Question 8001 – Theory-of-Computation

Extended transition function describes, what happens when we start in any state and follow any sequence of inputs.
If δ is our transition function, then the extended transition function is denoted by δ.
The extended transition function is a function that takes a state q and a string w and returns a state p (the state that the automaton reaches when starting in state q and processing the sequence of inputs w).
The starting state is q₂, from q₂, transition with input “a” is dead so we have to use epsilon transition to go to other state.

With epsilon transition we reach to q₀, at q₀ we have a transition with input symbol “a” so we reach to state q₁.

From q₁, we can take transition with symbol “b” and reach state q₂ but from q₂, again we have no further transition with symbol “a” as input, so we have to take another transition from state q₁, that is, the epsilon transition which goes to state q₂.

From q₂ we reach to state q₀ and read input “b” and then read input “a” and reach state q₁. So q₁ is one of the state of extended transition function.
From q₁ we can reach q₂ by using epsilon transition and from q₂ we can reach q₀ with epsilon move so state q₂ and q₀ are also part of extended transition function.

So state q₀,q₁,q₂.