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.

On 1-stable perfectly balanced Boolean functions

2017 ◽  
Vol 27 (2) ◽  
Author(s):  
Stanislav V. Smyshlyaev
Keyword(s):  
Boolean Function ◽  
Boolean Functions ◽  
The Other ◽  
Correlation Immunity ◽  
Other Hand

AbstractThe paper is concerned with relations between the correlation-immunity (stability) and the perfectly balancedness of Boolean functions. It is shown that an arbitrary perfectly balanced Boolean function fails to satisfy a certain property that is weaker than the 1-stability. This result refutes some assertions by Markus Dichtl. On the other hand, we present new results on barriers of perfectly balanced Boolean functions which show that any perfectly balanced function such that the sum of the lengths of barriers is smaller than the length of variables, is 1-stable.

Logical Analysis of Data

2011 ◽  
pp. 689-692 ◽  
Author(s):  
Endre Boros ◽  
Peter L. Hammer ◽  
Toshihide Ibaraki
Keyword(s):  
Data Mining ◽  
Boolean Function ◽  
Categorical Data ◽  
Boolean Functions ◽  
Binary Data ◽  
Logical Form ◽  
Logical Analysis ◽  
Logical Analysis Of Data ◽  
Primary Target ◽  
The Given

The logical analysis of data (LAD) is a methodology aimed at extracting or discovering knowledge from data in logical form. The first paper in this area was published as Crama, Hammer, & Ibaraki (1988) and precedes most of the data mining papers appearing in the 1990s. Its primary target is a set of binary data belonging to two classes for which a Boolean function that classifies the data into two classes is built. In other words, the extracted knowledge is embodied as a Boolean function, which then will be used to classify unknown data. As Boolean functions that classify the given data into two classes are not unique, there are various methodologies investigated in LAD to obtain compact and meaningful functions. As will be mentioned later, numerical and categorical data also can be handled, and more than two classes can be represented by combining more than one Boolean function.

scholarly journals Masking of Internal Nodes Faults Based on Applying of Incompletely Specified Boolean Functions

2020 ◽  
Vol 20 (4) ◽  
pp. 517-526
Author(s):  
Angela Yu. Matrosova ◽  
◽  
Victor A. Provkin ◽  
Valentina V. Andreeva ◽  
◽  
...  
Keyword(s):  
Boolean Function ◽  
Circuit Design ◽  
Boolean Functions ◽  
Internal Node ◽  
Sequential Circuits ◽  
Combinational Circuits ◽  
Suggested Approach ◽  
System A ◽  
Internal Fault ◽  
The Given

Combinational circuits (combinational parts of sequential circuits) are considered. Masking of internal nodes faults with applying sub-circuit, inputs of which are connected to the circuit inputs and outputs — to the circuit proper internal nodes, is suggested. The algorithm of deriving incompletely specified Boolean function for an internal node of the circuit based on using operations on ROBDDs is described. Masking circuit (patch circuit) design for the given internal fault nodes is reduced to covering of the system of incompletely specified Boolean functions corresponding to the fault nodes by the proper SoP system. Then the obtained system of completely specified Boolean functions is applied to derive masking circuit by using ABC system (A System for Sequential Synthesis and Verification). Experiments on bench marks show essential cutting of overhead in the frame of the suggested approach.

scholarly journals A Method of Two-Level Simplification of Boolean Functions

1967 ◽  
Vol 29 ◽  
pp. 201-210
Author(s):  
Toshio Umezawa
Keyword(s):  
Boolean Function ◽  
Boolean Functions ◽  
The Other ◽  
Product Form ◽  
Automatic Computation ◽  
Prime Implicants ◽  
Other Hand ◽  
Group 1

There are a number of methods to find minimal two-level forms for a given Boolean function, e g. Harvard’s group [1], Veitch [2], Quine [3], [4], Karnaugh [5], Nelson [6], [7] etc,. This paper presents an approach which is suitable for mechanical or automatic computation, as the Harvard method and the Quine method are so. On the other hand, it shares the same property as the Veitch method in the sense that some of essential prime implicants may be found before all prime implicants are computed. It also adopts the procedure to reduce the necessary steps for computation which is shown in Lawler [8]. The method described is applicable to the interval of Boolean functions f, g such that f implies g where for simplification of sum form the variables occurring in g also occur in f and for product form the variables in f also occur in g.

scholarly journals HEURISTIC METHODS FOR THE DESIGN OF CRYPTOGRAPHIC BOOLEAN FUNCTIONS

2019 ◽  
pp. 265-277
Author(s):  
Illarion Moskovchenko ◽  
Alexandr Kuznetsov ◽  
Sergii Kavun ◽  
Berik Akhmetov ◽  
Ivan Bilozertsev ◽  
...  
Keyword(s):  
Boolean Function ◽  
Computational Efficiency ◽  
Boolean Functions ◽  
Empirical Distribution ◽  
Heuristic Method ◽  
Distribution Functions ◽  
Hill Climbing ◽  
Heuristic Methods ◽  
Gradient Search ◽  
Empirical Distribution Functions

In this article, heuristic methods of hill climbing for cryptographic Boolean functions satisfying the required properties of balance, nonlinearity, autocorrelation, and other stability indicators are considered. A technique for estimating the computational efficiency of gradient search methods, based on the construction of selective (empirical) distribution functions characterizing the probability of the formation of Boolean functions with indices of stability not lower than required, is proposed. As an indicator of computational efficiency, an average number of attempts is proposed to be performed using a heuristic method to form a cryptographic Boolean function with the required properties. Comparative assessments of the effectiveness of the heuristic methods are considered. The results of investigations of the cryptographic properties of the formed Boolean functions in comparison with the best known assessments are given. On the basis of the conducted research, it can be concluded that the functions constructed in accordance with the developed method have high persistence indexes and exceed the known functions by these indicators.

