Shadow Minimization Boolean Function Reconstruction

This paper investigates the KK-MBF class of monotone Boolean functions coming from the finite set systems shadow minimization theory.Zeros of these functions correspond to the initial segments of the lexicographic order on the layers of binary cube.In particular, we focus on query-based recognition algorithms of KK-MBF functions, and consider the cardinality estimations of this class.

We consider the problem of learning monotone Boolean functions over {0, 1}/sup n/ under the uniform distribution. Specifically, given a polynomial number of uniform random samples for an unknown monotone Boolean function f, and given polynomial completing time, we would like to approximate f as well as possible. We describe a simple algorithm that we prove achieves error at most 1/2-/spl Omega/(1//spl radic/n), improving on the previous best bound of 1/2-/spl Omega/((log/sup 2/ n)/n). We also prove that no algorithm, given a polynomial number of samples, can guarantee error 1/2-/spl omega/((log n)//spl radic/n), improving on the previous best hardness bound of O(1//spl radic/n). These lower bounds hold even if the learning algorithm is allowed membership queries. Thus this paper settles to an O(log n) factor the question of the best achievable error for learning the class of monotone Boolean functions with respect to the uniform distribution.

We study the function evaluation problem in the priced information framework introduced in [Charikar et al. 2002]. We characterize the best possible extremal competitive ratio for the class of game tree functions. Moreover, we extend the above result to the case when the cost of reading a variable depends on the value of the variable. In this new value dependent cost variant of the problem, we also exactly evaluate the extremal competitive ratio for the whole class of monotone Boolean functions.

The number and probability of canalizing functions

We consider a learning model in which each element of a class of recursive functions is to be identified in the limit by a computable strategy. Given gradually growing initial segments of the graph of a function, the learner is supposed to generate a sequence of hypotheses converging to a correct hypothesis. The term correct means that the hypothesis is an index of the function to be learned in a given numbering. Restriction of the basic definition of learning in the limit yields several inference criteria, which have already been compared with respect to their learning power.The scope of uniform learning is to synthesize appropriate identification strategies for infinitely many classes of recursive functions by a uniform method, i.e. a kind of meta-learning is considered. In this concept we can also compare the learning power of several inference criteria. If we fix a single numbering to be used as a hypothesis space for all classes of recursive functions, we obtain results similar to the non-uniform case. This hierarchy of inference criteria changes, if we admit different hypothesis spaces for different classes of functions. Interestingly, in uniform identification most of the inference criteria can be separated by collections of finite classes of recursive functions.

This paper investigates the optimum OBDD representation problem based on two classes of Boolean functions. The first class is defined by OBDDs, in which the number of non-terminal nodes is equal to the number of input variables. We refer to such OBDDs and their corresponding Boolean functions as thin OBDDs and thin Boolean functions. The second class is the thin factored Boolean functions, which is defined by factored forms with each variable appearing exactly once. Many interesting properties of these two classes of functions are presented. Based on which, a revised dynamic shortest cube first OBDD variable ordering algorithm is developed. This algorithm is shown to be optimum for thin factored Boolean functions.

International audience

Several underlying structural and functional factors that determine the fault behavior of a network are not yet well understood. In this paper, we show that there exists a large class of Boolean functions, called root functions, which can never appear as faulty response in an irredundant two-level circuit even when any arbitrary multiple stuck-at faults are injected. Conversely, we show that any other Boolean function can appear as a faulty response in an irredundant realization of some root function under certain stuck-at faults. We characterize this new class of functions and show that for n variables, their number is exactly equal to the number of independent dominating sets (Harary and Livingston, Appl. Math. Lett., 1993) in a Boolean n-cube. Similar properties are observed for multiple-valued logic functions as well. Finally, we discuss its application to logic design and point out some open problems.

In this paper, we investigate the hardware circuit complexity of the class of Boolean functions recently introduced by Tang and Maitra (IEEE-TIT 64(1): 393–402, 2018). While this class of functions has very good cryptographic properties, the exact hardware requirement is an immediate concern as noted in the paper itself. In this direction, we consider different circuit architectures based on finite field arithmetic and Boolean optimization. An estimation of the circuit complexity is provided for such functions given any input size n. We study different candidate architectures for implementing these functions, all based on the finite field arithmetic. We also show different implementations for both ASIC and FPGA, providing further analysis on the practical aspects of the functions in question and the relation between these implementations and the theoretical bound. The practical results show that the Tang-Maitra functions are quite competitive in terms of area, while still maintaining an acceptable level of throughput performance for both ASIC and FPGA implementations.

We prove that polynomial size discrete synchronous Hopfield networks with hidden units compute exactly the class of Boolean functions PSPACE/poly, i.e., the same functions as are computed by polynomial space-bounded nonuniform Turing machines. As a corollary to the construction, we observe also that networks with polynomially bounded interconnection weights compute exactly the class of functions P/poly.

