The n-input function or a finite regular language can therefore be computed by a branching programme that consists of a constant size, according to the definitions of branching programme and regular language A as given above.
A circuit n is implemented as a Bubble-sort. Each branching programme in a group is chosen so that they all accept exactly the strings in A of length n.
After carefully comparing, it can be used to compare two bits. Let the inputs beand the outputs be.
A sub-circuit can be written which accomplishes this as &. This circuit contains a size of two.
Now, an array can be mimicked using the action of the bubble-sort algorithm. It can be implemented one step at position to be the n input, n-output subcircuit that passes through all the inputs taken as and are unchanged.
Now, the above described compare-swap sub-circuit, on and input is used for the generation of and output. This still has size two. Now, a pass can be implemented as the serial concatenation of steps for each of , which has a size.
A bubble-sort can be Proceed to implement as the serial concatenation of one pass. Therefore, this gives a size
This means that "a family of branching programmes in which a string, of length n in language A, is accepted by each member " may be inferred from the reasoning given above.