In both the rules shown on the facing page, the only replacement specified is for the block . And it is inevitably the case that in any sequence of 's and 's different blocks of the form do not overlap. If one had replacements for blocks such as , or then these could overlap. But there is an infinite sequence of blocks such as , or for which no overlap is possible, and thus for which different replacements can never interfere.

If a rule involves replacements for several distinct blocks, then to avoid the possibility of interference one must require that these blocks can never overlap either themselves or each other. The simplest non-trivial pair of blocks that has this property is , , while the simplest triple is , , . And any substitution system whose rules specify replacements only for blocks such as these is guaranteed to yield the same causal network regardless of the order in which replacements are performed.

In general the condition is in fact somewhat weaker. For it is not necessary that no overlaps exist at all in the replacements—only that no overlaps occur in whatever sequences of elements can actually be generated by the evolution of the substitution systems.

And in the end there are then all sorts of substitution systems which have the property that the causal networks they generate are always independent of the order in which their rules are applied.

So what does this mean for models of the universe?

In a system like a cellular automaton, the same underlying rule is in a sense always applied in exact synchrony to every cell at every step. But what we have seen in this section is that there also exist systems in which rules can in effect be applied whenever and wherever one wants—but the same definite causal network always emerges.

So what this means is that there is no need for any built-in global clock, or even for any mechanism like an active cell. Simply by choosing the appropriate underlying rules it is possible to ensure that any sequence of events consistent with these rules will yield the same causal network and thus in effect the same perceived history for the universe.

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From Stephen Wolfram: A New Kind of Science [citation]