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 A heuristic method for bi-decomposition of partial Boolean functions

Informatics ◽  
2020 ◽  
Vol 17 (3) ◽  
pp. 44-53
Author(s):  
Yu. V. Pottosin
Keyword(s):  
Boolean Function ◽  
Boolean Functions ◽  
Heuristic Method ◽  
Disjunctive Normal Form ◽  
The Other ◽  
Output Function ◽  
Minimal Rank ◽  
Boolean Space ◽  
Definitional Domain ◽  
The Given

The problem of decomposition of a Boolean function is to represent a given Boolean function in the form of a superposition of some Boolean functions whose number of arguments are less than the number of given function. The bi-decomposition represents a given function as a logic algebra operation, which is also given, over two Boolean functions. The task is reduced to specification of those two functions. A method for bi-decomposition of incompletely specified (partial) Boolean function is suggested. The given Boolean function is specified by two sets, one of which is the part of the Boolean space of the arguments of the function where its value is 1, and the other set is the part of the space where the function has the value 0. The complete graph of orthogonality of Boolean vectors that constitute the definitional domain of the given function is considered. In the graph, the edges are picked out, any of which has its ends corresponding the elements of Boolean space where the given function has different values. The problem of bi-decomposition is reduced to the problem of a weighted two-block covering the set of picked out edges of considered graph by its complete bipartite subgraphs (bicliques). Every biclique is assigned with a disjunctive normal form (DNF) in definite way. The weight of a biclique is a pair of certain parameters of   assigned DNF. According to each biclique of obtained cover, a Boolean function is constructed whose arguments are the variables from the term of minimal rank on the DNF. A technique for constructing the mentioned cover for two kinds of output function is described.

A generalization of Shannon function

2018 ◽  
Vol 28 (5) ◽  
pp. 309-318 ◽  
Author(s):  
Nikolay P. Redkin
Keyword(s):  
Boolean Function ◽  
Upper Bound ◽  
Boolean Functions ◽  
Shannon Function ◽  
Conceptual Content

Abstract When investigating the complexity of implementing Boolean functions, it is usually assumed that the basis inwhich the schemes are constructed and the measure of the complexity of the schemes are known. For them, the Shannon function is introduced, which associates with each Boolean function the least complexity of implementing this function in the considered basis. In this paper we propose a generalization of such a Shannon function in the form of an upper bound that is taken over all functionally complete bases. This generalization gives an idea of the complexity of implementing Boolean functions in the “worst” bases for them. The conceptual content of the proposed generalization is demonstrated by the example of a conjunction.

scholarly journals Two new results about quantum exact learning

Quantum ◽  
2021 ◽  
Vol 5 ◽  
pp. 587
Author(s):  
Srinivasan Arunachalam ◽  
Sourav Chakraborty ◽  
Troy Lee ◽  
Manaswi Paraashar ◽  
Ronald de Wolf
Keyword(s):  
Boolean Function ◽  
Upper Bound ◽  
Boolean Functions ◽  
Quantum Computers ◽  
Concept Class ◽  
Membership Queries ◽  
Exact Learning ◽  
Uniform Quantum

We present two new results about exact learning by quantum computers. First, we show how to exactly learn a k-Fourier-sparse n-bit Boolean function from O(k1.5(log⁡k)2) uniform quantum examples for that function. This improves over the bound of Θ~(kn) uniformly random classical examples (Haviv and Regev, CCC'15). Additionally, we provide a possible direction to improve our O~(k1.5) upper bound by proving an improvement of Chang's lemma for k-Fourier-sparse Boolean functions. Second, we show that if a concept class C can be exactly learned using Q quantum membership queries, then it can also be learned using O(Q2log⁡Qlog⁡|C|)classical membership queries. This improves the previous-best simulation result (Servedio and Gortler, SICOMP'04) by a log⁡Q-factor.

scholarly journals Quantum Random Access Codes for Boolean Functions

Quantum ◽  
2021 ◽  
Vol 5 ◽  
pp. 402
Author(s):  
João F. Doriguello ◽  
Ashley Montanaro
Keyword(s):  
Boolean Function ◽  
Upper Bound ◽  
Boolean Functions ◽  
Success Probability ◽  
Random Access ◽  
Fixed Size ◽  
Multiplicative Constant ◽  
Noise Stability ◽  
Access Code ◽  
Quantum Protocols

An n↦pm random access code (RAC) is an encoding of n bits into m bits such that any initial bit can be recovered with probability at least p, while in a quantum RAC (QRAC), the n bits are encoded into m qubits. Since its proposal, the idea of RACs was generalized in many different ways, e.g. allowing the use of shared entanglement (called entanglement-assisted random access code, or simply EARAC) or recovering multiple bits instead of one. In this paper we generalize the idea of RACs to recovering the value of a given Boolean function f on any subset of fixed size of the initial bits, which we call f-random access codes. We study and give protocols for f-random access codes with classical (f-RAC) and quantum (f-QRAC) encoding, together with many different resources, e.g. private or shared randomness, shared entanglement (f-EARAC) and Popescu-Rohrlich boxes (f-PRRAC). The success probability of our protocols is characterized by the noise stability of the Boolean function f. Moreover, we give an upper bound on the success probability of any f-QRAC with shared randomness that matches its success probability up to a multiplicative constant (and f-RACs by extension), meaning that quantum protocols can only achieve a limited advantage over their classical counterparts.

scholarly journals Most complex Boolean functions detected by the specialized normal form

2007 ◽  
Vol 20 (3) ◽  
pp. 259-279 ◽  
Author(s):  
Bernd Steinbach
Keyword(s):  
Normal Form ◽  
Boolean Function ◽  
Boolean Functions ◽  
Recursive Algorithm ◽  
Complexity Measure ◽  
Paper Properties ◽  
Sum Of Products ◽  
Adjacency Graphs

It is well known that Exclusive Sum-Of-Products (ESOP) expressions for Boolean functions require on average the smallest number of cubes. Thus, a simple complexity measure for a Boolean function is the number of cubes in its simplest ESOP. It will be shown that this structure-oriented measure of the complexity can be improved by a unique complexity measure which is based on the function. Thus, it is suggested to detect all most complex Boolean functions more precisely by means of the Specialized Normal Form (SNF). The SNF is a unique (canonical) ESOP representation of a Boolean function. In this paper properties of the most complex Boolean functions are studied. Adjacency graphs of the SNF will be used to calculate minimal ESOPs as well as to detect special properties of most complex Boolean functions. These properties affect the procedure of finding exact minimal ESOPs. A solution is given that overcomes these observed problems. Using the SNF, the number of most complex Boolean functions was found. A recursive algorithm will be given that calculates each most complex Boolean function for a given number of variables.

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.

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