Formal Languages and Automata - DFA Minimizationryan/cse4083/notes/min.pdf ·...
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Formal Languages and Automata DFA Minimization Ryan Stansifer Computer Sciences Florida Institute of Technology Melbourne, Florida USA 32901 http://www.cs.fit.edu/~ryan/ 2 July 2021 FL and Automata (DFA Minimization) © 2 July 2021 1/1
Transcript of Formal Languages and Automata - DFA Minimizationryan/cse4083/notes/min.pdf ·...
Formal Languages and AutomataDFA Minimization
Ryan Stansifer
Computer SciencesFlorida Institute of Technology
Melbourne, Florida USA 32901
http://www.cs.fit.edu/~ryan/
2 July 2021
FL and Automata (DFA Minimization) © 2 July 2021 1 / 1
Linz 6th, Section 2.4, pages 66–71HMU 3rd, Section 4.4, pages 155–165
Kozen 1997, Lecture 13 & 14, pages 77–88DFA minimization at Wikipedia
FL and Automata (DFA Minimization) © 2 July 2021 2 / 1
Any DFA defines one language, but a language may be defined by differentDFA. For a particular language there are many DFA’s that accept it.
For reasons of simplicity a DFA with the fewest number of state may bepreferred.
FL and Automata (DFA Minimization) © 2 July 2021 3 / 1
DefinitionA state p of a DFA ⟨Q, Σ, δ, q0, F ⟩ is said to be inaccessible if for all x ∈ Σ∗ itis the case the δ∗(q0, x) ̸= p.
FL and Automata (DFA Minimization) © 2 July 2021 4 / 1
Reach := {q0}New := {q0}loop
New := { delta(q,c) for q in New for c in Sigma } - ReachReach := Reach union Newexit when New is empty
end loop;UnReach = Q - Reach
FL and Automata (DFA Minimization) © 2 July 2021 5 / 1
DefinitionThe states p and q of a DFA ⟨Q, Σ, δ, q0, F ⟩ are said to be indistinguishable,written p ≈ q, if for all x ∈ Σ∗
δ∗(p, x) ∈ F ↔ δ∗(q, x) ∈ F
If two states are not indistinguishable, then they are distinguishable. Or,equivalently, we may define the following:
DefinitionThe states p and q of a DFA ⟨Q, Σ, δ, q0, F ⟩ are said to be distinguishable iffor some x ∈ Σ∗
δ∗(p, x) ∈ F xor δ∗(q, x) ∈ F
FL and Automata (DFA Minimization) © 2 July 2021 6 / 1
TheoremThe relation p ≈ q of indistinguishable states is an equivalence relation, i.e., itis reflexive, symmetric, and transitive.
Proof.It is obviously reflexive and symmetric.The proof is by contradiction. Suppose p ≈ q and q ≈ r , but p and r aredistinguishable by some string w . Suppose that δ∗(p, w) ∈ F and so δ∗(r , w)must not be in F . Since p ≈ q, δ∗(q, w) ∈ F . This contradicts the fact that qand r are indistinguishable. Similarly if δ∗(p, w) /∈ F . We conclude p ≈ r .
The significance of this is that the indistinguishable relation partitions the setof states of a DFA into equivalence classes.
[Notation] We write [p]≈ for the set {q ∈ Q | p ≈ q}
FL and Automata (DFA Minimization) © 2 July 2021 7 / 1
[Definintion] The DFA M/ ≈ is defined from a given DFA M as ⟨Q′, Σ, δ′, q′, F ′⟩where
• Q′ = {[p]≈ | p ∈ Q}• q′ = [q]≈• F ′ = {[p]≈ | p ∈ F}• δ′([p]≈, a) = [δ(p, a)] ≈ for all a ∈ Σ
The DFA M/ ≈ is well-defined since for all a ∈ Σ and all states in Q
p ≈ q ⇒ δ(p, a) ≈ δ(q, a)
FL and Automata (DFA Minimization) © 2 July 2021 8 / 1
TheoremThe language L(M/ ≈) = L(M).
Proof.
x ∈ L(M/ ≈) iff δ∗() ∈ F ′
IFF [δ∗(s, x)]≈ ∈ F ′
IFF δ∗(s, x) ∈ FIFF x ∈ L(M)
FL and Automata (DFA Minimization) © 2 July 2021 9 / 1
Minimization Algorithm
Two algorithms1 Hopcroft’s partition refinement2 Brzozowski: reverse edges, convert to DFA, and do it again
FL and Automata (DFA Minimization) © 2 July 2021 10 / 1
Minimization Algorithm
For a DFA ⟨Q, Σ, δ, q0, F ⟩:1 Remove inaccessible state2 For every pair of states, mark whether or not they are distinguishable.3 Collapse indistinguishable states.
The states of the minimized DFA are non-empty, pairwise-disjoint subsets of theoriginal DFA.
FL and Automata (DFA Minimization) © 2 July 2021 11 / 1
Marking Algorithm
For a DFA ⟨Q, Σ, δ, q0, F ⟩ systematically try longer and longer strings to establishthat a pair of states is distinguishable.
