Presentation
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Julius Richard Büchi (1924–1984)
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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Infinite words accepted by finite-state
automata.
The theory of automata on infinite words
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non-deterministic automata over infinite inputs
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more complex.
 more powerful.
Every language we consider either consists
exclusively of finite words or exclusively of
infinite words.
The set ∑ω denotes the set of infinite words
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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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The formal definition of Buchi automata is (K,
∑, Δ, S,A).
K is finite set of states
∑ is the input of alphabet
Δ 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 & εtransition is not allowed.
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Buchi (K, ∑, Δ, S,A).
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K is finite set of states
∑ is the input of alphabet
Δ 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.
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K is finite set of states
∑ is the input alphabet
δ is the transition Function. it maps from: K * ∑ to K.
S ϵ K is the start state.
A ⊆ K is the set of accepting states.
DFSM (K, ∑, δ, S,A).
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 Σω that contains
at least one occurrence of f.
This is example where e occurs ones.
This is an where c occurrence at least three
times.
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Let L ={ w ϵ {0, 1}ω): #1(w) is finite } Note that
every string in L must contain an infinite
number of 0’s.
The following nondeterministic Buchi
automaton accepts L:
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Rich, Elaine. Automata, Computability and
Complexity Theory and Applications. Upper
Saddle River (N. J.) Pearson Prentice Hall,
2008. Print.
http://www.math.uiuc.edu/~eid1/ba.pdf
Http://www.cmi.ac.in/~madhavan/papers/p
df/tcs-96-2.pdf. Web.