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2-Xor Revisited: Satisfiability and Probabilities of Functions

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Abstract

The problem 2-Xor-Sat asks for the probability that a random expression, built as a conjunction of clauses \(x \oplus y\), is satisfiable. We revisit this classical problem by giving an alternative, explicit expression of this probability. We then consider a refinement of it, namely the probability that a random expression computes a specific Boolean function. The answers to both problems involve a description of 2-Xor expressions as multigraphs and use classical methods of analytic combinatorics by expressing probabilities through coefficients of generating functions.

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Notes

  1. For the sake of brevity, in the sequel “(the set of) Boolean functions” is to be understood as either the set \({\mathcal {X}}_n\) or the set \( {\mathcal {X}}\), according to the context.

  2. Note that the relation \(\sim \) corresponds to an equivalence relation on the set of variables and therefore induces a partition on the set of variables. But as to the presence of negations, the formal structure is in fact a little bit richer than only a set with an equivalence relation.

  3. Here and in what follows, the constant denoted by K may vary and may depend on r—but it is always possible to get an explicit, though cumbersome, expression for it.

  4. It is also possible to extend the analysis to larger m, i.e. \(m\sim \alpha \cdot n\) with \(\alpha >\frac{g-1}{g}\), or \(m\gg n\).

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Acknowledgments

We thank Hervé Daudé and Vlady Ravelomanana for fruitful discussions.

Author information

Authors and Affiliations

  1. Alcatel-Lucent Bell Labs, Villarceau, France

    Élie de Panafieu

  2. DAVID Laboratory, University of Versailles Saint Quentin en Yvelines, Versailles, France

    Danièle Gardy

  3. Institute of Discrete Mathematics and Geometry, TU Wien, Wien, Austria

    Bernhard Gittenberger

  4. Institute of Applied Mathematics and Natural Sciences, FH-Technikum Wien, Höchstädtplatz 6, 1200, Wien, Austria

    Markus Kuba

Authors
  1. Élie de Panafieu
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  2. Danièle Gardy
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  3. Bernhard Gittenberger
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  4. Markus Kuba
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Corresponding author

Correspondence to Danièle Gardy.

Additional information

Élie de Panafieu: Supported by the ANR Projects BOOLE (2009–13) and MAGNUM (2010–14), the P.H.C. Amadeus project Boolean expressions: compactification, satisfiability and distributions of functions (2013–14), the PEPS HYDrATA, the Austrian Science Fund (FWF) Grant F5004, and the Complex Networks team, Sorbonne University. Danièle Gardy: Part of the work of this author was done during a long-term visit at the Institute of Discrete Mathematics and Geometry of the TU Wien. Supported by the P.H.C. Amadeus projects Random logical trees and related structures (2010–11) and Boolean expressions: compactification, satisfiability and distributions of functions (2013–14), and by the ANR project BOOLE (2009–13). Bernhard Gittenberger: Supported by the FWF (Austrian Science Foundation), Special Research Program F50, Grant F5003-N15, and by the ÖAD, grant Amadée F01/2015. Markus Kuba: Supported by ÖAD, Grant Amadée F01/2015.

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de Panafieu, É., Gardy, D., Gittenberger, B. et al. 2-Xor Revisited: Satisfiability and Probabilities of Functions. Algorithmica 76, 1035–1076 (2016). https://doi.org/10.1007/s00453-016-0119-x

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  • DOI: https://doi.org/10.1007/s00453-016-0119-x

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