1 Mark all unordered pairs {p, q} ∈ Q × Q as indistinguishable.2 Mark {p, q} as distinguishable, if p ∈ F xor q ∈ F .3 Repeat until no further changes: mark {p, q} as distinguishable, if
{δ(p, a), δ(q, a)} is distinguishable for some a ∈ Σ.
FL and Automata (DFA Minimization) © 2 July 2021 12 / 1
DFA MinimizationAn Example
Combining Indistinquishable StatesLinz 6th, Example 2.18, page 69
DFA Minimization 13 Linz 6th, Example 2.18—1 of 23
Minimize a DFAFind a DFA equivalent to the one below with the minimum number of states.
A
B
C
D
E
a
b
ab
a b
a
b
a, b
Q Σ QA a BA b DB a CB b EC a BC b ED a CD b EE a EE b E
An example DFA. Linz 6th, Figure 2.18, page 69.DFA Minimization 14 Linz 6th, Example 2.18—2 of 23
Minimize a DFA
A
B
C
D
E
a
b
ab
a b
a
b
a, b AB
CD
E
DFA Minimization 15 Linz 6th, Example 2.18—3 of 23
Minimize a DFA
A
B
C
D
E
a
b
ab
a b
a
b
a, b A ≈B
CD
E
A, B: A and B are both non-final states.
DFA Minimization 16 Linz 6th, Example 2.18—4 of 23
Minimize a DFA
A
B
C
D
E
a
b
ab
a b
a
b
a, b A ≈ ×B
CD
E
A, C : A is not final, but C is final.
DFA Minimization 17 Linz 6th, Example 2.18—5 of 23
Minimize a DFA
A
B
C
D
E
a
b
ab
a b
a
b
a, b A ≈ × ≈B
CD
E
A, D: A and D are both not final.
DFA Minimization 18 Linz 6th, Example 2.18—6 of 23
Minimize a DFA
A
B
C
D
E
a
b
ab
a b
a
b
a, b A ≈ × ≈ ×B
CD
E
A, E : A is not final, but E is final.
DFA Minimization 19 Linz 6th, Example 2.18—7 of 23
Minimize a DFA
A
B
C
D
E
a
b
ab
a b
a
b
a, b A ≈ × ≈ ×B ×
CD
E
B, C : B is not final, but C is final.
DFA Minimization 20 Linz 6th, Example 2.18—8 of 23
Minimize a DFA
A
B
C
D
E
a
b
ab
a b
a
b
a, b A ≈ × ≈ ×B × ≈
CD
E
B, D: B and D are both not final.
DFA Minimization 21 Linz 6th, Example 2.18—9 of 23
Minimize a DFA
A
B
C
D
E
a
b
ab
a b
a
b
a, b A ≈ × ≈ ×B × ≈ ×
CD
E
B, E : B is final, but E is not final.
DFA Minimization 22 Linz 6th, Example 2.18—10 of 23
Minimize a DFA
A
B
C
D
E
a
b
ab
a b
a
b
a, b A ≈ × ≈ ×B × ≈ ×
C ×D
E
C , D: C is final but D is not final.
DFA Minimization 23 Linz 6th, Example 2.18—11 of 23
Minimize a DFA
A
B
C
D
E
a
b
ab
a b
a
b
a, b A ≈ × ≈ ×B × ≈ ×
C × ≈D
E
C , E : Both C and E are final.
DFA Minimization 24 Linz 6th, Example 2.18—12 of 23
Minimize a DFA
A
B
C
D
E
a
b
ab
a b
a
b
a, b A ≈ × ≈ ×B × ≈ ×
C × ≈D ×
E
D, E : D is not final, but D is final.
DFA Minimization 25 Linz 6th, Example 2.18—13 of 23
Minimize a DFA
A
B
C
D
E
a
b
ab
a b
a
b
a, b A ≈ × ≈ ×B × ≈ ×
C × ≈D ×
E
The resulting partition: {A, B, D}, {C , E}
DFA Minimization 26 Linz 6th, Example 2.18—14 of 23
Minimize a DFA
All pairs which were marked distinguishable earlier, remain distinguishable from thenon. All the others are re-examined to see if they might become distinguishable.
A
B
C
D
E
a
b
ab
a b
a
b
a, b A × ×B ×
C ×D ×
E
DFA Minimization 27 Linz 6th, Example 2.18—15 of 23
Minimize a DFA
A
B
C
D
E
a
b
ab
a b
a
b
a, b A × × ×B ×
C ×D ×
E
A, B: a ∈ Σ distinguishes A from B, as δ(A, a) = B /∈ F but δ(B, a) = C ∈ F .
DFA Minimization 28 Linz 6th, Example 2.18—16 of 23
Minimize a DFA
A
B
C
D
E
a
b
ab
a b
a
b
a, b A × × × ×B ×
C ×D ×
E
A, C : a ∈ Σ distinguishes A from D, as δ(A, a) = B /∈ F but δ(D, a) = C ∈ F .
DFA Minimization 29 Linz 6th, Example 2.18—17 of 23
Minimize a DFA
A
B
C
D
E
a
b
ab
a b
a
b
a, b A × × × ×B × ≈
C ×D ×
E
B and D are indistinguishable, as δ(B, a) = δ(D, a) and δ(B, b) = δ(D, b).
