The shortest disjunctive normal form of a random Boolean function

2003 ◽  
Vol 22 (2) ◽  
pp. 161-186 ◽  
Author(s):  
Nicholas Pippenger
Keyword(s):  
Normal Form ◽  
Boolean Function ◽  
Disjunctive Normal Form

Canalyzing minors of Boolean functions

2020 ◽  
Vol 13 (08) ◽  
pp. 2050160
Author(s):  
Ivo Damyanov
Keyword(s):  
Normal Form ◽  
Boolean Function ◽  
Upper Bound ◽  
Boolean Functions ◽  
Disjunctive Normal Form ◽  
Function Identification

Canalyzing functions are a special type of Boolean functions. For a canalyzing function, there is at least one argument, in which taking a certain value can determine the value of the function. Identification of variables can also shrink the resulting function into constant or function depending on one variable. In this paper, we discuss a particular disjunctive normal form for representation of Boolean function with its identification minors. Then an upper bound of the number of canalyzing minors is obtained. Finally, the number of canalyzing minors for Boolean functions with five essential variables is discussed.

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.

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).

On closure under direct product

1958 ◽  
Vol 23 (2) ◽  
pp. 149-154 ◽  
Author(s):  
C. C. Chang ◽  
Anne C. Morel
Keyword(s):  
Normal Form ◽  
Direct Product ◽  
Disjunctive Normal Form ◽  
Negative Answer ◽  
Sufficient Condition ◽  
Natural Question ◽  
Existential Quantifier ◽  
Relational Systems ◽  
Image Position

In 1951, Horn obtained a sufficient condition for an arithmetical class to be closed under direct product. A natural question which arose was whether Horn's condition is also necessary. We obtain a negative answer to that question.We shall discuss relational systems of the formwhere A and R are non-empty sets; each element of R is an ordered triple 〈a, b, c〉, with a, b, c ∈ A.1 If the triple 〈a, b, c〉 belongs to the relation R, we write R(a, b, c); if 〈a, b, c〉 ∉ R, we write (a, b, c). If x0, x1 and x2 are variables, then R(x0, x1, x2) and x0 = x1 are predicates. The expressions (x0, x1, x2) and x0 ≠ x1 will be referred to as negations of predicates.We speak of α1, …, αn as terms of the disjunction α1 ∨ … ∨ αn and as factors of the conjunction α1 ∧ … ∧ αn. A sentence (open, closed or neither) of the formwhere each Qi (if there be any) is either the universal or the existential quantifier and each αi, l is either a predicate or a negation of a predicate, is said to be in prenex disjunctive normal form.

scholarly journals On the learnability of disjunctive normal form formulas

1995 ◽  
Vol 19 (3) ◽  
pp. 183-208 ◽  
Author(s):  
Howard Aizenstein ◽  
Leonard Pitt
Keyword(s):  
Normal Form ◽  
Disjunctive Normal Form
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