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Probably Approximately Correct Interpolants Generation

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Dependable Software Engineering. Theories, Tools, and Applications (SETTA 2020)

Part of the book series: Lecture Notes in Computer Science ((LNPSE,volume 12153))

Abstract

In this paper we propose a linear programming based method to generate interpolants for two Boolean formulas in the framework of probably approximately correct (PAC) learning. The computed interpolant is termed as a PAC interpolant with respect to a violation level \(\epsilon \in (0,1)\) and confidence level \(\beta \in (0,1)\): with at least \(1-\beta \) confidence, the probability that the PAC interpolant is a true interpolant is larger than \(1-\epsilon \). Unlike classical interpolants which are used to justify that two formulas are inconsistent, the PAC interpolant is proposed for providing a formal characterization of how inconsistent two given formulas are. This characterization is very important, especially for situations that the two formulas cannot be proven to be inconsistent. The PAC interpolant is computed by solving a scenario optimization problem, which can be regarded as a statistically sound formal method in the sense that it provides formal correct guarantees expressed using violation probabilities and confidences. The scenario optimization problem is reduced to a linear program in our framework, which is constructed by a family of independent and identically distributed samples of variables in the given two Boolean formulas. In this way we can synthesize interpolants for formulas that existing methods are not capable of dealing with. Three examples demonstrate the merits of our approach.

This work has been supported through grants by NSFC under grant No. 61836005, 61625206, the CAS Pioneer Hundred Talents Program under grant No. Y8YC235015, and the MoE, Singapore, Tier-2 grant #MOE2019-T2-2-040.

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Authors and Affiliations

  1. State Key Laboratory of Computer Science, Institute of Software, CAS, Beijing, China

    Bai Xue & Naijun Zhan

  2. University of Chinese Academy of Sciences, Beijing, China

    Bai Xue & Naijun Zhan

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  1. Bai Xue
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  2. Naijun Zhan
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Correspondence to Bai Xue .

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Editors and Affiliations

  1. University of Luxembourg, Esch-sur-Alzette, Luxembourg

    Jun Pang

  2. Chinese Academy of Sciences, Beijing, China

    Lijun Zhang

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Xue, B., Zhan, N. (2020). Probably Approximately Correct Interpolants Generation. In: Pang, J., Zhang, L. (eds) Dependable Software Engineering. Theories, Tools, and Applications. SETTA 2020. Lecture Notes in Computer Science(), vol 12153. Springer, Cham. https://doi.org/10.1007/978-3-030-62822-2_9

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  • DOI: https://doi.org/10.1007/978-3-030-62822-2_9

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