Theory of Computation Regular Languages
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1 Theory of Computtion Regulr Lnguges Bow-Yw Wng Acdemi Sinic Spring 2012 Bow-Yw Wng (Acdemi Sinic) Regulr Lnguges Spring / 38
2 Schemtic of Finite Automt control Figure: Schemtic of Finite Automt A finite utomton hs finite set of control sttes. A finite utomton reds input symbols from left to right. A finite utomton ccepts or rejects n input fter reding the input. Bow-Yw Wng (Acdemi Sinic) Regulr Lnguges Spring / 38
3 Finite Automton M q 1 q 2 q 3 0, 1 Figure: A Finite Automton M 1 Figure 2 shows the stte digrm of finite utomton M 1. M 1 hs 3 sttes: q 1, q 2, q 3 ; strt stte: q 1 ; ccept stte: q 2 ; trnsitions: q 1 q 1, q 1 q 2, q 2 q 2, q 2 q 3, q 3 q 2, 1 nd q 3 q 2. Bow-Yw Wng (Acdemi Sinic) Regulr Lnguges Spring / 38
4 Accepted nd Rejected String q 1 q 2 q 3 0, 1 Consider n input string M 1 processes the string from the strt stte q 1. It tkes the trnsition lbeled by the current symbol nd moves to the next stte. At the end of the string, there re two cses: If M1 is t n ccept stte, M 1 outputs ccept; Otherwise, M1 outputs reject. Strings ccepted by M 1 : 1, 01, 11, 1100, 1101,.... Strings rejected by M 1 : 0, 00, 10, 010, 1010,.... Bow-Yw Wng (Acdemi Sinic) Regulr Lnguges Spring / 38
5 Finite Automton Forml Definition A finite utomton is 5-tuple (Q, Σ, δ, q 0, F) where Q is finite set of sttes; Σ is finite set clled lphbet; δ : Q Σ Q is the trnsition function; q 0 Q is the strt stte; nd F Q is the set of ccept sttes. Accept sttes re lso clled finl sttes. The set of ll strings tht M ccepts is clled the lnguge of mchine M (written L(M)). Recll lnguge is set of strings. We lso sy M recognizes (or ccepts) L(M). Bow-Yw Wng (Acdemi Sinic) Regulr Lnguges Spring / 38
6 M 1 Forml Definition A finite utomton M 1 = (Q, Σ, δ, q 1, F) consists of Q = {q 1, q 2, q 3 }; Σ = {0, 1}; δ : Q Σ Q is 0 1 q1 is the strt stte; nd F = {q 2 }. Moreover, we hve q 1 q 1 q 2 q 2 q 3 q 2 q 3 q 2 q 2 L(M 1 ) = {w : w contins t lest one 1 nd n even number of 0 s follow the lst 1} Bow-Yw Wng (Acdemi Sinic) Regulr Lnguges Spring / 38
7 Finite Automton M q 1 q 2 0 Figure: Finite Automton M 2 Figure 3 shows M 2 = ({q 1, q 2 }, {0, 1}, δ, q 1, {q 2 }) where δ is Wht is L(M 2 )? L(M2 ) = {w : w ends in 1}. 0 1 q 1 q 1 q 2 q 2 q 2 q 2 Bow-Yw Wng (Acdemi Sinic) Regulr Lnguges Spring / 38
8 Finite Automton M q 1 q 2 0 Figure: Finite Automton M 2 Figure 3 shows M 2 = ({q 1, q 2 }, {0, 1}, δ, q 1, {q 2 }) where δ is Wht is L(M 2 )? L(M2 ) = {w : w ends in 1}. 0 1 q 1 q 1 q 2 q 2 q 2 q 2 Bow-Yw Wng (Acdemi Sinic) Regulr Lnguges Spring / 38
9 Finite Automton M q 1 q 2 0 Figure: Finite Automton M 3 Figure 4 shows M 3 = ({q 1, q 2 }, {0, 1}, δ, q 1, {q 1 }) where δ is Wht is L(M 3 )? 0 1 q 1 q 1 q 2 q 2 q 2 q 2 L(M3 ) = {w : w is the empty string ɛ or ends in 0}. Bow-Yw Wng (Acdemi Sinic) Regulr Lnguges Spring / 38
