布尔函数和伪布尔函数多项式表示的快速实现算法

布尔函数和伪布尔函数在不同的领域有着广泛的应用,利用多项式表示有利于刻划它们的一些特征属性。论文首先在已知输入都能得到输出的条件下给出了布尔函数多项式表示的快速实现算法,该算法仅用到模2加运算,运算次数少,具有简洁、易于编程实现、准确而快速的特点,而且该算法很易推广为伪布尔函数多项式表示的快速实现算法,只需把模2加运算换成实数加运算即可。接着通过比较说明了伪布尔函数多项式表示的快速实现算法,同时指出任何伪布尔函数都能通过多项式形式表示出来。最后通过实例进一步验证了算法的正确性。     

On Complexity of Computation of Partial Derivatives of Boolean Functions Realized by Zhegalkin Polynomials

The combinational complexity of a system of partial derivatives in the basis of linear functions is established for a Boolean function of n variables that is realized by a Zhegalkin polynomial. An algorithm whose complexity equals 3ⁿ – 2ⁿ modulo 2 additions is proposed for computation of all partial derivatives of such a function from the coefficients of its Zhegalkin polynomial.     

Bounds for parallel addition time of two numbers

The depth of a Boolean circuit that computes the sum of two n-digit binary numbers is shown to increase faster than log n + O(1). A similar result is proved for a circuit that evaluates a polynomial of n-th degree.Translated from Kibernetika i Sistemnyi Analiz, No. 4, pp. 11–24, July–August, 1991.     

On computation of Boolean involutive bases

Gröbner bases in Boolean rings can be calculated by an involutive algorithm based on the Janet or Pommaret division. The Pommaret division allows calculations immediately in the Boolean ring, whereas the Janet division implies use of a polynomial ring over field \(\mathbb{F}_2 \). In this paper, both divisions are considered, and distributive and recursive representations of Boolean polynomials are compared from the point of view of calculation effectiveness. Results of computer experiments with both representations for an algorithm based on the Pommaret division and for lexicographical monomial order are presented.     

布尔函数的复杂系数及其应用

本文利用线性复杂度相关理论,给出了布尔函数复杂系数的定义：得出任何布尔函数的线性复杂度均等于这个函数的复杂系数;给出了一种快速求解布尔函数多项式表示的算法;研究了Bent函数的线性复杂度特点,利用布尔函数的复杂系数,得出布尔函数为Bent函数的一个必要条件。     

Recognition of interval Boolean functions

Interval functions constitute a special class of Boolean functions for which it is very easy and fast to determine their functional value on a specified input vector. The value of an n-variable interval function specified by interval [a,b] (where a and b are n-bit binary numbers) is true if and only if the input vector viewed as an n-bit number belongs to the interval [a,b]. In this paper we study the problem of deciding whether a given disjunctive normal form represents an interval function and if so then we also want to output the corresponding interval. For general Boolean functions this problem is co-NP-hard. In our article we present a polynomial time algorithm which works for monotone functions. We shall also show that given a Boolean function f belonging to some class which is closed under partial assignment and for which we are able to solve the satisfiability problem in polynomial time, we can also decide whether f is an interval function in polynomial time. We show how to recognize a “renamable” variant of interval functions, i.e., their variable complementation closure. Another studied problem is the problem of finding an interval extension of partially defined Boolean functions. We also study some other properties of interval functions.      

Ordered binary decision diagrams as knowledge-bases

We consider the use of ordered binary decision diagrams (OBDDs) as a means of realizing knowledge-bases, and show that, from the view point of space requirement, the OBDD-based representation is more efficient and suitable in some cases, compared with the traditional CNF-based and/or model-based representations. We then present polynomial time algorithms for the two problems of testing whether a given OBDD represents a unate Boolean function, and of testing whether it represents a Horn function.     

