Abstract. An arithmetic circuit (McKenzie and Wagner [6]) is a labelled, directed 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 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.

Abstract. A numerical set-expression is a term specifying a cascade of arithmetic and logical operations to be performed on sets of non-negative integers. If these operations are confined to the usual Boolean operations together with the result of lifting addition to the level of sets, we speak of additive circuits. If they are confined to the usual Boolean operations together with the result of lifting addition and multiplication to the level of sets, we speak of arithmetic circuits. In this paper, we investigate the definability of sets and functions by means of additive and arithmetic circuits, occasionally augmented with additional operations.

Abstract. An arithmetic circuit (McKenzie and Wagner [6]) is a labelled, directed 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 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.