Buchi Automata 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.

History

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

What is Buchi Automata ?

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.

Where it is used?

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.  

Formal defination

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.

DFSM Vs Buchi

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 1

Example 2This is example where e occurs ones.

Example 3This is an where c occurrence at least three

times.

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:

Conversion From Deterministic to Nondeterministic

Thank You

?

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.

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