8/27/2009 sofya raskhodnikova intro to theory of computation l ecture 2 theory of computation finite...

8/27/2009

Sofya Raskhodnikova

Intro to Theory of Computation

LECTURE 2 Theory of Computation• Finite Automata• Operations on languages• Nondeterminism

L2.1Sofya Raskhodnikova; based on slides by Nick Hopper

Homework 0 dueHomework 1 handed out

JFLAP http://www.cs.duke.edu/csed/jflap/

L2.28/27/2009 Sofya Raskhodnikova; based on slides by Rakesh Verma

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Formal definition of FA

L2.38/27/2009 Sofya Raskhodnikova; based on slides by Nick Hopper

Q = {q0, q1, q2, q3}Σ = {0,1}

: Q Σ → Q transition function*q0 Q is start stateF = {q1, q2} Q accept states

M = (Q, Σ, , q0, F) where

M q2

00,1

00

1

1

1

q0

q1

q3

0 1

q0 q0 q1

q1 q2 q2

q2 q3 q2

q3 q0 q2

*

A language is regular if it is recognized by a finite automaton

L = { w | w contains 001} is regular

L = { w | w has an even number of 1s} is regular

The language of M, L(M), is the set of strings that M accepts

Language of FA

8/27/2009 L1.4Sofya Raskhodnikova; based on slides by Nick Hopper

8/27/2009 Sofya Raskhodnikova; based on slides by Nick Hopper L1.5

Regular languages

Many interesting programs accept regular languages

NETWORK PROTOCOLS

COMPILERS

GENETIC TESTING

ARITHMETIC

INTERNET TRANSMISSION CONTROL PROTOCOL

Let TCPS = { w | w is a complete TCP Session}Theorem. TCPS is regular


COMMENTS : Are delimited by /* */ Cannot have NESTED /* */ Must be CLOSED by */ */ is ILLEGAL outside a comment

COMMENTS = {strings with legal comments}

Theorem. COMMENTS is regular.

Compilers


DNA SEQUENCES are strings over the alphabet {A,C,G,T}

GENES are special substrings

A GENETIC TEST searches a DNA SEQUENCE for a GENE

Theorem. Every GENETIC TEST is regular.

Genetic testing


LET 3 =

• A string over 3 has three ROWS• Each ROW b0b1b2…bN represents the integer

b0 + 2b1 + … + 2NbN.• Let ADD = {S | ROW1 + ROW2 = ROW3 }

Theorem. ADD is regular.

{ [ ],[ ],[ ],[ ], [ ],[ ],[ ],[ ]}

000

001

010

011

100

101

110

111

Arithmetic


Sofya Raskhodnikova; based on slides by Nick Hopper

Operations on languages

L2.108/27/2009

Union: A B = { w | w A or w B }

Intersection: A B = { w | w A and w B }

Complement: A = { w | w A }

Reverse: AR = { w1 …wk | wk …w1 A }

Concatenation: A B = { vw | v A and w B }

Star: A* = { w1 …wk | k ≥ 0 and each wi A }

Regular


Concatenation: A B = { vw | v A and w B }

Star: A* = { w1 …wk | k ≥ 0 and each wi A }


Closure properties of the class of regular languages

L2.118/27/2009

THEOREM. The class of regular languages is closed under all 6 operations.

If A and B are regular, applying any of these operation yields a regular language.


Closure properties

L2.128/27/2009




Closure under union

L2.138/27/2009

Theorem. The union of two regular languages is also a regular language

Proof: Let

M1 = (Q1, Σ, 1, q0, F1) be finite automaton for L1

and

M2 = (Q2, Σ, 2, q0, F2) be finite automaton for L2

Construct a finite automaton M = (Q, Σ, , q0, F) that recognizes L = L1 L2

1

2


Example

L2.148/27/2009

q0 q1

00

1

1

p0 p1

11

0

0

M1

M2

M = ?


Proof (continued)

L2.158/27/2009

Idea: Run both M1 and M2 at the same time!

Q = pairs of states, one from M1 and one from M2

= { (q1, q2) | q1 Q1 and q2 Q2 }

= Q1 Q2

q0 = (q0, q0)1 2

F = { (q1, q2) | q1 F1 or q2 F2 }

( (q1,q2), ) = (1(q1, ), 2(q2, ))


Example (continued)

L2.168/27/2009

q0 q1

0 0

1

1

p0 p1

11

0

0

M1

M2

q0,p0 q1,p0

q0,p1 q1,p1

0000

1

1

1

1

M unionintersection


Closure properties

L2.178/27/2009


Intersection: A B = { w | w A and w B }


Reverse: AR = { w1 …wk | wk …w1 A }


Closure under reverse

L2.188/27/2009

Theorem. The reverse of a regular language is also regular language

Proof: Let L be a regular language and

M be a finite automaton that recognizes it.

Construct a finite automaton MR recognizing LR

Idea: Define MR as M with the arrows reversedSwap start and accept states


Closure under reverse

L2.198/27/2009

MR IS NOT ALWAYS A DFA!

It may have many start states

Some states may have too many outgoing edges, or none


Example

L2.208/27/2009

1 0

1

0 1

0,1

0

L2.218/27/2009 Sofya Raskhodnikova; based on slides by Nick Hopper

NONDETERMINISM

1 0

1

0 1

0,1

0

What happens with 100?

Nondeterministic Finite Automaton (NFA) accepts if there is a way to make it reach an accept state

Nondeterminism

Ways to think about nondeterminism

• parallel computation

• tree of possible computations

• guessing the “right” choice

L2.22

DeterministicComputation

NondeterministicComputation

accept or reject accept

reject

8/27/2009 Sofya Raskhodnikova; based on slides by Nick Hopper


Example

L2.238/27/2009

1

0

0

L(M)={1,00}

ε

ε


Example

L2.248/27/2009

0,1

0,ε 1

0,1

1

L(M)={w | w contains 101 or 11}

Formal Definition

8/25/2009

Q is the set of states

Σ is the alphabet

q0 Q is the start state

F Q is the set of accept states

An NFA is a 5-tuple M = (Q, Σ, , q0, F)

L1.25Sofya Raskhodnikova; based on slides by Nick Hopper

: Q Σε → P(Q) is the transition function

P(Q) is the set of subsets of Q and Σε = Σ {ε}


Example

L2.278/27/2009

0,1

0,ε 1

0,1

1

(q1,ε) =

N = (Q, Σ, , q0, F)

Q = {q0, q1, q2, q3}

Σ = {0,1}

F = {q3}

(q0,1) = {q0, q1}

{q1,q2}

(q2,0) =

q1 q2 q3q0

8/27/2009 sofya raskhodnikova intro to theory of computation l ecture 2 theory of computation finite...

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