10 Finite Automton M q 1 q 2 0 Figure: Finite Automton M 3 Figure 4 shows M 3 = ({q 1, q 2 }, {0, 1}, δ, q 1, {q 1 }) where δ is Wht is L(M 3 )? 0 1 q 1 q 1 q 2 q 2 q 2 q 2 L(M3 ) = {w : w is the empty string ɛ or ends in 0}. Bow-Yw Wng (Acdemi Sinic) Regulr Lnguges Spring / 38
11 Finite Automton M 5 0 2, RESET q q 0 q 2 0, RESET 1, RESET Figure: Finite Automton M 5 Figure 5 shows M 5 = ({q 0, q 1, q 2 }, {0, 1, 2, RESET }, δ, q 0, {q 0 }). Strings ccepted by M 5 : 0, 00, 12, 21, 012, 102, 120, 021, 201, 210, 111, 222,.... M 5 computes the sum of input symbols modulo 3. It resets upon the input symbol RESET. M 5 ccepts strings who sum is multiple of 3. Bow-Yw Wng (Acdemi Sinic) Regulr Lnguges Spring / 38
12 Computtion Forml Definition Let M = (Q, Σ, δ, q 0, F) be finite utomton nd w = w 1 w 2 w n string where w i Σ for every i = 1,..., n. We sy M ccepts w if there is sequence of sttes r 0, r 1,..., r n such tht r 0 = q 0 ; δ(r i, w i+1 ) = r i+1 for i = 0,..., n 1; nd r n F. M recognizes lnguge A if A = {w : M ccepts w}. Definition 1 A lnguge is clled regulr lnguge if some finite utomton recognizes it. Bow-Yw Wng (Acdemi Sinic) Regulr Lnguges Spring / 38
13 Regulr Opertions Definition 2 Let A nd B be lnguges. We define the following opertions: Union: A B = {x : x A or x B}. Conctention: A B = {xy : x A nd y B}. Str: A = {x 1 x 2 x k : k 0 nd every x i A}. Note tht ɛ A for every lnguge A. Bow-Yw Wng (Acdemi Sinic) Regulr Lnguges Spring / 38
14 Closure Property Union Theorem 3 The clss of regulr lnguges is closed under the union opertion. Tht is, A 1 A 2 is regulr if A 1 nd A 2 re. Proof. Let M i = (Q i, Σ, δ i, q i, F i ) recognize A i for i = 1, 2. Construct M = (Q, Σ, δ, q 0, F) where Q = Q 1 Q 2 = {(r 1, r 2 ) : r 1 Q 1, r 2 Q 2 }; δ((r 1, r 2 ), ) = (δ 1 (r 1, ), δ 2 (r 2, )); q 0 = (q 1, q 2 ); F = (F 1 Q 2 ) (Q 1 F 2 ) = {(r 1, r 2 ) : r 1 F 1 or r 2 F 2 }. Why is L(M) = A 1 A 2? Bow-Yw Wng (Acdemi Sinic) Regulr Lnguges Spring / 38
15 Nondeterminism When mchine is t given stte nd reds n input symbol, there is precisely one choice of its next stte. This is cll deterministic computtion. In nondeterministic mchines, multiple choices my exist for the next stte. A deterministic finite utomton is bbrevited s DFA; nondeterministic finite utomton is bbrevited s NFA. A DFA is lso n NFA. Since NFA llow more generl computtion, they cn be much smller thn DFA. Bow-Yw Wng (Acdemi Sinic) Regulr Lnguges Spring / 38
16 NFA N 4 q 1 b ɛ q 2 q 3, b Figure: NFA N 4 On input string b, N 4 hs severl possible computtion: q 1 b q 2 q 2 q 2 ; b q 1 q 2 q 2 q 3 ; or b q 1 q 2 q 3 q 1. Bow-Yw Wng (Acdemi Sinic) Regulr Lnguges Spring / 38
17 Nondeterministic Finite Automton Forml Definition For ny set Q, P(Q) = {R : R Q} denotes the power set of Q. For ny lphbet Σ, define Σ ɛ to be Σ {ɛ}. A nondeterministic finite utomton is 5-tuple (Q, Σ, δ, q 0, F) where Q is finite set of sttes; Σ is finite lphbet; δ : Q Σɛ P(Q) is the trnsition function; q0 Q is the strt stte; nd F Q is the ccept sttes. Note tht the trnsition function ccepts the empty string s n input symbol. Bow-Yw Wng (Acdemi Sinic) Regulr Lnguges Spring / 38
