Nondeterminism, Kleene’s Theorem

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Proof(Kleene’s Theorem Part II) >>

Theory of Automata

(CS402)

Theory of Automata

Lecture N0. 10

Reading Material

Chapter 6, 7

Introduction to Computer Theory

Summary

Examples of GTG accepting the languages of strings: containing aa or bb, beginning with and ending in same

letters, beginning with and ending in different letters, containing aaa or bbb,

Nondeterminism, Kleene's theorem (part I, part II, part III), proof of Kleene's theorem part I

Example

Consider the language L of strings, defined over Σ = {a,b}, containing double a or double b. The language L

can be expressed by the following regular expression (a+b)* (aa + bb) (a+b)*

The language L may be accepted by the following GTG.

Example

Consider the Language L of strings, defined over Σ = {a, b}, beginning with and ending in same letters.

The language L may be expressed by the following regular expression (a+b)+ a(a + b)*a + b(a + b)*b.

This language may be accepted by the following GTG

Example

Consider the language L of strings of, defined over Σ = {a, b}, beginning and ending in different letters.

a(a + b)*b + b(a + b)*a

The language L may be expressed by RE

The language L may be accepted by the following GTG

The language L may be accepted by the following GTG as well

Example

Consider the language L of strings, defined over Σ = {a, b}, having triple a or triple b. The language L may be

expressed by RE (a+b)* (aaa + bbb) (a+b)*

This language may be accepted by the following GTG

Theory of Automata

(CS402)

Nondeterminism

TGs and GTGs provide certain relaxations i.e. there may exist more than one path for a certain string or there

may not be any path for a certain string, this property creates nondeterminism and it can also help in

differentiating TGs or GTGs from FAs. Hence an FA is also called a Deterministic Finite Automaton (DFA).

Kleene's Theorem

If a language can be expressed by

FA or

TG or

then it can also be expressed by other two as well.

It may be noted that the theorem is proved, proving the following three parts

Kleene's Theorem Part I

If a language can be accepted by an FA then it can be accepted by a TG as well.

Kleene's Theorem Part II

If a language can be accepted by a TG then it can be expressed by an RE as well.

Kleene's Theorem Part III

If a language can be expressed by a RE then it can be accepted by an FA as well.

Proof(Kleene's Theorem Part I)

Since every FA can be considered to be a TG as well, therefore there is nothing to prove.

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