Buchi Automata

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Buchi Automata

• Presentation

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History

• Julius Richard Büchi (19241984)
• Swiss logician and mathematician.
• He received his Dr. sc. nat. in 1950 at the ETH
  Zürich
• Purdue University, Lafayette, Indiana
• had a major influence on the development of
  Theoretical Computer Science.

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What is Buchi Automata ?

• Infinite words accepted by finite-state automata.
• The theory of automata on infinite words
• more complex.
• non-deterministic automata over infinite inputs
• more powerful.
• Every language we consider either consists
  exclusively of finite words or exclusively of
  infinite words.
• The set ?? denotes the set of infinite words

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Where it is used?

• Many Systems including
• Operating system
• Air traffic control system
• A factory process control system
• What is common about these systems?
• such systems never halt.
• They should accept an infinite string of inputs
  and continue to function.

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Formal defination

• The formal definition of Buchi automata is (K, ?,
  ?, S,A).
• K is finite set of states
• ? is the input of alphabet
• ? is the transition relation it is finite set of
  (K ?) K.
• S ? K is the set of starting states.
• A ? K is the set of accepting states.
• Note could have more than start state
  e-transition is not allowed.  

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DFSM Vs Buchi

• Buchi (K, ?, ?, S,A).
• K is finite set of states
• ? is the input of alphabet
• ? is the transition relation it is finite subset
  of (K ?) K.
• S ? K is the set of starting states.
• A ? K is the set of accepting states.
• DFSM (K, ?, d, S,A).
• K is finite set of states
• ? is the input alphabet
• d is the transition Function. it maps from K
  ? to K.
• S ? K is the start state.
• A ? K is the set of accepting states.

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Example 1

• Suppose there are six events that can occur in a
  system that we wish to model. So let ? a, b,
  c, d, e, f in that case let us consider an event
  that f has to occur at least once, the Buchi
  automation accepts all and only the elements that
  S? that contains at least one occurrence of f.

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Example 2

• This is example where e occurs ones.

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Example 3

• This is an where c occurrence at least three
  times.

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Conversion From Deterministic to
Nondeterministic

• Let L w ? 0, 1?) 1(w) is finite Note
  that every string in L must contain an infinite
  number of 0s.
• The following nondeterministic Buchi automaton
  accepts L

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Thank You

• ?

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Resources

1. Rich, Elaine. Automata, Computability and
   Complexity Theory and Applications. Upper Saddle
   River (N. J.) Pearson Prentice Hall, 2008.
   Print. 
2. http//www.math.uiuc.edu/eid1/ba.pdf
3. Http//www.cmi.ac.in/madhavan/papers/pdf/tcs-96-2
   .pdf. Web.