18 NFA N 4 Forml Definition q 1 b ɛ q 2 q 3, b N 4 = (Q, Σ, δ, q 1, {q 1 }) is nondeterministic finite utomton where Q = {q 1, q 2, q 3 }; ɛ b q 1 {q 3 } {q 2 } q 2 {q 2, q 3 } {q 3 } q 3 {q 1 } Bow-Yw Wng (Acdemi Sinic) Regulr Lnguges Spring / 38
19 Nondeterministic Computtion Forml Definition Let N = (Q, Σ, δ, q 0, F) be n NFA nd w string over Σ. We sy N ccepts w if w cn be rewritten s w = y 1 y 2 y m with y i Σ ɛ nd there is sequence of sttes r 0, r 1,..., r m such tht r 0 = q 0 ; r i+1 δ(r i, y i+1 ) for i = 0,..., m 1; nd rm F. Note tht finitely mny empty strings cn be inserted in w. Also note tht one sequence stisfying the conditions suffices to show the cceptnce of n input string. Strings ccepted by N 4 :, b,.... Bow-Yw Wng (Acdemi Sinic) Regulr Lnguges Spring / 38
20 Equivlence of NFA s nd DFA s Theorem 4 Every nondeterministic finite utomton hs n equivlent deterministic finite utomton. Tht is, for every NFA N, there is DFA M such tht L(M) = L(N). Proof. Let N = (Q, Σ, δ, q 0, F) be n NFA. For R Q, define E(R) = {q : q cn be reched from R long 0 or more ɛ trnsitions }. Construct DFA M = (Q, Σ, δ, q 0, F ) where Q = P(Q); δ (R, ) = {q Q : q E(δ(r, )) for some r R}; q 0 = E({q 0}); F = {R Q : R F }. Why is L(M) = L(N)? Bow-Yw Wng (Acdemi Sinic) Regulr Lnguges Spring / 38
21 A DFA Equivlent to N 4 q 1 b ɛ, b q 2 q 3, b {q 1 } b {q 2 } {q 1, q 2 } b {q 3 } b {q 1, q 3 } b b {q 2, q 3 }, b b {q 1, q 2, q 3 } Bow-Yw Wng (Acdemi Sinic) Regulr Lnguges Spring / 38
22 Closure Properties Revisited Theorem 5 The clss of regulr lnguges is closed under the union opertion. Proof. Let N i = (Q i, Σ, δ i, q i, F i ) recognize A i for i = 1, 2. Construct N = (Q, Σ, δ, q 0, F) where Q = {q 0 } Q 1 Q 2 ; F = F 1 F 2 ; nd δ(q, ) = δ 1 (q, ) q Q 1 δ 2 (q, ) q Q 2 {q 1, q 2 } q = q 0 nd = ɛ q = q 0 nd ɛ Why is L(N) = L(N 1 ) L(N 2 )? Bow-Yw Wng (Acdemi Sinic) Regulr Lnguges Spring / 38
23 Closure Properties Revisited Theorem 6 The clss of regulr lnguges is closed under the conctention opertion. Proof. Let N i = (Q i, Σ, δ i, q i, F i ) recognize A i for i = 1, 2. Construct N = (Q, Σ, δ, q 1, F 2 ) where Q = Q 1 Q 2 ; nd δ 1 (q, ) q Q 1 nd q F 1 δ δ(q, ) = 1 (q, ) q F 1 nd ɛ δ 1 (q, ) {q 2 } q F 1 nd = ɛ δ 2 (q, ) q Q 2 Why is L(N) = L(N 1 ) L(N 2 )? Bow-Yw Wng (Acdemi Sinic) Regulr Lnguges Spring / 38
24 Closure Properties Revisited Theorem 7 The clss of regulr lnguges is closed under the str opertion. Proof. Let N 1 = (Q 1, Σ, δ 1, q 1, F 1 ) recognize A 1. Construct N = (Q, Σ, δ, q 0, F) where Q = {q 0 } Q 1 ; F = {q 0 } F 1 ; nd δ 1 (q, ) q Q 1 nd q F 1 δ 1 (q, ) q F 1 nd ɛ δ(q, ) = δ 1 (q, ) {q 1 } q F 1 nd = ɛ {q 1 } q = q 0 nd = ɛ q = q 0 nd ɛ Why is L(N) = [L(N 1 )]? Bow-Yw Wng (Acdemi Sinic) Regulr Lnguges Spring / 38
25 Closure Properties Revisited Theorem 8 The clss of regulr lnguges is closed under complementtion. Proof. Let M = (Q, Σ, δ, q 0, F) be DFA recognizing A. Consider M = (Q, Σ, δ, q 0, Q \ F). We hve w L(M) if nd only if w L(M). Tht is, L(M) = A s required. Bow-Yw Wng (Acdemi Sinic) Regulr Lnguges Spring / 38
