Regular Grammar and Regular Language [Def 3.1] Regular Grammar(use to in lexical analysis) Type 3...

Regular Grammar and Regular LanguageRegular Grammar and Regular Language[Def 3.1][Def 3.1] Regular GrammarRegular Grammar(use to in lexical analysis)(use to in lexical analysis)

Type 3 grammarType 3 grammar(regular grammar, RG)(regular grammar, RG)

1) A→tB1) A→tBt (right-linear, RLG)t (right-linear, RLG)

2) A→Bt2) A→Btt (left-linear, LLG)t (left-linear, LLG)

A, B V∈A, B V∈ NN, t V∈, t V∈ TT

[Ex] P : S→aA[Ex] P : S→aAbBbBb A→bAb A→bAa B→bSa B→bS

[Ex] P[Ex] P11 : S : S11→0S→0S11 SS11→0→0

PP22 : S : S22→1S→1S2 2 SS22→1→1

new P can be consist of P1 P2 {S→S1∪ ∪new P can be consist of P1 P2 {S→S1∪ ∪ S2}.S2}.

S →SS →S11SS22 (single production rule)(single production rule)

SS11→0S→0S110 0 SS22→1S→1S2211

∴∴ S →0SS →0S11001S1S221 1

SS11→0S→0S1100 SS22→1S→1S2211

Regular Grammar and Regular LanguageRegular Grammar and Regular Language[Def 3.1][Def 3.1] Regular GrammarRegular Grammar(use to in lexical analysis)(use to in lexical analysis)

Type 3 grammarType 3 grammar(regular grammar, RG)(regular grammar, RG)

1) A→tB1) A→tBt (right-linear, RLG)t (right-linear, RLG)

2) A→Bt2) A→Btt (left-linear, LLG)t (left-linear, LLG)

A, B V∈A, B V∈ NN, t V∈, t V∈ TT

[Ex] P : S→aA[Ex] P : S→aAbBbBb A→bAb A→bAa B→bSa B→bS

[Ex] P[Ex] P11 : S : S11→0S→0S11 SS11→0→0

PP22 : S : S22→1S→1S2 2 SS22→1→1

new P can be consist of P1 P2 {S→S1∪ ∪new P can be consist of P1 P2 {S→S1∪ ∪ S2}.S2}.

S →SS →S11SS22 (single production rule)(single production rule)

SS11→0S→0S110 0 SS22→1S→1S2211

∴∴ S →0SS →0S11001S1S221 1

SS11→0S→0S1100 SS22→1S→1S2211

Regular LanguageRegular LanguageRegular LanguageRegular Language

Regular Regular LanguageLanguage


▶ ▶ A Regular Grammar is used to scanner.A Regular Grammar is used to scanner.

simple token structuresimple token structure RG is more effective than CFG.RG is more effective than CFG. It can be consist of small module to process front-end part.It can be consist of small module to process front-end part.

Regular ExpressionRegular Expression[Def 3.2][Def 3.2] Regular ExpressionRegular Expression 1. 1. : empty set : empty set 2. 2. : { : {}} 3. 3. aa V∈V∈ TT : { : {aa}}

4. if e4. if e11 is R.E of L is R.E of L11 and e and e22 is R.E of L is R.E of L22,,

(e(e11)) ++ (e(e22)) : represent to L: represent to L22 L∪L∪ 22

(e(e11) ) • • (e(e22)) : represent to L : represent to L11LL22

(e(e11))* * : represent to { : represent to {} L∪} L∪ 1111 L∪L∪ 11

22∪∪...... L∪L∪ 11nn∪∪……

5. other cases are not a R.E.5. other cases are not a R.E.

▶ ▶ A Regular Grammar is used to scanner.A Regular Grammar is used to scanner.

simple token structuresimple token structure RG is more effective than CFG.RG is more effective than CFG. It can be consist of small module to process front-end part.It can be consist of small module to process front-end part.

Regular ExpressionRegular Expression[Def 3.2][Def 3.2] Regular ExpressionRegular Expression 1. 1. : empty set : empty set 2. 2. : { : {}} 3. 3. aa V∈V∈ TT : { : {aa}}

4. if e4. if e11 is R.E of L is R.E of L11 and e and e22 is R.E of L is R.E of L22,,

(e(e11)) ++ (e(e22)) : represent to L: represent to L22 L∪L∪ 22

(e(e11) ) • • (e(e22)) : represent to L : represent to L11LL22

(e(e11))* * : represent to { : represent to {} L∪} L∪ 1111 L∪L∪ 11

22∪∪...... L∪L∪ 11nn∪∪……

5. other cases are not a R.E.5. other cases are not a R.E.

[Ex][Ex] 1) R.E (0+1)1) R.E (0+1)**

2) R.E (a+b)2) R.E (a+b)**abbabb

[Def 3.3][Def 3.3] If two R.Es are represent to same set, there are same R.Es.If two R.Es are represent to same set, there are same R.Es.▶▶ algebraic propertyalgebraic property precedenceprecedence : : ++ < < •• < < ** ⓐ ⓐ ++ = = ++ (ⓑ (ⓑ )) = = (()) ⓒ ⓒ ((++)+)+ = = +( +(++) ) ⓓ ⓓ ((++) = ) = + + ⓔ ⓔ ((++)) = = ++ ⓕ ⓕ ++ = = ⓖ ⓖ ++ = = ⓗ ⓗ = = = = ⓘ ⓘ = = = = ⓙ ⓙ * * = = ++**

ⓚ ⓚ * * = (= (++))** (ⓛ (ⓛ **))* * = = **

ⓜ ⓜ **++ = = ** ⓝ ⓝ **+++ + = = ** ⓞ ⓞ ((++))* * = (= (****))** (ⓟ (ⓟ ))* * = (= (**++**))** ⓠ ⓠ * * = = ⓡ ⓡ * * = =

Regular Grammar Regular Grammar Regular Expression Regular Expression

[Def 3.4][Def 3.4] regular expression equationregular expression equation : The Equation that coefficient is R.E.: The Equation that coefficient is R.E.

