computational complexity

, Volume 5, Issue 2, pp 132–154

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

• Alain P. Hiltgen
• Mike S. Paterson
Article

## 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(n3/2). 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.

## Key words

Combinational complexity mass production slice functions set circuits upper bounds

68Q15 94C10

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