Let X = {a, b} and Y = {0, 1} and consider : X+ Y where:

(a) (x) = { 1 if the numbers of a's and b's in x are not equal, 0 otherwise.

(b) (x) = { 1 if x contains the subsequence abb and the subsequence bbab. 0 otherwise.

Note that the two subsequences could overlap. For example, both the strings bbabaabba and abbab contain both subsequences.

(i) For each of the two functions above, describe the Nerode equivalence classes [x] and the functions lx ( x X* ).

(ii) For each of the two functions above, if is not finite state realizable, prove it; if is finite state realizable give a transition graph for the reduced machine M or M(