[Def 3.3][Def 3.3] If two R.Es are represent to same set, there are same R.Es.If two R.Es are represent to same set, there are same R.Es.▶▶ algebraic propertyalgebraic property precedenceprecedence : : ++ < < •• < < ** ⓐ ⓐ ++ = = ++ (ⓑ (ⓑ )) = = (()) ⓒ ⓒ ((++)+)+ = = +( +(++) ) ⓓ ⓓ ((++) = ) = + + ⓔ ⓔ ((++)) = = ++ ⓕ ⓕ ++ = = ⓖ ⓖ ++ = = ⓗ ⓗ = = = = ⓘ ⓘ = = = = ⓙ ⓙ * * = = ++**

ⓚ ⓚ * * = (= (++))** (ⓛ (ⓛ **))* * = = **

ⓜ ⓜ **++ = = ** ⓝ ⓝ **+++ + = = ** ⓞ ⓞ ((++))* * = (= (****))** (ⓟ (ⓟ ))* * = (= (**++**))** ⓠ ⓠ * * = = ⓡ ⓡ * * = =

Regular Grammar Regular Grammar Regular Expression Regular Expression

[Def 3.4][Def 3.4] regular expression equationregular expression equation : The Equation that coefficient is R.E.: The Equation that coefficient is R.E.



[Theorem 3.1][Theorem 3.1] if if , , is R.E and is R.E and LL, then solution of X= , then solution of X= X+ X+ is X= is X= * * .. X= X= X+X+ = = ((**)+)+ = = ++++ = (= (++++)) = = **

▶▶ Search processing step to R.E of L(G) in LSearch processing step to R.E of L(G) in L

1.1. regular grammar → RE equation regular grammar → RE equation

2.2. X= X= X+X+ → X= → X= **

3.3. RE equation X is replace to RE equation X is replace to ** in RE equation,in RE equation, then construct to X= then construct to X= X+X+ form.form.

4.4. repeat step 3, then solve to X in X= repeat step 3, then solve to X in X= X+X+

[Ex] G=({S, R}, {a, b}, P, S) P: S→aS[Ex] G=({S, R}, {a, b}, P, S) P: S→aSbRbR R→aS ; R→aS ; What is a RE of L(G)?What is a RE of L(G)? ans)ans) 1.1. grammar → R.E equation grammar → R.E equation

S = aS + bR + S = aS + bR + ------ ①------ ① R = aSR = aS ------ ②------ ②

2.2. Non Non3.3. assign to ② ① assign to ② ① S = aS + b(aS) + S = aS + b(aS) + = aS + baS + = aS + baS + = (a + ba)S + = (a + ba)S + 4.4. L(G) is S = (a + ba)S + L(G) is S = (a + ba)S + = (a + ba) = (a + ba)** ∴∴ L(G) = (a + ba)L(G) = (a + ba)**

[Theorem 3.1][Theorem 3.1] if if , , is R.E and is R.E and LL, then solution of X= , then solution of X= X+ X+ is X= is X= * * .. X= X= X+X+ = = ((**)+)+ = = ++++ = (= (++++)) = = **

▶▶ Search processing step to R.E of L(G) in LSearch processing step to R.E of L(G) in L

1.1. regular grammar → RE equation regular grammar → RE equation

2.2. X= X= X+X+ → X= → X= **

3.3. RE equation X is replace to RE equation X is replace to ** in RE equation,in RE equation, then construct to X= then construct to X= X+X+ form.form.

4.4. repeat step 3, then solve to X in X= repeat step 3, then solve to X in X= X+X+

[Ex] G=({S, R}, {a, b}, P, S) P: S→aS[Ex] G=({S, R}, {a, b}, P, S) P: S→aSbRbR R→aS ; R→aS ; What is a RE of L(G)?What is a RE of L(G)? ans)ans) 1.1. grammar → R.E equation grammar → R.E equation

S = aS + bR + S = aS + bR + ------ ①------ ① R = aSR = aS ------ ②------ ②

2.2. Non Non3.3. assign to ② ① assign to ② ① S = aS + b(aS) + S = aS + b(aS) + = aS + baS + = aS + baS + = (a + ba)S + = (a + ba)S + 4.4. L(G) is S = (a + ba)S + L(G) is S = (a + ba)S + = (a + ba) = (a + ba)** ∴∴ L(G) = (a + ba)L(G) = (a + ba)**



Finite Automata(FA)Finite Automata(FA)▶▶ ConstructionConstruction

input/outputinput/output finite statefinite state control mechanism(set of transition)control mechanism(set of transition)

[Def 3.5][Def 3.5] FA M = (Q, FA M = (Q, , , , , qq00, F), F)

Q Q : finite set of states: finite set of states : set of input alphabet: set of input alphabet qq00 : initial state: initial state

F F : set of finial state, F Q⊆: set of finial state, F Q⊆ : transition function Q×Σ→2: transition function Q×Σ→2QQ

eg) eg) ((qq, , aa) = {) = {pp11, , pp22, ··· , , ··· , ppnn}}

⇒⇒ after scanning input symbol 'after scanning input symbol 'aa' in state ' in state qq,, next state select to state next state select to state pp11~ ~ ppnn

Finite Automata(FA)Finite Automata(FA)▶▶ ConstructionConstruction

input/outputinput/output finite statefinite state control mechanism(set of transition)control mechanism(set of transition)

[Def 3.5][Def 3.5] FA M = (Q, FA M = (Q, , , , , qq00, F), F)


