Petr Savický

Representations and rates of approximation of real-valued Boolean functions by neural networks

We give upper bounds on rates of approximation of real-valued functions of d Boolean variables by one-hidden-layer perceptron networks. Our bounds are of the form c/n where c depends on certain norms of the function being approximated and n is the number of hidden units. We describe sets of functions where these norms grow either polynomially or exponentially with d.

On Product Logic with Truth-constants

Product Logic Π is an axiomatic extension of Hajeks Basic Fuzzy Logic BL coping with the 1-tautologies when the strong conjunction & and implication → are interpreted by the product of reals in [0, 1] and its residuum respectively. In this paper we investigate expansions of Product Logic by adding into the language a countable set of truth-constants (one truth-constant for each r in a countable Π-subalgebra of [0, 1]) and by adding the corresponding book-keeping axioms for the truthconstants. We first show that the corresponding logics Π( ) are algebraizable, and hence complete with respect to the variety of Π( )-algebras. The main result of the paper is the canonical standard completeness of these logics, that is, theorems of Π( ) are exactly the 1-tautologies of the algebra defined over the real unit interval where the truth-constants are interpreted as their own values. It is also shown that they do not enjoy the canonical strong standard completeness, but they enjoy it for finite theories when restricted to evaluated Π-formulas of the kind → ψ, where is a truth-constant and ψ a formula not containing truth-constants. Finally we consider the logics ΠΔ( ), the expansion of Π( ) with the well-known Baazs projection connective Δ, and we show canonical finite strong standard completeness for them.

On P versus NP CO-NP for decision trees and read-once branching programs

Abstract. It is known that if a Boolean function f in n variables has a DNF and a CNF of size

Measures of Word Commonness

\le N

DNF tautologies with a limited number of occurences of every variable

then f also has a (deterministic) decision tree of size exp(O(log n log2N)). We show that this simulation cannot be made polynomial: we exhibit explicit Boolean functions f that require deterministic trees of size exp

Random Boolean formulas representing any Boolean function with asymptotically equal probability

(\Omega({\rm log^2} N))

On shifting networks

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.

Approximations by OBDDs and the Variable Ordering Problem

The main goal of this paper is to investigate methods of how to rank words in a way that corresponds to an intuitive notion of ‘commonness’. Since there is no formal definition of such a notion, our techniques may be considered as a suggestion for such a definition. The commonness of words is sometimes roughly substituted with their frequency in a language corpus. In order to suggest a better measure, we define a quantity, which we call corrected frequency. It depends not only on the frequency of a word in a corpus, but also on its distribution within the corpus. Unlike previous solutions of the same problem, we take the corpus as an uninterrupted sequence of words with no regard to borders between files, texts, genres, or any others. We introduce three different corrected frequencies. Their definitions are based on notions of information theory and analysis of random processes. Their values for individual words depend on the corpus. Hence, it is important to what extent they are stable with respect to the selection of the corpus. In order to investigate the suggested corrected frequencies from that point of view, we compare their values on five different subcorpora of the whole corpus. We present several examples of words taken from the Czech National Corpus that demonstrate in which way the corrected frequencies correspond to the intuitive commonness of these words.

Improved Boolean formulas for the Ramsey graphs

It is known that every DNF tautology with all monomials of length k contains a variable with at least Ω(2k/k) occurrences. It is not known, however, if this bound is tight, i.e. if there are tautologies with at most O(2k/k) occurrences of every variable. DNF tautologies with 2k monomials of length k and with at most 2k/kα occurrences of every variable, where α=log34−1⩾0.26 are presented. This has the following consequence. Let (k,s)-SAT be k-SAT restricted to instances with at most s occurrences of every variable. It is known that for every k, there is an sk such that (k,sk)-SAT is NP-complete and (k,sk−1)-SAT is trivial in the sense that every instance has positive answer. The above result implies that sk⩽2k/kα. This improves the previously known bound sk⩽13642k for all k⩾6.

A lower bound on branching programs reading some bits twice

Abstract We study the sequence of sets of Boolean formulas defined as follows: H 0 = {0, 1, x 1 , …, x n , ¬x 1 , …, ¬ x n }, H i+1 = {α(ϕ 1 , …, ϕ k ); ϕ j ∈ H i } , where α is a k-ary Boolean connective. We study the probability that a formula randomly chosen from Hi with the uniform distribution represents a given function f. We characterize the connectives α for which the limit of this probability when i→∞ is equal for all Boolean functions f of n variables.