Negated String Containment is Decidable (Technical Report)

2025-06-27

AI Summary

This paper resolves the long-standing open problem of decidability for the *notContains* predicate—i.e., whether a concatenation $y_1 cdots y_m$ is not a substring of $x_1 cdots x_n$—under regular constraints on string variables $x_1,ldots,x_n$ and $y_1,ldots,y_m$. Using automata-theoretic constructions and reasoning about word equations, the authors establish the first decidability result for *notContains* in the presence of regular membership constraints. They further extend this result to combined systems involving chain-free word equations and regular constraints. This work fills a fundamental gap in the theoretical foundations of string logic and provides new decidability guarantees and algorithmic foundations for precise string analysis in symbolic execution.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Security and Privacy: Large-scale security measurements

Abstract

We provide a positive answer to a long-standing open question of the decidability of the not-contains string predicate. Not-contains is practically relevant, for instance in symbolic execution of string manipulating programs. Particularly, we show that the predicate notContains(x1 ... xn, y1 ... ym), where x1 ... xn and y1 ... ym are sequences of string variables constrained by regular languages, is decidable. Decidability of a not-contains predicate combined with chain-free word equations and regular membership constraints follows.

Problem

• Decidability of not-contains string predicate
• Symbolic execution for string manipulation
• Combination with chain-free word equations

Innovation

• Decidable not-contains string predicate
• Symbolic execution for string programs
• Chain-free word equations with constraints