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Quadratic Word Equations with Length Constraints, Counter Systems, and Presburger Arithmetic with Divisibility

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Automated Technology for Verification and Analysis (ATVA 2018)

Part of the book series: Lecture Notes in Computer Science ((LNPSE,volume 11138))

Abstract

Word equations are a crucial element in the theoretical foundation of constraint solving over strings. A word equation relates two words over string variables and constants. Its solution amounts to a function mapping variables to constant strings that equate the left and right hand sides of the equation. While the problem of solving word equations is decidable, the decidability of the problem of solving a word equation with a length constraint (i.e., a constraint relating the lengths of words in the word equation) has remained a long-standing open problem. We focus on the subclass of quadratic word equations, i.e., in which each variable occurs at most twice. We first show that the length abstractions of solutions to quadratic word equations are in general not Presburger-definable. We then describe a class of counter systems with Presburger transition relations which capture the length abstraction of a quadratic word equation with regular constraints. We provide an encoding of the effect of a simple loop of the counter systems in the existential theory of Presburger Arithmetic with divisibility (PAD). Since PAD is decidable, we get a decision procedure for quadratic words equations with length constraints for which the associated counter system is flat (i.e., all nodes belong to at most one cycle). In particular, we show a decidability result (in fact, also an NP algorithm with a PAD oracle) for a recently proposed NP-complete fragment of word equations called regular-oriented word equations, when augmented with length constraints. Decidability holds when the constraints are extended with regular constraints with a 1-weak control structure.

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Notes

  1. 1.

    In fact, it is a NP algorithm with an oracle access to \(\mathcal {P}\mathcal {A}\mathcal {D}\). The best complexity bound for the latter is NEXP and NP-hardness [18].

  2. 2.

    Note that we mean polynomial in the size of the NFA, which can be exponential in |S|.

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Acknowledgment

We thank Jatin Arora, Dmitry Chistikov, Volker Diekert, Matthew Hague, Artur Jeż, Philipp Rümmer, and James Worrell for the helpful discussions. This research was partially funded by the ERC Starting Grant AV-SMP (grant agreement no. 759969) and the ERC Synergy Grant IMPACT (grant agreement no. 610150).

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Authors and Affiliations

  1. Oxford University, Oxford, UK

    Anthony W. Lin

  2. Max Planck Institute for Software Systems, Kaiserslautern, Germany

    Rupak Majumdar

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  1. Anthony W. Lin
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Correspondence to Anthony W. Lin .

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  1. Microsoft Research, Redmond, WA, USA

    Shuvendu K. Lahiri

  2. University of Southern California, Los Angeles, CA, USA

    Chao Wang

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Lin, A.W., Majumdar, R. (2018). Quadratic Word Equations with Length Constraints, Counter Systems, and Presburger Arithmetic with Divisibility. In: Lahiri, S., Wang, C. (eds) Automated Technology for Verification and Analysis. ATVA 2018. Lecture Notes in Computer Science(), vol 11138. Springer, Cham. https://doi.org/10.1007/978-3-030-01090-4_21

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  • DOI: https://doi.org/10.1007/978-3-030-01090-4_21

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