arXiv subjects

Naoki Nishida

Publications and source records attributed to Naoki Nishida.

At least 19 recordsLinked to original sources

arXiv2026

Termination of Innermost-Terminating Right-Linear Overlay Term Rewrite Systems

It has been shown that, regarding a terminating right-linear overlay term rewrite system (TRS), any rewrite sequence terminating in a normal form can be simulated by an innermost reduction. In this paper, using this simulation property, we show that for a right-linear overlay TRS, there is no infinite minimal dependency-pair chain if and only if there is no infinite innermost minimal dependency-pair chain. As a consequence, termination and innermost termination coincide for the class of right-linear overlay TRSs.

cs.LO↗

arXiv2026

Rewriting Induction for Existentially Quantified Equations in Logically Constrained Rewriting (Full Version)

Rewriting Induction (RI) is a principle to prove that an equation over terms is an inductive theorem of a rewrite system, i.e., that any ground instance of the equation is a theorem of the rewrite system. RI has been adapted to several kinds of rewrite systems, and RI for constrained rewrite systems has been extended to inequalities. In this paper, we extend RI for constrained equations to existentially quantified equations in logically constrained rewriting. To this end, we first extend constrained equations by introducing existential quantification to the equation part of constrained equations. Then, in applying a constrained rewrite rule to such extended constrained equations, we introduce existential quantification to extra variables of the applied rule. Finally, using the extended application of constrained rewrite rules, we extend RI for constrained equations to existentially quantified equations.

cs.LO↗

arXiv2026

Abstract Framework for All-Path Reachability Analysis toward Safety and Liveness Verification (Full Version)

An All-Path Reachability predicate over an object set is a pair of a source set and a target set, which are subsets of the object set. APR predicates have been defined for Abstract Reduction Systems and then extended to Logically Constrained Term Rewrite Systems as pairs of constrained terms that represent sets of terms modeling configurations, states, etc. An APR predicate is partially valid w.r.t. a rewrite system if every finite maximal reduction sequence of the system starting from any element in the source set includes an element in the target set. Partial validity of APR predicates w.r.t. ARSs is defined by means of two inference rules, which can be considered a proof system to construct (possibly infinite) derivation trees for partial validity. On the other hand, a proof system for LCTRSs consists of four inference rules, leaving a gap between the inference rules for ARSs and LCTRSs. In this paper, we revisit the framework for APR analysis and adapt it to verification of not only safety but also liveness properties. To this end, we first reformulate an abstract framework for partial validity w.r.t. ARSs so that there is a one-to-one correspondence between the inference rules for partial validity w.r.t. ARSs and LCTRSs. Secondly, we show how to apply APR analysis to safety verification. Thirdly, to apply APR analysis to liveness verification, we introduce a novel stronger validity of APR predicates, called total validity, which requires not only finite but also infinite execution paths to reach target sets. Finally, for a partially valid APR predicate with a cyclic-proof tree, we show that the acyclicity of the proof graph obtained from the cyclic-proof tree is a necessary and sufficient condition for total validity. The condition implies that if there exists a cyclic-proof tree for an APR predicate, the proof graph of which is acyclic, then the APR predicate is totally valid.

cs.LO↗

arXiv2026

Partial Rewriting and Value Interpretation of Logically Constrained Terms (Full Version)

Logically constrained term rewrite systems (LCTRSs) are a rewriting formalism that naturally supports built-in data structures, including integers and bit-vectors. The recent framework of existentially constrained terms and most general constrained rewriting on them (Takahata et al., 2025) has many advantages over the original approach of rewriting constrained terms. In this paper, we introduce partial constrained rewriting, a variant of rewriting existentially constrained terms whose underlying idea has already appeared implicitly in previous analyses and applications of LCTRSs. We examine the differences between these two notions of constrained rewriting. First, we establish a direct correspondence between them, leveraging subsumption and equivalence of constrained terms where appropriate. Then we give characterizations of each of them, using the interpretation of existentially constrained terms by instantiation. We further introduce the novel notion of value interpretation, that highlights subtle differences between partial and most general rewriting.

