Formal Methods in Computer Science CS1502 Equivalence of NFAs and DFAs

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Formal Methods in Computer ScienceCS1502Equivale
nce of NFAs and DFAs

• Patchrawat Uthaisombut
• University of Pittsburgh

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Goals

• Equivalence of NFAs and DFAs
• Closure properties of regular languages.
• General idea of closure property
• Closure of regular languages under regular
  operations.

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?? and P(Q)

• ?? ? ? ?
• If ? a, b
• Then ?? a, b, ?
• P(Q) power set of Q
• If Q A,B,C
• Then P(Q) , A, B, C, A,B, A,C,
  B,C, A,B,C
• Note, if Q has n members, then P(Q) has 2n
  members.

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Definition of DFA

• A (Deterministic) Finite Automaton(FA, DFA) is a
  5-tuple (Q,?,?,q0,F) where
• Q is a finite set of states
• ? is a finite alphabet
• ?Q???Q is the transition function
• q0?Q is the start state
• F?Q is the set of accept states

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Definition of NFA

• A Nondeterministic Finite Automaton (NFA) is a
  5-tuple (Q,?,?,q0,F) where
• Q is a finite set of states
• ? is a finite alphabet
• ?Q????P(Q) is the transition function
• q0?Q is the start state
• F?Q is the set of accept states

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• M (Q,?,?,q1,F) where
• Q q1,q2,q3,q4,q5
• ? 0, 1
• ? is given as the table
• q1 is the start state
• F q3, q5 .

?Q????P(Q)
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0,1
p
1, ?
0
p
q
r
0100
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DFAs vs. NFAs

• Are DFAs more powerful than NFAs?
• Is there a language L that is recognized by some
  DFA but is not recognized by any NFA?
• No. DFAs are special cases of NFAs.
• Are NFAs more powerful than DFAs?
• Unclear.
• We will show that they are not.

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0,1
p
1, ?
0
p
q
r
0100
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E(R)

• E(R) q q can be reached from R by traveling
  along zero or more ? transitions
• E( q1 ) ?
• q1, q2, q3, q4
• E( q2, q4 ) ?
• q2, q3, q4

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Equivalence of NFAs and DFAs

• Let N(Q,?,?,q0,F) be an NFA
• Construct a DFA M(Q,?,?, q0,F) as follows
• Q P(Q)
• for R?Q and a??,let ?(R,a) q ? Q q ?
  E(?(r,a)) for some r ? R .
• q0 E( q0 )
• F R?Q R contains an accept state of N
• We claim that L(M) L(N).

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Implications

• Are NFAs more powerful than DFAs?
• Is there a language L that is recognized by some
  NFA but is not recognized by any DFA?
• We can use NFAs to study the question
• If A and B are regular, is A . B regular?

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Union revisit

• If A and B are regular, is A ? B regular?
• Yes. We proved that by simulating 2 machines in
  parallel.
• A new proof
• Make use of the equivalence of DFAs and NFAs.

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Closure property

• A set X and an operation y.
• X is said to be closed under y if when we apply
  operation y to any member of X, the result is
  still a member of X.
• Examples
• The set of integers is closed under addition
• The set of integers is not closed under division
• The set of regular languages is closed under
  union.
• Take any union of two regular languages and the
  result is still a regular language.

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Closure under regular operations

• The class of regular languages is closed under
  union.
• The class of regular languages is closed under
  concatenation.
• The class of regular languages is closed under
  star.
• Proofs