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Some results on uniform arithmetic circuit complexity

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Abstract

We introduce a natural set of arithmetic expressions and define the complexity class AE to consist of all those arithmetic functions (over the fieldsF 2n) that are described by these expressions. We show that AE coincides with the class of functions that are computable with constant depth and polynomial-size unbounded fan-in arithmetic circuits satisfying a natural uniformity constraint (DLOGTIME-uniformity). A 1-input and 1-output arithmetic function over the fieldsF2n may be identified with ann-input andn-output Boolean function when field elements are represented as bit strings. We prove that if some such representation is X-uniform (where X is P or DLOGTIME), then the arithmetic complexity of a function (measured with X-uniform unbounded fan-in arithmetic circuits) is identical to the Boolean complexity of this function (measured with X-uniform threshold circuits). We show the existence of a P-uniform representation and we give partial results concerning the existence of representations with more restrictive uniformity properties.

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The research of G. S. Frandsen was partially carried out while visiting Dartmouth College, New Hampshire. He was partially supported by the Danish Natural Science Research Council (Grant No. 11-7991) and by the ESPRIT II Basic Research Actions Program of the EC under Contract No. 3075 (project ALCOM). D. A. M. Barrington's research was supported by NSF Computer and Computation Theory Grant CCR-8714714.

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Frandsen, G.S., Valence, M. & Mix Barrington, D.A. Some results on uniform arithmetic circuit complexity. Math. Systems Theory 27, 105–124 (1994). https://doi.org/10.1007/BF01195199

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  • DOI: https://doi.org/10.1007/BF01195199

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