Question

Find the final DFA (Deterministic Finite Automata) by performing the DFA minimization process.

Answer 1

The final minimized DFA is given by the transition table above.

Answer 2

The process of DFA minimization involves identifying and merging equivalent states. We use the table-filling algorithm (also known as the Myhill-Nerode theorem based approach) to achieve this.

Given DFA Transition Table:

State/input	0	1
->q₀	q₁	q₀
q₁	q₀	q₂
q₂	q₃	q₁
q₃*	q₃	q₀
q₄	q₃	q₅
q₅	q₆	q₄
q₆	q₅	q₆
q₇	q₆	q₃

Initial state: q₀ Final state: q₃

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DFA Minimization Process:

Step 1: Initial Partition (P₀) Separate final states from non-final states. P₀ = {{q₃}, {q₀, q₁, q₂, q₄, q₅, q₆, q₇}}

Step 2: First Refinement (P₁) Let F = {q₃} and NF = {q₀, q₁, q₂, q₄, q₅, q₆, q₇}. We examine transitions for states in NF based on which partition their next states fall into (F or NF).

• q₀: δ(q₀, 0) = q₁ ∈ NF, δ(q₀, 1) = q₀ ∈ NF
• q₁: δ(q₁, 0) = q₀ ∈ NF, δ(q₁, 1) = q₂ ∈ NF
• q₂: δ(q₂, 0) = q₃ ∈ F, δ(q₂, 1) = q₁ ∈ NF
• q₄: δ(q₄, 0) = q₃ ∈ F, δ(q₄, 1) = q₅ ∈ NF
• q₅: δ(q₅, 0) = q₆ ∈ NF, δ(q₅, 1) = q₄ ∈ NF
• q₆: δ(q₆, 0) = q₅ ∈ NF, δ(q₆, 1) = q₆ ∈ NF
• q₇: δ(q₇, 0) = q₆ ∈ NF, δ(q₇, 1) = q₃ ∈ F

Based on the above, the set NF splits:

• States that transition to F for input 0: {q₂, q₄}
• States that transition to F for input 1: {q₇}
• States that transition only to NF for both inputs: {q₀, q₁, q₅, q₆}

So, P₁ = {{q₃}, {q₂, q₄}, {q₇}, {q₀, q₁, q₅, q₆}}

Step 3: Second Refinement (P₂) Let the partitions in P₁ be: P₁_1 = {q₃} P₁_2 = {q₂, q₄} P₁_3 = {q₇} P₁_4 = {q₀, q₁, q₅, q₆}

• Check P₁_2 = {q₂, q₄}:

  • δ(q₂, 0) = q₃ ∈ P₁_1, δ(q₄, 0) = q₃ ∈ P₁_1
  • δ(q₂, 1) = q₁ ∈ P₁_4, δ(q₄, 1) = q₅ ∈ P₁_4 Since q₃ and q₃ are in the same partition (P₁_1), and q₁ and q₅ are in the same partition (P₁_4), q₂ and q₄ are not distinguishable. This partition remains {q₂, q₄}.

• Check P₁_3 = {q₇}: This is a single state, so it remains {q₇}.

• Check P₁_4 = {q₀, q₁, q₅, q₆}:

  • q₀: δ(q₀, 0) = q₁ ∈ P₁_4, δ(q₀, 1) = q₀ ∈ P₁_4
  • q₁: δ(q₁, 0) = q₀ ∈ P₁_4, δ(q₁, 1) = q₂ ∈ P₁_2 (Note: q₂ is in P₁_2, not P₁_4)
  • q₅: δ(q₅, 0) = q₆ ∈ P₁_4, δ(q₅, 1) = q₄ ∈ P₁_2 (Note: q₄ is in P₁_2, not P₁_4)
  • q₆: δ(q₆, 0) = q₅ ∈ P₁_4, δ(q₆, 1) = q₆ ∈ P₁_4

  Based on transitions for input 1:

  • States whose δ(s, 1) goes to P₁_2: {q₁, q₅}
  • States whose δ(s, 1) goes to P₁_4: {q₀, q₆} Thus, P₁_4 splits into {{q₁, q₅}, {q₀, q₆}}.

So, P₂ = {{q₃}, {q₂, q₄}, {q₇}, {q₁, q₅}, {q₀, q₆}}

Step 4: Third Refinement (P₃) Let the partitions in P₂ be: P₂_1 = {q₃} P₂_2 = {q₂, q₄} P₂_3 = {q₇} P₂_4 = {q₁, q₅} P₂_5 = {q₀, q₆}

• Check P₂_2 = {q₂, q₄}:

  • δ(q₂, 0) = q₃ ∈ P₂_1, δ(q₄, 0) = q₃ ∈ P₂_1
  • δ(q₂, 1) = q₁ ∈ P₂_4, δ(q₄, 1) = q₅ ∈ P₂_4 No split.

• Check P₂_4 = {q₁, q₅}:

  • δ(q₁, 0) = q₀ ∈ P₂_5, δ(q₅, 0) = q₆ ∈ P₂_5
  • δ(q₁, 1) = q₂ ∈ P₂_2, δ(q₅, 1) = q₄ ∈ P₂_2 No split.

• Check P₂_5 = {q₀, q₆}:

  • δ(q₀, 0) = q₁ ∈ P₂_4, δ(q₆, 0) = q₅ ∈ P₂_4
  • δ(q₀, 1) = q₀ ∈ P₂_5, δ(q₆, 1) = q₆ ∈ P₂_5 No split.

Since P₃ = P₂, the algorithm terminates.

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Final Minimized DFA:

We assign new names to the equivalent classes:

• A = {q₀, q₆} (Initial State, as q₀ is the original initial state)
• B = {q₁, q₅}
• C = {q₂, q₄}
• D = {q₃} (Final State, as q₃ is the original final state)
• E = {q₇}

Minimized DFA Transition Table:

State/input	0	1
->A	B	A
B	A	C
C	D	B
D*	D	A
E	A	D

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Solution:

The final minimized DFA is represented by the following transition table:

State/input	0	1
->A	B	A
B	A	C
C	D	B
D*	D	A
E	A	D

Where:

• Initial state is A.
• Final state is D.
• The new states correspond to the following sets of original states:
  • A = {q₀, q₆}
  • B = {q₁, q₅}
  • C = {q₂, q₄}
  • D = {q₃}
  • E = {q₇}

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Explanation of the solution:

1. Initial Partition: Group states into final and non-final sets (P₀).
2. Iterative Refinement: For each partition, check if any two states within it are distinguishable. States p and q are distinguishable if for some input a, δ(p, a) and δ(q, a) fall into different partitions of the previous step.
3. Splitting: If states are distinguishable, split the partition into smaller ones.
4. Termination: Repeat until no more partitions can be split (Pᵢ = Pᵢ₊₁).
5. Construct Minimized DFA: Each final partition forms a new state in the minimized DFA. The initial and final states are determined by which new states contain the original initial and final states.