cs.LO↗

arXiv2025

Recovering Commutation of Logically Constrained Rewriting and Equivalence Transformations (Full Version)

Logically constrained term rewriting is a relatively new rewriting formalism that naturally supports built-in data structures, such as integers and bit vectors. In the analysis of logically constrained term rewrite systems (LCTRSs), rewriting constrained terms plays a crucial role. However, this combines rewrite rule applications and equivalence transformations in a closely intertwined way. This intertwining makes it difficult to establish useful theoretical properties for this kind of rewriting and causes problems in implementations -- namely, that impractically large search spaces are often required. To address this issue, we propose in this paper a novel notion of most general constrained rewriting, which operates on existentially constrained terms, a concept recently introduced by the authors. We define a class of left-linear, left-value-free LCTRSs that are general enough to simulate all left-linear LCTRSs and exhibit the desired key property: most general constrained rewriting commutes with equivalence. This property ensures that equivalence transformations can be deferred until after the application of rewrite rules, which helps mitigate the issue of large search spaces in implementations. In addition to that, we show that the original rewriting formalism on constrained terms can be embedded into our new rewriting formalism on existentially constrained terms. Thus, our results are expected to have significant implications for achieving correct and efficient implementations in tools operating on LCTRSs.

cs.LO↗

arXiv2025

Difference of Constrained Patterns in Logically Constrained Term Rewrite Systems (Full Version)

Considering patterns as sets of their instances, a difference operator over patterns computes a finite set of two given patterns, which represents the difference between the dividend pattern and the divisor pattern. A complement of a pattern is a pattern set, the ground constructor instances of which comprise the complement of the ground constructor instances of the former pattern. Given finitely many unconstrained linear patterns, using a difference operator over linear patterns, a complement algorithm returns a finite set of linear patterns as a complement of the given patterns. In this paper, we extend the difference operator and complement algorithm to constrained linear patterns used in logically constrained term rewrite systems (LCTRSs, for short) that have no user-defined constructor term with a sort for built-in values. Then, as for left-linear term rewrite systems, using the complement algorithm, we show that quasi-reducibility is decidable for such LCTRSs with decidable built-in theories. For the single use of the difference operator over constrained patterns, only divisor patterns are required to be linear.

cs.LO↗

arXiv2025

Characterizing Equivalence of Logically Constrained Terms via Existentially Constrained Terms (Full Version)

Logically constrained term rewriting is a rewriting framework that supports built-in data structures such as integers and bit vectors. Recently, constrained terms play a key role in various analyses and applications of logically constrained term rewriting. A fundamental question on constrained terms arising there is how to characterize equivalence between them. However, in the current literature only limited progress has been made on this. In this paper, we provide several sound and complete solutions to tackle this problem. Our key idea is the introduction of a novel concept, namely existentially constrained terms, into which the original form of constrained terms can be embedded. We present several syntactic characterizations of equivalence between existentially constrained terms. In particular, we provide two different kinds of complete characterizations: one is designed to facilitate equivalence checking, while the other is intended for theoretical analysis.

cs.LO↗

arXiv2024

Equational Theories and Validity for Logically Constrained Term Rewriting (Full Version)

Logically constrained term rewriting is a relatively new formalism where rules are equipped with constraints over some arbitrary theory. Although there are many recent advances with respect to rewriting induction, completion, complexity analysis and confluence analysis for logically constrained term rewriting, these works solely focus on the syntactic side of the formalism lacking detailed investigations on semantics. In this paper, we investigate a semantic side of logically constrained term rewriting. To this end, we first define constrained equations, constrained equational theories and validity of the former based on the latter. After presenting the relationship of validity and conversion of rewriting, we then construct a sound inference system to prove validity of constrained equations in constrained equational theories. Finally, we give an algebraic semantics, which enables one to establish invalidity of constrained equations in constrained equational theories. This algebraic semantics derive a new notion of consistency for constrained equational theories.

