Chapter 4: Decidability I

Chapter 4: Decidability I

Now we have algorithms

We want to check problems solvable or not by computers

Need a TM to decide it

i.e., accept/reject in a finite number of steps We will show some examples

Acceptance Problems for DFA I

A_DFA = {hB, w i | B is a DFA that accepts w }

hB, w i is the input

Note that a DFA can be represented as a string (Q, Σ, . . .)

Is A_DFA decidable?

Idea: input hB, w i

1 simulate B on w

2 ends in an accept state ⇒ accept otherwise ⇒ reject

Proof of A

I

Put

B = hQ, Σ, δ, q₀, F i into a tape

Check if w ∈ Σ^∗ and B a valid DFA Simulate w according to δ

After processing the last element of w , check if in a final state

A

I

A_NFA = {hB, w i | B is an NFA that accepts w }

We can convert B to a DFA and use the procedure for A_DFA

It’s like to use the procedure for A_DFA as a subroutine

A

I

A_REX = {hR, w i | R : regular expression generates w }

It’s similar

We convert R to a DFA first

Recall that we had a procedure to convert R to an NFA. Then we can convert the NFA to a DFA The key is that the conversion is a finite procedure

E

I

E_DFA = {hAi | A : DFA, L(A) = ∅}

i.e. A accepts nothing Idea:

DFA accepts something

⇔ reaching a final state from q₀ after several links procedure

1 mark q0

E

II

2 repeat until no new state marked mark all

a → b, where a has been marked

3 if no q ∈ F marked, accept. otherwise, reject Example: a state diagram with 3 nodes and the following connections

1 → 2, 3

E

III

Marked states in running the procedure 1

12 12

Each iteration: at least one new state marked At most n iterations: n: # states

EQ

I

EQ_DFA = {hA, Bi | A, B : DFAs, L(A) = L(B)}

EQDFA is decidable Idea for the proof:

Let a DFA C be the exclusive or of A and B If

L(A) = L(B) then

L(C ) = ∅

EQ

II

A B

Exclusive or of A and B

(latex source from

https://texample.net/tikz/examples/

set-operations-illustrated-with-venn-diagrams/)

EQ

III

Formally

L(C ) = (L(A) ∩ L(B)) ∪ (L(A) ∩ L(B)) B DFA ⇒ so is B

A, B DFA ⇒ so is A ∪ B, A ∩ B

We then use E_DFA to check if L(C ) = ∅ or not