[Patterns from] arbitrary digit operations

If the operation on digit sequences that determines whether a square will be black can be performed by a finite automaton (see page 957) then the pattern generated must always be either repetitive or nested. The pictures below show examples with more general operations. Picture (a) in effect shows which words in a simple context-free language of parenthesis matching (see page 939) are syntactically correct. Scanning the digit sequences from the left, one starts with 0 open parentheses, then adds 1 whenever corresponding digits in the x and y coordinates differ, and subtracts 1 whenever they are the same. A square is black if no negative number ever appears. Picture (b) has a black square wherever digits at more than half the possible positions differ between the x and y coordinates. Picture (c) has a black square wherever the maximum run of either identical or different digits has a length which is an odd number. All the patterns shown have the kind of intricate substructure typical of nesting. But none of the patterns are purely nested.