Unrestricted vs restricted cut in a tableau method for Boolean circuits

This paper studies the relative efficiency of variations of a tableau method for Boolean circuit satisfiability checking. The considered method is a nonclausal generalisation of the Davis–Putnam–Logemann–Loveland (DPLL) procedure to Boolean circuits. The variations are obtained by restricting the use of the cut (splitting) rule in several natural ways. It is shown that the more restricted variations cannot polynomially simulate the less restricted ones. For each pair of methods T, T′, an infinite family of circuits is devised for which T has polynomial size proofs while in T′ the minimal proofs are of exponential size w.r.t. n, implying exponential separation of T and T′ w.r.t. n. The results also apply to DPLL for formulas in conjunctive normal form obtained from Boolean circuits by using Tseitin’s translation. Thus DPLL with the considered cut restrictions, such as allowing splitting only on the variables corresponding to the input gates, cannot polynomially simulate DPLL with unrestricted splitting.AMS subject classification 03B70, 03F20, 68T15, 68T20The financial support from Academy of Finland (project #53695) is gratefully acknowledged.Tommi Junttila: This work was partially done while visiting ITC-IRST (Trento, Italy), and has been sponsored by the CALCULEMUS! IHP-RTN EC project, contract code HPRN-CT-2000-00102.     

A Unifying Theoretical Background for Some Bdd-based Data Structures

In the paper, we propose a general concept (denoted by TBDD) for Boolean functions manipulation that is based on cube transformations. The basic idea is to manipulate a Boolean function by converting it by means of a cube transformation into a function that can be efficiently represented and manipulated in terms of ordered binary decision diagrams (OBDDs). We show that the new concept unifies several BDD–based data structures considerably, and simplifies their manipulation to work with the simple and well–understood data struture of OBDDs. This is especially important for practical applications.Further, to give an example of how TBDDs open new ways in the search for efficient data structures for Boolean functions, we discuss the data structure of typed kFBDDs.     

Algorithms for Synthesis of Polynomials Implementing Weakly Specified Boolean Functions and Systems

Weakly specified Boolean functions and systems are considered. A series of practically effective algorithms are proposed for their implementation by AND/EXOR circuits, which rely on the optimization of polynomial representations and the solutions of appropriate matrix logic equations. The obtained results are extended to multivalued logic.     

Error-Free and Best-Fit Extensions of Partially Defined Boolean Functions

In this paper, we address a fundamental problem related to the induction of Boolean logic: Given a set of data, represented as a set of binary “truen-vectors” (or “positive examples”) and a set of “falsen-vectors” (or “negative examples”), we establish a Boolean function (or an extension)f, so thatfis true (resp., false) in every given true (resp., false) vector. We shall further require that such an extension belongs to a certain specified class of functions, e.g., class of positive functions, class of Horn functions, and so on. The class of functions represents our a priori knowledge or hypothesis about the extensionf, which may be obtained from experience or from the analysis of mechanisms that may or may not cause the phenomena under consideration. The real-world data may contain errors, e.g., measurement and classification errors might come in when obtaining data, or there may be some other influential factors not represented as variables in the vectors. In such situations, we have to give up the goal of establishing an extension that is perfectly consistent with the given data, and we are satisfied with an extensionfhaving the minimum number of misclassifications. Both problems, i.e., the problem of finding an extension within a specified class of Boolean functions and the problem of finding a minimum error extension in that class, will be extensively studied in this paper. For certain classes we shall provide polynomial algorithms, and for other cases we prove their NP-hardness.     

Learning intersections and thresholds of halfspaces

We give the first polynomial time algorithm to learn any function of a constant number of halfspaces under the uniform distribution on the Boolean hypercube to within any constant error parameter. We also give the first quasipolynomial time algorithm for learning any Boolean function of a polylog number of polynomial-weight halfspaces under any distribution on the Boolean hypercube. As special cases of these results we obtain algorithms for learning intersections and thresholds of halfspaces. Our uniform distribution learning algorithms involve a novel non-geometric approach to learning halfspaces; we use Fourier techniques together with a careful analysis of the noise sensitivity of functions of halfspaces. Our algorithms for learning under any distribution use techniques from real approximation theory to construct low-degree polynomial threshold functions. Finally, we also observe that any function of a constant number of polynomial-weight halfspaces can be learned in polynomial time in the model of exact learning from membership and equivalence queries.     

