International journal of bifurcation and chaos vol:6 issue:12B pages:2507-2530
We consider the two-dimensional state-time evolution pattern (orbit) of a one-dimensional linear cellular automaton (CA) defined over a finite commutative ring R, and correlate this to shifted versions of itself by adding the original pattern and its shifted versions. The resulting pattern is called the shift-add (correlation) pattern. A dual counterpart of this pattern is also introduced: it forms a so-called pseudo-CA orbit. We show that the set of shift-add patterns for different shifts interrelate like a pseudo-CA whose states are two-dimensional patterns over R. The set of dual shift-add patterns forms likewise a regular CA. Complementary ways of viewing the shift-add correlation space are also presented. Finally, the nature of shift-add correlation patterns for the newly defined class of pseudo-CA is investigated.