cs.LO↗

arXiv2023

On Singleton Self-Loop Removal for Termination of LCTRSs with Bit-Vector Arithmetic

As for term rewrite systems, the dependency pair (DP, for short) framework with several kinds of DP processors is useful for proving termination of logically constrained term rewrite systems (LCTRSs, for short). However, the polynomial interpretation processor is not so effective against LCTRSs with bit-vector arithmetic (BV-LCTRSs, for short). In this paper, we propose a novel DP processor for BV-LCTRSs to solve a singleton DP problem consisting of a dependency pair forming a self-loop. The processor is based on an acyclic directed graph such that the nodes are bit-vectors and any dependency chain of the problem is projected to a path of the graph. We show a sufficient condition for the existence of such an acyclic graph, and simplify it for a specific case.

cs.LO↗

arXiv2019

On Transforming Functions Accessing Global Variables into Logically Constrained Term Rewriting Systems

In this paper, we show a new approach to transformations of an imperative program with function calls and global variables into a logically constrained term rewriting system. The resulting system represents transitions of the whole execution environment with a call stack. More precisely, we prepare a function symbol for the whole environment, which stores values for global variables and a call stack as its arguments. For a function call, we prepare rewrite rules to push the frame to the stack and to pop it after the execution. Any running frame is located at the top of the stack, and statements accessing global variables are represented by rewrite rules for the environment symbol. We show a precise transformation based on the approach and prove its correctness.

cs.LO↗

arXiv2019

On Transforming Narrowing Trees into Regular Tree Grammars Generating Ranges of Substitutions

The grammar representation of a narrowing tree for a syntactically deterministic conditional term rewriting system and a pair of terms is a regular tree grammar that generates expressions for substitutions obtained by all possible innermost-narrowing derivations that start with the pair and end with particular non-narrowable terms. In this paper, under a certain syntactic condition, we show a transformation of the grammar representation of a narrowing tree into another regular tree grammar that overapproximately generates the ranges of ground substitutions generated by the grammar representation. In our previous work, such a transformation is restricted to the ranges w.r.t. a given single variable, and thus, the usefulness is limited. We extend the previous transformation by representing the range of a ground substitution as a tuple of terms, which is obtained by the coding for finite trees. We show a precise definition of the transformation and prove that the language of the transformed regular tree grammar is an overapproximation of the ranges of ground substitutions generated by the grammar representation. We leave an experiment to evaluate the usefulness of the transformation as future work.

cs.LO↗

arXiv2018

A Theory of Reversibility for Erlang

In a reversible language, any forward computation can be undone by a finite sequence of backward steps. Reversible computing has been studied in the context of different programming languages and formalisms, where it has been used for testing and verification, among others. In this paper, we consider a subset of Erlang, a functional and concurrent programming language based on the actor model. We present a formal semantics for reversible computation in this language and prove its main properties, including its causal consistency. We also build on top of it a rollback operator that can be used to undo the actions of a process up to a given checkpoint.

cs.PL↗

arXiv2018

Transforming Proof Tableaux of Hoare Logic into Inference Sequences of Rewriting Induction

A proof tableau of Hoare logic is an annotated program with pre- and post-conditions, which corresponds to an inference tree of Hoare logic. In this paper, we show that a proof tableau for partial correctness can be transformed into an inference sequence of rewriting induction for constrained rewriting. We also show that the resulting sequence is a valid proof for an inductive theorem corresponding to the Hoare triple if the constrained rewriting system obtained from the program is terminating. Such a valid proof with termination of the constrained rewriting system implies total correctness of the program w.r.t. the Hoare triple. The transformation enables us to apply techniques for proving termination of constrained rewriting to proving total correctness of programs together with proof tableaux for partial correctness.

