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.

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 On the orthogonalization of arbitrary Boolean formulae

2005 ◽  
Vol 2005 (2) ◽  
pp. 61-74 ◽  
Author(s):  
Renato Bruni
Keyword(s):  
Normal Form ◽  
Boolean Function ◽  
General Procedure ◽  
Conjunctive Normal Form ◽  
Disjunctive Normal Form ◽  
Reliability Theory ◽  
Satisfiability Problem ◽  
Propositional Satisfiability ◽  
Strongly Connected ◽  
Great Relevance

The orthogonal conjunctive normal form of a Boolean function is a conjunctive normal form in which any two clauses contain at least a pair of complementary literals. Orthogonal disjunctive normal form is defined similarly. Orthogonalization is the process of transforming the normal form of a Boolean function to orthogonal normal form. The problem is of great relevance in several applications, for example, in the reliability theory. Moreover, such problem is strongly connected with the well-known propositional satisfiability problem. Therefore, important complexity issues are involved. A general procedure for transforming an arbitrary CNF or DNF to an orthogonal one is proposed. Such procedure is tested on randomly generated Boolean formulae.

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.

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).

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.

Chapter 30. Reasoning with Quantified Boolean Formulas

2021 ◽  
Author(s):  
Enrico Giunchiglia ◽  
Paolo Marin ◽  
Massimo Narizzano
Keyword(s):  
Computer Aided Design ◽  
Automated Planning ◽  
Conjunctive Normal Form ◽  
Knowledge Representation And Reasoning ◽  
Research Issue ◽  
Important Research ◽  
Variable Elimination ◽  
Quantified Boolean Formulas ◽  
Boolean Formulas ◽  
Aided Design

The implementation of effective reasoning tools for deciding the satisfiability of Quantified Boolean Formulas(QBFs) is an important research issue in Artificial Intelligence and Computer Science. Indeed, QBF solvers have already been proposed for many reasoning tasks in knowledge representation and reasoning, in automated planning and in formal methods for computer aided design. Even more, since QBF reasoning is the prototypical PSPACE problem, the reduction of many other decision problems in PSPACE are readily available. For these reasons, in the last few years several decision procedures for QBFs have been proposed and implemented, mostly based either on search or on variable elimination, or on a combination of the two. In this chapter, after a brief recap of the basic terminology and notation about QBFs, we briefly review various applications of QBF reasoning that have been recently proposed, and then we focus on the description of the main approaches which are at the basis of currently available solvers for prenex QBFs in conjunctive normal form (CNF). Other approaches and extensions to non prenex, non CNF QBFs are briefly reviewed at the end of the chapter.

scholarly journals Interval Valued Fuzzy Soft Sets and Fuzzy Connectives

2019 ◽  
Vol 8 (4) ◽  
pp. 20-23
Keyword(s):  
Normal Form ◽  
Conjunctive Normal Form ◽  
Disjunctive Normal Form ◽  
Soft Set ◽  
Fuzzy Subset ◽  
Fuzzy Soft Set ◽  
Soft Sets ◽  
Fuzzy Soft Sets ◽  
Interval Valued ◽  
Fuzzy Connectives

The Article, We Learning A Few Operations Of Interval Valued Fuzzy Soft Sets Of Connectives And Give Elementary Properties Of Interval Valued Fuzzy Soft Sets Of Principal Disjunctive Normal Form And Principal Conjunctive Normal Form. 2000 Ams Subject Classification: 03f55, 08a72, 20n25. Keywords: Interval Valued Fuzzy Subset, Interval Valued Fuzzy Soft Set, And Principal Conjunctive Normal Form And Principal Disjunctive Normal Form Interval Valued Fuzzy Soft Set ‘’∧ ‘’ Operator And ‘’∨’’ Operator.

On sentences which are true of direct unions of algebras

1951 ◽  
Vol 16 (1) ◽  
pp. 14-21 ◽  
Author(s):  
Alfred Horn
Keyword(s):  
Normal Form ◽  
Wide Class ◽  
Conjunctive Normal Form ◽  
Disjunctive Normal Form ◽  
Factor Algebra ◽  
The Matrix ◽  
Distinct Individual ◽  
Direct Union ◽  
Individual Variables ◽  
Image Position

It is well known that certain sentences corresponding to similar algebras are invariant under direct union; that is, are true of the direct union when true of each factor algebra. An axiomatizable class of similar algebras, such as the class of groups, is closed under direct union when each of its axioms is invariant. In this paper we shall determine a wide class of invariant sentences. We shall also be concerned with determining sentences which are true of a direct union provided they are true of some factor algebra. In the case where all the factor algebras are the same, a further result is obtained. In §2 it will be shown that these criteria are the only ones of their kind. Lemma 7 below may be of some independent interest.We adopt the terminology and notation of McKinsey with the exception that the sign · will be used for conjunction. Expressions of the form ∼∊, where ∊ is an equation, will be called inequalities. In accordance with the analogy between conjunction and disjunction with product and sum respectively, we shall call α1, …, αn the terms of the disjunctionand the factors of the conjunctionEvery closed sentence is equivalent to a sentence in prenez normal form,where x1, …, xm distinct individual variables, Q1, …, Qm are quantifiers, and the matrix S is an open sentence in which each of the variables x1, …, xm actually occurs. The sentence S may be written in either disjunctive normal form:where αi,j is either an equation or an inequality, or in conjunctive normal form:.

COMMUTATIVE IMPLICATIONS ON COMPLETE LATTICES

1994 ◽  
Vol 02 (03) ◽  
pp. 333-341 ◽  
Author(s):  
WANGMING WU
Keyword(s):  
Normal Form ◽  
Complete Lattice ◽  
Conjunctive Normal Form ◽  
Disjunctive Normal Form ◽  
Complete Lattices

This paper is devoted to the investigation of commutative implications on a complete lattice L. It is proved that the disjunctive normal form (DNF) of a linguistic composition * is included in the conjunctive normal form (CNF) of that *, i.e., DNF(*) ≤ CNF(*) holds, for a special family of t-norms, t-conorms and negations induced by commutative implications.

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