A New Approach to the Decomposition of Boolean Functions by the Method of q-Partitions. I. Separating Decomposition of Full and Partial Functions

A new approach to the decomposition of Boolean functions of n variables is considered; the functions being decomposed can be represented in various forms. The approach is based on the method of q-partitions of minterms and on the introduced concept of a decomposition clone. The theorem on simple separating decomposition of full and partial functions is formulated. The approach proposed is illustrated by examples.     

A New Approach to the Decomposition of Boolean Functions by the Method of q-Partitions. II. Repeated Decomposition

A new approach to the decomposition of Boolean functions that depend on n variables and are represented in various forms is considered. The approach is based on the method of #-partitioning of minterms and on the introduced concept of a decomposition clone. The theorem on simple disjunctive decomposition of full and partial functions is formulated. The approach proposed is illustrated by examples.     

Lower bounds for online makespan minimization on a small number of related machines

In online makespan minimization, the jobs characterized by their processing time arrive one-by-one and each has to be assigned to one of the m uniformly related machines. The goal is to minimize the length of the schedule. We prove new combinatorial lower bounds for m=4 and m=5, and computer-assisted lower bounds for m≤11.     

A method of finding simple disjunctive decomposition with one free variable

Simple disjunctive decomposition with one free variable can be obtained by using the prime implicants of the Boolean function. A function is decomposable with its redundant variable as a free variable. A necessary condition for decomposition with a non-redundant variable as free variable is that the number of minterms in an n variable function be 2^(n?1). A sufficient condition is that the variable remain present, in either true or complemented form, in all the prime implicants of the function.     

An efficient variable partitioning approach for functional decomposition of circuits

Functional decomposition is a process of splitting a complex circuit into smaller sub-circuits. There exist two major strategies in decomposition, namely, serial and parallel decomposition. In serial decomposition the problem the complex function represented as a truth table with support set variables and partitioned into free and bout set variables. The minterms corresponding to the bound set variables are represented as an equivalent function called the predecessor function. Equivalent minterms of the bound set variables are assigned an output code. The assigned output codes and the free set variable minterms are represented as the successor function. Serial decomposition is further categorized into disjoint and non-disjoint decomposition, when the free and bound set variables are disjoint and non-disjoint respectively. This paper deals with the problem of determining the set of best free and bound variables (variable partitioning problem) for disjoint serial decomposition. Variable partitioning is the first step in decomposition process. An efficient variable partition algorithm is one that determines the set of all free and bound set variables that satisfy the decomposition theorem in minimal time and by exploring the search space effectively. This will allow the decomposition algorithm to determine the best variable partition of a function that results in smaller decomposed functions and with maximum number of do not cares in these functions. Classical approaches to determine the best free and bound set use exhaustive search methods. The time and memory requirements for such approaches are exponential or super exponential.A novel heuristic search approach is proposed to determine the set of good variable partitions in minimal time by minimally exploring the search space. There are two heuristics employed in the proposed search approach, (1) r-admissibility based heuristic or pruned breadth first search (PBFS) approach and (2) Information relation based heuristic or improved pruned breadth first search (IPBFS) approach. The r-admissibility based heuristic is based on r-partition characteristics of the free and bound set variables. The information relation and measure based heuristic is based on information relationship of free and bound set variables that are expressed as r-partition heuristics. The proposed variable partition search approach has been successfully implemented and test with MCNC and Espresso benchmarks and the results indicate that the time complexity is comparable to r-admissible heuristic algorithm and the quality of solution is comparable to exact variable partitioning algorithm. A comparison of PBFS and IPBFS heuristics for certain benchmarks are also discussed in this paper.     

Minimization of Boolean functions of any number of variables using decimal labels

This paper deals with the problem of minimization of a Boolean function, particularly when the number of variables is very large. Using the decimal labels of the minterms, a table is worked out which helps to determine all the prime implicants and the essential prime implicants. A systematic method of sorting out the dominating and the dominated columns is then given, which helps to reduce the number of successive cover tables considerably.     

Connections among nonlinearity,avalanche and correlation immunity

Nonlinear Boolean functions play an important role in the design of block ciphers, stream ciphers and one-way hash functions. Over the years researchers have identified a number of indicators that forecast nonlinear properties of these functions. Studying the relationships among these indicators has been an area that has received extensive research. The focus of this paper is on the interplay of three notable nonlinear indicators, namely nonlinearity, avalanche and correlation immunity. We establish, for the first time, an explicit and simple lower bound on the nonlinearity N_f of a Boolean function f of n variables satisfying the avalanche criterion of degree p, namely, N_f⩾2ⁿ⁻¹−2^{n−1−(1/2)p}. We also identify all the functions whose nonlinearity attains the lower bound. As a further contribution of this paper, we prove that except for very few cases, the sum of the degree of avalanche and the order of correlation immunity of a Boolean function of n variables is at most n−2. The new results obtained in this work further highlight the significance of the fact that while avalanche property is in harmony with nonlinearity, both go against correlation immunity.     

