Unformatted text preview: THE PUMPING LEMMA THE PUMPING LEMMA
Theorem. For any regular language L there exists an integer n, such that for all x ∈ L with x ≥ n, there exist u, v, w ∈ Σ∗ , such that (1) x = uvw (2) uv  ≤ n (3) v  ≥ 1 (4) for all i ≥ 0: uv i w ∈ L. x x L THE PUMPING LEMMA
Theorem. For any regular language L there exists an integer n, such that for all x ∈ L with x ≥ n, there exist u, v, w ∈ Σ∗ , such that (1) x = uvw (2) uv  ≤ n (3) v  ≥ 1 (4) for all i ≥ 0: uv i w ∈ L. n x x L THE PUMPING LEMMA
Theorem. For any regular language L there exists an integer n, such that for all x ∈ L with x ≥ n, there exist u, v, w ∈ Σ∗ , such that (1) x = uvw (2) uv  ≤ n (3) v  ≥ 1 (4) for all i ≥ 0: uv i w ∈ L. n x L x THE PUMPING LEMMA
Theorem. For any regular language L there exists an integer n, such that for all x ∈ L with x ≥ n, there exist u, v, w ∈ Σ∗ , such that (1) x = uvw (2) uv  ≤ n (3) v  ≥ 1 (4) for all i ≥ 0: uv i w ∈ L. n L x x THE PUMPING LEMMA
Theorem. For any regular language L there exists an integer n, such that for all x ∈ L with x ≥ n, there exist u, v, w ∈ Σ∗ , such that (1) x = uvw (2) uv  ≤ n (3) v  ≥ 1 (4) for all i ≥ 0: uv i w ∈ L. n L x u vx w THE PUMPING LEMMA
Theorem. For any regular language L there exists an integer n, such that for all x ∈ L with x ≥ n, there exist u, v, w ∈ Σ∗ , such that (1) x = uvw (2) uv  ≤ n (3) v  ≥ 1 (4) for all i ≥ 0: uv i w ∈ L. n L x u u w vx w THE PUMPING LEMMA
Theorem. For any regular language L there exists an integer n, such that for all x ∈ L with x ≥ n, there exist u, v, w ∈ Σ∗ , such that (1) x = uvw (2) uv  ≤ n (3) v  ≥ 1 (4) for all i ≥ 0: uv i w ∈ L. n L
w x u u u w vx w vv THE PUMPING LEMMA
Theorem. For any regular language L there exists an integer n, such that for all x ∈ L with x ≥ n, there exist u, v, w ∈ Σ∗ , such that (1) x = uvw (2) uv  ≤ n (3) v  ≥ 1 (4) for all i ≥ 0: uv i w ∈ L. n L x u u u u w vx w vv w vvv w PROOF OF P.L. (SKETCH) PROOF OF P.L. (SKETCH)
Let M be a DFA for L. Take n be the number of states of M plus 1. PROOF OF P.L. (SKETCH)
Let M be a DFA for L. Take n be the number of states of M plus 1.
Take any x ∈ L with x ≥ n. Consider the path (from start state to an accepting state) in M that corresponds to x. The length of this path is x ≥ n. PROOF OF P.L. (SKETCH)
Let M be a DFA for L. Take n be the number of states of M plus 1.
Take any x ∈ L with x ≥ n. Consider the path (from start state to an accepting state) in M that corresponds to x. The length of this path is x ≥ n.
Since M has at most n − 1 states, some state must be visited twice or more in the ﬁrst n steps of the path. PROOF OF P.L. (SKETCH)
Let M be a DFA for L. Take n be the number of states of M plus 1.
Take any x ∈ L with x ≥ n. Consider the path (from start state to an accepting state) in M that corresponds to x. The length of this path is x ≥ n.
Since M has at most n − 1 states, some state must be visited twice or more in the ﬁrst n steps of the path. u w v PROOF OF P.L. (SKETCH)
Let M be a DFA for L. Take n be the number of states of M plus 1.
