We consider the classification problem to construct a classifier c:{0,1}n{0,1} from a given set of examples (training set), which (approximately) realizes the hidden oracle y:{0,1}n{0,1} describing the phenomenon under consideration. For this problem, a number of approaches are already known in computational learning theory; e.g., decision trees, support vector machines (SVM), and iteratively composed features (ICF). The last one, ICF, was proposed in our previous work (Haraguchi et al., (2004)). A feature, composed of a nonempty subset S of other features (including the original data attributes), is a Boolean function fS:{0,1}S{0,1} and is constructed according to the proposed rule. The ICF algorithm iterates generation and selection processes of features, and finally adopts one of the generated features as the classifier, where the generation process may be considered as embodying the idea of boosting, since new features are generated from the available features. In this paper, we generalize a feature to an extended Boolean function fS:{0,1,*}S{0,1,*} to allow partial knowledge, where * denotes the state of uncertainty. We then propose the algorithm ICF* to generate such generalized features. The selection process of ICF* is also different from that of ICF, in that features are selected so as to cover the entire training set. Our computational experiments indicate that ICF* is better than ICF in terms of both classification performance and computation time. Also, it is competitive with other representative learning algorithms such as decision trees and SVM.

Horn functions are Boolean functions where each of the prime implicants contains at most one negative literal. A class of Boolean functions is considered in this letter where a single term containing two negative literals is added by logical-or operation to a Horn function. We show that the function does not have any prime implicant containing three negative literals. We also show that if two terms containing two negative literals are added to a Horn function, then it may have many prime implicants all of which contain three negative literals. We show that it is P-complete to determine whether a given Boolean function in disjunctive normal form of the considered class is a tautology.

In this study, we construct balanced Boolean functions with a high nonlinearity and an optimum algebraic degree for both odd and even dimensions. Our approach is based on modifying functions from the Maiorana-McFarland's superclass, which has been introduced by Carlet. A drawback of Maiorana-McFarland's function is that their restrictions obtained by fixing some variables in their input are affine. Affine functions are cryptographically weak functions, so there is a risk that this property will be exploited in attacks. Due to the contribution of Carlet, our constructions do not have the potential weakness that is shared by the Maiorana-McFarland construction or its modifications.

In this paper, a new class of bent functions is constructed by combining class M and class C bent functions. Using the construction method of the class D bent functions defined on the binary vector space, new p-ary generalized bent functions are also introduced for odd prime p.

In cryptography, we want a Boolean function which satisfies PC(l) of order k for many (l,k). Let PCn(l,k) be a set of Boolean functions with n input bits satisfying PC(l) of order k. From a view point of construction, it is desirable that there exists (l0,k0) such that PCn(l0, k0) PCn(li,ki) for many i 1. In this paper, we show a negative result for this problem. We prove that PCn(l1,k1) PCn(l2,k2) for a large class of l1, k1, l2 and k2.

In this paper, we consider the complexity of recognizing ordered tree-shellable Boolean functions when Boolean functions are given as OBDDs. An ordered tree-shellable function is a positive Boolean function such that the number of prime implicants equals the number of paths from the root node to a 1-node in its ordered binary decision tree representation. We show that given an OBDD, it is possible to check within polynomial time if the function is ordered tree-shellable with respect to the variable ordering of the OBDD.

In several design methods for Pass-transistor Logic (PTL) circuits, Boolean functions are expressed as OBDDs in decomposed form and then the component OBDDs are directly mapped to PTL cells. The total size of OBDDs (number of nodes) corresponds to the circuit size. In this paper, we investigate a method for PTL synthesis based on exact minimization of Free BDDs (FBDDs). FBDDs are well-studied extension of OBDDs with free variable ordering on each path. We present statistics showing that more than 56% of 616126 NPN-equivalence classes of 5-variable Boolean functions have minimum FBDDs with less size than their OBDDs. This result can be used for PTL synthesis as libraries. We also applied the exact minimization algorithm of FBDDs to the minimization of subcircuits in the synthesis for MCNC benchmarks and found up to 5% size reduction.

