ON REDUCED SCALAR EQUATIONS FOR SYNCHRONOUS BOOLEAN NETWORKS

Abstract

A total description of a synchronous Boolean networ k is typically achieved by a matrix recurrence relation. A simpler alternative is to use a scalar equation which is a possibly nonlinear equation tha t involves two or more instances of a single scalar variable and some Bool ean operator(s). Further simplification is possible in terms of a linear reduced scalar equation which is the simplest two-term scalar equa tion that includes no Boolean operators and equates the value of a scalar variabl e at a latter instance t 2 to its value at an earlier instance t 1. This equation remains valid when the times t 1 and t2 are both augmented by any integral multiple of the underlying time period. In other words, there are i nfinitely many versions of a reduced scalar equatio n, any of which is useful for deducing information about the cyclic behavior of the network. However, to obtain correct information about the transient behavior of the network, one must find the true red uced scalar equation for which instances t1 and t 2 are minimal. This study investigates the nature, d erivation and utilization of reduced scalar equations. It relies on Boolean-algebraic ma nipulations for the derivation of such equations an d suggests that this derivation can be facilitated by seeking certain orthogonality relations among certain successive (albeit not necessarily consecutive) instances of t he same scalar variable. We demonstrate, contrary t o previously published assumptions or assertions, tha t there is typically no common reduced scalar equat ion for all the scalar variables. Each variable usually sat isfies its own distinct reduced scalar equation. We also demonstrate that the derivation of a reduced scalar equation is achieved not only by proving it but also by disproving an immediately preceding version of it when such a version might exist. We also demonstrate that, despite the useful insight supplied by the reduced scalar equations, they do not provide a total solut ion like the one offered by matrix methods and therefore they ne ed be supported by other techniques of mathematical reasoning. We present three classical examples to i llustrate our techniques. Two examples are tutorial s on the necessary Boolean-algebraic techniques. They present corrections of previously published results and r efute purported claims of discrepancies between scalar an d matrix methods. The third example illustrates how the reduced scalar equations can be supplemented by tec hniques of number theory, Diophantine equations and Boolean equations in making subtle inferences about Boolean networks. We achieved a better understanding of the nature of reduced scalar equations, demonstr ated Boolean-algebraic techniques for deriving them, presented other mathematical tools for utilizing th em and finally reconciled them with the more encompassing but more complex and less insightful matrix methods.