Functions Definable by Arithmetic Circuits

Abstract

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.

Circuit satisfiability and constraint satisfaction around Skolem Arithmetic

“…Apart from the research on circuit problems mentioned above there has been work on other variants like circuits over integers [26] and positive natural numbers [27], equivalence problems for circuits [28], functions computed by circuits [29], and equations over sets of natural numbers [30,31]. Typically, the complexity of membership of circuits is similar to the corresponding equivalence of circuits problem, though the latter may be slightly higher and belies some imperfect bounds 2 .…”

Complex Algebras of Arithmetic

“…The ordering ≤ on ω corresponds to the function defined by f (x) = {n ∈ ω : (∃m)[m ∈ x and n ≤ m}, and we have shown in [16] that this function is not circuit definable. In the same paper we have proved (ii).…”

“…Such set exists, since every every set definable by an arithmetic circuit is in the bounded hierarchy BH [16], and the bounded hierarchy is known to be contained within the zeroth Grzegorczyk class, E 0 * . By Lemma 4.1 there is a first order sentence (∃x)ϕ(x) such that Cm N |= ϕ(x/s) if and only if s = a.…”

“…It now makes sense to consider satisfiability and validity of (in-) equations of terms of this language under this interpretation. Furthermore, the operations f : (2 ω ) k → 2 ω definable from the given operators O can be studied [16]. In analogy to the membership problem, Glaßer et al [5] consider the complexity of…”

Circuit Satisfiability and Constraint Satisfaction Around Skolem Arithmetic

“…Apart from the research on circuit problems mentioned above there has been work on other variants like circuits over integers [26] and positive natural numbers [27], equivalence problems for circuits [28], functions computed by circuits [29], and equations over sets of natural numbers [30,31]. Typically, the complexity of membership of circuits is similar to the corresponding equivalence of circuits problem, though the latter may be slightly higher and belies some imperfect bounds 2 .…”