Some novel learning strategies based on set covering in Hamming geometrical space are presented and proved, which are related to the three-layer Boolean neural network (BNN) for implementing an arbitrary Boolean function with low-complexity. Each hidden neuron memorizes a set of learning patterns, then the output layer combines these hidden neurons for explicit output as a Boolean function. The network structure is simple, reliable and can be easily implemented by hardware.

In this paper we study n-input m-output Boolean functions (abbr. (n,m)-functions) with high nonlinearity. First, we present a basic construction method for a balanced (n,m)-function based on a primitive element in GF(2m). With an iterative procedure, we improve some lower bounds of the maximum nonlinearity of balanced (n,m)-functions. The resulting bounds are larger than the maximum nonlinearity achieved by any previous construction method for (n,m)-functions. Finally, our basic method is developed to construct an (n,m)-bent function and discuss its maximum algebraic degree.

This paper proposes a simple learning algorithm that can realize any boolean function using the three-layer binary neural networks. The algorithm has flexible learning functions. 1) moving "core" for the inputs separations,2) "don't care" settings of the separated inputs. The "don't care" inputs do not affect the successive separations. Performing numerical simulations on some typical examples, we have verified that our algorithm can give less number of hidden layer neurons than those by conventional ones.

An Ordered Binary Decision Diagram (OBDD) is a directed acyclic graph representing a Boolean function. The size of OBDDs largely depends on the variable ordering. In this paper, we show the size of the OBDD representing the i-th bit of the output of n-bit/n-bit integer division is Ω ( 2(n-i)/8 ) for any variable ordering. We also show that -OBDDs, -OBDDs and -OBDDs representing integer division has the same lower bounds on the size. We develop new methods for proving lower bounds on the size of -OBDDs, -OBDDs and -OBDDs.

We give an explicit formula for the number of n-variable clique function in terms of the parameters based upon the numbers of intersecting antichains of the lower half of the n-cube. We present the numbers of clique functions with up to seven variables through computer evaluation of the parameters.

Ordered Binary Decision Diagrams (OBDDs) are graph-based representations of Boolean functions which are widely used because of their good properties. In this paper, we introduce nondeterministic OBDDs (NOBDDs) and their restricted forms, and evaluate their expressive power. In some applications of OBDDs, canonicity, which is one of the good properties of OBDDs, is not necessary. In such cases, we can reduce the required amount of storage by using OBDDs in some non-canonical form. A class of NOBDDs can be used as a non-canonical form of OBDDs. In this paper, we focus on two particular methods which can be regarded as using restricted forms of NOBDDs. Our aim is to show how the size of OBDDs can be reduced in such forms from theoretical point of view. Firstly, we consider a method to solve satisfiability problem of combinational circuits using the structure of circuits as a key to reduce the NOBDD size. We show that the NOBDD size is related to the cutwidth of circuits. Secondly, we analyze methods that use OBDDs to represent Boolean functions as sets of product terms. We show that the class of functions treated feasibly in this representation strictly contains that in OBDDs and contained by that in NOBDDs.

An ordered binary decision diagram (OBDD) is a directed acyclic graph for representing a Boolean function. OBDDs are widely used in various areas which require Boolean function manipulation, since they can represent efficiently many practical Boolean functions and have other desirable properties. However, there is very little theoretical research on the complexity of constructing an OBDD. In this paper, we prove that the optimal variable ordering problem of a shared BDD is NP-complete, and briefly discuss the approximation hardness of this problem and related OBDD problems.

This paper presents implicit representation of binary decision diagrams (implicit BDDs) as a new effecient data structure for Boolean functions. A well-known method of representing graphs by binary decision diagrams (BDDs) is applied to BDDs themselves. Namely, it is a BDD representation of BDDs. Regularity in the structure of BDDs representing certain Boolean functions contributes to significant reduction in size of the resulting implicit BDD repersentation. Since the implicit BDDs also provide canonical forms for Boolean functions, the equivalence of the two implicit BDD forms is decided in time proportional to the representation size. We also show an algorithm to maniqulate Boolean functions on this implicit data structure.

