Functions Definable by Arithmetic Circuits

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 nonnegative 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. Keywords. Arithmetic circuit, integer expression, complex algebra, expressive power.

• Type: Conference
• Year: 2009
• Where: CIE
• Authors: Ian Pratt-Hartmann, Ivo Düntsch
• Keywords: Acyclic Graph, Applied Computing, Arithmetic Circuit, CIE 2009, Non-negative Integers