On affine (non)equivalence of Boolean functions

In this paper we construct a multiset S(f) of a Boolean function f consisting of the weights of the second derivatives of the function f with respect to all distinct two-dimensional subspaces of the domain. We refer to S(f) as the second derivative spectrum of f. The frequency distribution of the weights of these second derivatives is referred to as the weight distribution of the second derivative spectrum. It is demonstrated in this paper that this weight distribution can be used to distinguish affine nonequivalent Boolean functions. Given a Boolean function f on n variables we present an efficient algorithm having O(n2²ⁿ ) time complexity to compute S(f). Using this weight distribution we show that all the 6-variable affine nonequivalent bents can be distinguished. We study the subclass of partial-spreads type bent functions known as PS _ap type bents. Six different weight distributions are obtained from the set of PS _ap bents on 8-variables. Using the second derivative spectrum we show that there exist 6 and 8 variable bent functions which are not affine equivalent to rotation symmetric bent functions. Lastly we prove that no non-quadratic Kasami bent function is affine equivalent to Maiorana–MacFarland type bent functions.     

Horn minimization by iterative decomposition

Given a Horn CNF representing a Boolean function f, the problem of Horn minimization consists in constructing a CNF representation off which has a minimum possible number of clauses. This problem is the formalization of the problem of knowledge compression for speeding up queries to propositional Horn expert systems, and it is known to be NP-hard. In this paper we present a linear time algorithm which takes a Horn CNF as an input, and through a series of decompositions reduces the minimization of the input CNF to the minimization problem on a“shorter” CNF. The correctness of this decomposition algorithm rests on several interesting properties of Horn functions which, as we prove here, turn out to be independent of the particular CNF representations. This revised version was published online in June 2006 with corrections to the Cover Date.     

A subclass of Horn CNFs optimally compressible in polynomial time

The problem of Horn Minimization (HM) can be stated as follows: given a Horn CNF representing a Boolean function f, find a shortest possible (optimally compressed) CNF representation of f, i.e., a CNF representation of f which consists of the minimum possible number of clauses. This problem is the formalization of the problem of knowledge compression for speeding up queries to propositional Horn expert systems, and it is known to be NP-hard. There are two subclasses of Horn functions for which HM is known to be solvable in polynomial time: acyclic and quasi-acyclic Horn functions. In this paper we define a new class of Horn functions properly containing both of the known classes and design a polynomial time HM algorithm for this new class.     

Obtaining Online Ecological Colourings by Generalizing First-Fit

A colouring of a graph is ecological if every pair of vertices that have the same set of colours in their neighbourhood are coloured alike. We consider the following problem: if a graph G and an ecological colouring c of G are given, can further vertices added to G, one at a time, be coloured so that at each stage the current graph is ecologically coloured? If the answer is yes, then we say that the pair (G,c) is ecologically online extendible. By generalizing the well-known First-Fit algorithm, we are able to characterize when (G,c) is ecologically online extendible, and to show that deciding whether (G,c) is ecologically extendible can be done in polynomial time. We also describe when the extension is possible using only colours from a given finite set C. For the case where c is a colouring of G in which each vertex is coloured distinctly, we give a simple characterization of when (G,c) is ecologically online extendible using only the colours of c, and we also show that (G,c) is always online extendible using the colours of c plus one extra colour. We also study (off-line) ecological H-colourings (an H-colouring of G is a homomorphism from G to H). We study the problem of deciding whether G has an ecological H-colouring for some fixed H and give a characterization of its computational complexity in terms of the structure of H.     

Ranking-Based Black-Box Complexity

Randomized search heuristics such as evolutionary algorithms, simulated annealing, and ant colony optimization are a broadly used class of general-purpose algorithms. Analyzing them via classical methods of theoretical computer science is a growing field. While several strong runtime analysis results have appeared in the last 20 years, a powerful complexity theory for such algorithms is yet to be developed. We enrich the existing notions of black-box complexity by the additional restriction that not the actual objective values, but only the relative quality of the previously evaluated solutions may be taken into account by the black-box algorithm. Many randomized search heuristics belong to this class of algorithms. We show that the new ranking-based model can give more realistic complexity estimates. The class of all binary-value functions has a black-box complexity of O(logn) in the previous black-box models, but has a ranking-based complexity of Θ(n). On the other hand, for the class of all OneMax functions, we present a ranking-based black-box algorithm that has a runtime of Θ(n/logn), which shows that the OneMax problem does not become harder with the additional ranking-basedness restriction.     

