DFA Minimization

1
DFA Minimization
2
Strings over a,b with even number of as
S
Ea
Oa
Eb
Ob
Ob
Eb
b
Ea,Eb
Ea,Ob
b
a
a
a
a
b
Oa,Ob
Oa,Eb
b
3
Strings over a,b with even number of as
S
Ea
Oa
a
Oa
Ea
a
b
b
4
Observation

• The states among the state sets Ea,Eb,
  Ea,Ob and Oa,Eb, Oa,Ob differ on aspect
  immaterial for the problem at hand.
• Why not collapse these state sets into one state
  each, to get a smaller DFA?

5
Equivalent or Indistinguishable States

• Recall that a DFA state summarizes the substring
  consumed so far (that is, the past history).
• Two states qi and qj are equivalent or
  indistinguishable (for the future), if, when
  started in these states, every string causes the
  machine to either end up in an accepting state
  for both or end up in a non-accepting state for
  both.
• In other words, there is no string that can tell
  the equivalent states apart for the given
  language.

6
Main Idea
qi
qm
a
qn
qj
a
If qm and qn are distinguishable, then so are qi
and qj
7
Example DFA
b
a
b
a
q1
q2
q0
q3
a
a,b
b
a,b
a
q4
q7
q5
a,b
a
b
q6
(a u b)(a u b)
b
8
Refinement of State Partitions

• q0,q7 , q1,q2,q3,q4,q5,q6
• q0,q7, q1,q2,q3,q4,q5,q6
• On any transition
• q0,q7, q1,q2,q3,q4,q5,q6
• q0,q7, q1,q4, q2,q3,q5,q6
• On a transition
• q0,q7, q1,q4, q2,q5,q3,q6
• On b transition

9
Example DFA showing equivalent states
b
a
b
a
q1
q2
q0
q3
a
a,b
b
a,b
a
q4
q7
q5
a,b
a
b
q6
(a u b)(a u b)
b
10
Example Minimum DFA
q0
a,b
a,b
a
q7
q2,q5
q1,q4
a,b
a
b
q3,q6
(a u b)(a u b)
b
11

• For an efficient way of determining minimal DFA,
  see the details in Algorithm 5.7.2