Equivalence of Regular Language Representations

cs466(Prasad) L14Equiv 1

Equivalence of Regular Language Representations


Regular Languages: Grand UnificationGrand Unification

)( )()(

DFAsLNFAsLsNFAL

)()()()(

RELFALRELFAL

(Parallel Simulation) (Rabin and Scott’s work)

(Collapsing graphs; Structural Induction)(S. Kleene’s work)

)()( RGLFAL (Construction)(Solving linear equations))()( RELRGL


Role of various representations for Regular Languages

• Closure under complemention. (DFAs)• Closure under union, concatenation, and Kleene

star. (NFA-s, Regular expression.)• Consequence:

Closure under intersection by De Morgan’s Laws.

• Relationship to context-free languages. (Regular Grammars.)

• Ease of specification. (Regular expression.)

• Building tokenizers/lexical analyzers. (DFAs)


Application to Scanner (Lexer, Tokenizer)

• High-level view

Regularexpressions

NFA

DFA

LexicalSpecification

Table-driven Implementation of a minimal DFA


M(a)

Construction of Finite Automata from Regular Expressions

)()( FALREL

Show that there are FA for basis elements and there exist constructions on FA for capturing union, concatenation, and Kleene star operations.

Basis Case


Constructions on NFA-s

M(R1)

M(R1)

M(R2)

MM(R1 U R2)

MM(R1 R2)

MM(R*)

M(R2)

M(R)


Construction of Regular Expression from Finite Automaton

• Expression Graph is a labeled directed graph in which the arcs are labeled by regular expressions. An expression graph, like a state diagram, contains a distinguished start node and a set of accepting nodes.


Examples

ab

L(M) = (ab)*


Examples

ba

L(M) = (b+ a)* (a u b) (ba)*

b+ a

a u b


Examples

bb

L(M) = (b a)* b*( bb u (a+(ba)*b*) )*

ba

b*

a+


Main Idea

• To associate an RE with an FA, – reduce an arbitrary expression graph to one

containing at most two nodes, – by repeatedly removing nodes from the graph

and relabeling the arcs to preserve the language.• Without loss of generality, we can assume

one accepting state (because of the presence of the union operation).


Exampleqj qk

qj

qi

qk

Wj,i

Wj,i Wi,k

Wi,k


qj qk

qj

qi

qk

Wj,i

Wj,i (Wi,i)* Wi,k

Wi,k

Wi,i


Final Graph : Alternative 1

u

L(M) = (u)*


Final Graph : Alternative 2

w

L(M) = (u)* v( w u (x (u)* v) )*

u

v

x


Detailed Example

b

a ba

ab

bq0 q1

q2 q3


Delete node q1

b

a ba

ab

bq0 q1

q2 q3

bbab


Delete node q2

b

aa

b u bb

q0

q2 q3

ab

ab*ab


Finally

ab u bb

q0

q3

ab*ab

(ab*ab)*a ((bubb) (ab*ab)*a)*


• For precise details, see Algorithm 6.2.2 on Page 194 in Sudkamp’s Languages and Machines, 3rd Edition.


From Regular Expression to NFA to DFA to Regular Grammars

Via Examples


Exercise

• Construct a DFA for a+b+

q0b

q1 q2a

a b


Equivalent DFA

{q0} {q1,q2}

{q0,q1}

{}

a

a

a

a,b

b

b

b


Two Equivalent (Right-linear) Regular Grammars

<q0> -> a <q0> | a <q1>

<q1> -> b <q1> | b <q2>

<q2> -> λ

<{q0}> -> a <{q0,q1}> <{q0,q1}> ->

a <{q0,q1}> | b <{q1,q2}>

<{q1,q2}> -> λ | b <{q1,q2}>

• All productions involving <{}> can be deleted, as <{}> does not derive any terminal strings.


Two Equivalent (Left-linear) Regular Grammars

<q0> -> λ | <q0> a

<q1> -> <q1> b | <q0> a

<q2> -> <q1> b

<{q0}> -> λ<{q0,q1}> -> <{q0,q1}> a | <{q0}> a

<{q1,q2}> -> | <{q0,q1}> b | <{q1,q2}> b


From Grammars to Finite Automata

S -> aA | cA -> bB | bAB -> λ

S -> aA | cFA -> bB | bAB -> λF -> λ

SA

BF

a bb

c


From Grammars to Finite Automata

S -> aA | cA -> bB | bAB -> λ

S -> λA -> Sa | AbB -> AbF -> ScZZ -> B | F

SA

BF

a bb

c

Equivalence of Regular Language Representations

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