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

2026-02-16

AI Summary

This work addresses the challenge of inductive theorem proving for equations involving existential quantifiers within logically constrained rewriting systems. Existing rewrite induction methods are unable to handle such quantifiers, limiting their applicability. To overcome this limitation, the paper proposes an extended framework that explicitly incorporates existential quantifiers into equations and dynamically manages the auxiliary variables they introduce during rule application. This approach constitutes the first extension of rewrite induction to equations constrained by existential quantification, substantially broadening the class of provable formulas. The resulting framework significantly enhances the expressiveness and reasoning capabilities of automated inductive proof techniques in complex, constraint-rich systems.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Knowledge Representation and Reasoning: Automated Reasoning and Theorem Proving
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty

Application Category

• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

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.

Problem

• Rewriting Induction
• Existentially Quantified Equations
• Logically Constrained Rewriting
• Constrained Rewrite Systems
• Inductive Theorem

Innovation

• Rewriting Induction
• Existential Quantification
• Constrained Rewriting
• Inductive Theorem
• Logically Constrained Term Rewriting