26 Regulr Expressions Definition 9 R is regulr expression if R is for some Σ; ɛ; ; (R 1 R 2 ) where R i s re regulr expressions; (R 1 R 2 ) where R i s re regulr expressions; or (R 1 ) where R 1 is regulr expression. We write R + for R R. Hence R = R + ɛ. k {}}{ Moreover, write R k for R R R. Define R 0 = ɛ. We hve R = R 0 R 1 R n. L(R) denotes the lnguge described by the regulr expression R. Note tht {ɛ}. Bow-Yw Wng (Acdemi Sinic) Regulr Lnguges Spring / 38
27 Exmples of Regulr Expressions For convenience, we write RS for R S. We my lso write the regulr expression R to denote its lnguge L(R). L(0 10 ) = {w : w contins single 1}. L(Σ 1Σ ) = {w : w hs t lest one 1}. L((ΣΣ) ) = {w : w is string of even length }. (0 ɛ)(1 ɛ) = {ɛ, 0, 1, 01}. 1 =. = {ɛ}. For ny regulr expression R, we hve R = R nd R ɛ = R. Bow-Yw Wng (Acdemi Sinic) Regulr Lnguges Spring / 38
28 Exmples of Regulr Expressions For convenience, we write RS for R S. We my lso write the regulr expression R to denote its lnguge L(R). L(0 10 ) = {w : w contins single 1}. L(Σ 1Σ ) = {w : w hs t lest one 1}. L((ΣΣ) ) = {w : w is string of even length }. (0 ɛ)(1 ɛ) = {ɛ, 0, 1, 01}. 1 =. = {ɛ}. For ny regulr expression R, we hve R = R nd R ɛ = R. Bow-Yw Wng (Acdemi Sinic) Regulr Lnguges Spring / 38
29 Exmples of Regulr Expressions For convenience, we write RS for R S. We my lso write the regulr expression R to denote its lnguge L(R). L(0 10 ) = {w : w contins single 1}. L(Σ 1Σ ) = {w : w hs t lest one 1}. L((ΣΣ) ) = {w : w is string of even length }. (0 ɛ)(1 ɛ) = {ɛ, 0, 1, 01}. 1 =. = {ɛ}. For ny regulr expression R, we hve R = R nd R ɛ = R. Bow-Yw Wng (Acdemi Sinic) Regulr Lnguges Spring / 38
30 Exmples of Regulr Expressions For convenience, we write RS for R S. We my lso write the regulr expression R to denote its lnguge L(R). L(0 10 ) = {w : w contins single 1}. L(Σ 1Σ ) = {w : w hs t lest one 1}. L((ΣΣ) ) = {w : w is string of even length }. (0 ɛ)(1 ɛ) = {ɛ, 0, 1, 01}. 1 =. = {ɛ}. For ny regulr expression R, we hve R = R nd R ɛ = R. Bow-Yw Wng (Acdemi Sinic) Regulr Lnguges Spring / 38
31 Exmples of Regulr Expressions For convenience, we write RS for R S. We my lso write the regulr expression R to denote its lnguge L(R). L(0 10 ) = {w : w contins single 1}. L(Σ 1Σ ) = {w : w hs t lest one 1}. L((ΣΣ) ) = {w : w is string of even length }. (0 ɛ)(1 ɛ) = {ɛ, 0, 1, 01}. 1 =. = {ɛ}. For ny regulr expression R, we hve R = R nd R ɛ = R. Bow-Yw Wng (Acdemi Sinic) Regulr Lnguges Spring / 38
32 Exmples of Regulr Expressions For convenience, we write RS for R S. We my lso write the regulr expression R to denote its lnguge L(R). L(0 10 ) = {w : w contins single 1}. L(Σ 1Σ ) = {w : w hs t lest one 1}. L((ΣΣ) ) = {w : w is string of even length }. (0 ɛ)(1 ɛ) = {ɛ, 0, 1, 01}. 1 =. = {ɛ}. For ny regulr expression R, we hve R = R nd R ɛ = R. Bow-Yw Wng (Acdemi Sinic) Regulr Lnguges Spring / 38
