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Probabilistic Horn Clause Verification

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Static Analysis (SAS 2017)

Part of the book series: Lecture Notes in Computer Science ((LNPSE,volume 10422))

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Abstract

Constrained Horn clauses have proven to be a natural intermediate language for logically characterizing program semantics and reasoning about program behavior. In this paper, we present probabilistically constrained Horn clauses (pchc), which incorporate probabilistic variables inside otherwise traditional constrained Horn clauses. pchc enable reasoning about probabilistic programs by encoding them as Horn clauses. Encoding probabilistic program semantics as pchc allows us to seamlessly handle procedure calls and recursion, as well as angelic and demonic forms of nondeterminism. We formalize pchc semantics and present a verification algorithm that can prove probabilistic safety properties of programs. We present an implementation and evaluation of our approach on a number of probabilistic programs and properties.

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Notes

  1. 1.

    While in the non-probabilistic setting we can represent the function by two clauses (one representing the base case and one the recursive call), we need to combine the two clauses in the probabilistic setting. See Sect. 6 for a detailed explanation.

  2. 2.

    \(\mathbb {P}[\varphi ] = \mathbb {P}[x \geqslant 5] + \mathbb {P}[x < 0]*\mathbb {P}[x' \geqslant 5]\). Since \(x,x' \sim \texttt {gauss}(0,10)\), we have \(\mathbb {P}[x \geqslant 5] \approx 0.308\) and \(\mathbb {P}[x < 0] = 0.5\).

  3. 3.

    Since \(\mathbb {P}[x_1 \leqslant 1 \vee x_2 \geqslant 8] = 1 - \mathbb {P}[x_1 > 1 \wedge x_2 < 8] = 1 - 0.9 * 0.8 = 0.28\).

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  1. University of Wisconsin–Madison, Madison, USA

    Aws Albarghouthi

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  1. Dipartimento di Matematica, University of Padova, Padua, Italy

    Francesco Ranzato

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Albarghouthi, A. (2017). Probabilistic Horn Clause Verification. In: Ranzato, F. (eds) Static Analysis. SAS 2017. Lecture Notes in Computer Science(), vol 10422. Springer, Cham. https://doi.org/10.1007/978-3-319-66706-5_1

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