DFA Minimization 30 Linz 6th, Example 2.18—18 of 23
Minimize a DFA
A
B
C
D
E
a
b
ab
a b
a
b
a, b A × × × ×B × ≈ ×
C ×D ×
E
a ∈ Σ distinguishes C from E , as δ(A, a) = B /∈ F but δ(D, a) = C ∈ F .
DFA Minimization 31 Linz 6th, Example 2.18—19 of 23
Minimize a DFA
A
B
C
D
E
a
b
ab
a b
a
b
a, b A × × × ×B × ≈ ×
C × ×D ×
E
DFA Minimization 32 Linz 6th, Example 2.18—20 of 23
Minimize a DFA
A
B
C
D
E
a
b
ab
a b
a
b
a, b A × × × ×B × ≈ ×
C × ×D ×
E
The resulting partition: {A}, {B, D}, {C}, {E}
DFA Minimization 33 Linz 6th, Example 2.18—21 of 23
Minimize a DFA
Repeating the step creates no changes. The partition {A}, {B, D}, {C}, {E} cannotbe further refined. This can be seen from the fact the state B and D go to theidentical states on all symbols of the alphabet.
A
B
C
D
E
a
b
ab
a b
a
b
a, b A × × × ×B × ≈ ×
C × ×D ×
E
DFA Minimization 34 Linz 6th, Example 2.18—22 of 23
Minimize a DFA (Solution)The minimized DFA has a state for each equivalence class produced by the markingalgorithm. (One state fewer.) The equivalence class with the original start state isthe start state of the minimized DFA. Final states are those in which appears a finalstate of the original DFA.
A BD C E
a, b ab
a
b
a, b
Q Σ QA a BDA b BD
BD a CBD b EC a BDC b EE a EE b E
DFA Minimization 35 Linz 6th, Example 2.18—23 of 23
DFA MinimizationAn Example
Combining Indistinquishable StatesFrom notes from U of Innsbruck
DFA Minimization 36 Linz 6th, Example 2.18—1 of 40
Example
Design a DFA over {a, b} containing at least three occurrences of three consecutiveb’s, overlapping permitted.
Creating a DFA for this language is not so hard. Then minimize it.
DFA Minimization 37 Linz 6th, Example 2.18—2 of 40
Here is a DFA over {a, b} containing at least three occurrences of three consecutiveb’s, overlapping permitted.
A B C D
EFGH
I J K L
a
b
a
b
a
b
ab
b
aabb
aa b
ba
b b
aa, b
Now find an equivalent DFA with the minimum number of states.DFA Minimization 38 Linz 6th, Example 2.18—3 of 40
A B C D
EFGH
I J K L
a
b
a
b
a
b
ab
b
aabb
aa b
ba
b b
aa, b
