arXiv subjects

Misaki Kojima

Publications and source records attributed to Misaki Kojima.

4 recordsLinked to original sources

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↗

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↗

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↗