Presentation

 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.

 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

 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.

 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.

 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 K is the start state.  A ⊆ K is the set of accepting states.

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 Σ ω 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.

 Let L ={ w {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:

?

1. Rich, Elaine. Automata, Computability and Complexity Theory and Applications. Upper Saddle River (N. J.) Pearson Prentice Hall, Print df/tcs-96-2.pdf. Web.