Rule 136 and its equivalents are liberal, but become conservative in their conjugal life

Abstract

Rule 136 and its equivalents, rules 192, 238 and 252, are not number conserving rules. Out of these four, 136 and 192 are number nonincreasing rules, whereas 238 and 252 are number nondecreasing rules. However, when a non-uniform cellular automation (CA) is formed with rules 136 and 252, or 192 and 238 in alternate positions, the CA becomes conservative. This paper studies these non-uniform CAs. It is further shown that, except rules 136, 192, 238 and 252, no other number nonincreasing and nondecreasing rules can participate in a non-uniform CA to form a conservative system.