What is Inductive Hypothesis in TOC?

Induction is a powerful tool in mathematics. It is a way of proving propositions that hold for all natural numbers.

Hypothesis − The formal proof can be using deductive proof and inductive proof. The deductive proof consists of sequence of statements given with logical reasoning in order to prove the first or initial statement. The initial statement is called Hypothesis.

Suppose there exists a k > 0 such that for any regular expression r where 0

Inductive Step

Let r be a regular expression with k + 1 operators (OP(r) = k + 1), where k + 1 >= 1.

Case 1 − r = ri + r2

Since OP(r) = k +1, it follows that 0

Furthermore, both M1 and M2 have exactly one final state. We can construct M as given below −

Case 2 − r = r1r2

Since OP(r) = k+1, it follows that 0

Furthermore, both M1 and M2 have exactly one final state. We can now construct M as given below −

Case 3 − r = ri*

Since OP(r) = k+1, it follows that 0

Furthermore, M1 has exactly one final state. Hence, we can construct M as follows −