Learning of monotone functions with single error correction

2021 ◽  
Vol 31 (3) ◽  
pp. 193-205
Author(s):  
Svetlana N. Selezneva ◽  
Yongqing Liu
Keyword(s):  
Error Correction ◽  
Unknown Function ◽  
Boolean Functions ◽  
Monotone Function ◽  
Incorrect Response ◽  
Monotone Functions ◽  
Single Error ◽  
Monotone Boolean Functions

Abstract Learning of monotone functions is a well-known problem. Results obtained by V. K. Korobkov and G. Hansel imply that the complexity φM (n) of learning of monotone Boolean functions equals C n ⌊ n / 2 ⌋ $\begin{array}{} \displaystyle C_n^{\lfloor n/2\rfloor} \end{array}$ + C n ⌊ n / 2 ⌋ + 1 $\begin{array}{} \displaystyle C_n^{\lfloor n/2\rfloor+1} \end{array}$ (φM (n) denotes the least number of queries on the value of an unknown monotone function on a given input sufficient to identify an arbitrary n-ary monotone function). In our paper we consider learning of monotone functions in the case when the teacher is allowed to return an incorrect response to at most one query on the value of an unknown function so that it is still possible to correctly identify the function. We show that learning complexity in case of the possibility of a single error is equal to the complexity in the situation when all responses are correct.

scholarly journals Monotone Boolean Functions with s Zeros Farthest from Threshold Functions

2005 ◽  
Vol DMTCS Proceedings vol. AE,... (Proceedings) ◽  
Author(s):  
Kazuyuki Amano ◽  
Jun Tarui
Keyword(s):  
Boolean Function ◽  
Boolean Functions ◽  
Learning Algorithm ◽  
Monotone Function ◽  
Threshold Function ◽  
Binary Representation ◽  
Threshold Functions ◽  
Monotone Boolean Function ◽  
Monotone Boolean Functions ◽  
International Audience

International audience Let $T_t$ denote the $t$-threshold function on the $n$-cube: $T_t(x) = 1$ if $|\{i : x_i=1\}| \geq t$, and $0$ otherwise. Define the distance between Boolean functions $g$ and $h$, $d(g,h)$, to be the number of points on which $g$ and $h$ disagree. We consider the following extremal problem: Over a monotone Boolean function $g$ on the $n$-cube with $s$ zeros, what is the maximum of $d(g,T_t)$? We show that the following monotone function $p_s$ maximizes the distance: For $x \in \{0,1\}^n$, $p_s(x)=0$ if and only if $N(x) < s$, where $N(x)$ is the integer whose $n$-bit binary representation is $x$. Our result generalizes the previous work for the case $t=\lceil n/2 \rceil$ and $s=2^{n-1}$ by Blum, Burch, and Langford [BBL98-FOCS98], who considered the problem to analyze the behavior of a learning algorithm for monotone Boolean functions, and the previous work for the same $t$ and $s$ by Amano and Maruoka [AM02-ALT02].

Cryptographic properties of monotone Boolean functions

2016 ◽  
Vol 10 (1) ◽  
pp. 1-14 ◽  
Author(s):  
Claude Carlet ◽  
David Joyner ◽  
Pantelimon Stănică ◽  
Deng Tang
Keyword(s):  
Upper Bound ◽  
Boolean Functions ◽  
Monotone Functions ◽  
Monotone Boolean Functions

AbstractWe prove various results on monotone Boolean functions. In particular, we prove a conjecture proposed recently, stating that there are no monotone bent Boolean functions. Further, we give an upper bound on the nonlinearity of monotone functions in odd dimension, we describe the Walsh–Hadamard spectrum and investigate some other cryptographic properties of monotone Boolean functions.

scholarly journals On the Lyapunov Exponent of Monotone Boolean Networks †

Mathematics ◽  
2020 ◽  
Vol 8 (6) ◽  
pp. 1035
Author(s):  
Ilya Shmulevich
Keyword(s):  
Lyapunov Exponent ◽  
Dynamical Systems ◽  
Boolean Functions ◽  
Boolean Network ◽  
Discrete Dynamical Systems ◽  
Boolean Networks ◽  
Asymptotic Formulas ◽  
Small Perturbations ◽  
Monotone Boolean Functions ◽  
Almost All

Boolean networks are discrete dynamical systems comprised of coupled Boolean functions. An important parameter that characterizes such systems is the Lyapunov exponent, which measures the state stability of the system to small perturbations. We consider networks comprised of monotone Boolean functions and derive asymptotic formulas for the Lyapunov exponent of almost all monotone Boolean networks. The formulas are different depending on whether the number of variables of the constituent Boolean functions, or equivalently, the connectivity of the Boolean network, is even or odd.

scholarly journals Counting inequivalent monotone Boolean functions

2014 ◽  
Vol 167 ◽  
pp. 15-24 ◽  
Author(s):  
Tamon Stephen ◽  
Timothy Yusun
Keyword(s):  
Boolean Functions ◽  
Monotone Boolean Functions
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