CMPS 260
Another Example of Minimizing a DFA

As another illustration of the Hopcroft-like DFA minimization algorithm, here we apply it to this DFA:

		δ	Symbol		Transition Graph
Initial?	Final?	State	a	b
✓		q0	q2	q4
		q1	q3	q2
		q2	q1	q4
	✓	q3	q1	q1
		q4	q5	q0
		q5	q1	q0

I₀ has two equivalence classes, one containing the non-final states and the other containing the final states. For the given DFA, the latter contains only state q₃.

Round 1

[0] [3] ------------------- ----- 0 1 2 4 5 3 a [0] [3] [0] [0] [0] - b [0] [0] [0] [0] [0] -

As a result of Round 1, we find that state q₁ is 1-distinguishable from the other non-final states, all of which are related in I₁.

Round 2

[0] [1] [3] ---------------- ----- ----- 0 2 4 5 1 3 a [0] [1] [0] [1] - - b [0] [0] [0] [0] - -

Round 2 determines that states q₀ and q₄ are 2-indistinguishable from each other, as are q₂ and q₅, but q₀ and q₂ are not.

Round 3

[0] [2] [1] [3] ------- ------- ----- ----- 0 4 2 5 1 3 a [2] [2] [1] [1] - - b [0] [0] [0] [0] - -

Round 3 determines that there is no need for a Round 4, as, in each equivalence class, all states have the same signature.

The conclusion is that, to minimize the given DFA, we need to merge states q₀ and q₄ into a single state, and similarly for the pair of states q₂ and q₅.

Original DFA	Minimized DFA