A method for counting models on grid Boolean formulas1

2021 ◽  
pp. 1-8
Author(s):  
Marco A. López-Medina ◽  
J. Raymundo Marcial-Romero ◽  
Guillermo De Ita Luna ◽  
José A. Hernández
Keyword(s):  
Asymptotic Behavior ◽  
Normal Form ◽  
Time Complexity ◽  
Conjunctive Normal Form ◽  
Grid Graph ◽  
Boolean Formulas ◽  
Novel Algorithm

We present a novel algorithm based on combinatorial operations on lists for computing the number of models on two conjunctive normal form Boolean formulas whose restricted graph is represented by a grid graph Gm,n. We show that our algorithm is correct and its time complexity is O ( t · 1 . 618 t + 2 + t · 1 . 618 2 t + 4 ) , where t = n · m is the total number of vertices in the graph. For this class of formulas, we show that our proposal improves the asymptotic behavior of the time-complexity with respect of the current leader algorithm for counting models on two conjunctive form formulas of this kind.

METHODS OF REDUCING OF LARGE BOOLEAN FORMULAS REPRESENTED IN A CONJUNCTIVE NORMAL FORM FOR DETERMINING THEIR SATISFIABILITY

2020 ◽  
Author(s):  
N.I. Gdansky ◽  
◽  
A.A. Denisov ◽  
Keyword(s):  
Normal Form ◽  
Normal Forms ◽  
Conjunctive Normal Form ◽  
Boolean Formulas

The article explores the satisfiability of conjunctive normal forms used in modeling systems.The problems of CNF preprocessing are considered.The analysis of particular methods for reducing this formulas, which have polynomial input complexity is given.

NAE-resolution: A new resolution refutation technique to prove not-all-equal unsatisfiability

2020 ◽  
Vol 30 (7) ◽  
pp. 736-751
Author(s):  
Hans Kleine Büning ◽  
P. Wojciechowski ◽  
K. Subramani
Keyword(s):  
Normal Form ◽  
Conjunctive Normal Form ◽  
Principal Result ◽  
Satisfying Assignment ◽  
Boolean Formulas ◽  
Complete Procedure ◽  
Np Complete ◽  
Resolution Refutation ◽  
Original Formula ◽  
Cnf Formulas

AbstractIn this paper, we analyze Boolean formulas in conjunctive normal form (CNF) from the perspective of read-once resolution (ROR) refutation schemes. A read-once (resolution) refutation is one in which each clause is used at most once. Derived clauses can be used as many times as they are deduced. However, clauses in the original formula can only be used as part of one derivation. It is well known that ROR is not complete; that is, there exist unsatisfiable formulas for which no ROR exists. Likewise, the problem of checking if a 3CNF formula has a read-once refutation is NP-complete. This paper is concerned with a variant of satisfiability called not-all-equal satisfiability (NAE-satisfiability). A CNF formula is NAE-satisfiable if it has a satisfying assignment in which at least one literal in each clause is set to false. It is well known that the problem of checking NAE-satisfiability is NP-complete. Clearly, the class of CNF formulas which are NAE-satisfiable is a proper subset of satisfiable CNF formulas. It follows that traditional resolution cannot always find a proof of NAE-unsatisfiability. Thus, traditional resolution is not a sound procedure for checking NAE-satisfiability. In this paper, we introduce a variant of resolution called NAE-resolution which is a sound and complete procedure for checking NAE-satisfiability in CNF formulas. The focus of this paper is on a variant of NAE-resolution called read-once NAE-resolution in which each clause (input or derived) can be part of at most one NAE-resolution step. Our principal result is that read-once NAE-resolution is a sound and complete procedure for 2CNF formulas. Furthermore, we provide an algorithm to determine the smallest such NAE-resolution in polynomial time. This is in stark contrast to the corresponding problem concerning 2CNF formulas and ROR refutations. We also show that the problem of checking whether a 3CNF formula has a read-once NAE-resolution is NP-complete.

