86 research outputs found
A Lower Bound on the Constant in the Fourier Min-Entropy/Influence Conjecture
We describe a new construction of Boolean functions. A specific instance of
our construction provides a 30-variable Boolean function having
min-entropy/influence ratio to be which is presently
the highest known value of this ratio that is achieved by any Boolean function.
Correspondingly, is also presently the best known lower bound on the
universal constant of the Fourier min-entropy/influence conjecture
Influence of a Set of Variables on a Boolean Function
The influence of a variable is an important concept in the analysis of
Boolean functions. The more general notion of influence of a set of variables
on a Boolean function has four separate definitions in the literature. In the
present work, we introduce a new definition of influence of a set of variables
which is based on the auto-correlation function and develop its basic theory.
Among the new results that we obtain are generalisations of the Poincar\'{e}
inequality and the edge expansion property of the influence of a single
variable. Further, we obtain new characterisations of resilient and bent
functions using the notion of influence. We show that the previous definition
of influence due to Fischer et al. (2002) and Blais (2009) is half the value of
the auto-correlation based influence that we introduce. Regarding the other
prior notions of influence, we make a detailed study of these and show that
each of these definitions do not satisfy one or more desirable properties that
a notion of influence may be expected to satisfy
Achieving Maximum Utilization in Optimal Time for Learning or Convergence in the Kolkata Paise Restaurant Problem
The objective of the KPR agents are to learn themselves in the minimum
(learning) time to have maximum success or utilization probability (). A
dictator can easily solve the problem with in no time, by asking every
one to form a queue and go to the respective restaurant, resulting in no
fluctuation and full utilization from the first day (convergence time ). It has already been shown that if each agent chooses randomly the
restaurants, (where denotes the
Euler number) in zero time (). With the only available information
about yesterday's crowd size in the restaurant visited by the agent (as assumed
for the rest of the strategies studied here), the crowd avoiding (CA)
strategies can give higher values of but also of . Several numerical
studies of modified learning strategies actually indicated increased value of
for , with . We show here
using Monte Carlo technique, a modified Greedy Crowd Avoiding (GCA) Strategy
can assure full utilization () in convergence time ,
with of course non-zero probability for an even larger convergence time. All
these observations suggest that the strategies with single step memory of the
individuals can never collectively achieve full utilization () in finite
convergence time and perhaps the maximum possible utilization that can be
achieved is about eighty percent () in an optimal time of
order ten, even when the number of customers or of the restaurants goes to
infinity.Comment: 9 pages, 6 figures included in manuscript; Accepted for publication
in Indian Journal of Physic
Separation Results for Boolean Function Classes
We show (almost) separation between certain important classes of Boolean functions. The technique that we use is to show that the total influence of functions in one class is less than the total influence of functions in the other class. In particular, we show (almost) separation of several classes of Boolean functions which have been studied in the coding theory and cryptography from classes which have been studied in combinatorics and complexity theory
Influence of a Set of Variables on a Boolean Function
The influence of a set of variables on a Boolean function has three separate definitions in the literature, the first due to Ben-Or and Linial (1989), the second due to Fischer et al. (2002) and Blais (2009) and the third due to Tal (2017). The goal of the present work is to carry out a comprehensive study of the notion of influence of a set of variables on a Boolean function. To this end, we introduce a definition of this notion using the auto-correlation function. A modification of the definition leads to the notion of pseudo-influence. Somewhat surprisingly, it turns out that the auto-correlation based definition of influence is equivalent to the definition introduced by Fischer et al. (2002) and Blais (2009) and the notion of pseudo-influence is equivalent to the definition of influence considered by Tal (2017). Extensive analysis of influence and pseduo-influence as well as the Ben-Or and Linial notion of influence is carried out and the relations between these notions are established
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