cs480(prasad)l9fsa1 finite state automata. cs480(prasad)l9fsa2 formal specification of languages...

CS480(Prasad) L9FSA 1

Finite State Automata


Formal Specification of Languages

• Generators• Context-free Grammars

• Regular Expressions

• Recognizers• Parsers, Push-down Automata

• Finite State Automata

• FSA is a mechanism to recognize a set of valid inputs before carrying out an action.

• FSA is a notation for describing a family of language recognition algorithms.


OP

Parity Problem

)( )1(

)()0(

).(

1s. ofnumber even contains )(

:

}1,0{*

*

parityparity

parityparity

parity

parity

booleanparity

EP

1

10 0


Basic Features

• Consumes the entire input string.• Remembers the parity of the bit string by

abstracting from the number of 1s in the string.• Finite amount of memory required for this

purpose.• Observe that counting requires unbounded memory,

while computing the parity requires very small and fixed amount of memory.

• Accepts/Rejects the input in a deterministic fashion.


• State• Indicates the status of the machine after consuming

some portion of the input.

• Summarizes the history of the computation that is relevant to the future course of action.

• Initial / Start State• Final / Accepting state• State Transition

Even Parity Odd Parity

1

0


Q: Finite set of states

Finite Alphabet

: Transition function

total function from Qx to Q

: Initial/Start State

F : Set of final/accepting state

Deterministic Finite State Automaton (DFA)

),,,,( 0 FqQM

0q


Operation of the machine

• Read the current letter of input under the tape head.

• Transit to a new state depending on the current input and the current state, as dictated by the transition function.

• Halt after consuming the entire input.

0 0 1

0q

Input Tape

Finite Control


, where],[ Qqq• Machine configuration:

• Yields relation:

• Language:

Associating Language with the DFA

]),,([ ],[ *M aqaq

} ],[ ],[ |{ *M0

* Fqqq


Examples


• Set of strings over {a,b} that contain bb

• Design states by parititioning *.• Strings containing bb q2

• Strings not containing bb– Strings that end in b q1

– Strings that do not end in b q0

– Initial state: q0– Final state: q2

)()( babbba


q2

State Diagram and Table

q0 q1

ab

a

b

a

b a b

q0 q0 q1

q1 q0 q2

q2 q2 q2

}2{

},{

}2,1,0{

qF

ba

qqqQ

],1[],0[ * qaabq


q0 q2

Strings over {a,b} that do not contain bb

q1

ab

a

b

a

b a b

q0 q0 q1

q1 q0 q2

q2 q2 q2

}1,0{

},{

}2,1,0{

qqF

ba

qqqQ

],0[],0[ * qbaq


DFA for the complement of L given DFA for L

Let M = (Q,,,q0,F) be a DFA. Then,

M’ = (Q,,,q0,Q-F) is a DFA with

L(M’) = * - L(M).

Implication: Languages associated with DFAs are closed under complementation.

(Recall that languages associated with regular expressions are closed under union, concatenation, and Kleene Star operations, by definition.)


Strings over {a,b} containing even number of a’s and odd number of b’s.

*

Ea Oa

Eb Ob ObEb

b

b

b

b

aaa a

[Oa,Ob]

[Ea,Eb] [Ea,Ob]

[Oa,Eb]


(ab)*c

*

valid prefix invalid prefix

end_a

a

b

b

a,b,c

a,cc

Err

Eb Ea

Ec

end_b end_c

a,b,c

a

c

b


(ab)*c

*

valid prefix invalid prefix

end_a

a

b

b

a,b,c

a,cc

Err

Eb Ea

Ec

end_b end_c

a,b,c


Nondeterministic Finite Automata

qi qj

qkq

qi qja a

aDFA

NFA

a

relationsubset

function total)(:

function total :

QQ

QΡowQ

QQ

NFA

NFA

DFA


• How do we associate a language with an NFA?• Every DFA is an NFA. However, does non-

determinism make NFAs strictly more expressive (powerful) than DFAs?

DFA: Unique computation for a given string

NFA: Accept if there exists an accepting computation

)()( DFAsLNFAsL


q2

NFA State Diagram (Strings over {a,b} ending in bb)

q0 q1

a

b b

a b

q0 {q0} {q0,q1}

q1 {q2}

q2

}2{

},{

}2,1,0{

qF

ba

qqqQ

bbba )(

b


],2[],1[],0[],0[ qbqbbqabbq

Halts in accepting state after consuming the input.

],0[],0[],0[],0[ qbqbbqabbq

],1[],0[],0[],0[ qbqbbqabbq

Halts in non-accepting state after consuming the input.

)(NFALabb


NFA State Diagram (a U b)* bb (a U b)*

q2q0 q1

a

b b

b ba

q2q0 q1

a

b b

ba

a

DFA


NFA for (a U b)* (aa U bb) (a U b)*

q2q0 q1

a

a a

b ba

q22q11b b

ba


Introducing -transitions into NFA

• A -transition causes the machine to change its state non-deterministically, without consuming any input.

)(}){(: QPQ

)-()()( sNFALNFAsLDFAsL


Closure Properties of NFA-s

M1

M1

M

M2

M2

LL(M1) U LL(M2)

LL(M1) LL(M2)

LL(M)*


}|{for NFA niba ii

a0 a1

b1

a2

b2

a3

b3

a aa a

b

b b b

…

…

a3

b3

an

bn

This construction cannot be generalized to recognize

}0|{ iba ii

because the machine will have infinite number of states.

b b b

cs480(prasad)l9fsa1 finite state automata. cs480(prasad)l9fsa2 formal specification of languages...

Documents

c b slide

c slide

eaoa ebob eb b b b b

b q1 strings

b q0 initial state

eb slide

b ab q0 q1 q0q2 slide

current state