scholarly journals A Cardinal Improvement to Pseudo-Boolean Solving

2020 ◽  
Vol 34 (02) ◽  
pp. 1495-1503
Author(s):  
Jan Elffers ◽  
Jakob Nordstr”m
Keyword(s):  
Normal Form ◽  
Building Blocks ◽  
Conjunctive Normal Form ◽  
Conflict Analysis ◽  
Cardinality Constraints ◽  
Sat Solvers ◽  
Boolean Formulas ◽  
Clause Learning

Pseudo-Boolean solvers hold out the theoretical potential of exponential improvements over conflict-driven clause learning (CDCL) SAT solvers, but in practice perform very poorly if the input is given in the standard conjunctive normal form (CNF) format. We present a technique to remedy this problem by recovering cardinality constraints from CNF on the fly during search. This is done by collecting potential building blocks of cardinality constraints during propagation and combining these blocks during conflict analysis. Our implementation has a non-negligible but manageable overhead when detection is not successful, and yields significant gains for some SAT competition and crafted benchmarks for which pseudo-Boolean reasoning is stronger than CDCL. It also boosts performance for some native pseudo-Boolean formulas where this approach helps to improve learned constraints.

scholarly journals ADDMC: Weighted Model Counting with Algebraic Decision Diagrams

2020 ◽  
Vol 34 (02) ◽  
pp. 1468-1476
Author(s):  
Jeffrey Dudek ◽  
Vu Phan ◽  
Moshe Vardi
Keyword(s):  
Dynamic Programming ◽  
Data Structure ◽  
Normal Form ◽  
Standard Model ◽  
State Of The Art ◽  
Conjunctive Normal Form ◽  
Decision Diagrams ◽  
Boolean Formulas ◽  
Model Counting ◽  
Weighted Model

We present an algorithm to compute exact literal-weighted model counts of Boolean formulas in Conjunctive Normal Form. Our algorithm employs dynamic programming and uses Algebraic Decision Diagrams as the main data structure. We implement this technique in ADDMC, a new model counter. We empirically evaluate various heuristics that can be used with ADDMC. We then compare ADDMC to four state-of-the-art weighted model counters (Cachet, c2d, d4, and miniC2D) on 1914 standard model counting benchmarks and show that ADDMC significantly improves the virtual best solver.

scholarly journals Clause/Term Resolution and Learning in the Evaluation of Quantified Boolean Formulas

2006 ◽  
Vol 26 ◽  
pp. 371-416 ◽  
Author(s):  
E. Giunchiglia ◽  
M. Narizzano ◽  
A. Tacchella
Keyword(s):  
Normal Form ◽  
Automated Reasoning ◽  
Comparative Evaluation ◽  
State Of The Art ◽  
Conjunctive Normal Form ◽  
Disjunctive Normal Form ◽  
Propositional Satisfiability ◽  
Quantified Boolean Formulas ◽  
Boolean Formulas ◽  
Input Formula

Resolution is the rule of inference at the basis of most procedures for automated reasoning. In these procedures, the input formula is first translated into an equisatisfiable formula in conjunctive normal form (CNF) and then represented as a set of clauses. Deduction starts by inferring new clauses by resolution, and goes on until the empty clause is generated or satisfiability of the set of clauses is proven, e.g., because no new clauses can be generated. In this paper, we restrict our attention to the problem of evaluating Quantified Boolean Formulas (QBFs). In this setting, the above outlined deduction process is known to be sound and complete if given a formula in CNF and if a form of resolution, called ``Q-resolution'', is used. We introduce Q-resolution on terms, to be used for formulas in disjunctive normal form. We show that the computation performed by most of the available procedures for QBFs --based on the Davis-Logemann-Loveland procedure (DLL) for propositional satisfiability-- corresponds to a tree in which Q-resolution on terms and clauses alternate. This poses the theoretical bases for the introduction of learning, corresponding to recording Q-resolution formulas associated with the nodes of the tree. We discuss the problems related to the introduction of learning in DLL based procedures, and present solutions extending state-of-the-art proposals coming from the literature on propositional satisfiability. Finally, we show that our DLL based solver extended with learning, performs significantly better on benchmarks used in the 2003 QBF solvers comparative evaluation.