33 Exmples of Regulr Expressions For convenience, we write RS for R S. We my lso write the regulr expression R to denote its lnguge L(R). L(0 10 ) = {w : w contins single 1}. L(Σ 1Σ ) = {w : w hs t lest one 1}. L((ΣΣ) ) = {w : w is string of even length }. (0 ɛ)(1 ɛ) = {ɛ, 0, 1, 01}. 1 =. = {ɛ}. For ny regulr expression R, we hve R = R nd R ɛ = R. Bow-Yw Wng (Acdemi Sinic) Regulr Lnguges Spring / 38
34 Regulr Expressions nd Finite Automt Lemm 10 If lnguge is described by regulr expression, it is regulr. Proof. We prove by induction on the regulr expression R. R = for some Σ. Consider the NFA N = ({q 1 {, q 2 }, Σ, δ, q 1, {q 2 }) where {q2 } r = q δ(r, y) = 1 nd y = otherwise R = ɛ. Consider the NFA N ɛ = ({q 1 }, Σ, δ, q 1, {q 1 }) where δ(r, y) = for ny r nd y. R =. Consider the NFA N = ({q 1 }, Σ, δ, q 1, ) where δ(r, y) = for ny r nd y. R = R 1 R 2, R = R 1 R 2, or R = R 1. By inductive hypothesis nd the closure properties of finite utomt. Bow-Yw Wng (Acdemi Sinic) Regulr Lnguges Spring / 38
35 Regulr Expressions nd Finite Automt b b b ɛ b ɛ ɛ b b ɛ ɛ ɛ ɛ ɛ b ɛ (b ) ɛ Bow-Yw Wng (Acdemi Sinic) Regulr Lnguges Spring / 38
36 Regulr Expressions nd Finite Automt Lemm 11 If lnguge is regulr, it is described by regulr expression. For the proof, we introduce generliztion of finite utomt. Bow-Yw Wng (Acdemi Sinic) Regulr Lnguges Spring / 38
37 Generlized Nondeterministic Finite Automt Definition 12 A generlized nondeterministic finite utomton is 5-tuple (Q, Σ, q strt, q ccept ) where Q is the finite set of sttes; Σ is the input lphbet; δ : (Q {q ccept }) (Q {q strt }) R is the trnsition function, where R denotes the set of regulr expressions; q strt is the strt stte; nd q ccept is the ccept stte. A GNFA ccepts string w Σ if w = w 1 w 2 w k where w i Σ nd there is sequence of sttes r 0, r 1,..., r k such tht r 0 = q strt ; r k = q ccept ; nd for every i, w i L(R i ) where R i = δ(q i 1, q i ). Bow-Yw Wng (Acdemi Sinic) Regulr Lnguges Spring / 38
38 Regulr Expressions nd Finite Automt Proof of Lemm. Let M be the DFA for the regulr lnguge. Construct n equivlent GNFA G by dding q strt, q ccept nd necessry ɛ-trnsitions. CONVERT (G): 1 Let k be the number of sttes of G. 2 If k = 2, then return the regulr expression R lbeling the trnsition from q strt to q ccept. 3 If k > 2, select q rip Q \ {q strt, q ccept }. Construct G = (Q, Σ, δ, q strt, q ccept ) where Q = Q \ {q rip }; for ny q i Q \ {q ccept } nd q j Q \ {q strt }, define δ (q i, q j ) = (R 1 )(R 2 ) (R 3 ) R 4 where R 1 = δ(q i, q rip ), R 2 = δ(q rip, q rip ), R 3 = δ(q rip, q j ), nd R 4 = δ(q i, q j ). 4 return CONVERT (G ). Bow-Yw Wng (Acdemi Sinic) Regulr Lnguges Spring / 38
39 Regulr Expressions nd Finite Automt Lemm 13 For ny GNFA G, CONVERT (G) is equivlent to G. Proof. We prove by induction on the number k of sttes of G. k = 2. Trivil. Assume the lemm holds for k 1 sttes. We first show G is equivlent to G. Suppose G ccepts n input w. Let q strt, q 1, q 2,..., q ccept be n ccepting computtion of G. We hve w q 1 w strt q1 q i w i+1 w j 1 i 1 qi q rip q rip w i w i+1 w j qi q rip w j q j q ccept. w Hence q 1 strt q1 q i 1 q j q ccept is computtion of G. Conversely, ny string ccepted by G is lso ccepted by G since the trnsition between q i nd q j in G describes the strings tking q i to q j in G. Hence G is equivlent to G. By inductive hypothesis, CONVERT (G ) is equivlent to G. Bow-Yw Wng (Acdemi Sinic) Regulr Lnguges Spring / 38