cs.LO↗

arXiv2018

Transforming Dependency Chains of Constrained TRSs into Bounded Monotone Sequences of Integers

In the dependency pair framework for proving termination of rewriting systems, polynomial interpretations are used to transform dependency chains into bounded decreasing sequences of integers, and they play an important role for the success of proving termination, especially for constrained rewriting systems. In this paper, we show sufficient conditions of linear polynomial interpretations for transforming dependency chains into bounded monotone (i.e., decreasing or increasing) sequences of integers. Such polynomial interpretations transform rewrite sequences of the original system into decreasing or increasing sequences independently of the transformation of dependency chains. When we transform rewrite sequences into increasing sequences, polynomial interpretations have non-positive coefficients for reducible positions of marked function symbols. We propose four DP processors parameterized by transforming dependency chains and rewrite sequences into either decreasing or increasing sequences of integers, respectively. We show that such polynomial interpretations make us succeed in proving termination of the McCarthy 91 function over the integers.

cs.LO↗

arXiv2017

Reversible Computation in Term Rewriting

Essentially, in a reversible programming language, for each forward computation from state $S$ to state $S'$, there exists a constructive method to go backwards from state $S'$ to state $S$. Besides its theoretical interest, reversible computation is a fundamental concept which is relevant in many different areas like cellular automata, bidirectional program transformation, or quantum computing, to name a few. In this work, we focus on term rewriting, a computation model that underlies most rule-based programming languages. In general, term rewriting is not reversible, even for injective functions; namely, given a rewrite step $t_1 \rightarrow t_2$, we do not always have a decidable method to get $t_1$ from $t_2$. Here, we introduce a conservative extension of term rewriting that becomes reversible. Furthermore, we also define two transformations, injectivization and inversion, to make a rewrite system reversible using standard term rewriting. We illustrate the usefulness of our transformations in the context of bidirectional program transformation.

cs.PL↗

arXiv2017

Sound Structure-Preserving Transformation for Weakly-Left-Linear Deterministic Conditional Term Rewriting Systems

In this paper, we show that the SR transformation, a computationally equivalent transformation proposed by Serbanuta and Rosu, is a sound structure-preserving transformation for weakly-left-linear deterministic conditional term rewriting systems. More precisely, we show that every weakly-left-linear deterministic conditional term rewriting system can be converted to an equivalent weakly-left-linear and ultra-weakly-left-linear deterministic conditional term rewriting system and prove that the SR transformation is sound for weakly-left-linear and ultra-weakly-left-linear deterministic conditional term rewriting systems. Here, soundness for a conditional term rewriting system means that reduction of the transformed unconditional term rewriting system creates no undesired reduction sequence for the conditional system.

cs.FL↗

arXiv2016

Towards Reversible Computation in Erlang

In a reversible language, any forward computation can be undone by a finite sequence of backward steps. Reversible computing has been studied in the context of different programming languages and formalisms, where it has been used for debugging and for enforcing fault-tolerance, among others. In this paper, we consider a subset of Erlang, a concurrent language based on the actor model. We formally introduce a reversible semantics for this language. To the best of our knowledge, this is the first attempt to define a reversible semantics for Erlang.

cs.PL↗

arXiv2014

Verifying Procedural Programs via Constrained Rewriting Induction

This paper aims to develop a verification method for procedural programs via a transformation into Logically Constrained Term Rewriting Systems (LCTRSs). To this end, we extend transformation methods based on integer TRSs to handle arbitrary data types, global variables, function calls and arrays, as well as encode safety checks. Then we adapt existing rewriting induction methods to LCTRSs and propose a simple yet effective method to generalize equations. We show that we can automatically verify memory safety and prove correctness of realistic functions. Our approach proves equivalence between two implementations, so in contrast to other works, we do not require an explicit specification in a separate specification language.

cs.LO↗