PI k mass production and an optimal circuit for the Nečiporuk slice

Abstract

Letf: {0,1}n→{0,1}m be anm-output Boolean function inn variables.f is called ak-slice iff(x) equals the all-zero vector for allx with Hamming weight less thank andf(x) equals the all-one vector for allx with Hamming weight more thank. Wegener showed that “PI k -set circuits” (set circuits over prime implicants of lengthk) are at the heart of any optimum Boolean circuit for ak-slicef. We prove that, in PI k -set circuits, savings are possible for the mass production of anyFX, i.e., any collectionF ofm output-sets given any collectionX ofn input-sets, if their PI k -set complexity satisfiesSC m (FX)≥3n+2m. This PI k mass production, which can be used in monotone circuits for slice functions, is then exploited in different ways to obtain a monotone circuit of complexity 3n+o(n) for the Nečiporuk slice, thus disproving a conjecture by Wegener that this slice has monotone complexity θ(n 3/2). Finally, the new circuit for the Nečiporuk slice is proven to be asymptotically optimal, not only with respect to monotone complexity, but also with respect to combinational complexity.

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Hiltgen, A.P., Paterson, M.S. PI k mass production and an optimal circuit for the Nečiporuk slice. Comput Complexity 5, 132–154 (1995). https://doi.org/10.1007/BF01268142

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Key words

  • Combinational complexity
  • mass production
  • slice functions
  • set circuits
  • upper bounds

Subject classifications

  • 68Q15
  • 94C10