A ≈ ≈ ≈ ≈ ≈ ≈ ≈ ≈ ≈ ≈ ×B ≈ ≈ ≈ ≈ ≈ ≈ ≈ ≈ ≈ ×
C ≈ ≈ ≈ ≈ ≈ ≈ ≈ ≈ ×D ≈ ≈ ≈ ≈ ≈ ≈ ≈ ×
E ≈ ≈ ≈ ≈ ≈ ≈ ×F ≈ ≈ ≈ ≈ ≈ ×
G ≈ ≈ ≈ ≈ ×H ≈ ≈ ≈ ×
I ≈ ≈ ×J ≈ ×
K ×L
The first iteration merely separates the final from the non-final states. The resultingpartition:
{A, B, C , D, E , F , G , H, I, J , K}, {L}DFA Minimization 39 Linz 6th, Example 2.18—4 of 40
Second Iteration
A B C D
EFGH
I J K L
a
b
a
b
a
b
ab
b
aabb
aa b
ba
b b
aa, b
A ×B ≈ ≈ ×
C ≈ ≈ ×D ≈ ≈ ×
E ≈ ≈ ×F ≈ ≈ ×
G ≈ ≈ ×H ≈ ×
I ≈ ×J ≈ ×
K ×L
DFA Minimization 40 Linz 6th, Example 2.18—5 of 40
Second Iteration
A B C D
EFGH
I J K L
a
b
a
b
a
b
ab
b
aabb
aa b
ba
b b
aa, b
A ≈ ≈ ≈ ≈ ≈ ≈ ≈ ≈ ×B ≈ ≈ ≈ ≈ ≈ ≈ ≈ ≈ ≈ ×
C ≈ ≈ ≈ ≈ ≈ ≈ ≈ ≈ ×D ≈ ≈ ≈ ≈ ≈ ≈ ≈ ×
E ≈ ≈ ≈ ≈ ≈ ≈ ×F ≈ ≈ ≈ ≈ ≈ ×
G ≈ ≈ ≈ ≈ ×H ≈ ×
I ≈ ≈ ×J ≈ ×
K ×L
for all s, s ′ ∈ {B · · · G , I, H} and for ⋆ ∈ a, b, δ(s, ⋆) and δ(s ′, ⋆) are non-final.DFA Minimization 41 Linz 6th, Example 2.18—6 of 40
Second Iteration
A B C D
EFGH
I J K L
a
b
a
b
a
b
ab
b
aabb
aa b
ba
b b
aa, b
A ≈ ≈ ≈ ≈ ≈ ≈ × ≈ ≈ ×B ≈ ≈ ≈ ≈ ≈ ≈ ≈ ≈ ≈ ×
C ≈ ≈ ≈ ≈ ≈ ≈ ≈ ≈ ×D ≈ ≈ ≈ ≈ ≈ ≈ ≈ ×
E ≈ ≈ ≈ ≈ ≈ ≈ ×F ≈ ≈ ≈ ≈ ≈ ×
G ≈ ≈ ≈ ≈ ×H ≈ ×
I ≈ ≈ ×J ≈ ×
K ×L
A, H:DFA Minimization 42 Linz 6th, Example 2.18—7 of 40
Second Iteration
A B C D
EFGH
I J K L
a
b
a
b
a
b
ab
b
aabb
aa b
ba
b b
aa, b
A ≈ ≈ ≈ ≈ ≈ ≈ × ≈ ≈ × ×B ≈ ≈ ≈ ≈ ≈ ≈ ≈ ≈ ≈ ×
C ≈ ≈ ≈ ≈ ≈ ≈ ≈ ≈ ×D ≈ ≈ ≈ ≈ ≈ ≈ ≈ ×
E ≈ ≈ ≈ ≈ ≈ ≈ ×F ≈ ≈ ≈ ≈ ≈ ×
G ≈ ≈ ≈ ≈ ×H ≈ ×
I ≈ ≈ ×J ≈ ×
K ×L
A, D:DFA Minimization 43 Linz 6th, Example 2.18—8 of 40
Second Iteration
A B C D
EFGH
I J K L
a
b
a
b
a
b
ab
b
aabb
aa b
ba
b b
aa, b
A ≈ ≈ ≈ ≈ ≈ ≈ × ≈ ≈ × ×B ≈ ≈ ≈ ≈ ≈ ≈ ≈ ≈ ≈ ×
C ≈ ≈ ≈ ≈ ≈ ≈ ≈ ≈ ×D ≈ ≈ ≈ ≈ ≈ ≈ ≈ ×
E ≈ ≈ ≈ ≈ ≈ ≈ ×F ≈ ≈ ≈ ≈ ≈ ×
G ≈ ≈ ≈ ≈ ×H × ≈ ×
I ≈ ≈ ×J ≈ ×
K ×L
A, E : A and D are both non-final states.DFA Minimization 44 Linz 6th, Example 2.18—9 of 40
Second Iteration
A B C D
EFGH
I J K L
a
b
a
b
a
b
ab
b
aabb
aa b
ba
b b
aa, b
A ≈ ≈ ≈ ≈ ≈ ≈ × ≈ ≈ × ×B ≈ ≈ ≈ ≈ ≈ ≈ ≈ ≈ ≈ ×
C ≈ ≈ ≈ ≈ ≈ ≈ ≈ ≈ ×D ≈ ≈ ≈ ≈ ≈ ≈ ≈ ×
E ≈ ≈ ≈ ≈ ≈ ≈ ×F ≈ ≈ ≈ ≈ ≈ ×
G ≈ ≈ ≈ ≈ ×H × × ≈ ×
I ≈ ≈ ×J ≈ ×
K ×L
DFA Minimization 45 Linz 6th, Example 2.18—10 of 40
