Double Horn Functions

AbstractIn this paper, we define double Horn functions, which are the Boolean functionsfsuch that bothfand its complement (i.e., negation)fare Horn, and investigate their semantical and computational properties. Double Horn functions embody a balanced treatment of positive and negative information in the course of the extension problem of partially defined Boolean functions (pdBfs), where a pdBf is a pair (T,F) of disjoint setsT,F⊆{0,1}nof true and false vectors, respectively, and an extension of (T,F) is a Boolean functionfthat is compatible withTandF. We derive syntactic and semantic characterizations of double Horn functions, and determine the number of such functions. The characterizations are then exploited to give polynomial time algorithms (i) that recognize double Horn functions from Horn DNFs (disjunctive normal forms), and (ii) that compute the prime DNF from an arbitrary formula, as well as its complement and its dual. Furthermore, we consider the problem of determining a double Horn extension of a given pdBf. We describe a polynomial time algorithm for this problem and moreover an algorithm that enumerates all double Horn extensions of a pdBf with polynomial delay. However, finding a shortest double Horn extension (in terms of the size of a formulaϕrepresenting it) is shown to be intractable

Union-closed sets and Horn Boolean functions

A family of sets is union-closed if the union of any two sets from belongs to. The union-closed sets conjecture states that if is a finite union-closed family of finite sets, then there is an element that belongs to at least half of the sets in. The conjecture has several equivalent formulations in terms of other combinatorial structures such as lattices and graphs. In its whole generality the conjecture remains wide open, but it was verified for various important classes of lattices, such as lower semimodular lattices, and graphs, such as chordal bipartite graphs. In the present paper we develop a Boolean approach to the conjecture and verify it for several classes of Boolean functions, such as submodular functions and double Horn functions

Decision lists and related Boolean functions

AbstractWe consider Boolean functions represented by decision lists, and study their relationships to other classes of Boolean functions. It turns out that the elementary class of 1-decision lists has interesting relationships to independently defined classes such as disguised Horn functions, read-once functions, nested differences of concepts, threshold functions, and 2-monotonic functions. In particular, 1-decision lists coincide with fragments of the mentioned classes. We further investigate the recognition problem for this class, as well as the extension problem in the context of partially defined Boolean functions (pdBfs). We show that finding an extension of a given pdBf in the class of 1-decision lists is possible in linear time. This improves on previous results. Moreover, we present an algorithm for enumerating all such extensions with polynomial delay

Incremental polynomial time dualization of quadratic functions and a subclass of degree-k functions

Cataloged from PDF version of article.We consider the problem of dualizing a Boolean function f represented by a DNF. In its most general form, this problem is commonly believed not to be solvable by a quasi-polynomial total time algorithm.We show that if the input DNF is quadratic or is a special degree-k DNF, then dualization turns out to be equivalent to hypergraph dualization in hypergraphs of bounded degree and hence it can be achieved in incremental polynomial time

Boolean Functions: Theory, Algorithms, and Applications

This monograph provides the first comprehensive presentation of the theoretical, algorithmic and applied aspects of Boolean functions, i.e., {0,1}-valued functions of a finite number of {0,1}-valued variables. The book focuses on algebraic representations of Boolean functions, especially normal form representations. It presents the fundamental elements of the theory (Boolean equations and satisfiability problems, prime implicants and associated representations, dualization, etc.), an in-depth study of special classes of Boolean functions (quadratic, Horn, shellable, regular, threshold, read-once, etc.), and two fruitful generalizations of the concept of Boolean functions (partially defined and pseudo-Boolean functions). It features a rich bibliography of about one thousand items. Prominent among the disciplines in which Boolean methods play a significant role are propositional logic, combinatorics, graph and hypergraph theory, complexity theory, integer programming, combinatorial optimization, game theory, reliability theory, electrical and computer engineering, artificial intelligence, etc. The book contains applications of Boolean functions in all these areas

Lack of Correlations between Cold Molecular Gas and AGN Properties in Type 1 AGNs at z≲0.5z \lesssim 0.5

We present new NOrthern Extended Millimeter Array (NOEMA) observations of the CO(2--1) emission in eight of the brightest Palomar-Green quasars at $z \lesssim 0.5$ to investigate the role of active galactic nuclei (AGN) feedback in luminous quasars detected at low redshifts. We detect CO(2--1) emission in three objects, from which we derive CO luminosities, molecular gas masses and fractions, and gas depletion times. In combination with data available in the literature, we build a total sample of 138 local type 1 AGNs with CO(2--1) measurements. We compare the AGN properties with the host galaxy molecular gas properties, considering the systems non-detected in CO emission. We find that the CO luminosity does not correlate with AGN luminosity and Eddington ratio, while the molecular gas fraction is weakly correlated with Eddington ratio. The type 1 AGNs can be roughly separated into two populations in terms of infrared-to-CO luminosity ratio, one population presenting values typically found in normal star-forming systems, while the other have lower ratio values, comparable to those measured for starbursts. We find no evidence that AGN feedback rapidly quenches star formation in type 1 AGNs. Our results may imply an underlying the role of host galaxy gravitational instabilities or the fast inflow of cold gas in triggering AGN activity.Comment: 20 pages, 9 figures, 4 tables. Accepted for publication in Ap