F F : set of finial state, F Q⊆: set of finial state, F Q⊆ : transition function Q×Σ→2: transition function Q×Σ→2QQ

eg) eg) ((qq, , aa) = {) = {pp11, , pp22, ··· , , ··· , ppnn}}

⇒⇒ after scanning input symbol 'after scanning input symbol 'aa' in state ' in state qq,, next state select to state next state select to state pp11~ ~ ppnn



DFA(Deterministic FA)DFA(Deterministic FA) next state is determined to unique statenext state is determined to unique state

((qq, , aa) = ) = pp

[Def 3.6][Def 3.6] DFA M = (Q, DFA M = (Q, , , , , qq00, F), F)


F F : set of finial state, F Q⊆: set of finial state, F Q⊆ : transition function Q× : transition function Q× → Q ≡ → Q ≡ ((qq, , aa) = ) = pp

▶ ▶ DFA propertiesDFA properties 1.1. no no -transition-transition 2.2. determined to unique state determined to unique state

▶ ▶ Extensibility of transition functionExtensibility of transition function Q× Q× * * → Q→ Q ((qq, , ) = ) = qq ((qq, , xaxa) = ) = ((((qq, , xx), ), aa) x∈) x∈**, , aa∈∈ ⇒⇒ extensionextension : : symbolsymbol → → stringstring

DFA(Deterministic FA)DFA(Deterministic FA) next state is determined to unique statenext state is determined to unique state

((qq, , aa) = ) = pp

[Def 3.6][Def 3.6] DFA M = (Q, DFA M = (Q, , , , , qq00, F), F)


F F : set of finial state, F Q⊆: set of finial state, F Q⊆ : transition function Q× : transition function Q× → Q ≡ → Q ≡ ((qq, , aa) = ) = pp

▶ ▶ DFA propertiesDFA properties 1.1. no no -transition-transition 2.2. determined to unique state determined to unique state

▶ ▶ Extensibility of transition functionExtensibility of transition function Q× Q× * * → Q→ Q ((qq, , ) = ) = qq ((qq, , xaxa) = ) = ((((qq, , xx), ), aa) x∈) x∈**, , aa∈∈ ⇒⇒ extensionextension : : symbolsymbol → → stringstring



[Def 3.7][Def 3.7] L(M) that recognized DFA ML(M) that recognized DFA ML(M) = {xL(M) = {x(( q q00,, x x) F}∈) F}∈

[Ex] M=({[Ex] M=({pp, , qq, , rr}, {0, 1}, }, {0, 1}, δδ, , pp, {, {rr})}) : : ((pp, 0) = , 0) = qq ((pp, 1) = , 1) = pp ((qq, 0) = , 0) = rr ((qq, 1) = , 1) = pp

((rr, 0) = , 0) = rr ((rr, 1) = , 1) = rr Can be recognized sentence "1001", "0110"?Can be recognized sentence "1001", "0110"?

ans)ans) 1) 1) ((pp, 1001) = , 1001) = ((pp, 001) = , 001) = ((qq, 01) = , 01) = ((rr, 1) = , 1) = rr F ∈F ∈ ∴∴ acceptaccept 2) 2) ((pp, 0110) = , 0110) = ((qq, 110) = , 110) = ((pp, 10) = , 10) = ((pp, 0) = , 0) = qqF F ∴∴ rejectreject

※※ transition tabletransition table : transition function is represented to matrix : transition function is represented to matrix[Ex] Transition Table[Ex] Transition Table

00 11

pp qq p p

qq rr pp

rr rr rr

[Def 3.7][Def 3.7] L(M) that recognized DFA ML(M) that recognized DFA ML(M) = {xL(M) = {x(( q q00,, x x) F}∈) F}∈

[Ex] M=({[Ex] M=({pp, , qq, , rr}, {0, 1}, }, {0, 1}, δδ, , pp, {, {rr})}) : : ((pp, 0) = , 0) = qq ((pp, 1) = , 1) = pp ((qq, 0) = , 0) = rr ((qq, 1) = , 1) = pp

((rr, 0) = , 0) = rr ((rr, 1) = , 1) = rr Can be recognized sentence "1001", "0110"?Can be recognized sentence "1001", "0110"?

ans)ans) 1) 1) ((pp, 1001) = , 1001) = ((pp, 001) = , 001) = ((qq, 01) = , 01) = ((rr, 1) = , 1) = rr F ∈F ∈ ∴∴ acceptaccept 2) 2) ((pp, 0110) = , 0110) = ((qq, 110) = , 110) = ((pp, 10) = , 10) = ((pp, 0) = , 0) = qqF F ∴∴ rejectreject

※※ transition tabletransition table : transition function is represented to matrix : transition function is represented to matrix[Ex] Transition Table[Ex] Transition Table

00 11

pp qq p p

qq rr pp

rr rr rr



[Def 3.8][Def 3.8] transition diagramtransition diagram statestate : node : node transition functiontransition function : label, directed arc : label, directed arc eg) eg) ((qq, , aa) = ) = pp a a

≡ ≡ ⓠ ⓟ ⓠ ⓟ

final statefinal state : double circle : double circle

[Ex] What is a transition diagram for example before?[Ex] What is a transition diagram for example before?

[Def 3.9][Def 3.9] completely specifiedcompletely specified if ∀if ∀qq Q, ∈Q, ∈ aa∈∈ in M=(Q, in M=(Q, , , , , qq00, F), , F), ((qq, , aa) is existed uniquely next state) is existed uniquely next state

∴ ∴ DFA ≡ completely specifiedDFA ≡ completely specified

[Def 3.8][Def 3.8] transition diagramtransition diagram statestate : node : node transition functiontransition function : label, directed arc : label, directed arc eg) eg) ((qq, , aa) = ) = pp a a

≡ ≡ ⓠ ⓟ ⓠ ⓟ

final statefinal state : double circle : double circle

[Ex] What is a transition diagram for example before?[Ex] What is a transition diagram for example before?

