Summary

1
Summary

• Showing regular
• construct DFA, NFA
• construct regular expression
• show L is the union, concatenation, intersection
  (regular operations) of regular languages.
• Showing non-regular
• pumping lemma
• assume regular, apply closure properties of
  regular languages and obtain a known non-regular
  language.

2
Equivalence MinimizationofAutomata

3
Outline

• Equivalence of
• a state in a DFA
• regular languages
• Minimization of DFA
• smallest possible DFA for a given regular language

4
Equivalent States.Example
Consider the accept states c and g. They are
both sinks meaning that any string which ever
reaches them is guaranteed to be accepted
later. Q Do we need both states?
0,1
0
b
c
1
0,1
0
1
0
a
d
e
0,1
1
1
0
g
f
5
Example (cont)
A No, they can be unified as illustrated
below. Q Can any other states be unified
because any subsequent string suffixes produce
identical results?
b
0,1
0
0
1
1
0
a
d
0,1
e
cg
1
1
0
f
6
Example (cont)
A Yes, b and f. Notice that if youre in b or
f then if string ends, reject in both cases if
next character is 0, forever accept in both
cases if next character is 1, forever reject in
both cases So unify b with f.
b
0,1
0
0
1
1
0
a
d
0,1
e
cg
1
1
0
f
7
Example (cont)
Intuitively two states are equivalent if all
subsequent behavior from those states is the
same. Come up with a formal characterization of
state equivalence. Q Any other ways to simplify
the automaton?
0,1
1
0
a
d
0,1
e
cg
0,1
1
0
bf
8
Example (cont)
A Get rid of d. i.e. Getting rid of unreachable
useless states doesnt affect the accepted
language.
0,1
a
0,1
e
cg
0,1
1
0
bf
9
Equivalent States.Definition
DEF Two states q and q in a DFA M (Q, S,
d, q0, F ) are said to be equivalent (or
indistinguishable) if for all strings u ? S, the
states on which u ends on when read from q and q
are both accept, or both non-accept. Equivalent
states may be glued together without affecting M
s behavior.
10
Minimization Algorithm.Goals

• DEF An automaton is irreducible if
• it contains no useless states, and
• no two distinct states are equivalent.
• The goal of minimization algorithm is to create
  irreducible automata from arbitrary ones. Later
  remarkably, the algorithm actually produces
  smallest possible DFA for the given language,
  hence the name minimization.

11
Minimization of DFAs
Example
The DFA has equivalence classes A,E, B,H,
C, D,F, G
12
Equivalence Class
Transitivity If p q and q r, then p
r Proof Suppose to the contrary that p ?
r Then or vice versa. Now is
either accepting or not
If Otherwise The vice versa case is proved
symmetrically
13
Minimization of DFAs (cont)
14
Example
Can be minimized to
15
Cannot apply the TF-algo to NFAs For Example, to
minimize
Simply remove state C However, A ? C
16
Minimized DFA Cannot be Beaten
Let B be the minimized DFA obtained by applying
the TF-Algo to DFA A. We know that L(A)
L(B) What if there existed a DFA C with L(C)
L(B) and fewer states than B? If we run the
TF-Algo on B union C Also If we could
distinguish successors then
17
Minimized DFA Cannot be Beaten (cont)
Claim For each state p in B there is at least
one state q in C, s.t. Proof There are no
inaccessible states, so Now
and Since C has fewer states
than B, there must be two states r s of B s.t.
For some state t of C. But
CONTRADICTION!!