Second Iteration
A B C D
EFGH
I J K L
a
b
a
b
a
b
ab
b
aabb
aa b
ba
b b
aa, b
A ≈ ≈ ≈ ≈ ≈ ≈ × ≈ ≈ × ×B ≈ ≈ ≈ ≈ ≈ ≈ ≈ ≈ ≈ ×
C ≈ ≈ ≈ ≈ ≈ ≈ ≈ ≈ ×D ≈ ≈ ≈ ≈ ≈ ≈ ≈ ×
E ≈ ≈ ≈ ≈ ≈ ≈ ×F ≈ ≈ ≈ ≈ ≈ ×
G ≈ ≈ ≈ ≈ ×H × × ≈ ×
I ≈ ≈ ×J ≈ ×
K ×L
The resulting partition: {A, B, C , D, E , F , G , I, J}, {H, K}, {L}DFA Minimization 46 Linz 6th, Example 2.18—11 of 40
Third Iteration
A B C D
EFGH
I J K L
a
b
a
b
a
b
ab
b
aabb
aa b
ba
b b
aa, b
A ×B ≈ ≈ ×
C ≈ ≈ ×D ≈ ≈ ×
E ≈ ≈ ×F ≈ ≈ ×
G ≈ ≈ ×H ≈ ×
I ≈ ×J ≈ ×
K ×L
DFA Minimization 47 Linz 6th, Example 2.18—12 of 40
Third Iteration
A B C D
EFGH
I J K L
a
b
a
b
a
b
ab
b
aabb
aa b
ba
b b
aa, b
A ≈ ≈ ≈ ≈ ≈ ≈ ≈ ≈ ×B ≈ ≈ ≈ ≈ ≈ ≈ ≈ ≈ ≈ ×
C ≈ ≈ ≈ ≈ ≈ ≈ ≈ ≈ ×D ≈ ≈ ≈ ≈ ≈ ≈ ≈ ×
E ≈ ≈ ≈ ≈ ≈ ≈ ×F ≈ ≈ ≈ ≈ ≈ ×
G ≈ ≈ ≈ ≈ ×H ≈ ×
I ≈ ≈ ×J ≈ ×
K ×L
for all s, s ′ ∈ {B · · · G , I, H} and for ⋆ ∈ a, b, δ(s, ⋆) and δ(s ′, ⋆) are non-final.DFA Minimization 48 Linz 6th, Example 2.18—13 of 40
Third Iteration
A B C D
EFGH
I J K L
a
b
a
b
a
b
ab
b
aabb
aa b
ba
b b
aa, b
A ≈ ≈ ≈ ≈ ≈ ≈ × ≈ ≈ ×B ≈ ≈ ≈ ≈ ≈ ≈ ≈ ≈ ≈ ×
C ≈ ≈ ≈ ≈ ≈ ≈ ≈ ≈ ×D ≈ ≈ ≈ ≈ ≈ ≈ ≈ ×
E ≈ ≈ ≈ ≈ ≈ ≈ ×F ≈ ≈ ≈ ≈ ≈ ×
G ≈ ≈ ≈ ≈ ×H ≈ ×
I ≈ ≈ ×J ≈ ×
K ×L
A, H:DFA Minimization 49 Linz 6th, Example 2.18—14 of 40
Third Iteration
A B C D
EFGH
I J K L
a
b
a
b
a
b
ab
b
aabb
aa b
ba
b b
aa, b
A ≈ ≈ ≈ ≈ ≈ ≈ × ≈ ≈ × ×B ≈ ≈ ≈ ≈ ≈ ≈ ≈ ≈ ≈ ×
C ≈ ≈ ≈ ≈ ≈ ≈ ≈ ≈ ×D ≈ ≈ ≈ ≈ ≈ ≈ ≈ ×
E ≈ ≈ ≈ ≈ ≈ ≈ ×F ≈ ≈ ≈ ≈ ≈ ×
G ≈ ≈ ≈ ≈ ×H ≈ ×
I ≈ ≈ ×J ≈ ×
K ×L
A, D:DFA Minimization 50 Linz 6th, Example 2.18—15 of 40
Third Iteration
A B C D
EFGH
I J K L
a
b
a
b
a
b
ab
b
aabb
aa b
ba
b b
aa, b
A ≈ ≈ ≈ ≈ ≈ ≈ × ≈ ≈ × ×B ≈ ≈ ≈ ≈ ≈ ≈ ≈ ≈ ≈ ×
C ≈ ≈ ≈ ≈ ≈ ≈ ≈ ≈ ×D ≈ ≈ ≈ ≈ ≈ ≈ ≈ ×
E ≈ ≈ ≈ ≈ ≈ ≈ ×F ≈ ≈ ≈ ≈ ≈ ×
G ≈ ≈ ≈ ≈ ×H × ≈ ×
I ≈ ≈ ×J ≈ ×
K ×L
DFA Minimization 51 Linz 6th, Example 2.18—16 of 40
Third Iteration
A B C D
EFGH
I J K L
a
b
a
b
a
b
ab
b
aabb
aa b
ba
b b
aa, b
A ≈ ≈ ≈ ≈ ≈ ≈ × ≈ ≈ × ×B ≈ ≈ ≈ ≈ ≈ ≈ ≈ ≈ ≈ ×
C ≈ ≈ ≈ ≈ ≈ ≈ ≈ ≈ ×D ≈ ≈ ≈ ≈ ≈ ≈ ≈ ×
E ≈ ≈ ≈ ≈ ≈ ≈ ×F ≈ ≈ ≈ ≈ ≈ ×
G ≈ ≈ ≈ ≈ ×H × × ≈ ×
I ≈ ≈ ×J ≈ ×
K ×L
DFA Minimization 52 Linz 6th, Example 2.18—17 of 40