[Def 3.9][Def 3.9] completely specifiedcompletely specified if ∀if ∀qq Q, ∈Q, ∈ aa∈∈ in M=(Q, in M=(Q, , , , , qq00, F), , F), ((qq, , aa) is existed uniquely next state) is existed uniquely next state

∴ ∴ DFA ≡ completely specifiedDFA ≡ completely specified



startstartp q r

11

11

00 00

0, 10, 1

▶▶ AAlgorithm that string is recognized by DFA M = (Q, lgorithm that string is recognized by DFA M = (Q, , , , , qq00, F), F)

Algorithm_Recognize()Algorithm_Recognize() { { current_statecurrent_state = = qq00;;

get(get(next_symbolnext_symbol);); while(while(input string not exhaustedinput string not exhausted) {) { current_statecurrent_state = = ((current_statecurrent_state, , next_symbolnext_symbol);); get(get(next_symbolnext_symbol);); }} if(if(current_statecurrent_state F) printf("∈F) printf("∈ AcceptAccept");"); else printf("else printf("RejectReject");"); }}

NFA(Non-deterministic FA)NFA(Non-deterministic FA)

[Def 3.10][Def 3.10] NFA M = (Q, NFA M = (Q, , , , , qq00, F), F)


F F : set of finial state, F Q⊆: set of finial state, F Q⊆ : transition function Q×: transition function Q× → 2 → 2QQ, , ((qq, , aa) = {) = {pp11, , pp22, ···, , ···, ppnn}}

▶▶ AAlgorithm that string is recognized by DFA M = (Q, lgorithm that string is recognized by DFA M = (Q, , , , , qq00, F), F)

Algorithm_Recognize()Algorithm_Recognize() { { current_statecurrent_state = = qq00;;

get(get(next_symbolnext_symbol);); while(while(input string not exhaustedinput string not exhausted) {) { current_statecurrent_state = = ((current_statecurrent_state, , next_symbolnext_symbol);); get(get(next_symbolnext_symbol);); }} if(if(current_statecurrent_state F) printf("∈F) printf("∈ AcceptAccept");"); else printf("else printf("RejectReject");"); }}

NFA(Non-deterministic FA)NFA(Non-deterministic FA)

[Def 3.10][Def 3.10] NFA M = (Q, NFA M = (Q, , , , , qq00, F), F)


F F : set of finial state, F Q⊆: set of finial state, F Q⊆ : transition function Q×: transition function Q× → 2 → 2QQ, , ((qq, , aa) = {) = {pp11, , pp22, ···, , ···, ppnn}}



[Ex] [Ex] What is transition diagram of M=({What is transition diagram of M=({qq00, q, q11, q, q22, q, q33, q, qff}, {0, 1}, }, {0, 1}, , , qq00, {, {qqff})?})?

0 0 1 1qq0 0 {{ q q11, q, q2 2 } {} { q q11, q, q3 3 }}

qq1 1 {{ q q11, q, q2 2 } {} { q q11, q, q3 3 }}

qq2 2 {{ q qf f }} qq3 3 { { qqf f }}

qqff { { q qf f } } { { q qf f }} ※※ ((qq, , aa) = ) = : : not definednot defined

[Ex] [Ex] What is transition diagram of M=({What is transition diagram of M=({qq00, q, q11, q, q22, q, q33, q, qff}, {0, 1}, }, {0, 1}, , , qq00, {, {qqff})?})?

0 0 1 1qq0 0 {{ q q11, q, q2 2 } {} { q q11, q, q3 3 }}

qq1 1 {{ q q11, q, q2 2 } {} { q q11, q, q3 3 }}

qq2 2 {{ q qf f }} qq3 3 { { qqf f }}

qqff { { q qf f } } { { q qf f }} ※※ ((qq, , aa) = ) = : : not definednot defined



startstartqq

00

1111

0000

0, 10, 1

qq11

0, 10, 1qq

ff

qq22

qq33

11

0, 10, 1

▶▶ Extension of NFAExtension of NFA

[step 1][step 1] Q×Q× → 2 → 2QQ Q× Q× * * → 2→ 2QQ

((qq, , ) = {) = {qq}} ((qq, , xaxa) = ∪) = ∪ ((pp, , aa)) p∈p∈((qq, , xx))

xx∈∈**, , aa∈∈ ∴∴ symbolsymbol → → extend to stringextend to string

[step 2][step 2] Q×Q× → 2 → 2QQ 2 2QQ××* * → 2→ 2QQ

({({ p p11, , pp22, ···, , ···, ppkk}, }, xx) = ) = ((ppi i , , xx))

∴∴ one stateone state → → extended to more stateextended to more state

[Ex] [Ex] ((qq00, 1001), 1001) = = ((qq11, 001)∪, 001)∪((qq33, 001) , 001)

= = ((qq11, 01) ∪, 01) ∪ ((qq22, 01)∪, 01)∪

= = ((qq11, 1)∪, 1)∪((qq22, 1)∪, 1)∪((qqff, 1), 1)

= {= {qq11, q, q33, q, qff } }

▶▶ Extension of NFAExtension of NFA

[step 1][step 1] Q×Q× → 2 → 2QQ Q× Q× * * → 2→ 2QQ

((qq, , ) = {) = {qq}} ((qq, , xaxa) = ∪) = ∪ ((pp, , aa)) p∈p∈((qq, , xx))

xx∈∈**, , aa∈∈ ∴∴ symbolsymbol → → extend to stringextend to string

[step 2][step 2] Q×Q× → 2 → 2QQ 2 2QQ××* * → 2→ 2QQ

({({ p p11, , pp22, ···, , ···, ppkk}, }, xx) = ) = ((ppi i , , xx))

∴∴ one stateone state → → extended to more stateextended to more state

[Ex] [Ex] ((qq00, 1001), 1001) = = ((qq11, 001)∪, 001)∪((qq33, 001) , 001)

