Finite Automata and Non Determinism

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Finite Automata andNon Determinism

• http//cis.k.hosei.ac.jp/yukita/

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Definition 1.1 Finite Automaton
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State Diagram for M1
1
0
q3
0
q1
q2
1
0, 1
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Data Representation for M1
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Task 01DFA

• Implement M1 with your favorite programming
  language.
• GUI
• Two buttons for input 0 and 1
• State chart with the current state highlighted

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Language of M1
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State Diagram for M5
0
q1
1
2, ltRESETgt
0, ltRESETgt
0
1
2
q2
q0
2
1, ltRESETgt
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Data Representation for M5
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Informal Description of M5

• M5 keeps a running count of the sum of the
  numerical symbols it reads, modulo 3.
• Every time it receives the ltRESETgt symbol it
  resets the count to 0.
• M5 accepts if the sum is 0, modulo 3.

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Definition 1.7 Regular Language

• A language is called a regular language if some
  finite automaton recognizes it.

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Example 1.9 A finite automaton E2

• E2 recognizes the regular language of all strings
  that contain the string 001 as a substring.
• 0010, 1001, 001, and 1111110011110 are all
  accepted,
• but 11 and 0000 are not.

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Find a set of states of E2

• You
• havent just seen any symbols of the pattern,
• have just seen a 0,
• have just seen 00 or,
• have just seen the entire pattern 001.
• Assign the states q,q0,q00, and q001 to these
  possibilities.

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Draw a State Diagram for E2
0, 1
0
1
q00
q001
q0
0
q
1
0
1
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Regular Operations on Languages
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Example 1.11
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Theorem 1.12 Closedness for Union
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Proof of Theorem 1.12
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Construction of M
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Correctness

• You should check the following.
• For any string recognized by M1 is recognized by
  M.
• For any string recognized by M2 is recognized by
  M.
• For any string recognized by M is recognized by
  M1 or M2.

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Theorem 1.13 Closedness for concatenation
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Nondeterminism

• To prove Theorem 1.13, we need nondeterminism.
• Nondeterminism is a generalization of
  determinism. So, every deterministic automaton is
  automatically a nondeterministic automaton.

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Nondetermistic Finite Automata

• A nondeterministic finite automaton can be
  different from a deterministic one in that
• for any input symbol, nondeterministic one can
  transit to more than one states.
• epsilon transition
• NFA and DFA stand for nondeterministic finite
  automaton and deterministic finite automaton,
  respectively.

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NFA N1
0,1
0,1
q3
q4
1
q1
q2
1
0,e
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Parallel world and NFA
...
...
accept
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Example 1.14 NFA N2
0,1
q3
q4
q1
q2
0,1
1
0,1
Let language A consist of all strings over 0,1
containing a 1 in the third position from the
end. N2 recognizes A.
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A DFA equivalent to N2
0
0
q010
q110
q000
q100
0
0
1
0
1
0
1
0
0
1
1
q011
q111
q001
q101
1
1
1
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Example 1.15 NFA N3
0
0
e
0
e
0
0
Let language A consist of all strings 0k , where
k is a multiple of 2 or 3. N3 recognizes A.
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A DFA equivalent to N3
0, 1
q-1
1
1
1
1
1
1
q5
q4
q2
q1
q3
q0
0
0
0
0
0
0
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Example 1.16 NFA N4
q1
a
b
e
a
q3
q2
a,b
N4 accepts e, a, baba, and baa. N4 does not
accept b, nor babba.
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Definition 1.17 NFA
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Example 1.18 NFA N1
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In what situation is Non Determinism relevant?

• Von Neumann machines are deterministic.
• However, there are many cases where machine
  specification is all we need.

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Theorem 1.19

• Every nondeterministic finite automaton has an
  equivalent deterministic finite automaton.
• Def. The two machines are equivalent is they
  recognize the same language.

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Proof of Th. 1.19
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Incorporate e arrows
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Corollary 1.20

• A language is regular if and only if some
  nondeterministic finite automaton recognizes it.

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Example 1.21 NFA N4 to DFA
1
b
a
e
a
3
2
a,b
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Task 02Parallel World

• Write a program that simulates N4.
• GUI
• Three buttons for input 0, 1, and epsion.
• State chart that reflect the branching of the
  world.

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Start and Accept states
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The state diagram of D
2
1,2
a,b
a
b
f
1
a,b
b
a
b
b
a
a
2,3
1,2,3
3
a
1,3
a
b
b
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Theorem 1.22 The class of regular languages is
closed under the union operation.
N
N1
e
e
N2
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Proof of Th. 1.22
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Theorem 1.23 The class of regular languages is
closed under the concatenation operation.
N
e
e
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Proof of Th. 1.23
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Theorem 1.24 The class of regular languages is
closed under the star operation.
N
e
e
e
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Proof of Th. 1.24