Question

Show using a cross-product construction that the class of regular languages is closed under set difference. You do not need an inductive proof, but you should convincingly explain why your construction works.

Answer 1

Set difference operation refers to the operation which includes the first operation but should not perform the second operation.

Suppose there are 2 regular languages A and B.

The DFA for A be :

S1 S2 Sm

The DFA for B be :

T1 T2 Tn

There are m states in A and n states in B. Then, the cross product will have m*n states.

The cross product construction takes place as follows :

The transition in the resultant DFA of the A - B will be :

If in the first DFA, state Si on seeing symbol p goes to state Sj and in the second DFA, state Tk on seeing symbol p goes to state Th.

Then in the new DFA of set differenece, the state [ Si Tk ] on seeing symbol p goes to state [ Sj Th ].

The initial state will be the state comprising the initial state of first and second DFA that is state [ S1 T1 ].

As per definition of set difference, the regular language A - B will accept those strings which are part of A but not B. Then, in the resultant DFA, the final state will be [ Sm Tq ] where Tq is any non final state of the second DFA.

Since, in the resultant DFA, the states are defined using the cross product of the 2 DFAs, the initial state, final state and the transition function is also defined and the alphabet input remains the same, the resultant language is a regular language.

Hence, using a cross-product construction it can be proved that the class of regular languages is closed under set difference.