= = ((qq11, 01) ∪, 01) ∪ ((qq22, 01)∪, 01)∪

= = ((qq11, 1)∪, 1)∪((qq22, 1)∪, 1)∪((qqff, 1), 1)

= {= {qq11, q, q33, q, qff } }





[Def 3.11][Def 3.11] L(M) that is recognized by NFA ML(M) that is recognized by NFA M L(M) = {L(M) = {xx((qq00, , xx)∩F≠)∩F≠}}

[Ex] Can be recognized to 1011?[Ex] Can be recognized to 1011? ((qq00, 1011) = , 1011) = ({({qq11, q, q33}, 011) = }, 011) = ({({qq11, q, q22}, 11)}, 11)

= = ({({qq11, q, q33}, 1) = {}, 1) = {qq11, q, q33, q, qff}}

{{qq11, q, q33, q, qff}∩{}∩{qqff}≠}≠

∴∴ 10111011 is accepted by Mis accepted by M

if one more than final state is exist, the string is accepted.if one more than final state is exist, the string is accepted.

state of NFA M : when number of state of NFA M : when number of mm, length(string) = , length(string) = nn number of maximum nodes in tree = number of maximum nodes in tree = mmnn

∴∴ string that is recognized in NFA, excessively time-consumedstring that is recognized in NFA, excessively time-consumed

[Def 3.11][Def 3.11] L(M) that is recognized by NFA ML(M) that is recognized by NFA M L(M) = {L(M) = {xx((qq00, , xx)∩F≠)∩F≠}}

[Ex] Can be recognized to 1011?[Ex] Can be recognized to 1011? ((qq00, 1011) = , 1011) = ({({qq11, q, q33}, 011) = }, 011) = ({({qq11, q, q22}, 11)}, 11)

= = ({({qq11, q, q33}, 1) = {}, 1) = {qq11, q, q33, q, qff}}

{{qq11, q, q33, q, qff}∩{}∩{qqff}≠}≠

∴∴ 10111011 is accepted by Mis accepted by M

if one more than final state is exist, the string is accepted.if one more than final state is exist, the string is accepted.

state of NFA M : when number of state of NFA M : when number of mm, length(string) = , length(string) = nn number of maximum nodes in tree = number of maximum nodes in tree = mmnn

∴∴ string that is recognized in NFA, excessively time-consumedstring that is recognized in NFA, excessively time-consumed

Transformation from NFA to DFATransformation from NFA to DFA

A NFA represent language structure simply(RE→NFA), A NFA represent language structure simply(RE→NFA), but difficult to implementation.but difficult to implementation.

[Theorem 3.2][Theorem 3.2] Let NFA M=(Q, Let NFA M=(Q, , , , , qq00, F)., F).

if L(M) is accepted by NFA M, DFA M'=(Q', if L(M) is accepted by NFA M, DFA M'=(Q', ', ', ', ', qq00', F') is existed.', F') is existed.

≡ ≡ construct DFA M'=(Q', construct DFA M'=(Q', , , ', ', qq00', F')', F')

① ① Q' = 2Q' = 2Q Q (power set of Q), [presented a state Q : [(power set of Q), [presented a state Q : [qq11, q, q22,···, q,···, qii], ], qqii Q]∈Q]∈

② ② qq00'=['=[qq00]]

③ ③ F'={F'={qq Q'∈Q'∈ qq F}∈F}∈ ④ ④ if if ({({ q q11, q, q22,···, q,···, qii}, }, aa)={)={pp11, p, p22,···, p,···, pjj}, then }, then '(['([qq11, q, q22,···, q,···, qii], ], aa)=[)=[ p p11, p, p22,···, p,···, pjj]]

Transformation from NFA to DFATransformation from NFA to DFA

A NFA represent language structure simply(RE→NFA), A NFA represent language structure simply(RE→NFA), but difficult to implementation.but difficult to implementation.

[Theorem 3.2][Theorem 3.2] Let NFA M=(Q, Let NFA M=(Q, , , , , qq00, F)., F).

if L(M) is accepted by NFA M, DFA M'=(Q', if L(M) is accepted by NFA M, DFA M'=(Q', ', ', ', ', qq00', F') is existed.', F') is existed.

≡ ≡ construct DFA M'=(Q', construct DFA M'=(Q', , , ', ', qq00', F')', F')

① ① Q' = 2Q' = 2Q Q (power set of Q), [presented a state Q : [(power set of Q), [presented a state Q : [qq11, q, q22,···, q,···, qii], ], qqii Q]∈Q]∈

② ② qq00'=['=[qq00]]

③ ③ F'={F'={qq Q'∈Q'∈ qq F}∈F}∈ ④ ④ if if ({({ q q11, q, q22,···, q,···, qii}, }, aa)={)={pp11, p, p22,···, p,···, pjj}, then }, then '(['([qq11, q, q22,···, q,···, qii], ], aa)=[)=[ p p11, p, p22,···, p,···, pjj]]



[Ex 21] NFA M=({[Ex 21] NFA M=({qq00, , qq11}, {0, 1}, }, {0, 1}, , , qq00, {, {qq11})})

state name change :state name change : [ [qq00]=A, []=A, [qq11]=B, []=B, [qq00, , qq11]=C]=C

[Ex 22], [Ex 23] [Ex 22], [Ex 23] Reference of Text Reference of Text pp.78 ~ pp.80pp.78 ~ pp.80

[Ex 21] NFA M=({[Ex 21] NFA M=({qq00, , qq11}, {0, 1}, }, {0, 1}, , , qq00, {, {qq11})})

state name change :state name change : [ [qq00]=A, []=A, [qq11]=B, []=B, [qq00, , qq11]=C]=C

[Ex 22], [Ex 23] [Ex 22], [Ex 23] Reference of Text Reference of Text pp.78 ~ pp.80pp.78 ~ pp.80



’ ’ 0 10 1

[[qq00] [] [qq00, , qq11] [] [qq00] ]

[[qq00] ] [ [qq00, , qq11] ]

[[qq00, , qq11] [] [qq00, , qq11] ] [ [qq00, , qq11] ]

0 10 1

qq00 { {qq00, , qq11} {} {qq00} }

qq11 { {qq00, , qq11}}

AAAA BBBB

CCCC

startstartstartstart 1111

0000

1111

0,10,1

AAAAstartstartstartstart

1111

CCCC0,10,10000



[Def 3.12][Def 3.12] -NFA M=(Q, -NFA M=(Q, , , , , qq00, F) :, F) :

: Q(: Q( {∪{∪ }) → 2}) → 2QQ

other cases are same to NFA. other cases are same to NFA.

