A method for simplifying the disjunctive normal form (DNF) of boolean functions

Cybernetics ◽  
1969 ◽  
Vol 3 (1) ◽  
pp. 17-20
Author(s):  
F. I. Andon
Keyword(s):  
Normal Form ◽  
Boolean Functions ◽  
Disjunctive Normal Form

Switching function based on hypergraphs with algorithm and python programming

2020 ◽  
Vol 39 (3) ◽  
pp. 2845-2859
Author(s):  
Mohammad Hamidi ◽  
Marzieh Rahmati ◽  
Akbar Rezaei
Keyword(s):  
Normal Form ◽  
Boolean Functions ◽  
Disjunctive Normal Form ◽  
Truth Table ◽  
Boolean Logic ◽  
Switching Function ◽  
Logical Formula ◽  
Boolean Expression ◽  
Switching Functions ◽  
Python Programming

According to Boolean logic, a disjunctive normal form (DNF) is a canonical normal form of a logical formula consisting of a disjunction of conjunctions (it can also be described as an OR of AND’s). For each table an arbitrary T.B.T is given (total binary truth table) Boolean expression can be written as a disjunctive normal form. This paper considers a notation of a T.B.T, introduces a new concept of the hypergraphable Boolean functions and the Boolean functionable hypergraphs with respect to any given T.B.T. This study defines a notation of unitors set on switching functions and proves that every T.B.T corresponds to a minimum Boolean expression via unitors set and presents some conditions on a T.B.T to obtain a minimum irreducible Boolean expression from switching functions. Indeed, we generate a switching function in different way via the concept of hypergraphs in terms of Boolean expression in such a way that it has a minimum irreducible Boolean expression, for every given T.B.T. Finally, an algorithm is presented. Therefore, a Python programming(with complete and original codes) such that for any given T.B.T, introduces a minimum irreducible switching expression.

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.

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.

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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