40 Regulr Expressions nd Finite Automt q 1 q strt ɛ q 1 b b q 2, b q ccept ɛ q 2, b () DFA M (b) GNFA G q strt ɛ q 1 q strt b( b) b( b) q ccept q ccept (c) GNFA (d) GNFA Bow-Yw Wng (Acdemi Sinic) Regulr Lnguges Spring / 38
41 Regulr Expressions nd Finite Automt Theorem 14 A lnguge is regulr if nd only if some regulr expression describes it. Bow-Yw Wng (Acdemi Sinic) Regulr Lnguges Spring / 38
42 Pumping Lemm Lemm 15 If A is regulr lnguge, then there is number p such tht for ny s A of length t lest p, there is prtition s = xyz with 1 for ech i 0, xy i z A; 2 y > 0; nd 3 xy p. Proof. Let M = (Q, Σ, δ, q 1, F) be DFA recognizing A nd p = Q. Consider ny string s = s 1 s 2 s n of length n p. Let r 1 = q 1,..., r n+1 be the sequence of sttes such tht r i+1 = δ(r i, s i ) for 1 i n. Since n + 1 p + 1 = Q + 1, there re 1 j < l p + 1 such tht r j = r l (why?). Choose x = s 1 s j 1, y = s j s l 1, nd z = s l s n. x y z Note tht r 1 r j, r j r l, nd r l r n+1 F. Thus M ccepts xy i z for i 0. Since j l, y > 0. Finlly, xy p for l p + 1. Bow-Yw Wng (Acdemi Sinic) Regulr Lnguges Spring / 38
43 Applictions of Pumping Lemm Exmple 16 B = {0 n 1 n : n 0} is not regulr lnguge. Proof. Suppose B is regulr. Let p be the pumping length given by the pumping lemm. Choose s = 0 p 1 p. Then s B nd s p, there is prtition s = xyz such tht xy i z B for i 0. y 0 + or y 1 +. xz B. A contrdiction. y xyyz B. A contrdiction. Corollry 17 C = {w : w hs n equl number of 0 s nd 1 s} is not regulr lnguge. Proof. Suppose C is regulr. Then B = C 0 1 is regulr. Bow-Yw Wng (Acdemi Sinic) Regulr Lnguges Spring / 38
44 Applictions of Pumping Lemm Exmple 18 F = {ww : w {0, 1} } is not regulr lnguge. Proof. Suppose F is regulr lnguge nd p the pumping length. Choose s = 0 p 10 p 1. By the pumping lemm, there is prtition s = xyz such tht xy p nd xy i z F for i 0. Since xy p, y 0 +. But then xz F. A contrdiction. Bow-Yw Wng (Acdemi Sinic) Regulr Lnguges Spring / 38
45 Applictions of Pumping Lemm Exmple 19 D = {1 n2 : n 0} is not regulr lnguge. Proof. Suppose D is regulr lnguge nd p the pumping length. Choose s = 1 p2. By the pumping lemm, there is prtition s = xyz such tht y > 0, xy p, nd xy i z D for i 0. Consider the strings xyz nd xy 2 z. We hve xyz = p 2 nd xy 2 z = p 2 + y p 2 + p < p 2 + 2p + 1 = (p + 1) 2. Since y > 0, we hve p 2 = xyz < xy 2 z < (p + 1) 2. Thus xy 2 z D. A contrdiction. Bow-Yw Wng (Acdemi Sinic) Regulr Lnguges Spring / 38
46 Applictions of Pumping Lemm Exmple 20 E = {0 i 1 j : i > j} is not regulr lnguge. Proof. Suppose E is regulr lnguge nd p the pumping length. Choose s = 0 p+1 1 p. By the pumping lemm, there is prtition s = xyz such tht y > 0, xy p, nd xy i z E for i 0. Since xy p, y 0 +. But then xz E for y > 0. A contrdiction. Bow-Yw Wng (Acdemi Sinic) Regulr Lnguges Spring / 38
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