[Def 3.13][Def 3.13] -CLOSURE(-CLOSURE(ss)) case of case of ss that is one state that is one state -CLOSURE(-CLOSURE(ss)) = include itself state = include itself state ss and and all reachable statesall reachable states by label by label from state from state ss (repeat until not changes elements of (repeat until not changes elements of -CLOSURE(-CLOSURE(ss))))

case of case of TT that is one or more states that is one or more states -CLOSURE(-CLOSURE(TT)) = each state in = each state in T, T, executing step executing step and union its all results and union its all results ((-CLOSURE(-CLOSURE(TT)) = ∪)) = ∪ -CLOSURE(-CLOSURE(xx) ) qq∈∈TT

[Def 3.12][Def 3.12] -NFA M=(Q, -NFA M=(Q, , , , , qq00, F) :, F) :

: Q(: Q( {∪{∪ }) → 2}) → 2QQ

other cases are same to NFA. other cases are same to NFA.

[Def 3.13][Def 3.13] -CLOSURE(-CLOSURE(ss)) case of case of ss that is one state that is one state -CLOSURE(-CLOSURE(ss)) = include itself state = include itself state ss and and all reachable statesall reachable states by label by label from state from state ss (repeat until not changes elements of (repeat until not changes elements of -CLOSURE(-CLOSURE(ss))))

case of case of TT that is one or more states that is one or more states -CLOSURE(-CLOSURE(TT)) = each state in = each state in T, T, executing step executing step and union its all results and union its all results ((-CLOSURE(-CLOSURE(TT)) = ∪)) = ∪ -CLOSURE(-CLOSURE(xx) ) qq∈∈TT



[Ex 24] [Ex 24] What is CLOSURE in following What is CLOSURE in following -NFA?-NFA?

CLOSURE(A)={A, B, D}CLOSURE(A)={A, B, D} CLOSURE({A, C})=CLOSURE(A) CLOSURE({A, C})=CLOSURE(A) CLOSURE(C)={A, B, C, D} CLOSURE(C)={A, B, C, D}

[Ex 25] [Ex 25] Transform fromTransform from -NFA to DFA-NFA to DFA

[Ex 24] [Ex 24] What is CLOSURE in following What is CLOSURE in following -NFA?-NFA?

CLOSURE(A)={A, B, D}CLOSURE(A)={A, B, D} CLOSURE({A, C})=CLOSURE(A) CLOSURE({A, C})=CLOSURE(A) CLOSURE(C)={A, B, C, D} CLOSURE(C)={A, B, C, D}

[Ex 25] [Ex 25] Transform fromTransform from -NFA to DFA-NFA to DFA

aaaaaaaa

AAAAstartstartstartstart

BBBB

bbbb

aaaa

CCCC DDDD

aaaa

startstartstartstart

cccc

3333

bbbb

1111 4444

2222



state name :state name : A=[1,3,4], B=[2], C=[3,4], D=[4] A=[1,3,4], B=[2], C=[3,4], D=[4]state name :state name : A=[1,3,4], B=[2], C=[3,4], D=[4] A=[1,3,4], B=[2], C=[3,4], D=[4]

a b ca b c

CL(1)={1,3,4}CL(1)={1,3,4} [1,3,4][1,3,4]

[2][2]

[3,4][3,4]

[4] [4]

CL(2)={2} CL(3)={3,4}CL(2)={2} CL(3)={3,4} [2] [2] [3,4] [3,4]

CL(4)={4} CL(4)={4} [4][4]

CL(3)={3,4}CL(3)={3,4} [3,4][3,4]

AA

CC

startstartstartstart aaaaDDBB bbbb

cccc cccc

DFA's state minimizationDFA's state minimization

use of equivalence relation use of equivalence relation →→ state merge state merge

▶▶ ≡≡(equivalence relation)(equivalence relation) p≡q iff p≡q iff δδ((pp, , xx) is an accepting state) is an accepting state iff iff δδ((qq, , xx) is an accepting state) is an accepting state

[Def 3.14][Def 3.14] distinguishdistinguish if an ∃if an ∃ xx such that such that δδ((pp, , xx)=)=rr, , δδ((qq, , xx)=)=tt, a final state is only one of , a final state is only one of rr and and tt..

▶▶ inaccessible stateinaccessible state : only existed out arc, except start state : only existed out arc, except start state

▶▶ minimizing methodminimizing method 1. eliminate inaccessible state1. eliminate inaccessible state 2. grouping final states and non-final states2. grouping final states and non-final states 3. divide state that going to another equivalence class by same input symbol.3. divide state that going to another equivalence class by same input symbol. 4. repeat step 3 when can not divided4. repeat step 3 when can not divided

DFA's state minimizationDFA's state minimization

use of equivalence relation use of equivalence relation →→ state merge state merge

▶▶ ≡≡(equivalence relation)(equivalence relation) p≡q iff p≡q iff δδ((pp, , xx) is an accepting state) is an accepting state iff iff δδ((qq, , xx) is an accepting state) is an accepting state

[Def 3.14][Def 3.14] distinguishdistinguish if an ∃if an ∃ xx such that such that δδ((pp, , xx)=)=rr, , δδ((qq, , xx)=)=tt, a final state is only one of , a final state is only one of rr and and tt..