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.

CLASS OF BOOLEAN FUNCTIONS CONSTRUCTED USING SIGNIFICANT BITS OF LINEAR RECURRENCES OVER THE RING ℤ2n

2019 ◽  
Vol 6 (2) ◽  
pp. 90-94
Author(s):  
Hernandez Piloto Daniel Humberto
Keyword(s):  
Linear Function ◽  
Boolean Functions ◽  
Hamming Distance ◽  
Linear Recurrences ◽  
Maximum Period ◽  
Non Linear ◽  
The Given ◽  
Class Of Functions

In this work a class of functions is studied, which are built with the help of significant bits sequences on the ring ℤ2n. This class is built with use of a function ψ: ℤ2n → ℤ2. In public literature there are works in which ψ is a linear function. Here we will use a non-linear ψ function for this set. It is known that the period of a polynomial F in the ring ℤ2n is equal to T(mod 2)2α, where α∈ , n01- . The polynomials for which it is true that T(F) = T(F mod 2), in other words α = 0, are called marked polynomials. For our class we are going to use a polynomial with a maximum period as the characteristic polyomial. In the present work we show the bounds of the given class: non-linearity, the weight of the functions, the Hamming distance between functions. The Hamming distance between these functions and functions of other known classes is also given.

The acquisition of Hungarian recursive PPs

2018 ◽  
Vol 4 (1) ◽  
pp. 105-123
Author(s):  
Ágnes Langó-Tóth
Keyword(s):  
Language Acquisition ◽  
Functional Element ◽  
The Other ◽  
Functional Head ◽  
Experiment Subject ◽  
The Given

Abstract In this study an experiment is presented on how Hungarian children interpret two word orders of recursive PPs (subject-PP-verb and PP-subject-verb order). According to the research of Roeper (2011) and Hollebrandse and Roeper (2014), children tend to give conjunctive interpretation to multiple embedded sentences at the beginning of language acquisition. This interpretation later turns into an adult-like, recursive interpretation. Our aim is to discover (i) whether Hungarian children start with conjunction as well, and whether (ii) the apparently more salient functional head lévő appearing in Hungarian recursive PPs can help them to acquire the correct, recursive interpretation early. We also want to find out whether (iii) the word orders in recursive PPs have an influence on the acquisition of children. In this paper two experiments are presented conducted with 6 and 8-year-olds and adults, in which the participants were asked to choose between two pictures. One of the pictures depicted recursive and the other one depicted conjunctive interpretation of the given sentence. In the first experiment subject-PP-verb order was tested, but in the second one sentences were tested with PP-subject-verb order. We will claim that lévő, which is (arguably) a more salient Hungarian functional element than -i, does not help children to acquire the embedded reading of recursive sentences, because both of them are overt functional heads. However, the two types of word orders affect the acquisition of recursive PPs. PP-subject-verb order is easier to compute because the order of the elements in the sentences and the order of the elements in the pictures matches.

Growth Characteristics of Activated Sludges Acclimated to para-Nitrophenol in Batch and Continuous Modes

1994 ◽  
Vol 29 (7) ◽  
pp. 327-333
Author(s):  
Y. Matsui ◽  
F. Yamaguchi ◽  
Y. Suwa ◽  
Y. Urushigawa
Keyword(s):  
Growth Characteristics ◽  
The Other ◽  
Operational Mode ◽  
Continuous Mode ◽  
Relative Resistance ◽  
Batch Mode ◽  
The Given ◽  
Operational Modes ◽  
Very High ◽  
Sensitive Group

Activated sludges were acclimated to p-nitrophenol (PNP) in two operational modes, a batch and a continuous. The operational mode of the PNP acclimation of activated sludges strongly affected the physiological characteristics of predominant microorganisms responsible for PNP degradation. Predominant PNP degraders in the sludge in batch mode (Sludge B) had lower PNP affinity and were relatively insensitive to PNP concentration. Those of the sludge in continuous mode (Sludge C), on the other hand, had very high PNP affinity and were sensitive to PNP. MPN enumeration of PNP degraders in sludge B and C using media with different PNP concentrations (0.05, 0.2,0.5 and 2.0 mM) supported the above results. Medium with 0.2 mM of PNP did not recover PNP degraders in sludge C well, while it recovered PNP degraders in sludge B as well as the medium with 0.05 mM did. When switching from one operational mode to the other, the predominant population in sludge B shifted to the sensitive group, but that of sludge C did not shift at the given loading of PNP, showing relative resistance to inhibitive concentration.

Giving and Promising

2018 ◽  
Author(s):  
Jean-Yves Lacoste ◽  
Oliver O’Donovan
Keyword(s):  
The Other ◽  
Face To Face ◽  
Present Experience ◽  
The World ◽  
The Given

Giving and promise must be thought together. Being-in-the world entails being-with the other, who is both “given” and bearer of a gift promised. But any disclosure may be understood as a gift; it is not anthropomorphic to speak of “self-giving” with a wider reference than person-to-person disclosure. Which implies that no act of giving can exhaust itself in its gift. Present experience never brings closure to self-revealing. Yet giving is crystallized into “the given,” the closure of gift. “The given” is what it is, needing no gift-event to reveal it. But the given, too, is precarious, and can be destabilized when giving brings us face to face with something unfamiliar. Nothing appears without a promise of further appearances, and God himself can never be “given.”

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