A reduction class containing formulas with one monadic predicate and one binary function symbol

1976 ◽  
Vol 41 (1) ◽  
pp. 45-49
Author(s):  
Charles E. Hughes
Keyword(s):  
Normal Form ◽  
Predicate Symbol ◽  
Conjunctive Normal Form ◽  
Function Symbol ◽  
Predicate Calculus ◽  
Binary Function ◽  
First Order ◽  
The Matrix ◽  
Reduction Class ◽  
Order Predicate Calculus

AbstractA new reduction class is presented for the satisfiability problem for well-formed formulas of the first-order predicate calculus. The members of this class are closed prenex formulas of the form ∀x∀yC. The matrix C is in conjunctive normal form and has no disjuncts with more than three literals, in fact all but one conjunct is unary. Furthermore C contains but one predicate symbol, that being unary, and one function symbol which symbol is binary.

Chapter 29. Theory of Quantified Boolean Formulas

2021 ◽  
Author(s):  
Hans Kleine Büning ◽  
Uwe Bubeck
Keyword(s):  
Normal Form ◽  
Boolean Function ◽  
Expressive Power ◽  
Modeling Language ◽  
Complex Problems ◽  
Quantified Boolean Formulas ◽  
Horn Formulas ◽  
Boolean Formulas ◽  
Function Models ◽  
Resolution Calculus

Quantified Boolean formulas (QBF) are a generalization of propositional formulas by allowing universal and existential quantifiers over variables. This enhancement makes QBF a concise and natural modeling language in which problems from many areas, such as planning, scheduling or verification, can often be encoded in a more compact way than with propositional formulas. We introduce in this chapter the syntax and semantics of QBF and present fundamental concepts. This includes normal form transformations and Q-resolution, an extension of the propositional resolution calculus. In addition, Boolean function models are introduced to describe the valuation of formulas and the behavior of the quantifiers. We also discuss the expressive power of QBF and provide an overview of important complexity results. These illustrate that the greater capabilities of QBF lead to more complex problems, which makes it interesting to consider suitable subclasses of QBF. In particular, we give a detailed look at quantified Horn formulas (QHORN) and quantified 2-CNF (Q2-CNF).

Chapter 13. Symmetry and Satisfiability

2021 ◽  
Author(s):  
Karem A. Sakallah
Keyword(s):  
Group Theory ◽  
Normal Form ◽  
Boolean Functions ◽  
Search Algorithm ◽  
Conjunctive Normal Form ◽  
Search Space ◽  
Decision Problems ◽  
Mathematical Language ◽  
Sat Solvers ◽  
Speed Up

Symmetry is at once a familiar concept (we recognize it when we see it!) and a profoundly deep mathematical subject. At its most basic, a symmetry is some transformation of an object that leaves the object (or some aspect of the object) unchanged. For example, a square can be transformed in eight different ways that leave it looking exactly the same: the identity “do-nothing” transformation, 3 rotations, and 4 mirror images (or reflections). In the context of decision problems, the presence of symmetries in a problem’s search space can frustrate the hunt for a solution by forcing a search algorithm to fruitlessly explore symmetric subspaces that do not contain solutions. Recognizing that such symmetries exist, we can direct a search algorithm to look for solutions only in non-symmetric parts of the search space. In many cases, this can lead to significant pruning of the search space and yield solutions to problems which are otherwise intractable. This chapter explores the symmetries of Boolean functions, particularly the symmetries of their conjunctive normal form (CNF) representations. Specifically, it examines what those symmetries are, how to model them using the mathematical language of group theory, how to derive them from a CNF formula, and how to utilize them to speed up CNF SAT solvers.

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