On SAT Representations of XOR Constraints

Abstract

We study the problem of finding good CNF-representations F of systems of linear equations S over the two-element field, also known as systems of XORconstraints x1 ⊕ · · · ⊕ x k = ε, ε ∈ {0, 1}, or systems of parity-constraints. The number of equations in S is m, the number of variables is n. These representations are used as parts of SAT problems F * ⊃ F , such that F has "good" properties for SAT solving in the context of F * ; here F * may for example represent the problem of finding the key for a cryptographic cipher. The basic quality criterion is "arc consistency" (AC), that is, for every partial assignment ϕ to the variables of S, all assignments xi = ε forced by ϕ are determined by unit-clause propagation on the result ϕ * F of the application.We show there is no AC-representation of polynomial size for arbitrary S. We use the lower bound on monotone circuits for monotone span programs from [2], and we show a close relation between monotone circuits and ACrepresentations, based on the work in [9]. We then turn to constructing good representations. We analyse the basic translation F = X1(S), which translates each constraint on its own, by splitting up x1 ⊕ · · · ⊕ x k into sums x1 ⊕ x2 = y2, y2 ⊕ x3 = y3, . . . , y k−1 ⊕ x k = ε, introducing auxiliary variables yi. We show that X1(S * ), where S * is obtained from S by considering all derived equations, is an AC-representation of S. The derived equations are obtained by adding up the equations of all sub-systems S ′ ⊆ S. There are 2 m such S ′ , and computing an AC-representation is fixed-parameter tractable (fpt) in the parameter m, improving [61], which showed fpt in n.To obtain stronger representations, instead of mere AC we consider the class PC of propagation-complete clause-sets, as introduced in [12]. The stronger criterion is F ∈ PC, which requires for all partial assignments, possibly involving also the auxiliary (new) variables in F , that forced assignments can be determined by unit-clause propagation. Using "propagation hardness" phd(F ) ∈ N0 as introduced in [37], we have F ∈ PC ⇔ phd(F ) ≤ 1. We show that X1 applied to a single equation (m = 1) yields a translation in PC, i.e., phd(X1(S)) ≤ 1. Then we study m = 2. Now S * has two equations more, and X1(S * ) is an AC-representation, but the "distance" to PC is arbitrarily high, i.e., phd(X1(S * )) is unbounded (using results from [54]). We show two possibilities to remedy this (for m = 2). On the one hand, if instead of unit-clause propagation we allow (arbitrary) resolution with clauses of length at most 3 (i.e., 3-resolution), and only require refutation of inconsistencies after (arbitrary, partial) instantiations, then even just X1(S) suffices. On the other hand, with a more intelligent translation, which avoids duplication of equivalent auxiliary variables yi, we obtain a (short) representation in PC. We conjecture that also the general case can be handled this way, that is, computing a representation F ∈ PC of S is fpt in m.Recall that the two-element field Z 2 has elements 0, 1, where ad...

DRAT Proofs for XOR Reasoning

“…We assume from now on that variables in XOR constraints are sorted. The natural splitting [14] of X = [A 1 , . .…”

Unified Characterisations of Resolution Hardness Measures

“…Based on the notion of "k-resolution" introduced in [39], the "asymmetric width" was introduced in [44,47,48] (and further studied in [33,32,34]). 5) Different from the symmetric width, only one parent clause needs to have size at most k (while there is no restriction on the other parent clause nor on the resolvent):…”

“…Perhaps the main aim of measuring the complexity of satisfiable clausesets is to obtain SAT representations of boolean functions of various quality ("hardness") and sizes; see [31,34] for investigations into XOR-constraints. The goal is to obtain "good" representations F of boolean functions (like cardinality or XOR-constraints) in the context of a larger SAT problem representations.…”

Simulating Parity Reasoning

“…Using the translation to simulate full Gauss-Jordan elimination with plain unit propagation requires an exponential number of additional xor-constraints in the worst case, but we show that the translation is polynomial for instance families of bounded treewidth. Recently, it has been shown in [21] that a conjunction of xor-constraints does not have a polynomialsize "arc consistent" CNF-representation, which implies it is not feasible to simulate Gauss-Jordan elimination by unit propagation in the general case. On many instances, though, better solver performance can be obtained by simulating a weaker parity reasoning system as it reduces the size of the translation substantially.…”