The present paper is concerned with an algorithm for the minimization of multiple-valued input, binary-valued output functions. The algorithm is an extension to muitiple-valued logic of an algorithm for the minimization of ordinary single-output Boolean functions. It is based on a local covering approach. Basically, it uses a "divide and conquer" technique, consisting of two steps called expansion and selection. The present algorithm preserves two important features of the original one. First, a lower bound on the number of prime implicants in the minimum cover of the given function is furnished as a by-product of the minimization. Second, all the essential primes of the function are identified and selected during the expansion process. That usually improves efficiency when handling functions with many essential primes. Results of a comparison of the proposed algorithm with the program ESPRESSO-IIC developed at Berkeley are presented.

In this paper, a new optimum input encoding algorithm with m-out-of-2m code which is called Half-Hot Code is presented. By applying Half-Hot Code to the input encoding in PLA-based digital system, the logic functions of the system turn out to be unate functions, thus, the number of bit-lines of PLA may be reduced. The proposed method further reduces the number of product-lines of PLA optimally. In this code assignment procedure, computed Boolean subspaces satisfying suggeset two conditions are assigned to each partitioned subset of digital input variables which are obtained by disjoint minimization or other techniques. As an experiment to evaluate the method, the state assignment for finite state machines of two-lavel implementation is considered. Specifically, the proposed Half-Hot Code assignment is compared with arbitrary Half-Hot Code assignment. The results show that the optimum encoding is superior to an arbitrary assignment up to about 24% in the number of product-lines of PLA.

Complexity of Boolean functions satisfying the propagation criterion (PC), an extended notion of the perfect nonlinearity, is discussed on several computation models. The following topics are investigated: (i) relationships between the unateness and the degree of the PC, (ii) the inversion complexity of perfectly nonlinear Boolean functions, (iii) the formula size of Boolean functions that satisfy the PC of degree 1, (iv) the area-time-square complexity of VLSI circuits computing perfectly nonlinear Boolean functions, (v) the OBDD size perfectly nonlinear Boolean functions.

For symmetric cryptosystems, their transformations should have nonlinear elements to be secure against various attacks. Several nonlinearity criteria have been defined and their properties have been made clear. This paper focuses on, among these criteria, the propagation criterion (PC) and the strict avalanche criterion (SAC), and makes a further investigation of them. It discusses the sets of Boolean functions satisflying the PC of higher degrees, the sets of those satisfying the SAC of higher orders and their relationships. We give a necessary and sufficient condition for an n-input Boolean function to satisfy the PC with respect to a set of all but one or two elements in {0,1}n{(0,...,0)}. From this condition, it follows that, for every even n 2, an n-input Boolean function satisfies the PC of degree n 1 if and only if it satisfies the PC of degree n. We also show a method that constructs, for any odd n 3, n-input Boolean functions that satisfy the PC with respect to a set of all but one elements in {0,1}n{(0,...,0)}. This method is a generalized version of a previous one. Concerned with the SAC of higher orders, it is shown that the previously proved upper bound of the nonlinear order of Boolean functions satisfying the criterion is tight. The relationships are discussed between the set of n-input Boolean functions satisfying the PC and the sets of those satisfying the SAC.

This paper discusses Boolean functions satisfying the propagation criterion (PC) and their balancedness. Firstly, we discuss Boolean functions with n variables that satisfy the PC with respect to all but three elements in {0,1}n-{(0,...,0)}. For even n4, a necessary and sufficient condition is presented for Boolean functions with n variables to satisfy the PC with respect to all but three elements in {0,1}n-{(0,...,0)}. From this condition, it is proved that all of these Boolean functions are constructed from all perfectly nonlinear Boolean functions with n-2 variables. For odd n3, it is shown that Boolean functions with n variables satisfying the PC with respect to all but three elements in {0,1}n-{(0,...,0)} satisfy the PC with respect to all but one elements in it. Secondly, Boolean functions satisfying the PC of degree n-2 and their balancedness are considered. For even n4, it is proved that an upper bound on the degree of the PC is n-3 for balanced Boolean functions with n variables. This bound is optimal for n=4,6. It is also proved that, for odd n3, balanced Boolean functions with n variables satisfying the PC of degree n-2 satisfy the PC with respect to all but one elements in {0,1}n-{(0,...,0)}.