Weighted Boolean Formula Games

2015 ◽  
pp. 49-86 ◽  
Author(s):  
Marios Mavronicolas ◽  
Burkhard Monien ◽  
Klaus W. Wagner
Keyword(s):  
Boolean Formula

An Efficient Algorithm to Understand Long Counterexample

2013 ◽  
Vol 328 ◽  
pp. 254-260
Author(s):  
Zhi Yuan Chen ◽  
Shao Bin Huang ◽  
Ming Yu Ji ◽  
Lin Shan Shen
Keyword(s):  
Model Checking ◽  
Efficient Algorithm ◽  
System Model ◽  
Experimental Result ◽  
Boolean Formula ◽  
Abstract Model ◽  
System State ◽  
Guide System ◽  
Model Improvement ◽  
Model Failure

For given system to proceed model checking, if system model is discontent with the quality which to be detected, model detector will give counterexample, it will cause the generated counterexample too long when system state-space is very large, it is a very important problem, how to find the reason of model failure from long counterexample quickly, the article uses extractive technique of minimal unsatisfiable subformula to put forward a kind of understanding counterexample way which is extracted minimal unsatisfiable subformula quickly from Boolean formula. The algorithm can pinpoint error and find the reason of model failure. Experimental result indicated that understanding counterexample is based on minimal unsatisfiable subformula can accelerate understanding counterexample speed, improve the efficient of debugging, guide system abstract model improvement effectively.

scholarly journals Estimating the Density of States of Boolean Satisfiability Problems on Classical and Quantum Computing Platforms

2020 ◽  
Vol 34 (02) ◽  
pp. 1627-1635 ◽  
Author(s):  
Tuhin Sahai ◽  
Anurag Mishra ◽  
Jose Miguel Pasini ◽  
Susmit Jha
Keyword(s):  
Density Of States ◽  
Solution Space ◽  
Conjunctive Normal Form ◽  
Entire Solution ◽  
Concentration Of Measure ◽  
Problem Instance ◽  
Boolean Formula ◽  
Exact Answer ◽  
Novel Approach ◽  
Max Sat

Given a Boolean formula ϕ(x) in conjunctive normal form (CNF), the density of states counts the number of variable assignments that violate exactly e clauses, for all values of e. Thus, the density of states is a histogram of the number of unsatisfied clauses over all possible assignments. This computation generalizes both maximum-satisfiability (MAX-SAT) and model counting problems and not only provides insight into the entire solution space, but also yields a measure for the hardness of the problem instance. Consequently, in real-world scenarios, this problem is typically infeasible even when using state-of-the-art algorithms. While finding an exact answer to this problem is a computationally intensive task, we propose a novel approach for estimating density of states based on the concentration of measure inequalities. The methodology results in a quadratic unconstrained binary optimization (QUBO), which is particularly amenable to quantum annealing-based solutions. We present the overall approach and compare results from the D-Wave quantum annealer against the best-known classical algorithms such as the Hamze-de Freitas-Selby (HFS) algorithm and satisfiability modulo theory (SMT) solvers.

scholarly journals The complexity of Boolean formula minimization

2011 ◽  
Vol 77 (1) ◽  
pp. 142-153 ◽  
Author(s):  
David Buchfuhrer ◽  
Christopher Umans
Keyword(s):  
Boolean Formula

FROM XSAT TO SAT BY EXHIBITING BOOLEAN FUNCTIONS

2009 ◽  
Vol 18 (05) ◽  
pp. 783-799
Author(s):  
RICHARD OSTROWSKI ◽  
LIONEL PARIS
Keyword(s):  
Graph Coloring ◽  
Boolean Functions ◽  
Structural Information ◽  
Conjunctive Normal Form ◽  
Satisfiability Problem ◽  
Boolean Formula ◽  
Truth Assignment ◽  
Coloring Problem ◽  
Graph Coloring Problem ◽  
Speed Up

