Functions Definable by Arithmetic Circuits

Authors:
Ian Pratt-Hartmann;Ivo Düntsch
Affiliations:
School of Computer Science, University of Manchester, Manchester, U.K. M13 9PL;Department of Computer Science, Brock University, St. Catharines, Canada L2S 3A1
Venue:
CiE '09 Proceedings of the 5th Conference on Computability in Europe: Mathematical Theory and Computational Practice
Year:
2009

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Abstract

An arithmetic circuit is a labelled, directed, acyclic graph specifying a cascade of arithmetic and logical operations to be performed on sets of non-negative integers. In this paper, we consider the definability of functions from tuples of sets of non-negative integers to sets of non-negative integers by means of arithmetic circuits. We prove two negative results: the first shows, roughly, that a function is not circuit-definable if it has an infinite range and sub-linear growth; the second shows, roughly, that a function is not circuit-definable if it has a finite range and fails to converge on certain `sparse' chains under inclusion. We observe that various functions of interest fall under these descriptions.