Third Iteration
A B C D
EFGH
I J K L
a
b
a
b
a
b
ab
b
aabb
aa b
ba
b b
aa, b
A ≈ ≈ ≈ ≈ ≈ ≈ × ≈ ≈ × ×B ≈ ≈ ≈ ≈ ≈ ≈ ≈ ≈ ≈ ×
C ≈ ≈ ≈ ≈ ≈ ≈ ≈ ≈ ×D ≈ ≈ ≈ ≈ ≈ ≈ ≈ ×
E ≈ ≈ ≈ ≈ ≈ ≈ ×F ≈ ≈ ≈ ≈ ≈ ×
G ≈ ≈ ≈ ≈ ×H × × ≈ ×
I ≈ ≈ ×J ≈ ×
K ×L
The resulting partition: {A, B, C , E , F , I}, {D, G , J}, {H, K}, {L}DFA Minimization 53 Linz 6th, Example 2.18—18 of 40
Fouth Iteration
A B C D
EFGH
I J K L
a
b
a
b
a
b
ab
b
aabb
aa b
ba
b b
aa, b
A ×B ≈ ≈ ×
C ≈ ≈ ×D ≈ ≈ ×
E ≈ ≈ ×F ≈ ≈ ×
G ≈ ≈ ×H ≈ ×
I ≈ ×J ≈ ×
K ×L
DFA Minimization 54 Linz 6th, Example 2.18—19 of 40
Fouth Iteration
A B C D
EFGH
I J K L
a
b
a
b
a
b
ab
b
aabb
aa b
ba
b b
aa, b
A ≈ ≈ ≈ ≈ ≈ ≈ ≈ ≈ ×B ≈ ≈ ≈ ≈ ≈ ≈ ≈ ≈ ≈ ×
C ≈ ≈ ≈ ≈ ≈ ≈ ≈ ≈ ×D ≈ ≈ ≈ ≈ ≈ ≈ ≈ ×
E ≈ ≈ ≈ ≈ ≈ ≈ ×F ≈ ≈ ≈ ≈ ≈ ×
G ≈ ≈ ≈ ≈ ×H ≈ ×
I ≈ ≈ ×J ≈ ×
K ×L
for all s, s ′ ∈ {B · · · G , I, H} and for ⋆ ∈ a, b, δ(s, ⋆) and δ(s ′, ⋆) are non-final.DFA Minimization 55 Linz 6th, Example 2.18—20 of 40
Fouth Iteration
A B C D
EFGH
I J K L
a
b
a
b
a
b
ab
b
aabb
aa b
ba
b b
aa, b
A ≈ ≈ ≈ ≈ ≈ ≈ × ≈ ≈ ×B ≈ ≈ ≈ ≈ ≈ ≈ ≈ ≈ ≈ ×
C ≈ ≈ ≈ ≈ ≈ ≈ ≈ ≈ ×D ≈ ≈ ≈ ≈ ≈ ≈ ≈ ×
E ≈ ≈ ≈ ≈ ≈ ≈ ×F ≈ ≈ ≈ ≈ ≈ ×
G ≈ ≈ ≈ ≈ ×H ≈ ×
I ≈ ≈ ×J ≈ ×
K ×L
A, H:DFA Minimization 56 Linz 6th, Example 2.18—21 of 40
Fouth Iteration
A B C D
EFGH
I J K L
a
b
a
b
a
b
ab
b
aabb
aa b
ba
b b
aa, b
A ≈ ≈ ≈ ≈ ≈ ≈ × ≈ ≈ × ×B ≈ ≈ ≈ ≈ ≈ ≈ ≈ ≈ ≈ ×
C ≈ ≈ ≈ ≈ ≈ ≈ ≈ ≈ ×D ≈ ≈ ≈ ≈ ≈ ≈ ≈ ×
E ≈ ≈ ≈ ≈ ≈ ≈ ×F ≈ ≈ ≈ ≈ ≈ ×
G ≈ ≈ ≈ ≈ ×H ≈ ×
I ≈ ≈ ×J ≈ ×
K ×L
DFA Minimization 57 Linz 6th, Example 2.18—22 of 40
Fouth Iteration
A B C D
EFGH
I J K L
a
b
a
b
a
b
ab
b
aabb
aa b
ba
b b
aa, b
A ≈ ≈ ≈ ≈ ≈ ≈ × ≈ ≈ × ×B ≈ ≈ ≈ ≈ ≈ ≈ ≈ ≈ ≈ ×
C ≈ ≈ ≈ ≈ ≈ ≈ ≈ ≈ ×D ≈ ≈ ≈ ≈ ≈ ≈ ≈ ×
E ≈ ≈ ≈ ≈ ≈ ≈ ×F ≈ ≈ ≈ ≈ ≈ ×
G ≈ ≈ ≈ ≈ ×H × ≈ ×
I ≈ ≈ ×J ≈ ×
K ×L
DFA Minimization 58 Linz 6th, Example 2.18—23 of 40
Fouth Iteration
A B C D
EFGH
I J K L
a
b
a
b
a
b
ab
b
aabb
aa b
ba
b b
aa, b
A ≈ ≈ ≈ ≈ ≈ ≈ × ≈ ≈ × ×B ≈ ≈ ≈ ≈ ≈ ≈ ≈ ≈ ≈ ×
C ≈ ≈ ≈ ≈ ≈ ≈ ≈ ≈ ×D ≈ ≈ ≈ ≈ ≈ ≈ ≈ ×
E ≈ ≈ ≈ ≈ ≈ ≈ ×F ≈ ≈ ≈ ≈ ≈ ×
G ≈ ≈ ≈ ≈ ×H × × ≈ ×
I ≈ ≈ ×J ≈ ×
K ×L
DFA Minimization 59 Linz 6th, Example 2.18—24 of 40
Fouth Iteration
