Satisfiability of Algebraic Circuits over Sets of Natural Numbers

Functions Definable by Arithmetic Circuits

“…This contrasts with the circuit-definability of the function d(x) given in (2). Let the function ↓: 2 N → 2 N be given by ↓ (x) = {n ∈ N | ∃m ∈ x, n < m}.…”

“…It is easy to see that these problems are reducible to one another; however, the question of their decidability is currently still open. The complexity of the membership problem (and related problems) for variable-free arithmetic circuits with operators chosen from various proper subsets of {∪, ∩, − , +, •} are studied in Glaßer et al [1,2], building on the work of Meyer and Stockmeyer, op. cit., McKenzie and Wagner, op.…”

Circuit Satisfiability and Constraint Satisfaction Around Skolem Arithmetic

“…We revisit results of Glaßer et al [1] in the context of CSPs and settle the major open question from that paper, finding a certain satisfiability problem on circuits-involving complement, intersection, union and multiplication-to be decidable. This we prove using the decidability of Skolem Arithmetic.…”

“…Meanwhile, a literature existed on satisfiability of circuit problems over sets of integers involving work of the first author [1], itself continuing a line of investigation begun in [22] and pursued in [23,24,25]. The circuits typically compute some set of integers at their unique output node and one asks for satisfiability in terms of evaluations of free set-variables at their input nodes.…”

“…The circuits typically compute some set of integers at their unique output node and one asks for satisfiability in terms of evaluations of free set-variables at their input nodes. The problems in [1] can be seen as variants of certain functional CSPs whose domain is all singleton sets of the non-negative integers and whose relations are set operations of the form: complement, intersection, union, addition and multiplication (the latter two are defined set-wise, e.g. A × B := {ab : a ∈ A ∧ b ∈ B}).…”