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Register a new userLanguage of Boolean functions its Grammar and Machine
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Added by
Sudhakar Sahoo
Publication date
2008
fields
Informatics Engineering
and research's language is
English
Authors
Birendra Kumar Nayak
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In this paper an algorithm is designed which generates in-equivalent Boolean functions of any number of variables from the four Boolean functions of single variable. The grammar for such set of Boolean function is provided. The Turing Machine that accepts such set is constructed.
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Classification of Non-linear Boolean functions is a long-standing problem in the area of theoretical computer science. In this paper, effort has been made to achieve a systematic classification of all n-variable Boolean functions, where only one affine Boolean function belongs to each class. Two different methods are proposed to achieve this classification. The first method is a recursive procedure that uses the Cartesian product of sets starting from the set of 1-variable Boolean function and in the second method classification is achieved through a set of invariant bit positions with respect to an affine function belonging to that class. The invariant bit positions also provide information concerning the size and symmetry properties of the classes/sub-classes, such that the members of classes/sub-classes satisfy certain similar properties.
A pairing function J associates a unique natural number z to any two natural numbers x,y such that for two unpairing functions K and L, the equalities K(J(x,y))=x, L(J(x,y))=y and J(K(z),L(z))=z hold. Using pairing functions on natural number representations of truth tables, we derive an encoding for Binary Decision Diagrams with the unique property that its boolean evaluation faithfully mimics its structural conversion to a a natural number through recursive application of a matching pairing function. We then use this result to derive {em ranking} and {em unranking} functions for BDDs and reduced BDDs. The paper is organized as a self-contained literate Prolog program, available at http://logic.csci.unt.edu/tarau/research/2008/pBDD.zip Keywords: logic programming and computational mathematics, pairing/unpairing functions, encodings of boolean functions, binary decision diagrams, natural number representations of truth tables
We study the volatility of the output of a Boolean function when the input bits undergo a natural dynamics. For $n = 1,2,ldots$, let $f_n:{0,1}^{m_n} ra {0,1}$ be a Boolean function and $X^{(n)}(t)=(X_1(t),ldots,X_{m_n}(t))_{t in [0,infty)}$ be a vector of i.i.d. stationary continuous time Markov chains on ${0,1}$ that jump from $0$ to $1$ with rate $p_n in [0,1]$ and from $1$ to $0$ with rate $q_n=1-p_n$. Our object of study will be $C_n$ which is the number of state changes of $f_n(X^{(n)}(t))$ as a function of $t$ during $[0,1]$. We say that the family ${f_n}_{nge 1}$ is volatile if $C_n ra iy$ in distribution as $ntoinfty$ and say that ${f_n}_{nge 1}$ is tame if ${C_n}_{nge 1}$ is tight. We study these concepts in and of themselves as well as investigate their relationship with the recent notions of noise sensitivity and noise stability. In addition, we study the question of lameness which means that $Pro(C_n =0)ra 1$ as $ntoinfty$. Finally, we investigate these properties for a number of standard Boolean functions such as the majority function, iterated 3-majority, the AND/OR function on the binary tree and percolation on certain trees at various levels of the parameter $p_n$.
The subject of this textbook is the analysis of Boolean functions. Roughly speaking, this refers to studying Boolean functions $f : {0,1}^n to {0,1}$ via their Fourier expansion and other analytic means. Boolean functions are perhaps the most basic object of study in theoretical computer science, and Fourier analysis has become an indispensable tool in the field. The topic has also played a key role in several other areas of mathematics, from combinatorics, random graph theory, and statistical physics, to Gaussian geometry, metric/Banach spaces, and social choice theory. The intent of this book is both to develop the foundations of the field and to give a wide (though far from exhaustive) overview of its applications. Each chapter ends with a highlight showing the power of analysis of Boolean functions in different subject areas: property testing, social choice, cryptography, circuit complexity, learning theory, pseudorandomness, hardness of approximation, concrete complexity, and random graph theory. The book can be used as a reference for working researchers or as the basis of a one-semester graduate-level course. The author has twice taught such a course at Carnegie Mellon University, attended mainly by graduate students in computer science and mathematics but also by advanced undergraduates, postdocs, and researchers in adjacent fields. In both years most of Chapters 1-5 and 7 were covered, along with parts of Chapters 6, 8, 9, and 11, and some additional material on additive combinatorics. Nearly 500 exercises are provided at the ends of the books chapters.
We consider the extensions of modal transition systems (MTS), namely Boolean MTS and parametric MTS and we investigate the refinement problems over both classes. Firstly, we reduce the problem of modal refinement over both classes to a problem solvable by a QBF solver and provide experimental results showing our technique scales well. Secondly, we extend the algorithm for thorough refinement of MTS providing better complexity then via reductions to previously studied problems. Finally, we investigate the relationship between modal and thorough refinement on the two classes and show how the thorough refinement can be approximated by the modal refinement.
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