A B C D
EFGH
I J K L
a
b
a
b
a
b
ab
b
aabb
aa b
ba
b b
aa, b
A ≈ ≈ ≈ ≈ ≈ ≈ × ≈ ≈ × ×B ≈ ≈ ≈ ≈ ≈ ≈ ≈ ≈ ≈ ×
C ≈ ≈ ≈ ≈ ≈ ≈ ≈ ≈ ×D ≈ ≈ ≈ ≈ ≈ ≈ ≈ ×
E ≈ ≈ ≈ ≈ ≈ ≈ ×F ≈ ≈ ≈ ≈ ≈ ×
G ≈ ≈ ≈ ≈ ×H × × ≈ ×
I ≈ ≈ ×J ≈ ×
K ×L
The resulting partition: {A, B, E}, {C , F , I}, {D, G , J}, {H, K}, {L}DFA Minimization 60 Linz 6th, Example 2.18—25 of 40
Firth Iteration
A B C D
EFGH
I J K L
a
b
a
b
a
b
ab
b
aabb
aa b
ba
b b
aa, b
A ×B ≈ ≈ ×
C ≈ ≈ ×D ≈ ≈ ×
E ≈ ≈ ×F ≈ ≈ ×
G ≈ ≈ ×H ≈ ×
I ≈ ×J ≈ ×
K ×L
DFA Minimization 61 Linz 6th, Example 2.18—26 of 40
Firth Iteration
A B C D
EFGH
I J K L
a
b
a
b
a
b
ab
b
aabb
aa b
ba
b b
aa, b
A ≈ ≈ ≈ ≈ ≈ ≈ ≈ ≈ ×B ≈ ≈ ≈ ≈ ≈ ≈ ≈ ≈ ≈ ×
C ≈ ≈ ≈ ≈ ≈ ≈ ≈ ≈ ×D ≈ ≈ ≈ ≈ ≈ ≈ ≈ ×
E ≈ ≈ ≈ ≈ ≈ ≈ ×F ≈ ≈ ≈ ≈ ≈ ×
G ≈ ≈ ≈ ≈ ×H ≈ ×
I ≈ ≈ ×J ≈ ×
K ×L
for all s, s ′ ∈ {B · · · G , I, H} and for ⋆ ∈ a, b, δ(s, ⋆) and δ(s ′, ⋆) are non-final.DFA Minimization 62 Linz 6th, Example 2.18—27 of 40
Firth Iteration
A B C D
EFGH
I J K L
a
b
a
b
a
b
ab
b
aabb
aa b
ba
b b
aa, b
A ≈ ≈ ≈ ≈ ≈ ≈ × ≈ ≈ ×B ≈ ≈ ≈ ≈ ≈ ≈ ≈ ≈ ≈ ×
C ≈ ≈ ≈ ≈ ≈ ≈ ≈ ≈ ×D ≈ ≈ ≈ ≈ ≈ ≈ ≈ ×
E ≈ ≈ ≈ ≈ ≈ ≈ ×F ≈ ≈ ≈ ≈ ≈ ×
G ≈ ≈ ≈ ≈ ×H ≈ ×
I ≈ ≈ ×J ≈ ×
K ×L
A, H:DFA Minimization 63 Linz 6th, Example 2.18—28 of 40
Firth Iteration
A B C D
EFGH
I J K L
a
b
a
b
a
b
ab
b
aabb
aa b
ba
b b
aa, b
A ≈ ≈ ≈ ≈ ≈ ≈ × ≈ ≈ × ×B ≈ ≈ ≈ ≈ ≈ ≈ ≈ ≈ ≈ ×
C ≈ ≈ ≈ ≈ ≈ ≈ ≈ ≈ ×D ≈ ≈ ≈ ≈ ≈ ≈ ≈ ×
E ≈ ≈ ≈ ≈ ≈ ≈ ×F ≈ ≈ ≈ ≈ ≈ ×
G ≈ ≈ ≈ ≈ ×H ≈ ×
I ≈ ≈ ×J ≈ ×
K ×L
DFA Minimization 64 Linz 6th, Example 2.18—29 of 40
Firth Iteration
A B C D
EFGH
I J K L
a
b
a
b
a
b
ab
b
aabb
aa b
ba
b b
aa, b
A ≈ ≈ ≈ ≈ ≈ ≈ × ≈ ≈ × ×B ≈ ≈ ≈ ≈ ≈ ≈ ≈ ≈ ≈ ×
C ≈ ≈ ≈ ≈ ≈ ≈ ≈ ≈ ×D ≈ ≈ ≈ ≈ ≈ ≈ ≈ ×
E ≈ ≈ ≈ ≈ ≈ ≈ ×F ≈ ≈ ≈ ≈ ≈ ×
G ≈ ≈ ≈ ≈ ×H × ≈ ×
I ≈ ≈ ×J ≈ ×
K ×L
DFA Minimization 65 Linz 6th, Example 2.18—30 of 40
Firth Iteration
A B C D
EFGH
I J K L
a
b
a
b
a
b
ab
b
aabb
aa b
ba
b b
aa, b
A ≈ ≈ ≈ ≈ ≈ ≈ × ≈ ≈ × ×B ≈ ≈ ≈ ≈ ≈ ≈ ≈ ≈ ≈ ×
C ≈ ≈ ≈ ≈ ≈ ≈ ≈ ≈ ×D ≈ ≈ ≈ ≈ ≈ ≈ ≈ ×
E ≈ ≈ ≈ ≈ ≈ ≈ ×F ≈ ≈ ≈ ≈ ≈ ×
G ≈ ≈ ≈ ≈ ×H × × ≈ ×
I ≈ ≈ ×J ≈ ×
K ×L
DFA Minimization 66 Linz 6th, Example 2.18—31 of 40
Firth Iteration
A B C D
