The regular grammar for the language L = {w|na(w) and nb(w) are both even, w ∈…

Solution overview: Construct a grammar that tracks the parity (even/odd) of the number of a's and b's. Use four states to represent the four parity combinations.

State mapping:

• p: even number of a's and even number of b's (start and accepting state)

• q: odd number of a's and even number of b's

• r: even number of a's and odd number of b's

• s: odd number of a's and odd number of b's

Productions (regular grammar):

• p → a q | b r | λ

• q → b s | a p

• r → a s | b p

• s → a r | b q

Why this works:

• Reading an 'a' moves between states that differ in the a-parity (p↔q or r↔s), and reading a 'b' moves between states that differ in the b-parity (p↔r or q↔s). Each production flips exactly the affected parity.

• The λ production at p ensures the empty string (zero a's and zero b's) is accepted, as required.

• Only p is accepting because only p represents even counts for both letters. Accepting any other state would allow strings with odd counts to be accepted.

Common mistakes to avoid:

• Omitting λ from the start state will reject the empty string, which should be accepted.

• Marking an odd-parity state as accepting will incorrectly accept strings where one or both letters occur an odd number of times.