But the details of how one does this tend to have a great effect on the results one gets. … But now the color of that square is reversed not only if doing so actually changes the total number of squares that violate the constraint, but also if it leaves this number the same.

And it is inevitably the case that in any sequence of 's and 's different blocks of the form do not overlap. … So what does this mean for models of the universe?

But whenever there is significant randomness produced by the evolution of the multiway system, this does not typically appear to occur. … But while this means that it might be possible for there to be arbitrariness in the causal network for the universe, it still tends to be my suspicion that there is not—and that in fact the particular rules followed by the universe do in the end have the property that they always yield the same causal network.

But the first set of pictures below show two simple examples where it does not. … But to what extent does this non-planarity correspond to a definite structure in the network?

So given such an encrypting sequence, is there any easy way to do cryptanalysis and go backwards and work out the key? … But what happens if the encrypting sequence does not include every single cell in a particular column?

But now, by making use of the Principle of Computational Equivalence, I am finally able to do this. … But one of the consequences of the Principle of Computational Equivalence is that it does not.

But the string BAAB that is produced at the second step does not contain ABA , so now the first replacement cannot be used. Nevertheless, since the string does contain the single element A , the second replacement can still be used.

And indeed if one starts with almost any other initial conditions—say for example ones that do not allow any pair of black cells together, then as the second set of pictures below illustrate, rapidly increasing complexity in the sets of sequences that are allowed is again observed.

Picture (b) shows the system with sequences on successive steps rearranged so that they do not shift to the left when the first element is removed.

Symbolic representations [and networks] Expressions in which common subexpressions are shared correspond to networks, as do collections of relations between objects representing nodes.