1. Kleene’s Theorem (Part-3)
CSC312
Automata Theory
Lecture # 14
Chapter # 7 by Cohen
Kleene’s Theorem (Part-3)

2. Proof of Part-3 (Cont…) Rule 4: (Closure of an FA) Proof of Rule 4:
If r is a RE and FA1 is a finite automaton that accepts exactly the language defined by r, then there is an FA called FA2 that will accepts exactly the language defined by RE r*.
Proof of Rule 4:
The language defined by r* must always contained the null word. The closure of an FA1 is same as concatenation of an FA with itself, excepts that the initial stat of the required FA is a final state as well.

3. Proof of Part-3 (Cont…)
Proof of Rule 4 (Cont…) In order to decide initial and final states of new FA we have following rule - IF the start state of FA1 is also a final state OR the start state of FA1 is not re-enterable, THEN the start state of FA2 is z1 = x1 , which must be a final state of FA2 i.e. ±z1 = x1

4. Proof of Part-3 (Cont…) Proof of Rule 4 (Cont…) ELSE
we make two states from x1 as
z1 = x1 ; z1 as initial and final state i.e. ±z1 = x1 and
z2 = x1 ; z2 as non-initial and non-final state
Note:
- whenever x1 appears under ‘new states’ columns of the transition table we treat it as z2.
Other than the start state of FA2 (which is always also a final state), the final states of FA2 are the states zk = {xi’s} for which a number of the subset {xi’s} is a final state of FA1.
(Solve exercise Q No. 6)