We investigate the long-term behavior of solutions of the difference equation[ x_{n+1}=x_{n}x_{n-3}-1 ,, n=0 ,, 1 ,, ldots ,, ]noindent where the initial conditions $x_{-3} ,, x_{-2} ,, x_{-1} ,, x_{0}$ are real numbers.  In particular, we look at the periodicity and asymptotic periodicity of solutions, as well as the existence of unbounded solutions.

A derivation of the Dirac equation in ‘3 + 1’ dimensions is presented based on a master equation approach originally developed for the ‘1+ 1’ problem by McKeon and Ord. The method of derivation presented here suggests a mechanism by which the work of Knuth and Bahrenyi on causal sets may be extended to a derivation of the Dirac equation in the context of an inference problem.