Complex Stochastic Boolean Systems: Comparing Bitstrings with the Same Hamming Weight

A complex stochastic Boolean system (CSBS) is a complex system depending on an arbitrarily large number of random Boolean variables. CSBSs arise in many different areas of science and engineering. A proper mathematical model for the analysis of such systems is based on the intrinsic order: a partial order relation defined on the set of all binary -tuples of 0s and 1s. The intrinsic order enables one to compare the occurrence probabilities of two given binary -tuples with no need to compute them, simply looking at the relative positions of their 0s and 1s. Regarding the analysis of CSBSs, the intrinsic order reduces the complexity of the problem from exponential ( binary -tuples) to linear ( Boolean variables). In this paper, using the intrinsic ordering, we compare the occurrence probabilities of any two binary -tuples having the same number of 1-bits (i.e., the same Hamming weight). Our results can be applied to any CSBS with mutually independent Boolean variables. 1. Introduction This paper deals with the mathematical modeling of a special kind of complex systems, namely, those depending on an arbitrary number of random Boolean variables. That is, the basic variables of the system are assumed to be stochastic and they only take two possible values, or , with probabilities where the values will be referred to as the basic probabilities or parameters of the system. We call such a system a complex stochastic Boolean system (CSBS). These systems can be found in many different scientific or engineering areas like mechanical engineering, meteorology and climatology, nuclear physics, complex systems analysis, operations research, and so forth. CSBSs also arise very often when analyzing system safety in reliability engineering and risk analysis; see, for example, [1–3]. Each one of the possible outcomes associated with a CSBS is given by a binary -tuple (or bitstring of length ) , and it has its own occurrence probability . Throughout this paper, the Boolean variables of the CSBS are assumed to be mutually independent, so that the occurrence probability of a given binary string of length can be easily computed as that is, is the product of factors if , if . As an example of CSBS, we can consider a technical system like the accumulator system of a pressured water reactor in a nuclear power plant, taken from [4]. This technical system depends on mutually independent basic components . Assuming that if component fails, otherwise; then the failure and working probabilities of component will be , , respectively. The probability of failure of each basic component