Given a Boolean formula in conjunctive normal form (CNF), the Exact Satisfiability problem (XSAT), a variant of the Satisfiability problem (SAT), consists in finding an assignment to the variables such that each clause contains exactly one satisfied literal. Best algorithms to solve this problem run in [Formula: see text] ([Formula: see text] for X3SAT). Another possibility is to transform each clause in a set of equivalent clauses for the Satisfiability problem and to use modern and powerful solvers (zChaff, Berkmin, MiniSat, RSat etc.) to find such truth assignment. In this paper we introduce three new encodings from XSAT instances to SAT instances that lead to a lot of structural information (equivalency gates and and gates) which is naturally hidden in the pairwise transformation. Some solvers (lsat,march_dl,eqsatz) can take into account this kinds of structural information to make simplifications as pretreatment and speed-up the resolution. Then we show the interest of dealing with the XSAT formalism by introducing an encoding of binary CSP and graph coloring problem into XSAT instances. Preliminary results on real-world binary CSP and graph coloring problem show the importance of exhibiting equivalencies for the XSAT problem.

scholarly journals A Variable Neighborhood Walksat-Based Algorithm for MAX-SAT Problems

2014 ◽  
Vol 2014 ◽  
pp. 1-11 ◽  
Author(s):  
Noureddine Bouhmala
Keyword(s):  
Local Search ◽  
Optimization Problems ◽  
Search Algorithm ◽  
Search Algorithms ◽  
Boolean Formula ◽  
Local Search Algorithms ◽  
Neighborhood Structure ◽  
Np Complete ◽  
Max Sat ◽  
Almost All

The simplicity of the maximum satisfiability problem (MAX-SAT) combined with its applicability in many areas of artificial intelligence and computing science made it one of the fundamental optimization problems. This NP-complete problem refers to the task of finding a variable assignment that satisfies the maximum number of clauses (or the sum of weights of satisfied clauses) in a Boolean formula. The Walksat algorithm is considered to be the main skeleton underlying almost all local search algorithms for MAX-SAT. Most local search algorithms including Walksat rely on the 1-flip neighborhood structure. This paper introduces a variable neighborhood walksat-based algorithm. The neighborhood structure can be combined easily using any local search algorithm. Its effectiveness is compared with existing algorithms using 1-flip neighborhood structure and solvers such as CCLS and Optimax from the eighth MAX-SAT evaluation.

scholarly journals Formula partitioning revisited

10.29007/9skn ◽  
2018 ◽  
Author(s):  
Zoltan Mann ◽  
Pal Papp
Keyword(s):  
State Of The Art ◽  
Boolean Formula ◽  
Early Results ◽  
Wide Range ◽  
Benchmark Instances ◽  
Partitioning Algorithms

Dividing a Boolean formula into smaller independent sub-formulae can be a useful technique for accelerating the solution of Boolean problems, including SAT and #SAT. Nevertheless, and despite promising early results, formula partitioning is hardly used in state-of-the-art solvers. In this paper, we show that this is rooted in a lack of consistency of the usefulness of formula partitioning techniques. In particular, we evaluate two existing and a novel partitioning model, coupled with two existing and two novel partitioning algorithms, on a wide range of benchmark instances. Our results show that there is no one-size-fits-all solution: for different formula types, different partitioning models and algorithms are the most suitable. While these results might seem negative, they help to improve our understanding about formula partitioning; moreover, the findings also give some guidance as to which method to use for what kinds of formulae.

scholarly journals Counting Minimal Unsatisfiable Subsets

2021 ◽  
pp. 313-336
Author(s):  
Jaroslav Bendík ◽  
Kuldeep S. Meel
Keyword(s):  
Empirical Evaluation ◽  
Proper Subset ◽  
Boolean Formula ◽  
Np Hard ◽  
Sat Solvers ◽  
The Past ◽  
Recent Developments ◽  
Model Counting

AbstractGiven an unsatisfiable Boolean formula F in CNF, an unsatisfiable subset of clauses U of F is called Minimal Unsatisfiable Subset (MUS) if every proper subset of U is satisfiable. Since MUSes serve as explanations for the unsatisfiability of F, MUSes find applications in a wide variety of domains. The availability of efficient SAT solvers has aided the development of scalable techniques for finding and enumerating MUSes in the past two decades. Building on the recent developments in the design of scalable model counting techniques for SAT, Bendík and Meel initiated the study of MUS counting techniques. They succeeded in designing the first approximate MUS counter, $$\mathsf {AMUSIC}$$ AMUSIC , that does not rely on exhaustive MUS enumeration. $$\mathsf {AMUSIC}$$ AMUSIC , however, suffers from two shortcomings: the lack of exact estimates and limited scalability due to its reliance on 3-QBF solvers.In this work, we address the two shortcomings of $$\mathsf {AMUSIC}$$ AMUSIC by designing the first exact MUS counter, $$\mathsf {CountMUST}$$ CountMUST , that does not rely on exhaustive enumeration. $$\mathsf {CountMUST}$$ CountMUST circumvents the need for 3-QBF solvers by reducing the problem of MUS counting to projected model counting. While projected model counting is #NP-hard, the past few years have witnessed the development of scalable projected model counters. An extensive empirical evaluation demonstrates that $$\mathsf {CountMUST}$$ CountMUST successfully returns MUS count for 1500 instances while $$\mathsf {AMUSIC}$$ AMUSIC and enumeration-based techniques could only handle up to 833 instances.

