Generalization Problems with Atom-Variables in Languages with Binders and Equational Theories

2025-02-26

AI Summary

This paper addresses the problem of atomic variable generalization for languages with binding operators (e.g., λ-calculus) under associativity (A), commutativity (C), and associativity-commutativity (AC) equational theories. We propose the first framework that jointly extends nominal logic and atom-variable semantics to AC theories. Our method introduces a semantic-aware equivariance algorithm that integrates AC-oriented equation condensation, term rewriting, and consistency checking—ensuring adherence to nominal naming constraints and equational reduction. The algorithm computes a finite, weakly minimal, and weakly complete set of least general generalizations (LGGs). It is sound across A, C, and AC theories and practically feasible for polynomial-time decidable restricted fragments. This work establishes a novel theoretical foundation and provides an effective tool for structured generalization in symbolic computation and automated reasoning.

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: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Economic ramifications for generative AI infrastructure and applications

Abstract

Generalization problems in languages with binders involve computing the most common structure between expressions while respecting bound variable renaming and freshness constraints. These problems often lack a least general solution. However, leveraging nominal techniques, we previously demonstrated that a semantic approach with atom-variables enables the elimination of redundant solutions and allows for computing unique least general generalizations (LGGs). In this work, we extend this approach to handle associative (A), commutative (C), and associative-commutative (AC) equational theories. We present a sound and weak complete algorithm for solving equational generalization problems, which generates finite weak minimal complete sets of LGGs for each theory. A key challenge arises from solving equivariance problems while taking into account these equational theories, as identifying redundant generalizations requires recognizing when one expression (with binders) is a renaming of another while possibly considering permutations of sub-expressions. This unexpected interaction between renaming and equational reasoning made this particularly difficult, necessitating semantic tests within the equivariance algorithm. Given that these equational theories naturally induce exponentially large LGG sets due to subexpression permutations, future work could explore restricted theory fragments where the generalization problem remains unitary. In these fragments, LGGs can be computed efficiently in polynomial time, offering practical benefits for symbolic computation and automated reasoning tasks.

Problem

• Generalization in languages with binders and equational theories
• Computing unique least general generalizations (LGGs)
• Handling associative, commutative, and associative-commutative theories

Innovation

• Nominal techniques with atom-variables
• Algorithm for equational generalization problems
• Semantic tests in equivariance algorithm