EFGH
I J K L
a
b
a
b
a
b
ab
b
aabb
aa b
ba
b b
aa, b
A ≈ ≈ ≈ ≈ ≈ ≈ × ≈ ≈ × ×B ≈ ≈ ≈ ≈ ≈ ≈ ≈ ≈ ≈ ×
C ≈ ≈ ≈ ≈ ≈ ≈ ≈ ≈ ×D ≈ ≈ ≈ ≈ ≈ ≈ ≈ ×
E ≈ ≈ ≈ ≈ ≈ ≈ ×F ≈ ≈ ≈ ≈ ≈ ×
G ≈ ≈ ≈ ≈ ×H × × ≈ ×
I ≈ ≈ ×J ≈ ×
K ×L
The resulting partition: {A}, {B, E}, {C , F}, {D, G}, {I}, {J}, {H, K}, {L}DFA Minimization 67 Linz 6th, Example 2.18—32 of 40
Sixth And Final Iteration
A B C D
EFGH
I J K L
a
b
a
b
a
b
ab
b
aabb
aa b
ba
b b
aa, b
A × × × × × × × × × × ×B ≈ ≈ ×
C ≈ ≈ ×D ≈ ≈ ×
E ≈ ≈ ×F ≈ ≈ ×
G ≈ ≈ ×H ≈ ×
I × × ×J × ×
K ×L
DFA Minimization 68 Linz 6th, Example 2.18—33 of 40
Sixth And Final Iteration
A B C D
EFGH
I J K L
a
b
a
b
a
b
ab
b
aabb
aa b
ba
b b
aa, b
A × × × × × × × × × × ×B ≈ ≈ ≈ ≈ ≈ ≈ ≈ ≈ ≈ ×
C ≈ ≈ ≈ ≈ ≈ ≈ ≈ ≈ ×D ≈ ≈ ≈ ≈ ≈ ≈ ≈ ×
E ≈ ≈ ≈ ≈ ≈ ≈ ×F ≈ ≈ ≈ ≈ ≈ ×
G ≈ ≈ ≈ ≈ ×H × ≈ ×
I × × ×J × ×
K ×L
for all s, s ′ ∈ {B · · · G , I, H} and for ⋆ ∈ a, b, δ(s, ⋆) and δ(s ′, ⋆) are non-final.DFA Minimization 69 Linz 6th, Example 2.18—34 of 40
Sixth And Final Iteration
A B C D
EFGH
I J K L
a
b
a
b
a
b
ab
b
aabb
aa b
ba
b b
aa, b
A × × × × × × × × × × ×B ≈ ≈ ≈ ≈ ≈ ≈ ≈ ≈ ≈ ×
C ≈ ≈ ≈ ≈ ≈ ≈ ≈ ≈ ×D ≈ ≈ ≈ ≈ ≈ ≈ ≈ ×
E ≈ ≈ ≈ ≈ ≈ ≈ ×F ≈ ≈ ≈ ≈ ≈ ×
G ≈ ≈ ≈ ≈ ×H × × ≈ ×
I × × ×J × ×
K ×L
DFA Minimization 70 Linz 6th, Example 2.18—35 of 40
Sixth And Final Iteration
A B C D
EFGH
I J K L
a
b
a
b
a
b
ab
b
aabb
aa b
ba
b b
aa, b
A × × × × × × × × × × ×B ≈ ≈ ≈ ≈ ≈ ≈ ≈ ≈ ≈ ×
C ≈ ≈ ≈ ≈ ≈ ≈ ≈ ≈ ×D ≈ ≈ ≈ ≈ ≈ ≈ ≈ ×
E ≈ ≈ ≈ ≈ ≈ ≈ ×F ≈ ≈ ≈ ≈ ≈ ×
G ≈ ≈ ≈ ≈ ×H × × ≈ ×
I × × ×J × ×
K ×L
The resulting partition: {A}, {B}, {C}, {F}, {D, G}?, {I}, {J}, {H, K}, {L}DFA Minimization 71 Linz 6th, Example 2.18—36 of 40
DFA MinimizationAn Example
Combining Indistinquishable StatesHMU 3rd, Example 4,18, page 156
The example ignores an inaccessible state which, in this case, goes way
DFA Minimization 72 HMU 3rd, Example 4.18—1 of 22
A B C D
E
F G H
0 1 0
10 1
0
11 0
0 1
10
1
0
DFA Minimization 73 HMU 3rd, Example 4.18—1 of 22
AE BH C
FD G
0 1
10
0
10
1
1
0
The final partition {A, E}, {B, H}, {C}, {F , D}, {G}.
DFA Minimization 74 HMU 3rd, Example 4.18—1 of 22
