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A Decidable Logic for Tree Data-Structures with Measurements

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Verification, Model Checking, and Abstract Interpretation (VMCAI 2019)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 11388))

Abstract

We present \({\textsc {Dryad}}_\textit{dec}\), a decidable logic that allows reasoning about tree data-structures with measurements. This logic supports user-defined recursive measure functions based on Max or Sum, and recursive predicates based on these measure functions, such as AVL trees or red-black trees. We prove that the logic’s satisfiability is decidable. The crux of the decidability proof is a small model property which allows us to reduce the satisfiability of \({\textsc {Dryad}}_\textit{dec}\) to quantifier-free linear arithmetic theory which can be solved efficiently using SMT solvers. We also show that \({\textsc {Dryad}}_\textit{dec}\) can encode a variety of verification and synthesis problems, including natural proof verification conditions for functional correctness of recursive tree-manipulating programs, legality conditions for fusing tree traversals, synthesis conditions for conditional linear-integer arithmetic functions. We developed the decision procedure and successfully solved 220+ \({\textsc {Dryad}}_\textit{dec}\) formulae raised from these application scenarios, including verifying functional correctness of programs manipulating AVL trees, red-black trees and treaps, checking the fusibility of height-based mutually recursive tree traversals, and counterexample-guided synthesis from linear integer arithmetic specifications. To our knowledge, \({\textsc {Dryad}}_\textit{dec}\) is the first decidable logic that can solve such a wide variety of problems requiring flexible combination of measure-related, data-related and shape-related properties for trees.

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Notes

  1. 1.

    Intuitively, a \({\textsc {Dryad}}_\textit{dec}\) function is increasing/decreasing if its value monotonically increases/decreases when the input tree expands. The monotonicity will be formally defined in Sect. 3.1.

  2. 2.

    Let \(n_{l+1}\) be \({\textit{nil}}\) if \(|\mathcal {N}| \le l\).

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Acknowledgments

This material is based upon work supported by the National Science Foundation under Grant No. 1837023.

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  1. Purdue University, West Lafayette, USA

    Xiaokang Qiu & Yanjun Wang

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  1. Xiaokang Qiu
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Correspondence to Xiaokang Qiu or Yanjun Wang .

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  1. IRIF, University Paris Diderot and CNRS, Paris, France

    Constantin Enea

  2. Yale University, New Haven, CT, USA

    Ruzica Piskac

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Qiu, X., Wang, Y. (2019). A Decidable Logic for Tree Data-Structures with Measurements. In: Enea, C., Piskac, R. (eds) Verification, Model Checking, and Abstract Interpretation. VMCAI 2019. Lecture Notes in Computer Science(), vol 11388. Springer, Cham. https://doi.org/10.1007/978-3-030-11245-5_15

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