The number of pessimistic guesses in Generalized Mastermind

Mastermind is a famous two-player game, where the codemaker has to choose a secret code and the codebreaker has to guess it in as few questions as possible. The code consists of 4 pegs, each of which is one of 6 colors. In Generalized Mastermind a general number p of pegs and a general number c of colors is considered. Let f(p,c) be the pessimistic number of questions for the generalization of Mastermind with an arbitrary number p of pegs and c of colors. By a computer program we compute ten new values of f(p,c). Combining this program with theoretical methods, we compute all values f(3,c) and a tight lower and upper bound for f(4,c). For f(p,2) we give an upper bound and a lower bound. Finally, combining results for fixed p and c, we give bounds for the general case f(p,c).     

On P versus NP $ \cap $ co-NP for decision trees and read-once branching programs

It is known that if a Boolean function f in n variables has a DNF and a CNF of size then f also has a (deterministic) decision tree of size exp(O(log n log² N)). We show that this simulation cannot be made polynomial: we exhibit explicit Boolean functions f that require deterministic trees of size exp where N is the total number of monomials in minimal DNFs for f and ?f. Moreover, we exhibit new examples of explicit Boolean functions that require deterministic read-once branching programs of exponential size whereas both the functions and their negations have small nondeterministic read-once branching programs. One example results from the Bruen—Blokhuis bound on the size of nontrivial blocking sets in projective planes: it is remarkably simple and combinatorially clear. Other examples have the additional property that f is in AC⁰. Received: June 5 1997.     

Aperiodic propagation criteria for Boolean functions

We characterise the aperiodic autocorrelation for a Boolean function, f, and define the Aperiodic Propagation Criteria (APC) of degree l and order q. We establish the strong similarity between APC and the Extended Propagation Criteria (EPC) as defined by Preneel et al. in 1991, although the criteria are not identical. We also show how aperiodic autocorrelation can be related to the first derivative of f. We further propose the metric APC distance and show that quantum error correcting codes (QECCs) are natural candidates for Boolean functions with favourable APC distance.     

Fixed-Parameter Evolutionary Algorithms and the Vertex Cover Problem

In this paper, we consider multi-objective evolutionary algorithms for the Vertex Cover problem in the context of parameterized complexity. We consider two different measures for the problem. The first measure is a very natural multi-objective one for the use of evolutionary algorithms and takes into account the number of chosen vertices and the number of edges that remain uncovered. The second fitness function is based on a linear programming formulation and proves to give better results. We point out that both approaches lead to a kernelization for the Vertex Cover problem. Based on this, we show that evolutionary algorithms solve the vertex cover problem efficiently if the size of a minimum vertex cover is not too large, i.e., the expected runtime is bounded by O(f(OPT)?n ^c ), where c is a constant and f a function that only depends on OPT. This shows that evolutionary algorithms are randomized fixed-parameter tractable algorithms for the vertex cover problem.     

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.      

Harmonic Analysis, Real Approximation, and the Communication Complexity of Boolean Functions

The two-party communication complexity of Boolean function f is known to be at least log rank (M _f ), i.e., the logarithm of the rank of the communication matrix of f [19]. Lovász and Saks [17] asked whether the communication complexity of f can be bounded from above by (log rank (M _f )) ^c , for some constant c . The question was answered affirmatively for a special class of functions f in [17], and Nisan and Wigderson proved nice results related to this problem [20], but, for arbitrary f , it remained a difficult open problem. We prove here an analogous polylogarithmic upper bound in the stronger multiparty communication model of Chandra et al. [6], which, instead of the rank of the communication matrix, depends on the L ₁ norm of function f , for arbitrary Boolean function f . Received August 24, 1996; revised October 15, 1997.     

Block sensitivity of minterm-transitive functions

Boolean functions with a high degree of symmetry are interesting from a complexity theory perspective: extensive research has shown that these functions, if nonconstant, must have high complexity according to various measures.In a recent work of this type, Sun (2007) [9] gave lower bounds on the block sensitivity of nonconstant Boolean functions invariant under a transitive permutation group. Sun showed that all such functions satisfy bs(f)=Ω(N^1/3). He also showed that there exists such a function for which bs(f)=O(N^3/7lnN). His example belongs to a subclass of transitively invariant functions called “minterm-transitive” functions, defined by Chakraborty (2005) [3].We extend these results in two ways. First, we show that nonconstant minterm-transitive functions satisfy bs(f)=Ω(N^3/7). Thus, Sun’s example has nearly minimal block sensitivity for this subclass. Second, we improve Sun’s example: we exhibit a minterm-transitive function for which bs(f)=O(N^3/7ln^1/7N).     

Computation of signal output probability for Boolean functions represented by OBDD

In [l] and [2], two algorithms have been proposed to calculate the output probability of Boolean functions represented by OBDDs, assuming that the input variables are equiprobable and each variable is statistically independent from others. In this paper, we point out that under these assumptions, the output probability calculation is equivalent to counting the number of minterms of the corresponding Boolean functions. An algorithm is proposed to compute the output probability using simple integer arithmetic as opposed to floating point arithmetic involved in [1,2]. To compute output probability of Boolean functions represented by shared OBDI)s and OBDDs with edge negation, we further propose a generalized algorithm.