and AC-MPO, a new AC compatible reduction order that can be seen as a much weakened version of AC-RPO.

cs.LO↗

arXiv2025

Left-Linear Completion with AC Axioms

We revisit completion modulo equational theories for left-linear term rewrite systems where unification modulo the theory is avoided and the normal rewrite relation can be used in order to decide validity questions. To that end, we give a new correctness proof for finite runs and establish a simulation result between the two inference systems known from the literature. Given a concrete reduction order, novel canonicity results show that the resulting complete systems are unique up to the representation of their rules' right-hand sides. Furthermore, we show how left-linear AC completion can be simulated by general AC completion. In particular, this result allows us to switch from the former to the latter at any point during a completion process.

cs.LO↗

arXiv2024

Compositional Confluence Criteria

We show how confluence criteria based on decreasing diagrams are generalized to ones composable with other criteria. For demonstration of the method, the confluence criteria of orthogonality, rule labeling, and critical pair systems for term rewriting are recast into composable forms. We also show how such a criterion can be used for a reduction method that removes rewrite rules unnecessary for confluence analysis. In addition to them, we prove that Toyama's parallel closedness result based on parallel critical pairs subsumes his almost parallel closedness theorem.

cs.LO↗

arXiv2023

Generalizing Weighted Path Orders

We show that weighted path orders are special instances of a variant of semantic path orders. Exploiting this fact, we introduce a generalization of weighted path orders that goes beyond the realm of simple termination. Experimental data show that generalized weighted path orders are viable.

cs.LO↗

arXiv2019

Abstract Completion, Formalized

Completion is one of the most studied techniques in term rewriting and fundamental to automated reasoning with equalities. In this paper we present new correctness proofs of abstract completion, both for finite and infinite runs. For the special case of ground completion we present a new proof based on random descent. We moreover extend the results to ordered completion, an important extension of completion that aims to produce ground-complete presentations of the initial equations. We present new proofs concerning the completeness of ordered completion for two settings. Moreover, we revisit and extend results of Métivier concerning canonicity of rewrite systems. All proofs presented in the paper have been formalized in Isabelle/HOL.

cs.LO↗

arXiv2019

Confluence by Critical Pair Analysis Revisited (Extended Version)

We present two methods for proving confluence of left-linear term rewrite systems. One is hot-decreasingness, combining the parallel/development closedness theorems with rule labelling based on a terminating subsystem. The other is critical-pair-closing system, allowing to boil down the confluence problem to confluence of a special subsystem whose duplicating rules are relatively terminating.

cs.LO↗

arXiv2017

Critical Peaks Redefined - $Φ\sqcup Ψ= \top$

Let a cluster be a term with a number of patterns occurring in it. We give two accounts of clusters, a geometric one as sets of (node and edge) positions, and an inductive one as pairs of terms with gaps (2nd order variables) and pattern-substitutions for the gaps. We show both notions of cluster and the corresponding refinement/coarsening orders on them, to be isomorphic. This equips clusters with a lattice structure which we lift to (parallel/multi) steps to yield an alternative account of the notion of critical peak.

cs.LO↗

arXiv2015

AC-KBO Revisited

Equational theories that contain axioms expressing associativity and commutativity (AC) of certain operators are ubiquitous. Theorem proving methods in such theories rely on well-founded orders that are compatible with the AC axioms. In this paper we consider various definitions of AC-compatible Knuth-Bendix orders. The orders of Steinbach and of Korovin and Voronkov are revisited. The former is enhanced to a more powerful version, and we modify the latter to amend its lack of monotonicity on non-ground terms. We further present new complexity results. An extension reflecting the recent proposal of subterm coefficients in standard Knuth-Bendix orders is also given. The various orders are compared on problems in termination and completion.

cs.LO↗

arXiv2011

Automated Complexity Analysis Based on the Dependency Pair Method

This article is concerned with automated complexity analysis of term rewrite systems. Since these systems underlie much of declarative programming, time complexity of functions defined by rewrite systems is of particular interest. Among other results, we present a variant of the dependency pair method for analysing runtime complexities of term rewrite systems automatically. The established results significantly extent previously known techniques: we give examples of rewrite systems subject to our methods that could previously not been analysed automatically. Furthermore, the techniques have been implemented in the Tyrolean Complexity Tool. We provide ample numerical data for assessing the viability of the method.

cs.LO↗

arXiv2011

Uncurrying for Innermost Termination and Derivational Complexity

First-order applicative term rewriting systems provide a natural framework for modeling higher-order aspects. In earlier work we introduced an uncurrying transformation which is termination preserving and reflecting. In this paper we investigate how this transformation behaves for innermost termination and (innermost) derivational complexity. We prove that it reflects innermost termination and innermost derivational complexity and that it preserves and reflects polynomial derivational complexity. For the preservation of innermost termination and innermost derivational complexity we give counterexamples. Hence uncurrying may be used as a preprocessing transformation for innermost termination proofs and establishing polynomial upper and lower bounds on the derivational complexity. Additionally it may be used to establish upper bounds on the innermost derivational complexity while it neither is sound for proving innermost non-termination nor for obtaining lower bounds on the innermost derivational complexity.

cs.LO↗

arXiv2009

Decreasing Diagrams and Relative Termination

In this paper we use the decreasing diagrams technique to show that a left-linear term rewrite system R is confluent if all its critical pairs are joinable and the critical pair steps are relatively terminating with respect to R. We further show how to encode the rule-labeling heuristic for decreasing diagrams as a satisfiability problem. Experimental data for both methods are presented.

cs.LO↗