Partitioning 1-variable Boolean functions for various classification of n-variable Boolean functions

This paper addresses all possible equivalence classes of 1-variable Boolean functions and from these classes using recursion and Cartesian product of sets, 15 different ways of classifications of n-variable Boolean functions are obtained. The properties with regard to the size and the number of classes for these 15 different ways are also elaborated.     

Probabilistic verification of Boolean functions

We present a novel method for verifying the equivalence of two Boolean functions. Each function is hashed to an integer code by assigning random integer values to the input variables and evaluating an integer-valued transformation of the original function. The hash codes for two equivalent functions are always equal. Thus the equivalence of two functions can be verified with a very low probability of error, which arises from unlikely collisions between inequivalent functions. An upper bound, , on the probability of error is known a priori. The bound can be decreased exponentially by making multiple runs. Results indicate significant time and space advantages for this method over techniques that represent each function as a single OBDD. Some functions known to require space (and time) exponential in the number of input variables for these techniques require only polynomial resources using our method. Experimental results indicate that probabilistic verification can provide an attractive alternative for verifying functions too large to be handled using these OBDD-based techniques.     

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.      

基于plateaued函数的平衡布尔函数构造

布尔函数在对称密码的设计和分析中起着重要的作用。通过对谱不相交函数集中子函数平衡性的问题的研究给出了包含4个plateaued函数的函数集中有3个为平衡函数的充分条件。在此基础上,基于3个平衡的谱不相交plateaued函数,一类特殊的布尔置换以及一个高非线性度平衡函数,提出了一个构造高非线性度平衡布尔函数的方法。通过分析可知,利用该方法可以构造代数次数达到最优、非线性度不小于2^2k-1-2^k-1-2^k/2-2^{⌈(k-1)/2⌉}的2k元平衡函数。     

Complexity hierarchies for Boolean functions

Summary Circuit size and depth are two important complexity measures for a Boolean function. Uniform hierarchies are shown to exist with respect to each of these measures.     

Detection of symmetries of Boolean functions

A method is proposed to determine various symmetries of a completely or incompletly specified Boolean function. Equivalence, nonequivalence and single-variable symmetries are detected from their respective symmetry tables. A symmetry table of n variables can be used to detect symmetries of functions of n, or less, variables. A procedure is given for obtaining total or partial symmetries, with all primed (or all unprimed), mixed or multiform variables of symmetry, from equivalence and nonequivalence symmetries of the function.     

Decision tree approximations of Boolean functions

Decision trees are popular representations of Boolean functions. We show that, given an alternative representation of a Boolean function f, say as a read-once branching program, one can find a decision tree T which approximates f to any desired amount of accuracy. Moreover, the size of the decision tree is at most that of the smallest decision tree which can represent f and this construction can be obtained in quasi-polynomial time. We also extend this result to the case where one has access only to a source of random evaluations of the Boolean function f instead of a complete representation. In this case, we show that a similar approximation can be obtained with any specified amount of confidence (as opposed to the absolute certainty of the former case.) This latter result implies proper PAC-learnability of decision trees under the uniform distribution without using membership queries.     

Analysis of affinely equivalent Boolean functions

By some basic transforms and invariant theory, we give two results: 1) an algorithm, which can be used to judge if two Boolean functions are affinely equivalent and to obtain the equivalence relationship if they are equivalent. This is useful in studying Boolean functions and in engineering. For example, we classify all 8-variable homogeneous bent functions of degree 3 into two classes; 2) Reed-Muller codes R(4,6)/R(1,6), R(3,7)/R(1,7) are classified efficiently.     

On cross-correlation properties of Boolean functions

The relationship among cross-correlation of arbitrary four Boolean functions is presented. Several known cross-correlation properties of Boolean functions are generalized. Based on them, a lower bound for the maximal cross-correlation (in absolute value) of two Boolean functions (if one of the functions is bent) is obtained.     

On the independence of Boolean functions

Boolean functions are widely used because they can be used to precisely describe logical circuits. Properties of Boolean functions with respect to their applications to cryptography have been studied, but relationship between Boolean functions are rarely studied. This paper studies the independence of Boolean functions, gives necessary and sufficient conditions for judging whether two given Boolean functions are independent. Some enumeration formulae are given.     

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.