▶▶ inaccessible stateinaccessible state : only existed out arc, except start state : only existed out arc, except start state

▶▶ minimizing methodminimizing method 1. eliminate inaccessible state1. eliminate inaccessible state 2. grouping final states and non-final states2. grouping final states and non-final states 3. divide state that going to another equivalence class by same input symbol.3. divide state that going to another equivalence class by same input symbol. 4. repeat step 3 when can not divided4. repeat step 3 when can not divided



▶▶ new DFA M'= (Q', Σ, new DFA M'= (Q', Σ, δδ', ', qq00', F')', F')

1. Q1. Q'' : set of equivalence class : set of equivalence class eg)eg) [[qq] : equivalence class include ] : equivalence class include qq

2. let [2. let [pp], [], [qq] are equivalence class, if ] are equivalence class, if δδ((pp, , aa) = ) = qq, then , then δδ'(['([qq], ], aa) = [) = [qq].].

3. 3. qq00' is [' is [qq00]]

4. F'= {[4. F'= {[qq]] qq F}∈F}∈

[Ex 26] [Ex 26] DFA's minimizationDFA's minimization

▶▶ new DFA M'= (Q', Σ, new DFA M'= (Q', Σ, δδ', ', qq00', F')', F')

1. Q1. Q'' : set of equivalence class : set of equivalence class eg)eg) [[qq] : equivalence class include ] : equivalence class include qq

2. let [2. let [pp], [], [qq] are equivalence class, if ] are equivalence class, if δδ((pp, , aa) = ) = qq, then , then δδ'(['([qq], ], aa) = [) = [qq].].

3. 3. qq00' is [' is [qq00]]

4. F'= {[4. F'= {[qq]] qq F}∈F}∈

[Ex 26] [Ex 26] DFA's minimizationDFA's minimization



aaaa

aaaa

AAstartstartstartstart

DD

BB

bbbb

aaaa

EEbbbb

aaaaCC

aaaa

bbbbbbbb

bbbb



1 : {A, B, D}1 : {A, B, D} 2 : {C, E}2 : {C, E}

aa

bb

1 1 11 1 11 2 21 2 2

1 11 12 22 2

1 : {A}1 : {A} 2 : {B, D}2 : {B, D}

aa

bb

2222

2 22 23 33 3

3 : {C, E}3 : {C, E}

2 22 23 33 3

state name :state name : X=[A], Y=[B, D], Z=[C, E] X=[A], Y=[B, D], Z=[C, E]state name :state name : X=[A], Y=[B, D], Z=[C, E] X=[A], Y=[B, D], Z=[C, E]

XXXX YYYY ZZZZstartstartstartstart

aaaa

aaaa

a, ba, ba, ba, b bbbb

bbbb

aa bb

XX

YY

ZZ

YY

YY

YY

δδ''

YY

ZZ

ZZ

A Close Properties of Finite AutomataA Close Properties of Finite Automata

[Theorem 3.3][Theorem 3.3] if Lif L11, L, L22 are R.L. are R.L.

1. L1. L11 L∪ L∪ 22

2. L2. L11·L·L22

3. L3. L11** are R.L are R.L

Proof)Proof) case 1.case 1. LL11, L, L22 are R.L, such that exist G are R.L, such that exist G11=(V=(VN1N1, V, VT1T1, P, P11, S, S11), G), G22=(V=(VN2N2, V, VT2T2, P, P22, S, S22))

∴ ∴ LL11 = L(G = L(G11), L), L22 = L(G = L(G22))

It can be consist of new Grammar G.It can be consist of new Grammar G. G=(G=(VVN1N1 V∪V∪ N2N2 {S}∪{S}∪ , , VVT1T1 V∪V∪ T2T2, , PP11 P∪P∪ 22 {S→S∪{S→S∪ 11 ｜｜ SS22}}, , SS),),

S S⇒S S⇒ 11⇒⇒11 L(G∈L(G∈ 11) ) oror S S⇒ S S⇒ 22⇒⇒22 L(G∈L(G∈ 22).).

∴ ∴ L(G) = L(GL(G) = L(G11) L(G∪) L(G∪ 22) = L) = L11 L∪L∪ 22

G is R.G L⇒G is R.G L⇒ 11 L∪ L∪ 22 is R.L. is R.L.

case 2, 3.case 2, 3. Same. Same. ∴∴ R.L is closed to union, concatenation, closureR.L is closed to union, concatenation, closure

A Close Properties of Finite AutomataA Close Properties of Finite Automata

[Theorem 3.3][Theorem 3.3] if Lif L11, L, L22 are R.L. are R.L.

1. L1. L11 L∪ L∪ 22

2. L2. L11·L·L22

3. L3. L11** are R.L are R.L

Proof)Proof) case 1.case 1. LL11, L, L22 are R.L, such that exist G are R.L, such that exist G11=(V=(VN1N1, V, VT1T1, P, P11, S, S11), G), G22=(V=(VN2N2, V, VT2T2, P, P22, S, S22))

∴ ∴ LL11 = L(G = L(G11), L), L22 = L(G = L(G22))

It can be consist of new Grammar G.It can be consist of new Grammar G. G=(G=(VVN1N1 V∪V∪ N2N2 {S}∪{S}∪ , , VVT1T1 V∪V∪ T2T2, , PP11 P∪P∪ 22 {S→S∪{S→S∪ 11 ｜｜ SS22}}, , SS),),

S S⇒S S⇒ 11⇒⇒11 L(G∈L(G∈ 11) ) oror S S⇒ S S⇒ 22⇒⇒22 L(G∈L(G∈ 22).).

