Malign distributions for average case circuit complexity

  • Andreas Jakoby
  • Rüdiger Reischuk
  • Christian Schindelhauer
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 900)


In contrast to machine models like Turing machines or random access machines, circuits are a rigid computational model. The internal information flow of a computation is fixed in advance, independent of the actual input. Therefore, in complexity theory only worst case complexity measures have been used to analyse this model. In [JRS94] we have defined an average case measure for the time complexity of circuits. Using this notion tight upper and lower bounds could be obtained for the average case complexity of several basic Boolean functions.

In this paper we will examine the asymptotic average case complexity of the set of n-input Boolean functions. In contrast to the worst case a simple counting argument does not work. We prove that almost all Boolean function require at least n— log n— log n expected time even for the uniform probability distribution. On the other hand, we show that there are significant subsets of functions that can be computed with a constant average delay.

Finally, the worst case and average case complexity of a Boolean function will be compared. We show that for each function that requires circuit depth d, the expected time complexity will be at least d— log n— log d with respect to an explicitely defined probability distribution. A nontrivial bound on the complexity of such a distribution is obtained.

Key words

average case complexity circuit complexity delay distribution 


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Copyright information

© Springer-Verlag Berlin Heidelberg 1995

Authors and Affiliations

  • Andreas Jakoby
    • 1
  • Rüdiger Reischuk
    • 1
  • Christian Schindelhauer
    • 1
  1. 1.Med. Universität zu LübeckGermany

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