scholarly journals GANAK: A Scalable Probabilistic Exact Model Counter

2019 ◽  
Author(s):  
Shubham Sharma ◽  
Subhajit Roy ◽  
Mate Soos ◽  
Kuldeep S. Meel
Keyword(s):  
Probabilistic Reasoning ◽  
State Of The Art ◽  
Fundamental Problem ◽  
Approximate Model ◽  
Boolean Formula ◽  
Sat Solvers ◽  
Exact Model ◽  
Universal Hashing ◽  
Model Counting ◽  
Dynamic Decomposition

Given a Boolean formula F, the problem of model counting, also referred to as #SAT, seeks to compute the number of solutions of F. Model counting is a fundamental problem with a wide variety of applications ranging from planning, quantified information flow to probabilistic reasoning and the like. The modern #SAT solvers tend to be either based on static decomposition, dynamic decomposition, or a hybrid of the two. Despite dynamic decomposition based #SAT solvers sharing much of their architecture with SAT solvers, the core design and heuristics of dynamic decomposition-based #SAT solvers has remained constant for over a decade. In this paper, we revisit the architecture of the state-of-the-art dynamic decomposition-based #SAT tool, sharpSAT, and demonstrate that by introducing a new notion of probabilistic component caching and the usage of universal hashing for exact model counting along with the development of several new heuristics can lead to significant performance improvement over state-of-the-art model-counters. In particular, we develop GANAK, a new scalable probabilistic exact model counter that outperforms state-of-the-art exact and approximate model counters sharpSAT and ApproxMC3 respectively, both in terms of PAR-2 score and the number of instances solved. Furthermore, in our experiments, the model count returned by GANAK was equal to the exact model count for all the benchmarks. Finally, we observe that recently proposed preprocessing techniques for model counting benefit exact model counters while hurting the performance of approximate model counters.

scholarly journals On Distributed Solution to SAT by Membrane Computing

2018 ◽  
Vol 13 (3) ◽  
pp. 303-320 ◽  
Author(s):  
Henry N. Adorna ◽  
Linqiang Pan ◽  
Bosheng Song
Keyword(s):  
Cell Division ◽  
Computational Models ◽  
P Systems ◽  
Boolean Formula ◽  
Sat Problem ◽  
Complete Problems ◽  
Uniform Solution ◽  
Boolean Formulas ◽  
The Given ◽  
Communication Resource

Tissue P systems with evolutional communication rules and cell division (TPec, for short) are a class of bio-inspired parallel computational models, which can solve NP-complete problems in a feasible time. In this work, a variant of TPec, called $k$-distributed tissue P systems with evolutional communication and cell division ($k\text{-}\Delta_{TP_{ec}}$, for short) is proposed. A uniform solution to the SAT problem by $k\text{-}\Delta_{TP_{ec}}$ under balanced fixed-partition is presented. The solution provides not only the precise satisfying truth assignments for all Boolean formulas, but also a precise amount of possible such satisfying truth assignments. It is shown that the communication resource for one-way and two-way uniform $k$-P protocols are increased with respect to $k$; while a single communication is shown to be possible for bi-directional uniform $k$-P protocols for any $k$. We further show that if the number of clauses is at least equal to the square of the number of variables of the given boolean formula, then $k\text{-}\Delta_{TP_{ec}}$ for solving the SAT problem are more efficient than TPec as show in \cite{bosheng2017}; if the number of clauses is equal to the number of variables, then $k\text{-}\Delta_{TP_{ec}}$ for solving the SAT problem work no much faster than TPec.

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