Classifying and Propagating Parity Constraints

Abstract

Parity constraints, common in application domains such as circuit verification, bounded model checking, and logical cryptanalysis, are not necessarily most efficiently solved if translated into conjunctive normal form. Thus, specialized parity reasoning techniques have been developed in the past for propagating parity constraints. This paper studies the questions of deciding whether unit propagation or equivalence reasoning is enough to achieve full propagation in a given parity constraint set. Efficient approximating tests for answering these questions are developed. It is also shown that equivalence reasoning can be simulated by unit propagation by adding a polynomial amount of redundant parity constraints to the problem. It is proven that without using additional variables, an exponential number of new parity constraints would be needed in the worst case. The presented classification and propagation methods are evaluated experimentally. ⋆ The original version of the paper [1] has been presented in the 18th International Conference on Principle and Practice of Constraint Programming, CP 2012. An earlier version of the extended version has been presented for CP 2012 reviewers. This revised version uses proof techniques from [2].

DRAT Proofs for XOR Reasoning

“…In the future, we plan to implement both translations in CoProcessor and apply similar ideas to obtain proofs for cardinality resolution [26], as suggested in [16]. Another interesting problem is adapting the presented approaches to SAT solvers where XOR reasoning takes place within the CDCL procedure [20,21].…”

“…The satisfiability problem (SAT) is a paramount problem in computer science and artificial intelligence. Modern SAT solvers based on the DPLL algorithm [10] use many advanced techniques such as clause learning [27], clause removal [2,12], formula simplifications [11,19] and specialized reasoning procedures such as XOR reasoning [20,21,29,31]. These improvements led to a spectacular performance of conflict-driven satisfiability solvers.…”

Simulating Parity Reasoning

“…The size of the GE-simulation formula for φ xor may be reduced considerably if φ xor is partitioned into disjoint xor-constraint conjunctions φ 1 xor ∧ · · · ∧ φ n xor according to the connected components of the xor-constraint graph, and then combining the componentwise GE-simulation formulas k 1 -Ge(φ 1 xor ) ∧ · · · ∧ k n -Ge(φ n xor ). Efficient structural tests for deciding whether unit propagation or equivalence reasoning is enough to achieve full propagation in an xor-constraint conjunction, presented in [17], can indicate appropriate values for some of the parameters k 1 , . .…”

Towards a Semantics of Unsatisfiability Proofs with Inprocessing

“…This is very convenient from the perspective of SAT solving: state-of-the-art solvers can perform exponentially better than pure CDCL solvers due to the use of inprocessing techniques such as Gaussian elimination, cardinality resolution and symmetry breaking [44,45,31,32,6,8,1]. Expressing such techniques as RUP inferences can lead to much longer proofs, both in theory [17,48,49] and in practice [40].…”

“…One of the main reasons for this efficiency leap was the extension of the conflict-driven clause learning (CDCL) algorithm [38] by a number of inprocessing techniques [27], such as clause elimination [12,3,2,26], variable elimination [46,11], bounded variable addition [33], cardinality resolution [6], symmetry breaking [8,1] and parity reasoning [44,45,31,32]. These techniques modify the CNF formula in the SAT solver and replace it by a satisfiability-equivalent one, and often semantic equivalence is not preserved.…”