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International Colloquium on Automata, Languages, and Programming

ICALP 2009: Automata, Languages and Programming pp 701–712Cite as

Limiting Negations in Formulas

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Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 5555))

Abstract

Negation-limited circuits have been studied as a circuit model between general circuits and monotone circuits. In this paper, we consider limiting negations in formulas. The minimum number of NOT gates in a Boolean circuit computing a Boolean function f is called the inversion complexity of f. In 1958, Markov determined the inversion complexity of every Boolean function and particularly proved that ⌈log2(n + 1) ⌉ NOT gates are sufficient to compute any Boolean function on n variables. We determine the inversion complexity of every Boolean function in formulas, i.e., the minimum number of NOT gates (NOT operators) in a Boolean formula computing (representing) a Boolean function, and particularly prove that ⌈n/2 ⌉ NOT gates are sufficient to compute any Boolean function on n variables. Moreover we show that if there is a polynomial-size formula computing a Boolean function f, then there is a polynomial-size formula computing f with at most ⌈n/2 ⌉ NOT gates. We consider also the inversion complexity in formulas of negation normal form and prove that the inversion complexity is at most polynomials of n for every Boolean function on n variables.

Some results of this paper have appeared in the preliminary version [12].

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Author information

Authors and Affiliations

  1. Graduate School of Information Sciences, Tohoku University, Sendai, 980-8579, Japan

    Hiroki Morizumi

Authors
  1. Hiroki Morizumi
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Editors and Affiliations

  1. Department of Computer Science, University of Freiburg, Georges Köhler Allee 79, 79110, Freiburg, Germany

    Susanne Albers

  2. Department of Computer and Systems Sciences, Sapienza University of Rome, Via Ariosto 25, 00184, Roma, Italy

    Alberto Marchetti-Spaccamela

  3. Google R&D Center, School of Computer Science, Tel Aviv University, 69978, Tel Aviv, Israel

    Yossi Matias

  4. University of Patras and CTI, N. Kazantzaki Street 1, 26504 Rion, Patras, Greece

    Sotiris Nikoletseas

  5. RWTH Aachen, Lehrstuhl Informatik 7, Ahornstraße 55, 52056, Aachen, Germany

    Wolfgang Thomas

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Morizumi, H. (2009). Limiting Negations in Formulas. In: Albers, S., Marchetti-Spaccamela, A., Matias, Y., Nikoletseas, S., Thomas, W. (eds) Automata, Languages and Programming. ICALP 2009. Lecture Notes in Computer Science, vol 5555. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-02927-1_58

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  • DOI: https://doi.org/10.1007/978-3-642-02927-1_58

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-02926-4

  • Online ISBN: 978-3-642-02927-1

  • eBook Packages: Computer ScienceComputer Science (R0)

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