Formal languages over GF(2)

Variants of the union and concatenation operations on formal languages are investigated, in which Boolean logic in the definitions (that is, conjunction and disjunction) is replaced with the operations in the two-element field GF(2) (conjunction and exclusive OR). Union is thus replaced with symmetric difference, whereas concatenation gives rise to a new GF(2)-concatenation operation, which is notable for being invertible. All operations preserve regularity, and for a pair of languages recognized by an m-state and an n-state DFA, their GF(2)-concatenation is recognized by a DFA with m ⋅ 2 n states, and this number of states is in the worst case necessary. Similarly, the state complexity of GF(2)-inverse is 2 n + 1 . Next, a new class of formal grammars based on GF(2)-operations is defined, and it is shown to have the same computational complexity as ordinary grammars with union and concatenation: in particular, simple parsing in time O ( n 3 ) , fast parsing in the time of matrix multiplication, and parsing in NC2.