Lower Bounds for DeMorgan Circuits of Bounded Negation Width

Authors Stasys Jukna, Andrzej Lingas

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Stasys Jukna
  • Institute of Computer Science, Goethe University Frankfurt, Frankfurt am Main, Germany
  • Institute of Data Science and Digital Technologies, Vilnius University, Lithuania
Andrzej Lingas
  • Department of Computer Science, Lund University, Box 118, 22100 Lund, Sweden

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Stasys Jukna and Andrzej Lingas. Lower Bounds for DeMorgan Circuits of Bounded Negation Width. In 36th International Symposium on Theoretical Aspects of Computer Science (STACS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 126, pp. 41:1-41:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)


We consider Boolean circuits over {or, and, neg} with negations applied only to input variables. To measure the "amount of negation" in such circuits, we introduce the concept of their "negation width". In particular, a circuit computing a monotone Boolean function f(x_1,...,x_n) has negation width w if no nonzero term produced (purely syntactically) by the circuit contains more than w distinct negated variables. Circuits of negation width w=0 are equivalent to monotone Boolean circuits, while those of negation width w=n have no restrictions. Our motivation is that already circuits of moderate negation width w=n^{epsilon} for an arbitrarily small constant epsilon>0 can be even exponentially stronger than monotone circuits. We show that the size of any circuit of negation width w computing f is roughly at least the minimum size of a monotone circuit computing f divided by K=min{w^m,m^w}, where m is the maximum length of a prime implicant of f. We also show that the depth of any circuit of negation width w computing f is roughly at least the minimum depth of a monotone circuit computing f minus log K. Finally, we show that formulas of bounded negation width can be balanced to achieve a logarithmic (in their size) depth without increasing their negation width.

Subject Classification

ACM Subject Classification
  • Theory of computation → Circuit complexity
  • Boolean circuits
  • monotone circuits
  • lower bounds


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