Finite-State Machines with No Output Ying Lu Based on Slides by Elsa L Gunter, NJIT, Costas Busch...

Finite-State Machines with No OutputYing Lu

Based on Slides by Elsa L Gunter, NJIT,

Costas Busch, and Longin Jan Latecki, Temple University

http://www.csc.lsu.edu/~busch/

Kleene closure

• A and B are two sets of strings. The concatenation of A and B isAB={xy: x string in A and y string in B}

• Example: A={0, 11} and B={1, 10, 110}AB={01,010,0110,111,1110,11110}

• A0={λ}, where λ represents the empty stringAn+1=AnA for n=0,1,2,…

Let A be any set of strings formed of characters in V.Kleene closure of A, denoted by A*, is

0

*

k

kAA

Examples:If C={11}, then C*={12n: n=0,1,2,…}

If B={0,1}, then B*={all binary strings}.

Finite State Automata

• A FSA is similar to a compiler in that: – A compiler recognizes legal programs in some (source) language.

– A finite-state machine recognizes legal strings in some language.

• Example: Pascal Identifiers– sequences of one or more letters or digits,

starting with a letter:

letterletter | digit

S A

Finite Automaton

• Input

“Accept” or“Reject”

String

FiniteAutomaton

Output


• A finite state automaton over an alphabet is illustrated by a state diagram:

– a directed graph– edges are labeled with elements of alphabet,– some nodes (or states), marked as final– one node marked as start state

Transition Graph

initialstate

accepting state

statetransition

0q 1q 2q 3q 4qa b b a

5q

a a bb

ba,

ba,

Initial Configuration

•

1q 2q 3q 4qa b b a

5q

a a bb

ba,

Input Stringa b b a

ba,

0q

Reading the Input

•


5q

a a bb

ba,

a b b a

ba,

•


5q

a a bb

ba,

a b b a

ba,


accept

5q

a a bb

ba,

a b b a

ba,

Input finished

Rejection

•

1q 2q 3q 4qa b b a

5q

a a bb

ba,

a b a

ba,

0q

•


5q

a a bb

ba,

a b a

ba,


5q

a a bb

ba,

reject

a b a

ba,

Input finished

Another Rejection

•

1q 2q 3q 4qa b b a

5q

a a bb

ba,

ba,

0q

•

1q 2q 3q 4qa b b a

5q

a a bb

ba,

ba,

0q

reject

Another Example

a

b ba,

ba,

0q 1q 2q

a ba

a

b ba,

ba,

0q 1q 2q

a ba

a

b ba,

ba,

0q 1q 2q

a ba

accept

Input finished

Rejection Example

a

b ba,

ba,

0q 1q 2q

ab b

a

b ba,

ba,

0q 1q 2q

ab b

a

b ba,

ba,

0q 1q 2q

ab b

reject

Input finished


• A finite state automaton M=(S,Σ,δ,s0,F) consists of

• a finite set S of states,

• a finite input alphabet Σ,

• a state transition function δ: S x Σ S,

• an initial state s0,

• F subset of S that represent the final states.

Finite Automata

• Transition

• Is read ‘In state s1 on input “a” go to state s2’

• At the end of input– If in accepting state => accept

– Otherwise => reject

• If no transition possible (got stuck) => reject

• FSA = Finite State Automata

21 ss a

Input Alphabet

•


5q

a a bb

ba,

ba,

ba,

Set of States

•

Q


5q

a a bb

ba,

543210 ,,,,, qqqqqqQ

ba,

Initial State

•

0q

1q 2q 3q 4qa b b a

5q

a a bb

ba,

ba,

0q

Set of Accepting States

•

F

0q 1q 2q 3qa b b a

5q

a a bb

ba,

4qF

ba,

4q

Transition Function

•


5q

a a bb

ba,

SS :

ba,

10 , qaq

2q 3q 4qa b b a

5q

a a bb

ba,

ba,

0q 1q

50 , qbq

1q 2q 3q 4qa b b a

5q

a a bb

ba,

ba,

0q


5q

a a bb

ba,

ba,

32 , qbq

Transition Function

•


5q

a a bb

ba,

a b

0q

1q

2q

3q

4q

5q

1q 5q

5q 2q

5q 3q

4q 5q

ba,5q5q5q5q

Language accepted by FSA

• The language accepted by a FSA is the set of strings accepted by the FSA.

• in the language of the FSM shown below: x, tmp2, XyZzy, position27.

