A Quantifier Elimination Algorithm for Linear Modular Equations and Disequations

John, Ajith K.; Chakraborty, Supratik

doi:10.1007/978-3-642-22110-1_39

Cited by 9 publications

(5 citation statements)

References 8 publications

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“…In all, our benchmark suite consisted of 773 benchmarks arising from a wide range of real-world applications. Specifically, we used constraints arising from DQMR networks, bit-blasted versions of SMT-LIB (SMT) benchmarks, and ISCAS89 circuits [8] with parity conditions on randomly chosen subsets of outputs and nextstate variables [34,48]. We assigned random weights to literals wherever weights were not already available in our benchmarks.…”

Section: Discussionmentioning

confidence: 99%

$$\mathsf {WAPS}$$ : Weighted and Projected Sampling

Gupta

Sharma

Roy

et al. 2019

Lecture Notes in Computer Science

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Given a set of constraints F and a user-defined weight function W on the assignment space, the problem of constrained sampling is to sample satisfying assignments of F conditioned on W. Constrained sampling is a fundamental problem with applications in probabilistic reasoning, synthesis, software and hardware testing. Consequently, the problem of sampling has been subject to intense theoretical and practical investigations over the years. Despite such intense investigations, there still remains a gap between theory and practice. In particular, there has been significant progress in the development of sampling techniques when W is a uniform distribution, but such techniques fail to handle general weight functions W. Furthermore, we are, often, interested in Σ 1 1 formulas, i.e., G(X) := ∃Y F (X, Y) for some F ; typically the set of variables Y are introduced as auxiliary variables during encoding of constraints to F. In this context, one wonders whether it is possible to design sampling techniques whose runtime performance is agnostic to the underlying weight distribution and can handle Σ 1 1 formulas? The primary contribution of this work is a novel technique, called WAPS, for sampling over Σ 1 1 whose runtime is agnostic to W. WAPS is based on our recently discovered connection between knowledge compilation and uniform sampling. WAPS proceeds by compiling F into a well studied compiled form, d-DNNF, which allows sampling operations to be conducted in linear time in the size of the compiled form. We demonstrate that WAPS can significantly outperform existing state-of-the-art weighted and projected sampler WeightGen, by up to 3 orders of magnitude in runtime while achieving a geometric speedup of 296× and solving 564 more instances out of 773. The distribution generated by WAPS is statistically indistinguishable from that generated by an ideal weighted and projected sampler. Furthermore, WAPS is almost oblivious to the number of samples requested.

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Section: Discussionmentioning

confidence: 99%

$$\mathsf {WAPS}$$ : Weighted and Projected Sampling

Gupta

Sharma

Roy

et al. 2019

Lecture Notes in Computer Science

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show abstract

“…The suite of benchmarks was made up of problems arising from various practical domains as well as problems of theoretical interest. Specifically, we used bit-level unweighted versions of constraints arising from grid networks, plan recognition, DQMR networks, bounded model checking of circuits, bit-blasted versions of SMT-LIB (SMT) benchmarks, and ISCAS89 (Brglez, Bryan, and Kozminski 1989) circuits with parity conditions on randomly chosen subsets of outputs and nextstate variables (Sang, Beame, and Kautz 2005;John and Chakraborty 2011). While our algorithms are agnostic to the weight oracle, other tools that we used for comparison require the weight of an assignment to be the product of the weights of its literals.…”

Section: Resultsmentioning

confidence: 99%

Distribution-Aware Sampling and Weighted Model Counting for SAT

Chakraborty

Fremont

Meel

et al. 2014

AAAI

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Given a CNF formula and a weight for each assignment of values tovariables, two natural problems are weighted model counting anddistribution-aware sampling of satisfying assignments. Both problems have a wide variety of important applications. Due to the inherentcomplexity of the exact versions of the problems, interest has focusedon solving them approximately. Prior work in this area scaled only tosmall problems in practice, or failed to provide strong theoreticalguarantees, or employed a computationally-expensive most-probable-explanation ({\MPE}) queries that assumes prior knowledge of afactored representation of the weight distribution. We identify a novel parameter,\emph{tilt}, which is the ratio of the maximum weight of satisfying assignment to minimum weightof satisfying assignment and present anovel approach that works with a black-box oracle for weights ofassignments and requires only an {\NP}-oracle (in practice, a {\SAT}-solver) to solve both thecounting and sampling problems when the tilt is small. Our approach provides strong theoretical guarantees, and scales toproblems involving several thousand variables. We also show that theassumption of small tilt can be significantly relaxed while improving computational efficiency if a factored representation of the weights is known.

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“…By a result of Jerrum, Valiant and Vazirani (Jerrum, Valiant, and Vazirani 1986), we also know that approximate model counting and almost uniform sampling (a special case of approximate weighted sampling) are polynomially inter-reducible. Therefore, it is unlikely that there exist polynomial-time algorithms for either approximate weighted model counting or approximate weighted sampling (Karp, Luby, and Madras 1989). Recently, a new class of algorithms that use pairwise independent random parity constraints and a MAP (maximum a posteriori probability)-oracle have been proposed for solving both problems (Ermon et al 2013a;Ermon et al 2014;Ermon et al 2013b).…”

Section: Introductionmentioning

confidence: 99%

Distribution-Aware Sampling and Weighted Model Counting for SAT

Chakraborty¹,

Fremont²,

Meel³

et al. 2014

Preprint

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Given a CNF formula and a weight for each assignment of values to variables, two natural problems are weighted model counting and distribution-aware sampling of satisfying assignments. Both problems have a wide variety of important applications. Due to the inherent complexity of the exact versions of the problems, interest has focused on solving them approximately. Prior work in this area scaled only to small problems in practice, or failed to provide strong theoretical guarantees, or employed a computationally-expensive maximum a posteriori probability (MAP) oracle that assumes prior knowledge of a factored representation of the weight distribution. We present a novel approach that works with a black-box oracle for weights of assignments and requires only an NPoracle (in practice, a SAT-solver) to solve both the counting and sampling problems. Our approach works under mild assumptions on the distribution of weights of satisfying assignments, provides strong theoretical guarantees, and scales to problems involving several thousand variables. We also show that the assumptions can be significantly relaxed while improving computational efficiency if a factored representation of the weights is known.

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A Quantifier Elimination Algorithm for Linear Modular Equations and Disequations

Cited by 9 publications

References 8 publications

$$\mathsf {WAPS}$$ : Weighted and Projected Sampling

$$\mathsf {WAPS}$$ : Weighted and Projected Sampling

Distribution-Aware Sampling and Weighted Model Counting for SAT

Distribution-Aware Sampling and Weighted Model Counting for SAT

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