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Lecture 16

• FSA’s– Defining FSA’s– Computing with FSA’s

• Defining L(M)

– Defining language class LFSA– Comparing LFSA to set of solvable languages

(REC)

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Quick Review

• What is the functional definition of a configuration?

• What is a computation?

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Finite State Automata

New Computational Model

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Finite State Automata (FSA)

1 2

a,b

a,b

1 2

a;a;Rb;b;R

a;a;Rb;b;R

•Written character is identical to read character•Tape head moves right one cell•Model can be simplified by removing these elements

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Formal Definition

• FSA M=(Q,,q0,A,)– Q = set of states = {1,2}

– = character set = {a,b}

– q0 = initial state = 1

– A = set of accepting (final) states = {1}

– = state transition function

• Comments:– Read only

– Tape head moves right

1 2

a,b

a,b

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Exercise

• FSA M = (Q, , q0, A, )

– Q = {1, 2, 3}– = {a, b}

– q0 = 1

– A = {2,3}– : {(1,a) = 1, (1,b) = 2, (2,a)= 2, (2,b) = 3,

(3,a) = 3, (3,b) = 1}

• Draw this FSA as a transition diagram

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Transition Diagram

1

2

3

a

a

a

b

b

b

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Computing with FSA’s

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Computation Example

• (1, aabbaa)• (1,abbaa)• (1,bbaa)• (2,baa)• (3,aa)• (3,a)• (3

1

2

3

a

a

a

b

b

b

Input: aabbaa

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Computation of FSA’s in detail

• A computation of an FSA M on an input x is a complete sequence of configurations

• We need to define– Initial configuration of the computation– How to determine the next configuration given

the current configuration– Halting or final configurations of the

computation

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• Given an FSA M and an input string x, what is the initial configuration of the computation of M on x?– (q0,x)

– Examples• x = aabbaa

• (1, aabbaa)

• x = abab

• (1, abab)

• x = • (1, )

Initial Configuration

1

2

3

a

a

a

b

b

b

FSA M

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• (1, aabbaa) |-M (1, abbaa)

– config 1 “yields” config 2 in one step using FSA M

• (1,aabbaa) |-M (2, baa)

– config 1 “yields” config 2 in 3 steps using FSA M

• (1, aabbaa) |-M (2, baa)

– config 1 “yields” config 2 in 0 or more steps using FSA M

• Comment:

– |-M determined by transition function

– There must always be one and only one next configuration

• If not, M is not an FSA

Definition of |-M

1

2

3

a

a

a

b

b

b

3

FSA M

*

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• Halting configuration– (q, )

– Examples• (1, )

• (3, )

• Accepting Configuration– State in halting configuration

is in A

• Rejecting Configuration– State in halting configuration

is not in A

Halting Configurations

1

2

3

a

a

a

b

b

b

FSA M

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• Two possibilities for M running on x– M accepts x

• M accepts x iff the computation of M on x ends up in an accepting configuration

• (q0, x) |-M (q, ) where q is in A

– M rejects x• M rejects x iff the computation of M on x ends up in a rejecting

configuration

• (q0, x) |-M (q, ) where q is not in A

– M does not loop or crash on x• There is always one and only one next configuration.

• Tape head always moves right one cell until EOF

FSA M on xb

bb

aa

a

FSA M

*

*

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– For the following input strings, does M accept or reject?• • aa• aabba• aab• babbb

Examplesb

bb

aa

a

FSA M

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• Notation from the book

• (q, c) = p

• k(q, x) = p

• *(q, cx) = p– q, p in Q

– k is any integer >= 1

– c in – x in * or k

• Examples– (1, a) = 1

– (1, b) = 2

– 4(1, abbb) = 1

– *(1, abbb) = 1

– (2, baaaaa) = 1

Definition of *(q, x)

1

2

3

a

a

a

b

b

b

FSA M

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• L(M) or Y(M)– The set of strings M accepts

• Basically the same as Y(P) from previous unit

– We say that M accepts/decides/recognizes/solves L(M)• Remember an FSA will not loop or crash

– What is L(M) (or Y(M)) for the FSA M above?

• N(M)– Rarely used, but it is the set of strings M rejects

• LFSA– L is in LFSA iff there exists an FSA M such that L(M) = L.

L(M) and LFSA

bb

b

aa

a

FSA M

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LFSA Unit Overview

• Study limits of LFSA– Understand what languages are in LFSA

• Develop techniques for showing L is in LFSA

– Understand what languages are not in LFSA• Develop techniques for showing L is not in LFSA

• Prove Closure Properties of LFSA• Identify relationship of LFSA to other

language classes

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Comparing language classes

Showing LFSA is a subset of REC, the set of solvable languages

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LFSA subset REC

• Proof– Let L be an arbitrary language in LFSA– Let M be an FSA such that L(M) = L

• M exists by definition of L in LFSA

– Construct C++ program P from FSA M– Argue P solves L– There exists a C++ program P which solves L– L is solvable

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Visualization

LFSAREC

FSA’s C++ Programs

L L

M P

•Let L be an arbitrary language in LFSA•Let M be an FSA such that L(M) = L

•M exists by definition of L in LFSA•Construct C++ program P from FSA M•Argue P solves L•There exists a program P which solves L•L is solvable

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Construction

• The construction is an algorithm which solves a problem with a program as input– Input to A: FSA M– Output of A: C++ program P such that P solves

L(M)– Informal description of algorithm

• Implement FSA M as C++ program using switch statements, etc.

Construction Algorithm

FSA M

Program P

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Comparing computational models

• The previous slides show one method for comparing the relative power of two different computational models– Computational model CM1 is at least as general or

powerful as computational model CM2 if

• Any program P2 from computational model CM2 can be converted into an equivalent program P1 in computational model CM1.

– Question: How can we show two computational models are equivalent?