Some lower bounds on the algebraic immunity of functions given by their trace forms

The algebraic immunity of a Boolean function is a parameter that characterizes the possibility to bound this function from above or below by a nonconstant Boolean function of a low algebraic degree. We obtain lower bounds on the algebraic immunity for a class of functions expressed through the inversion operation in the field GF(2 ⁿ ), as well as for larger classes of functions defined by their trace forms. In particular, for n ≥ 5, the algebraic immunity of the function Tr _n (x ^?1) has a lower bound ?2√n + 4? ? 4, which is close enough to the previously obtained upper bound ?√n? + ?n/?√n?? ? 2. We obtain a polynomial algorithm which, give a trace form of a Boolean function f, computes generating sets of functions of degree ≤ d for the following pair of spaces. Each function of the first (linear) space bounds f from below, and each function of the second (affine) space bounds f from above. Moreover, at the output of the algorithm, each function of a generating set is represented both as its trace form and as a polynomial of Boolean variables.     

On the degree of boolean functions as real polynomials

Every Boolean function may be represented as a real polynomial. In this paper, we characterize the degree of this polynomial in terms of certain combinatorial properties of the Boolean function. Our first result is a tight lower bound of Ω(logn) on the degree needed to represent any Boolean function that depends onn variables. Our second result states that for every Boolean functionf, the following measures are all polynomially related:

o The decision tree complexity off.

o The degree of the polynomial representingf.

o The smallest degree of a polynomialapproximating f in theL _max norm.

     

Learning Read-Constant Polynomials of Constant Degree Modulo Composites

Boolean functions that have constant degree polynomial representation over a fixed finite ring form a natural and strict subclass of the complexity class ACC⁰. They are also precisely the functions computable efficiently by programs over fixed and finite nilpotent groups. This class is not known to be learnable in any reasonable learning model. In this paper, we provide a deterministic polynomial time algorithm for learning Boolean functions represented by polynomials of constant degree over arbitrary finite rings from membership queries, with the additional constraint that each variable in the target polynomial appears in a constant number of monomials. Our algorithm extends to superconstant but low degree polynomials and still runs in quasipolynomial time.     

最优代数免疫布尔函数的完全构造

任意的布尔函数可以唯一地表示成有限域上的单变元多项式函数,利用布尔函数的单变元多项式表示和代数编码理论,讨论了布尔函数的代数免疫达到最优的判别条件,得到了布尔函数的变元个数为奇数时,布尔函数具有最优代数免疫(MAI)的等价判别条件。利用该等价判别条件,给出3元布尔函数满足MAI的等价判别条件,进而构造出所有3元的MAI布尔函数。     

Boolean functions with long prime implicants

In this short note we introduce a hierarchy of classes of Boolean functions, where each class is defined by the minimum allowed length of prime implicants of the functions in the class. We show that for a given DNF and a given class in the hierarchy, it is possible to test in polynomial time whether the DNF represents a function from the given class. For the first class in the hierarchy we moreover present a polynomial time algorithm which for a given input DNF outputs a shortest logically equivalent DNF, i.e. a shortest DNF representation of the underlying function. This class is therefore a new member of a relatively small family of classes for which the Boolean minimization problem can be solved in polynomial time. For the second class and higher classes in the hierarchy we show that the Boolean minimization problem can be approximated within a constant factor.     

零化多项式与仿射空间关系研究

为把流密码的代数攻击问题转化为求解布尔函数的低次数零化多项式问题,讨论布尔函数的性质,介绍{0,1}上矩阵的特殊结构,研究两者间的关系,在此基础上探讨n元布尔函数f的零化多项式次数与f的支撑点集之间的关系,实验结果表明,寻找布尔函数零化多项式等价于在布尔函数的零点集合中寻找最大的仿射空间。     

A Calculus of Terms for Coalgebras of Polynomial Functors

A syntax and semantics of types, terms and formulas for coalgebras of polynomial functors is developed, extending earlier work [4] on monomial coalgebras to include functors constructed using coproducts. A modified ultrapower construction for polynomial coalgebras is introduced, adapting the conventional ultrapower to retain only those states that evaluate observable terms in a standard way.A special role is played by terms that take observable values and are “rigid”: their free variables do not occur in any state-valued subterm. The following “co-Birkhoff” theorem is proved: a class of polynomial coalgebras is definable by Boolean combinations of equations between rigid terms iff the class is closed under disjoint unions, images of bisimulations, and observable ultrapowers.