Boomerang uniformity of some classes of functions over finite fields

Let f be a function on a set of variables V. For each x e V, let c(x) be the cost of reading the value of x. An algorithm for evaluating f is a strategy for adaptively identifying and reading a set of variables U ⊆ V whose values uniquely determine the value of f. We are interested in finding algorithms which minimize the cost incurred to evaluate f in the above sense. Competitive analysis is employed to measure the performance of the algorithms. We study two variants of the above problem. First we consider the classical setting in which one assumes that the algorithm knows the cost c(x), for each x e V. For the case where f is a monotone boolean function which is representable by a threshold tree, we provide a polynomial time algorithm with the best possible competitive ratio γfc for each fixed cost function c(·). Remarkably, the best known result for the same class of functions is a pseudo-polynomial algorithm with competitiveness 2γfc. For the class of game tree functions our polynomial time algorithm attains, for each fixed cost function, the same competitiveness as the best known algorithm for the same class of functions, which instead runs in pseudo-polynomial time.In the second part of the paper, we study a novel variant of the problem. Here, we assume that the cost function is not known in advance and some preemption is allowed in the reading operations. This model has applications, e.g., when reading a variable coincides with obtaining the output of a job on a CPU and the cost is the CPU time. In such a case, it is reasonable to assume that no exact knowledge of the cost is available. We define a new algorithm for this problem based on the solution of a Linear Program. We show the optimality of our algorithm for the class of monotone boolean functions representable by AND-OR-trees. We also show a sub-optimal implementation for general monotone boolean functions.

An analysis of root functions—A subclass of the Impossible Class of Faulty Functions (ICFF)

The total influence of a function is a central notion in analysis of Boolean functions, and characterizing functions that have small total influence is one of the most fundamental questions associated with it. The KKL theorem and the Friedgut junta theorem give a strong characterization of such functions whenever the bound on the total influence is \(o(\log n)\). However, both results become useless when the total influence of the function is \(\omega (\log n)\). The only case in which this logarithmic barrier has been broken for an interesting class of functions was proved by Bourgain and Kalai, who focused on functions that are symmetric under large enough subgroups of \(S_n\). In this paper, we build and improve on the techniques of the Bourgain–Kalai paper and establish new concentration results on the Fourier spectrum of Boolean functions with small total influence. Our results include: 1. A quantitative improvement of the Bourgain–Kalai result regarding the total influence of functions that are transitively symmetric. 2. A slightly weaker version of the Fourier-Entropy Conjecture of Friedgut and Kalai. Our result establishes new bounds on the Fourier entropy of a Boolean function f, as well as stronger bounds on the Fourier entropy of low-degree parts of f. In particular, it implies that the Fourier spectrum of a constant variance, Boolean function f is concentrated on \(2^{O(I[f]\log I[f])}\) characters, improving an earlier result of Friedgut. Removing the \(\log I[f]\) factor would essentially resolve the Fourier-Entropy Conjecture, as well as settle a conjecture of Mansour regarding the Fourier spectrum of polynomial size DNF formulas. Our concentration result for the Fourier spectrum of functions with small total influence also has new implications in learning theory. More specifically, we conclude that the class of functions whose total influence is at most K is agnostically learnable in time \(2^{O(K\log K)}\) using membership queries. Thus, the class of functions with total influence \(O(\log n/\log \log n)\) is agnostically learnable in \(\mathsf{poly}(n)\) time.

Given a Boolean function f, the f-isomorphism testing problem requires a randomized algorithm to distinguish functions that are identical to f up to relabeling of the input variables from functions that are far from being so. An important open question in property testing is to determine for which functions f we can test f-isomorphism with a constant number of queries. Despite much recent attention to this question, essentially only two classes of functions were known to be efficiently isomorphism testable: symmetric functions and juntas. We unify and extend these results by showing that all partially symmetric functions -- functions invariant to the reordering of all but a constant number of their variables -- are efficiently isomorphism-testable. This class of functions, first introduced by Shannon, includes symmetric functions, juntas, and many other functions as well. We conjecture that these functions are essentially the only functions efficiently isomorphism-testable. To prove our main result, we also show that partial symmetry is efficiently testable. In turn, to prove this result we had to revisit the junta testing problem. We provide a new proof of correctness of the nearly-optimal junta tester. Our new proof replaces the Fourier machinery of the original proof with a purely combinatorial argument that exploits the connection between sets of variables with low influence and intersecting families. Another important ingredient in our proofs is a new notion of symmetric influence. We use this measure of influence to prove that partial symmetry is efficiently testable and also to construct an efficient sample extractor for partially symmetric functions. We then combine the sample extractor with the testing-by-implicit-learning approach to complete the proof that partially symmetric functions are efficiently isomorphism-testable.