Boolean functional synthesis: hardness and practical algorithms

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 ar...

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Published in:	Formal methods in system design Vol. 57; no. 1; pp. 53 - 86
Main Authors:	Akshay, S., Chakraborty, Supratik, Goel, Shubham, Kulal, Sumith, Shah, Shetal
Format:	Journal Article
Language:	English
Published:	New York Springer US 01.07.2021 Springer Nature B.V
Subjects:	Algorithms Benchmarks Boolean Boolean algebra CAE) and Design Circuits and Systems Complexity theory Computer-Aided Engineering (CAD Electrical Engineering Engineering Functionals Hardness Software Engineering/Programming and Operating Systems Specifications Surface hardness Synthesis Skolem functions Boolean functional synthesis Expansion-based algorithms
ISSN:	0925-9856, 1572-8102
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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.
AbstractList	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. 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.
Author	Kulal, Sumith Akshay, S. Chakraborty, Supratik Goel, Shubham Shah, Shetal
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References_xml	– reference: BryantREGraph-based algorithms for Boolean function manipulationIEEE Trans Comput198635867769110.1109/TC.1986.1676819 – reference: BoudetAJouannaudJPSchmidt-SchaussMUnification in Boolean rings and Abelian groupsJ Symb Comput198985449477103192910.1016/S0747-7171(89)80054-9 – reference: Chakraborty S, Fremont DJ, Meel KS, Seshia SA, Vardi MY (2015) On parallel scalable uniform SAT witness generation. In: Proceedings of international conference on tools and algorithms for the construction and analysis of systems (TACAS), pp 304–319 – reference: QBFLib (2018) QBFEval 2018. http://www.qbflib.org/qbfeval18.php. Accessed July 2018 – reference: Chandrasekaran V, Srebro N, Harsha P (2008) Complexity of inference in graphical models. In: Proceedings of international conference on uncertainty in artificial intelligence (UAI), pp 70–78 – reference: BalabanovVJiangJHRUnified QBF certification and its applicationsForm Methods Syst Des2012411456510.1007/s10703-012-0152-6 – reference: GanianRHlinenýPLangerAObdrzálekJRossmanithPSikdarSLower bounds on the complexity of MSO1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$_{\text{1}}$$\end{document} model-checkingJ Comput Syst Sci2014801180194310591510.1016/j.jcss.2013.07.005 – reference: HellermanLA catalog of three-variable Or-Invert and And-Invert logical circuitsIEEE Trans Electron Comput196312319822310.1109/PGEC.1963.263531 – reference: Akshay S, Chakraborty S, John AK, Shah S (2017) Towards parallel Boolean functional synthesis. In: Proceedings of international conference on tools and algorithms for construction and analysis of systems (TACAS), part I, pp 337–353 – reference: Golia P, Roy S, Meel KS (2020) Manthan: a data-driven approach for Boolean function synthesis. In: Proceedings of international conference on computer-aided verification (CAV), pp 611–633 – reference: Chakraborty S, Fried D, Tabajara LM, Vardi MY (2018) Functional synthesis via input–output separation. In: Proceedings of international conference on formal methods in computer-aided design (FMCAD), pp 1–9 – reference: Kuehlmann A, Krohm F (1997) Equivalence checking using cuts and heaps. In: Proceedings of design automation conference (DAC), pp 263–268 – reference: MartinUNipkowTBoolean unification: the story so farJ Symb Comput198973–427529399366710.1016/S0747-7171(89)80013-6 – reference: Rabe MN, Tentrup L, Rasmussen C, Seshia SA (2018) Understanding and extending incremental determinization for 2QBF. 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Snippet	Given a relational specification between Boolean inputs and outputs, Boolean functional synthesis seeks to synthesize each output as a function of the inputs...
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SubjectTerms	Algorithms Benchmarks Boolean Boolean algebra CAE) and Design Circuits and Systems Complexity theory Computer-Aided Engineering (CAD Electrical Engineering Engineering Functionals Hardness Software Engineering/Programming and Operating Systems Specifications Surface hardness Synthesis
Title	Boolean functional synthesis: hardness and practical algorithms
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