Complex stochastic boolean systems: generating and counting the binary n-tuples intrinsically less or greater than u

Abstract

A complex stochastic Boolean system (CSBS) is a system depending on an arbitrary number n of random Boolean variables. The behavior of a CSBS is determined by the ordering between the occurrence probabilities Pr {u} of the 2(n) associated binary n-tuples u = (u(1), ..., u(n)) is an element of {0, 1}(n). In this context, for every fixed binary n-tuple u, this paper presents two simple algorithms-exclusively based on the vector of positions of the 1-bits (0-bits, respectively) in u for rapidly generating (and counting) all the binary n-tuples v whose occurrence probabilities Pr {v} are always less than or equal to (greater than or equal to, respectively) Pr{u}. Results are illustrated with the intrinsic order graph and they obey to a nice duality property (interchange 0s by 1s and "more probable" by "less probable").