• not in the language of the FSM shown below: • 123, a?, 13apples.

letterletter | digit

S A

Example

•


5q

a a bb

ba,

ba,

abbaML M

accept

Example


5q

a a bb

ba,

ba,

abbaabML ,, M

acceptacceptaccept

•

Example•

a

b ba,

ba,

0q 1q 2q

}0:{ nbaML n

accept trap state

Extended Transition Function

•

*

QQ *:*


5q

a a bb

ba,

ba,

20 ,* qabq

3q 4qa b b a

5q

a a bb

ba,

ba,

0q 1q 2q

40 ,* qabbaq


5q

a a bb

ba,

ba,

50 ,* qabbbaaq

1q 2q 3q 4qa b b a

5q

a a bb

ba,

ba,

0q

qwq ,*

Observation: if there is a walk from to with label then

q qw

q qw

q qkw 21

1 2 k

50 ,* qabbbaaq

1q 2q 3q 4qa b b a

5q

a a bb

ba,

ba,

0q

Example: There is a walk from to with label

0qabbbaa

5q

Recursive Definition )),,(*(,*

,*

wqwq

qq

q qw1q

qq

qwq

),(

,*

1

1

1

,*

),(,*

qwq

qwq

)),,(*(,* wqwq

Language Accepted by FSAs

• For a FSA

• Language accepted by :

•

FqQM ,,,, 0

M

FwqwML ,*:* 0

0q qw Fq

Observation

• Language rejected by :

FwqwML ,*:* 0

M

0q qw Fq

Example

• ML = { all strings with prefix }ab

a b

ba,

0q 1q 2q

accept

ba,3q

ab

Example

• ML = { all strings without substring }001

0 00 001

1

0

1

10

0 1,0

Example

• *,:)( bawawaML

a

b

ba,

a

b

ba

0q 2q 3q

4q

Deterministic FSAs

• If a FSA has for every state exactly one edge for each letter in alphabet then FSA is deterministic

• In general a FSA is non-deterministic.

• Deterministic FSA special kind of non-deterministic FSA

Example FSA

• Deterministic FSA

• Regular expression: (0 1)* 1

0 1

1

0

Example DFSA


• Accepts string 0 1 1 0 1

0 1

1

0

Example NFSA


• Non-deterministic FSA

0

1

1

Example NFSA



0

1

1



Example NFSA



• Guess

0

1

1



Example NFSA



• Backtrack

0

1

1



Example NFSA



• Guess again

0

1

1



Example NFSA



• Guess0

1

1



Example NFSA



• Backtrack

0

1

1



Example NFSA



• Guess again

0

1

1



Example NFSA



0

1

1



Example NFSA



• Guess (Hurray!!)

0

1

1



If a language L is recognized by a nondeterministic FSA, then L is recognized by a deterministic FSA

NFSA FSA

How to Implement an FSA

A table-driven approach:• table:

– one row for each state in the machine, and

– one column for each possible character.

• Table[j][k] – which state to go to from state j on character k,

– an empty entry corresponds to the machine getting stuck.

The table-driven program for a Deterministic FSA

state = S // S is the start state

repeat {k = next character from the input

if (k == EOF) // the end of inputif state is a final state then acceptelse reject

state = T[state,k]

if state = empty then reject // got stuck

}

In-Class Exercise

• Construct a finite-state automaton that recognizes the set of bit strings consisting of a 0 followed by a string with an odd number of 1s.

Appendix

Regular Expressions

Regular expressions

describe regular languages

Example:

describes the language

*)( cba

,...,,,,,*, bcaabcaabcabca

Recursive Definition,,

1

1

21

21

*

r

r

rr

rr

Are regular expressions

Primitive regular expressions:

2r1rGiven regular expressions and

Examples

)(* ccbaA regular expression:

baNot a regular expression:

Languages of Regular Expressions

: language of regular expression

Example

rL r

,...,,,,,*)( bcaabcaabcacbaL

Definition

For primitive regular expressions:

aaL

L

L

Definition (continued)

For regular expressions and

1r 2r

2121 rLrLrrL

2121 rLrLrrL

** 11 rLrL

11 rLrL

ExampleRegular expression: *aba

*abaL *aLbaL *aLbaL *aLbLaL

*aba ,...,,,, aaaaaaba

,...,,,...,,, baababaaaaaa

Example

Regular expression bbabar *

,...,,,,, bbbbaabbaabbarL

Example

Regular expression bbbaar **

}0,:{ 22 mnbbarL mn

Example

Regular expression *)10(00*)10( r

)(rL = { all strings with at least two consecutive 0 }

Example

Regular expression )0(*)011( r

)(rL = { all strings without two consecutive 0 }

Equivalent Regular Expressions

• Definition:

• Regular expressions and

• are equivalent if

1r 2r

)()( 21 rLrL

Example

L= { all strings without two consecutive 0 }

)0(*)011(1 r

)0(*1)0(**)011*1(2 r

LrLrL )()( 211r 2rand

are equivalentregular expr.

Example: Lexing

• Regular expressions good for describing lexemes (words) in a programming language– Identifier = (a b … z A B … Z) (a

b … z A B … Z 0 1 … 9 _ ‘ )*

– Digit = (0 1 … 9)

Implementing Regular Expressions

• Regular expressions, regular grammars reasonable way to generates strings in language

• Not so good for recognizing when a string is in language

• Regular expressions: which option to choose, how many repetitions to make

• Answer: finite state automata

Finite-State Machines with No Output Ying Lu Based on Slides by Elsa L Gunter, NJIT, Costas Busch...

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