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Introduction to the Theory of Computation. John Paxton Montana State University Summer 2003. Humor.

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Introduction to the theory of computation

Introduction to the Theory of Computation

John Paxton

Montana State University

Summer 2003


Humor
Humor

  • A man walks up to the Marine guard at the White House front gate and asked to see President Clinton. The guard says "Sorry sir but Clinton is no longer president."       The man comes back the next day and again approaches the same guard and asks to see President Clinton. Again the guard says "Sir, Clinton is no longer president." The man leaves but comes back the third day and again asked to see President Clinton.       The guard says "Sir...you have been here three days in a row asking to see President Clinton...and I have told you each time that Clinton is no longer president...what gives?"       The man says "I know... I just like hearing it."


1 3 regular expression
1.3 Regular Expression

  • Pattern Recogniton

    • Unix (GREP)

    • Perl

    • Text Editors


Formal definition
Formal Definition

R is a regular expression if R is

  • a for some a in the alphabet S

  • e

  • {}

  • (R1 U R2) where R1 and R2 are both regular expressions

  • (R1 R2)

  • (R1*)


Exercise describe the language
Exercise: Describe the Language

  • 0*10*

  • S*1S*

  • (SS)*

  • 01 U 10

  • (0 U e)1*

  • 0S*0 U 1S*1 U 0 U 1

  • 1*{}

  • {}*


Exercise give the regular expression
Exercise: Give the Regular Expression

  • {w | w begins with a 1 and ends with a 0}

  • {w | w contains at least 3 0s}

  • {w | w starts with a 0 and has odd length or starts with a 1 and has even length}

  • {w | the length of w is at most 5}

  • {w | w contains at least two 0s and at most one 1}

  • All strings except the empty string


Theorem
Theorem

  • A language is regular if and only if some regular expression describes it.

  • Part I. If a language is described by a regular expression, then it is regular.

  • Part II. If a language is regular, then it is described by a regular expression.


Part 1 show constructions
Part 1: Show constructions

  • R = a

  • R = e

  • R = {}

  • R = R1 U R2

  • R = R1 R2

  • R = R1*

#1

a


Exercises
Exercises

  • Build an NFA from the regular expression (ab U a)*.

  • Build an NFA from the regular expression (0 U 1)*000(0 U 1)*.


Part ii
Part II

  • A generalized nondeterministic finite automaton (GNFA):

    • The start state has transition arrows going to every other state but no arrows coming in from any other state.

    • There is only a single accept state, and it has arrows coming in from every other state but no arrows going to any other state. Furthermore, the accept state is not the same as the start state


GNFA

  • Except for the start state and accept state, one arrow goes from every state to every other state and also from each state to itself.

a U b

a

e

qstart

q1

qaccept

a*


Gnfa definition
GNFA Definition

  • A generalized nondeterministic finite automaton, (Q, S, d, qstart, qaccept) is a 5-tuple where

    • Q is the finite set of states

    • S is the input alphabet

    • d : (Q – {qaccept}) x (Q – {qstart})  R is the transition function

    • qstart is the start state

    • qaccept is the accept state


Gnfa acceptance
GNFA Acceptance

  • A GNFA accepts a string w in S* if w = w1w2…wk where each wi is in S* and a sequence of states q0,q1,…qk exists such that

    • q0 = qstart

    • qk = qaccept

    • for each i, wie L(Ri) where Ri = d(qi-1,qi)


Part ii proof process
Part II: Proof Process

  • Start with a k state DFA

  • Construct a k+2 state GNFA

  • Map to a k+1 state GNFA

  • …..

  • Map to a 2 state GNFA

  • Read off the regular expression!


Conversion example
Conversion Example

  • 2 state DFA

a

a, b

b

1

2


Conversion example1
Conversion Example

  • 4 state GNFA

a

a U b

b

1

2

{}

e

e

{}

{}


Conversion example2
Conversion Example

  • 3 state GNFA

a U b

ea*b

2

e

ea*{} U {}


Conversion example3
Conversion Example

  • 3 state GNFA (simplified)

a U b

a*b

2

e

{}


Conversion example4
Conversion Example

  • 2 state GNFA

Regular Expression: a*b(a U b)*

{} U a*b(a U b)*e


Exercise
Exercise

  • Convert to a regular expression

a

b

1

3

2

e


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