Simulating Parity Reasoning

Laitinen, Tero; Junttila, Tommi; Niemelä, Ilkka

doi:10.1007/978-3-642-45221-5_38

Cited by 2 publications

(7 citation statements)

References 22 publications

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“…Experimental evaluation on SAT 2005 benchmarks instances showed that, when "not too large", such CNF translations outperform dedicated XOR reasoning modules. The successor [61] provides several comparisons of special-reasoning machinery with resolution-based methods, and in Theorem 4 there we find a general ACtranslation; our Theorem 9.2 yields a better upper bound, but the heuristic reasoning of [59,61] seems valuable, and should be explored further.…”

Section: Translations To Cnfmentioning

confidence: 83%

“…Conjecture 11.1 There exists X * : CLS → PC, which computes for an XORclause-set F ∈ CLS a CNF-representation X * (F ) in time 2 O(tw(F )) ·ℓ(F ) O(1) , where tw(F ) is the treewidth of the incidence graph of F . Theorem 7 in [62] shows a weaker form of Conjecture 11.1, where instead of the incidence graph the variable-interaction graph (or "primal graph"; recall Lemma 4.7) is considered, and where instead of (absolute) propagation-completeness only AC is achieved. Altogether there seem to be interesting general theorems and methods waiting to be discovered, targeting representations in PC.…”

Section: Open Problems and Future Research Directionsmentioning

confidence: 99%

“…So we have (modulo equivalence) X 1 (C) ⊆ F * , where due to X 1 (C) ∈ PC (Lemma 8.4) the forced literal x for ϕ * X 1 (C) is set by r 1 , contradicting the assumption. Theorem 4 in [61] yields the weaker bound O(4 n(F ) ) for the number of clauses in an AC-representation of F (note that w.l.o.g. c(F ) ≤ n(F )).…”

Section: Representing Monotone Span Programsmentioning

confidence: 99%

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On SAT Representations of XOR Constraints

Gwynne

Kullmann

2014

Language and Automata Theory and Applications

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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...

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Section: Translations To Cnfmentioning

confidence: 83%

Section: Open Problems and Future Research Directionsmentioning

confidence: 99%

Section: Representing Monotone Span Programsmentioning

confidence: 99%

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On SAT Representations of XOR Constraints

Gwynne

Kullmann

2014

Language and Automata Theory and Applications

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“…, y k−1 ⊕ x k = ε, introducing auxiliary variables y i . We show that X 1 (S * ), where S * is obtained from S by considering all derived equations, is a GAC-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 a GAC-representation is fixed-parameter tractable (fpt) in the parameter m, improving Laitinen et al 2013 [14], which showed fpt in n.…”

Section: Introductionmentioning

confidence: 81%

“…Namely the general "meta theorem" is, that search for a solution, when done properly, and the underlying (hyper-)graph is "sufficiently acyclic" together with sufficient "local consistency" of the constraints (the F i ), can proceed without backtracking. 14 In [63,64] this is studied for binary CSPs (i.e., n(F i ) = 2 for all i ∈ I), while in [65,66] these considerations are generalised to non-binary CSPs; see [67] for an overview. Due to the importance of this basic result, we provide a self-contained (complete) proof.…”

Section: Acyclicitymentioning

confidence: 99%

A framework for good SAT translations, with applications to CNF representations of XOR constraints

Gwynne¹,

Kullmann²

2014

Preprint

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We present a general framework for "good CNF-representations" of boolean constraints, to be used for translating decision problems into SAT problems (i.e., deciding satisfiability for conjunctive normal forms). We apply it to the representation of systems of XOR-constraints ("exclusive-or"), also known as systems of linear equations over the two-element field, or systems of parity constraints, or as systems of equivalences (XOR is the negation of an equivalence).The general framework defines the notion of "representation", and provides several methods to measure the quality of the representation, by measuring the complexity ("hardness") needed for making implicit "knowledge" of the representation explicit (to a SAT-solving mechanism). We obtain general upper and lower bounds.Applied to systems of XOR-constraints, we show a super-polynomial lower bound on "good" representations under very general circumstances. A corresponding upper bound shows fixed-parameter tractability in the number of constraints.The measurement underlying this upper bound ignores the auxiliary variables needed for shorter representations of XOR-constraints. Improved upper bounds for special cases take them into account, and a rich picture begins to emerge, under the various hardness measurements.

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Simulating Parity Reasoning

Cited by 2 publications

References 22 publications

On SAT Representations of XOR Constraints

On SAT Representations of XOR Constraints

A framework for good SAT translations, with applications to CNF representations of XOR constraints

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