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Predicate Abstraction and CEGAR for \(\nu \mathrm {HFL}_\mathbb {Z}\) Validity Checking

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Static Analysis (SAS 2020)

Part of the book series: Lecture Notes in Computer Science ((LNPSE,volume 12389))

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Abstract

We propose an automated method for \(\nu \mathrm {HFL}_\mathbb {Z}\) validity checking. \(\mathrm {HFL}_{\mathbb {Z}}\) is an extension of the higher-order fixpoint logic HFL with integers, and \(\nu \mathrm {HFL}_\mathbb {Z}\) is a restriction of it to the fragment without the least fixpoint operator. The validity checking problem for \(\mathrm {HFL}_{\mathbb {Z}}\) has recently been shown to provide a uniform approach to higher-order program verification. The restriction to \(\nu \mathrm {HFL}_\mathbb {Z}\) studied in this paper already provides an automated method for a large class of program verification problems including safety and non-termination verification, and also serves as a key building block for solving the validity checking problem for full \(\mathrm {HFL}_{\mathbb {Z}}\) . Our approach is based on predicate abstraction and counterexample-guided abstraction refinement (CEGAR). We have implemented the proposed method, and applied it to program verification. According to experiments, our tool outperforms a closely related tool called Horus in terms of precision, and is competitive with a more specialized program verification tool called MoCHi despite the generality of our approach.

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Notes

  1. 1.

    Kobayashi et al. [13] actually considered model checking problems, but it is actually sufficient to consider validity checking problems for formulas without modal operators, as shown in a follow-up paper [23]; thus, throughout this paper, we shall consider only validity checking for formulas without modal operators.

  2. 2.

    It is possible to further extend \(\mathrm {HFL}_{\mathbb {Z}}\) with other data structures such as lists and trees, and extend our predicate abstraction method accordingly, as long as the background solvers (such as SMT and CHC solvers) support them.

  3. 3.

    In contrast, the existential quantifier over integers is not definable in this logic.

  4. 4.

    We can prove the validity of \((\lambda x.\lambda k.k(x+x))1\,(\lambda r.r\ge 2)\) if we use a different abstraction type, like \(x\mathbin {:}\mathtt {int}[\,]\rightarrow (r\mathbin {:}\mathtt {int}[r\ge 2x]\rightarrow \bullet )\rightarrow \bullet \) for \(\lambda x.\lambda k.k(x+x)\).

  5. 5.

    More precisely there exists a formula and \( \varphi ' =_{\beta \eta } [(b_{x \le 0} \wedge b_{x \ge 0})/b_{x=0}]\varphi \).

  6. 6.

    Note that the case where both \( x \le 0 \) and \( x \ge 0 \) hold (i.e. \(x=0\)) is not problematic because of monotonicity: if \( [\mathtt {true}/b_{x \le 0}, \mathtt {false}/b_{x \ge 0}]\varphi \) is true, then \( [\mathtt {true}/b_{x \le 0}, \mathtt {true}/b_{x \ge 0}]\varphi \) is true as well.

  7. 7.

    This is not the case for full \(\mathrm {HFL}_{\mathbb {Z}}\) .

  8. 8.

    This procedure does not terminate in general. An example is \( \mathtt {false}\rhd (\nu f. \lambda x. f\,x)\,0 \).

  9. 9.

    Existential quantifiers that arise from the original programs have been replaced with finite disjunctions.

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Acknowledgments

We would like to thank anonymous referees for useful comments. This work was supported by JSPS KAKENHI Grant Number JP15H05706, JP20H00577 and JP20H05703.

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Authors and Affiliations

  1. The University of Tokyo, Tokyo, Japan

    Naoki Iwayama, Naoki Kobayashi & Ryota Suzuki

  2. Chiba University, Chiba, Japan

    Takeshi Tsukada

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Correspondence to Naoki Kobayashi .

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  1. Inria, IRISA, ENS Rennes, Bruz, France

    David Pichardie

  2. LSV, ENS Paris-Saclay, Gif-sur-Yvette, France

    Mihaela Sighireanu

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Iwayama, N., Kobayashi, N., Suzuki, R., Tsukada, T. (2020). Predicate Abstraction and CEGAR for \(\nu \mathrm {HFL}_\mathbb {Z}\) Validity Checking. In: Pichardie, D., Sighireanu, M. (eds) Static Analysis. SAS 2020. Lecture Notes in Computer Science(), vol 12389. Springer, Cham. https://doi.org/10.1007/978-3-030-65474-0_7

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