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Buchi Automata

Buchi Automata . Presentation. History . 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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Buchi Automata

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  1. Buchi Automata Presentation

  2. History • 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.

  3. 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 ∑ω denotes the set of infinite words

  4. 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.

  5. Formal defination • 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 & ε-transition is not allowed.  

  6. DFSM Vs Buchi • Buchi (K, ∑, Δ, S,A). • 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. • DFSM (K, ∑, δ, S,A). • 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.

  7. 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 Σω that contains at least one occurrence of f.

  8. Example 2 This is example where e occurs ones.

  9. Example 3 This is an where c occurrence at least three times.

  10. Conversion From Deterministic to Nondeterministic • 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:

  11. Thank You ?

  12. Resources • 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/pdf/tcs-96-2.pdf. Web.

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