Encoding and Reasoning About Arrays in Set Theory

2025-08-15

AI Summary

Existing set-theoretic automated reasoning tools—such as {log}—lack native support for arrays, limiting their applicability to programs involving mixed data structures. Method: We propose encoding arrays as functions represented by sets of ordered pairs, thereby reducing array reasoning to pure set-theoretic reasoning. To formalize this, we define a decidable fragment of set theory extended with function and array semantics, and implement its solver using constraint logic programming. Contribution/Results: This work introduces the first unified formalization and automated reasoning framework for arrays, sets, and relations within {log}, overcoming the prior decidability barrier posed by array constructs. Experimental evaluation demonstrates that our approach effectively encodes and solves programs featuring nontrivial array operations—including indexing, update, and length constraints—thereby substantially enhancing {log}’s capability to reason about heterogeneous data structures.

Technology Category

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

Application Category

• Graph Algorithms and Modeling for the Web: Querying, indexing, and retrieval in Web-related graphs
• 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
• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications

Abstract

We encode arrays as functions which, in turn, are encoded as sets of ordered pairs. The set cardinality of each of these functions coincides with the length of the array it is representing. Then we define a fragment of set theory that is used to give the specifications of a non-trivial class of programs with arrays. In this way, array reasoning becomes set reasoning. Furthermore, a decision procedure for this fragment is also provided and implemented as part of the {log} (read 'setlog') tool. {log} is a constraint logic programming language and satisfiability solver where sets and binary relations are first-class citizens. The tool already implements a few decision procedures for different fragments of set theory. In this way, arrays are seamlessly integrated into {log} thus allowing users to reason about sets, functions and arrays all in the same language and with the same solver. The decision procedure presented in this paper is an extension of decision procedures defined in earlier works not supporting arrays.

Problem

• Encode arrays as sets for set theory reasoning
• Define set theory fragment for array program specifications
• Extend decision procedure in {log} to support arrays

Innovation

• Encode arrays as sets of ordered pairs
• Define set theory fragment for array specifications
• Extend {log} tool with array decision procedure