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Boolean functional synthesis: hardness and practical algorithms

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Abstract

Given a relational specification between Boolean inputs and outputs, Boolean functional synthesis seeks to synthesize each output as a function of the inputs such that the specification is met. Despite significant algorithmic advances in Boolean functional synthesis over the past few years, there are relatively small specifications that have remained beyond the reach of all state-of-the-art tools. In trying to understand this behaviour, we show that unless some hard conjectures in complexity theory are falsified, Boolean functional synthesis must generate large Skolem functions in the worst-case. Given this inherent hardness, what does one do to solve the problem? We present a two-phase algorithm, where the first phase is efficient in practice both in terms of time and size of synthesized functions, and solves a large fraction of our benchmarks. This phase is also guaranteed to solve the problem when the representation of the input specification satisfies some structural requirements. For those cases where the first phase doesn’t suffice, we present a second phase of our synthesis algorithm that uses a special class of algorithms, called expansion-based algorithms, to generate correct Skolem functions. This may require exponential time and generate exponential-sized Skolem functions in the worst-case. Detailed experimental evaluation shows that our overall synthesis algorithm performs better than other techniques for a large number of benchmarks.

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Notes

  1. Otherwise, we could efficiently factorize products of n-bit prime numbers, rendering cryptographic systems vulnerable to attacks.

  2. We use the standard definition for \({\mathsf {ETH}}_{\mathsf {nu}}\) see e.g., [15, 20]. We note however that in [16] the authors consider an alternate definition of this notion.

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Acknowledgements

We thank Ajith K. John for many technical discussions. We also thank the anonymous reviewers for several pertinent remarks and suggestions.

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Authors and Affiliations

  1. Indian Institute of Technology Bombay, Mumbai, India

    S. Akshay, Supratik Chakraborty & Shetal Shah

  2. University of California, Berkeley, USA

    Shubham Goel

  3. Stanford University, Stanford, USA

    Sumith Kulal

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  1. S. Akshay
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  2. Supratik Chakraborty
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  3. Shubham Goel
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Correspondence to Supratik Chakraborty.

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The authors wish to acknowledge funding support from DST/CEFIPRA/INRIA Project EQuaVE and DST/SERB Matrices Grant MTR/2018/000744 for S. Akshay, and from MHRD/IMPRINT-1/Project 5496(FMSAFE) for Supratik Chakraborty and Shetal Shah. Most of this work was done when Shubham Goel and Sumith Kulal were at Indian Institute of Technology Bombay, India.

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Akshay, S., Chakraborty, S., Goel, S. et al. Boolean functional synthesis: hardness and practical algorithms. Form Methods Syst Des 57, 53–86 (2021). https://doi.org/10.1007/s10703-020-00352-2

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