Combinatorial and Algorithmic Properties of One Matrix Structure at Monotone Boolean Functions
Valentin Bakoev
备注：
Note (Feb. 15, 2019). This manuscript was written in 2005 and has not been published till now. This is its original version where few misprints have been corrected and the Internet references have been updated
One matrix structure in the area of monotone Boolean functions is defined here. Some of its combinatorial, algebraic and algorithmic properties are derived. On the base of these properties, three algorithms are built. First of them generates all monotone Boolean functions of $n$ variables in lexicographic order. The second one determines the first (resp. the last) lexicographically minimal true (resp. maximal false) vector of an unknown monotone function $f$ of $n$ variables. The algorithm uses at most $n$ membership queries and its running time is $Θ(n)$. It serves the third algorithm, which identifies an unknown monotone Boolean function $f$ of $n$ variables by using membership queries only. The experimental results show that for $1\leq n\leq 6$, the algorithm determines $f$ by using at most $m.n$ queries, where $m$ is the combined size of the sets of minimal true and maximal false vectors of $f$.
We prove that the fully asynchronous dynamics of a Boolean network $f:\{0,1\}^n\to\{0,1\}^n$ without negative loop can be simulated, in a very specific way, by a monotone Boolean network with $2n$ components. We then use this result to prove that, for every even $n$, there exists a monotone Boolean network $f:\{0,1\}^n\to\{0,1\}^n$, an initial configuration $x$ and a fixed point $y$ of $f$ such that: (i) $y$ can be reached from $x$ with a fully asynchronous updating strategy, and (ii) all such strategies contains at least $2^{\frac{n}{2}}$ updates. This contrasts with the following known prope…
The asynchronous automaton associated with a Boolean network $f:\{0,1\}^n\to\{0,1\}^n$ is considered in many applications. It is the finite deterministic automaton with set of states $\{0,1\}^n$, alphabet $\{1,\dots,n\}$, where the action of letter $i$ on a state $x$ consists in either switching the $i$th component if $f_i(x)\neq x_i$ or doing nothing otherwise. This action is extended to words in the natural way. We then say that a word $w$ fixes $f$ if, for all states $x$, the result of the action of $w$ on $x$ is a fixed point of $f$. In this paper, we ask for the existence of fixing words,…
Inversion Complexity of Functions of Multi-Valued Logic
The minimum number of NOT gates in a logic circuit computing a Boolean function is called the inversion complexity of the function. In 1957, A. A. Markov determined the inversion complexity of every Boolean function and proved that $\lceil\log_{2}(d(f)+1)\rceil$ NOT gates are necessary and sufficient to compute any Boolean function $f$ (where $d(f)$ is the maximum number of value changes from greater to smaller over all increasing chains of tuples of variables values). This result is extended to $k$ -valued functions computing in this paper. Thereupon one can use monotone functions "for free" …
Recently, Deshpande et al. introduced a new measure of the complexity of a Boolean function. We call this measure the "goal value" of the function. The goal value of $f$ is defined in terms of a monotone, submodular utility function associated with $f$. As shown by Deshpande et al., proving that a Boolean function $f$ has small goal value can lead to a good approximation algorithm for the Stochastic Boolean Function Evaluation problem for $f$. Also, if $f$ has small goal value, it indicates a close relationship between two other measures of the complexity of $f$, its average-case decision tree…
The asynchronous dynamics associated with a Boolean network $f : \{0,1\}^n \to \{0,1\}^n$ is a finite deterministic automaton considered in many applications. The set of states is $\{0,1\}^n$, the alphabet is $[n]$, and the action of letter $i$ on a state $x$ consists in either switching the $i$th component if $f_i(x)\neq x_i$ or doing nothing otherwise. This action is extended to words in the natural way. We then say that a word $w$ {\em fixes} $f$ if, for all states $x$, the result of the action of $w$ on $x$ is a fixed point of $f$. A whole family of networks is fixable if its members are a…
Given a digraph $G$, a lot of attention has been deserved on the maximum number $ϕ(G)$ of fixed points in a Boolean network $f:\{0,1\}^n\to\{0,1\}^n$ with $G$ as interaction graph. In particular, a central problem in network coding consists in studying the optimality of the classical upper bound $ϕ(G)\leq 2^τ$, where $τ$ is the minimum size of a feedback vertex set of $G$. In this paper, we study the maximum number $ϕ_m(G)$ of fixed points in a {\em monotone} Boolean network with interaction graph $G$. We establish new upper and lower bounds on $ϕ_m(G)$ that depends on the cycle structure of $…
We show improved monotonicity testers for the Boolean hypercube under the $p$-biased measure, as well as over the hypergrid $[m]^n$. Our results are: 1. For any $p\in (0,1)$, for the $p$-biased hypercube we show a non-adaptive tester that makes $\tilde{O}(\sqrt{n}/\varepsilon^2)$ queries, accepts monotone functions with probability $1$ and rejects functions that are $\varepsilon$-far from monotone with probability at least $2/3$. 2. For all $m\in\mathbb{N}$, we show an $\tilde{O}(\sqrt{n}m^3/\varepsilon^2)$ query monotonicity tester over $[m]^n$. We also establish corresponding directed isoper…
Logical models have been successfully used to describe regulatory and signaling networks without requiring quantitative data. However, existing data is insufficient to adequately define a unique model, rendering the parametrization of a given model a difficult task. Here, we focus on the characterization of the set of Boolean functions compatible with a given regulatory structure, i.e. the set of all monotone nondegenerate Boolean functions. We then propose an original set of rules to locally explore the direct neighboring functions of any function in this set, without explicitly generating th…
Division of Science, NYU Abu Dhabi（纽约大学阿布扎比分校科学系）；Center for Interdisciplinary Data Science and AI, NYUAD Research Institute（NYUAD研究院跨学科数据科学与人工智能中心）；Courant Institute of Mathematical Sciences, New York University（纽约大学库朗数学科学研究所）
Motivated by practical applications, recent works have considered maximization of sums of a submodular function $g$ and a linear function $\ell$. Almost all such works, to date, studied only the special case of this problem in which $g$ is also guaranteed to be monotone. Therefore, in this paper we systematically study the simplest version of this problem in which $g$ is allowed to be non-monotone, namely the unconstrained variant, which we term Regularized Unconstrained Submodular Maximization (RegularizedUSM). Our main algorithmic result is the first non-trivial guarantee for general Regular…
In this paper we consider a variant of the well-known Achlioptas process for graphs adapted to monotone Boolean functions. Fix a number of choices $r\in \mathbb N$ and a sequence of increasing functions $(f_n)_{n\ge 1}$ such that, for every $n\ge 1$, $f_n:\{0,1\}^n\mapsto \{0,1\}$. Given $n$ bits which are all initially equal to 0, at each step $r$ 0-bits are sampled uniformly at random and are proposed to an agent. Then, the agent selects one of the proposed bits and turns it from 0 to 1 with the goal to reach the preimage of 1 as quickly as possible. We nearly characterize the conditions und…
