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

2025-07-12

AI Summary

In Logic Constraint Term Rewriting Systems (LCTRSs), the tight coupling between rule application and equivalence transformations hinders theoretical analysis and causes combinatorial explosion in the search space. Method: This paper introduces a most-general constrained rewriting mechanism based on existential constrained terms, which for the first time achieves commutativity between constrained rewriting and equivalence transformations. The mechanism supports built-in data structure operations and enables deferred equivalence transformations. By restricting attention to left-linear, left-value-irrelevant LCTRSs, the framework ensures full embedding preservation during rewriting. Contribution/Results: The resulting framework enjoys desirable theoretical properties—including decidable confluence and termination—and offers strong potential for efficient implementation. It provides a rigorous foundation for developing practical analysis tools for LCTRSs, bridging the gap between theoretical soundness and computational feasibility.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Knowledge Representation and Reasoning: Automated Reasoning and Theorem Proving
• Machine Learning: Statistical Relational/Logic Learning

Application Category

• Semantics and Knowledge: Data modeling to support human-machine intelligence, including LLMs agents, intelligent system behavior, explanations, and user-friendly interactions
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs

Abstract

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.

Problem

• Address intertwining of rewrite rules and equivalence transformations
• Reduce impractically large search spaces in LCTRS implementations
• Establish commutation of rewriting and equivalence for efficiency

Innovation

• Most general constrained rewriting on existentially constrained terms
• Left-linear, left-value-free LCTRSs for simulation
• Commutation of rewriting and equivalence transformations