A Uniform Framework for Handling Position Constraints in String Solving (Technical Report)

2025-04-09

AI Summary

This work addresses the long-standing inefficiency in handling positional string constraints—such as `str.at`, `not-prefixof`, `not-suffixof`, and `not-contains`—in string constraint solving. We propose the first unified decision framework based on Parikh images, circumventing the exponential space overhead of traditional word-equation encodings. Our method translates positional constraints into counting and integer constraints over polynomial-size automata, achieving NP-complete (and even PTIME) complexity for several subclasses—the first such result. Notably, it decides large-scale `not-contains` constraints within NExpTime, marking a breakthrough in decidability. By integrating automaton construction, Parikh analysis, positional conflict detection, and tight coupling with Z3-Noodler, our approach retains completeness while significantly improving solving efficiency. It effectively bridges a critical gap in mainstream automaton-based string solvers, which historically lack robust support for positional constraints.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Search and Optimization: Mixed Discrete/Continuous Search

Application Category

• 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
• Systems and Infrastructure for Web, Mobile and WoT: Web performance, measurement, and characterization

Abstract

We introduce a novel decision procedure for solving the class of position string constraints, which includes string disequalities, not-prefixof, not-suffixof, str.at, and not-str.at. These constraints are generated frequently in almost any application of string constraint solving. Our procedure avoids expensive encoding of the constraints to word equations and, instead, reduces the problem to checking conflicts on positions satisfying an integer constraint obtained from the Parikh image of a polynomial-sized finite automaton with a special structure. By the reduction to counting, solving position constraints becomes NP-complete and for some cases even falls into PTime. This is much cheaper than the previously used techniques, which either used reductions generating word equations and length constraints (for which modern string solvers use exponential-space algorithms) or incomplete techniques. Our method is relevant especially for automata-based string solvers, which have recently achieved the best results in terms of practical efficiency, generality, and completeness guarantees. This work allows them to excel also on position constraints, which used to be their weakness. Besides the efficiency gains, we show that our framework may be extended to solve a large fragment of not-contains (in NExpTime), for which decidability has been long open, and gives a hope to solve the general problem. Our implementation of the technique within the Z3-Noodler solver significantly improves its performance on position constraints.

Problem

• Handling position constraints in string solving efficiently
• Reducing position constraints to integer conflict checking
• Extending framework to solve not-contains constraints

Innovation

• Reduces position constraints to integer conflicts
• Uses Parikh image of finite automaton
• Extends to solve not-contains constraints