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

2025-07-05

AI Summary

Existing difference and complement algorithms for term rewriting operate only on unconstrained linear patterns, failing to handle patterns with logical constraints (e.g., equalities, ordering relations) in Logic Constraint Term Rewriting Systems (LCTRSs). Method: This paper systematically extends difference and complement operations to constrained linear patterns within left-linear LCTRSs endowed with decidable built-in theories and no user-defined built-in constructors. We provide formal definitions and constructive algorithms for both operators under this setting. Contributions/Results: (1) First difference algorithm supporting non-purely-linear divisor patterns; (2) Complement closure achieved without user-defined built-in constructors; (3) Decidability of quasi-reducibility established in this framework. These results significantly enhance the expressiveness and automation level of constrained pattern analysis and program verification.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Machine Learning: Statistical Relational/Logic Learning

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments

Abstract

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 set of patterns, the ground constructor instances of which comprise the complement set of the ground constructor instances of the former pattern. Given finitely many unconstrained linear patterns, using a difference operator over linear patterns, a known 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 TRSs, 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.

Problem

• Extend difference operator to constrained linear patterns in LCTRSs
• Develop complement algorithm for constrained linear patterns in LCTRSs
• Decide quasi-reducibility for left-linear LCTRSs with decidable theories

Innovation

• Extends difference operator to constrained patterns
• Develops complement algorithm for LCTRSs
• Decides quasi-reducibility via complement algorithm