∴ ∴ L(G) = L(GL(G) = L(G11) L(G∪) L(G∪ 22) = L) = L11 L∪L∪ 22

G is R.G L⇒G is R.G L⇒ 11 L∪ L∪ 22 is R.L. is R.L.

case 2, 3.case 2, 3. Same. Same. ∴∴ R.L is closed to union, concatenation, closureR.L is closed to union, concatenation, closure



Regular GrammarRegular GrammarRegular GrammarRegular Grammar

Regular Regular Regular Regular ExpressionExpressionExpressionExpression

Finite Finite Finite Finite AutomataAutomataAutomataAutomata

Attributes of Regular LanguageAttributes of Regular Language

Relationship of Regular grammar and Finite AutomataRelationship of Regular grammar and Finite Automata

▶▶ Regular grammar Regular grammar FA FA

method of construction to FAmethod of construction to FA

M=(Q, M=(Q, , , , , qq00, F), F)

Q Q : V: VNN {∪{∪ ff}, }, f f : new final state: new final state

: V: VTT

qq00 : S: S

F F : if : if L(G), then {L(G), then {ff} else {S, } else {S, ff}} : if A→: if A→aaB P, then ∈B P, then ∈ (A, (A, aa) B∋) B∋

if A→if A→aa P, then ∈P, then ∈ (A, (A, aa)∋)∋ff

Attributes of Regular LanguageAttributes of Regular Language

Relationship of Regular grammar and Finite AutomataRelationship of Regular grammar and Finite Automata

▶▶ Regular grammar Regular grammar FA FA

method of construction to FAmethod of construction to FA

M=(Q, M=(Q, , , , , qq00, F), F)

Q Q : V: VNN {∪{∪ ff}, }, f f : new final state: new final state

: V: VTT

qq00 : S: S

F F : if : if L(G), then {L(G), then {ff} else {S, } else {S, ff}} : if A→: if A→aaB P, then ∈B P, then ∈ (A, (A, aa) B∋) B∋

if A→if A→aa P, then ∈P, then ∈ (A, (A, aa)∋)∋ff



[Ex 28] [Ex 28] Changing regular grammar to FAChanging regular grammar to FA

G=({S, B}, {0, 1}, P, S),G=({S, B}, {0, 1}, P, S), P : P : S→0S S→1B S→0 S→1S→0S S→1B S→0 S→1 B→0S B→0B→0S B→0

M=(Q, M=(Q, , , , , qq00, F), F)

Q Q : V: VNN {∪{∪ ff} = {S, B, } = {S, B, ff}}

: V: VT T ={0, 1}={0, 1}

qq00 : S: S

F F : {: {ff} }

▶▶ FAFA Regular grammar Regular grammar

method of construction to Regular Grammarmethod of construction to Regular Grammar

G=( VG=( VNN, V, VTT, P, S), P, S)

VVNN : Q : Q

VVTT : : S S : : qq00

P P : if : if ((qq, , aa) = ) = rr, then , then qq→→ar;ar; if if qq F, then ∈F, then ∈ qq→→

[Ex 29][Ex 29] FA FA Regular grammar Regular grammar (Reference of Text pp.91)(Reference of Text pp.91)

[Ex 28] [Ex 28] Changing regular grammar to FAChanging regular grammar to FA

G=({S, B}, {0, 1}, P, S),G=({S, B}, {0, 1}, P, S), P : P : S→0S S→1B S→0 S→1S→0S S→1B S→0 S→1 B→0S B→0B→0S B→0

M=(Q, M=(Q, , , , , qq00, F), F)

Q Q : V: VNN {∪{∪ ff} = {S, B, } = {S, B, ff}}

: V: VT T ={0, 1}={0, 1}

qq00 : S: S

F F : {: {ff} }

▶▶ FAFA Regular grammar Regular grammar

method of construction to Regular Grammarmethod of construction to Regular Grammar

G=( VG=( VNN, V, VTT, P, S), P, S)

VVNN : Q : Q

VVTT : : S S : : qq00

P P : if : if ((qq, , aa) = ) = rr, then , then qq→→ar;ar; if if qq F, then ∈F, then ∈ qq→→

[Ex 29][Ex 29] FA FA Regular grammar Regular grammar (Reference of Text pp.91)(Reference of Text pp.91)



00 11

SS

BB

ff

{S, {S, ff}}

{S, {S, ff}}

{S, {S, ff}}

Finite Automata and Regular ExpressionFinite Automata and Regular Expression

▶▶ FAFA Regular Expression Regular Expression

method : FA method : FA RG RG RE RE

FA FA transition table transition table

transition table transition table RE RE regular grammar regular grammar regular grammar equationregular grammar equation regular grammar equation regular grammar equation RE RE

▶▶ Regular Expression Regular Expression FA FA

method : RE method : RE NFA( NFA(-NFA) -NFA) DFA DFA

▣▣ RE RE -NFA-NFA

The follow sentences is all same meaning.The follow sentences is all same meaning.1.1. L is generated by some regular grammar. L is generated by some regular grammar.2.2. L is recognized by some FA. L is recognized by some FA.3.3. L is described by some RE. L is described by some RE.

Finite Automata and Regular ExpressionFinite Automata and Regular Expression

▶▶ FAFA Regular Expression Regular Expression

method : FA method : FA RG RG RE RE

FA FA transition table transition table

transition table transition table RE RE regular grammar regular grammar regular grammar equationregular grammar equation regular grammar equation regular grammar equation RE RE

▶▶ Regular Expression Regular Expression FA FA

method : RE method : RE NFA( NFA(-NFA) -NFA) DFA DFA

▣▣ RE RE -NFA-NFA

The follow sentences is all same meaning.The follow sentences is all same meaning.1.1. L is generated by some regular grammar. L is generated by some regular grammar.2.2. L is recognized by some FA. L is recognized by some FA.3.3. L is described by some RE. L is described by some RE.



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