Buchi Automata Presentation History Julius Richard Bchi 1924

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.

What is Buchi Automata ? § Infinite words accepted by finite-state automata. The theory of automata on infinite words § non-deterministic automata over infinite inputs § § § Ømore complex. Ø 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

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.

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 & εtransition is not allowed.

DFSM Vs Buchi � Buchi (K, ∑, Δ, S, A). � DFSM (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. � 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.

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.

Example 2 This is example where e occurs ones.

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

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:

Thank You ?

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