Take any x ∈ L with x ≥ n. Consider the path (from start state to an accepting state) in M that corresponds to x. The length of this path is x ≥ n.
Since M has at most n − 1 states, some state must be visited twice or more in the ﬁrst n steps of the path. u w v uw ∈ L x = uvw ∈ L uvvw ∈ L uvvvw ∈ L ... USING PUMPING LEMMA TO PROVE NONREGULARITY
L regular =⇒ L satisﬁes P.L. L nonregular =⇒ ? L nonregular ⇐= L doesn’t satisfy P.L. Negation:
∃ n ∈ N ∀ x ∈ L with x ≥ n ∃ u, v, w ∈ Σ∗ all of these hold: (1) x = uvw (2) uv  ≤ n (3) v  ≥ 1 (4) ∀ i ≥ 0: uv i w ∈ L. USING PUMPING LEMMA TO PROVE NONREGULARITY
L regular =⇒ L satisﬁes P.L. L nonregular =⇒ ? L nonregular ⇐= L doesn’t satisfy P.L. Negation:
∃ n ∈ N ∀ x ∈ L with x ≥ n ∃ u, v, w ∈ Σ∗ all of these hold: (1) x = uvw (2) uv  ≤ n (3) v  ≥ 1 (4) ∀ i ≥ 0: uv i w ∈ L. USING PUMPING LEMMA TO PROVE NONREGULARITY
L regular =⇒ L satisﬁes P.L. L nonregular =⇒ ? L nonregular ⇐= L doesn’t satisfy P.L. Negation:
∃ n ∈ N ∀ x ∈ L with x ≥ n ∃ u, v, w ∈ Σ∗ all of these hold: (1) x = uvw (2) uv  ≤ n (3) v  ≥ 1 (4) ∀ i ≥ 0: uv i w ∈ L. ∀ n ∈ N ∃ x ∈ L with x ≥ n ∀ u, v, w ∈ Σ∗ not all of these hold: (1) x = uvw (2) uv  ≤ n (3) v  ≥ 1 (4) ∀ i ≥ 0: uv i w ∈ L. USING PUMPING LEMMA TO PROVE NONREGULARITY
L regular =⇒ L satisﬁes P.L. L nonregular =⇒ ? L nonregular ⇐= L doesn’t satisfy P.L. Negation:
∃ n ∈ N ∀ x ∈ L with x ≥ n ∃ u, v, w ∈ Σ∗ all of these hold: (1) x = uvw Equivalently: (2) uv(1) ∧ (2) ∧ (3) ⇒ not(4) ≤n where not(4) is: (3) v  ≥ 1 i ∃ i : uv w ∈ L (4) ∀ i ≥ 0: uv i w ∈ L. ∀ n ∈ N ∃ x ∈ L with x ≥ n ∀ u, v, w ∈ Σ∗ not all of these hold: (1) x = uvw (2) uv  ≤ n (3) v  ≥ 1 (4) ∀ i ≥ 0: uv i w ∈ L. EXAMPLE 1
Prove that L = {0i 1i : i ≥ 0} is NOT regular. EXAMPLE 1
Prove that L = {0i 1i : i ≥ 0} is NOT regular. Proof. Show that P.L. doesn’t hold (note: showing P.L. holds doesn’t mean regularity ). EXAMPLE 1
Prove that L = {0i 1i : i ≥ 0} is NOT regular. Proof. Show that P.L. doesn’t hold (note: showing P.L. holds doesn’t mean regularity ). If L is regular, then by P.L. ∃ n such that . . . EXAMPLE 1
Prove that L = {0i 1i : i ≥ 0} is NOT regular. Proof. Show that P.L. doesn’t hold (note: showing P.L. holds doesn’t mean regularity ). If L is regular, then by P.L. ∃ n such that . . . Now let x = 0n 1n EXAMPLE 1
Prove that L = {0i 1i : i ≥ 0} is NOT regular. Proof. Show that P.L. doesn’t hold (note: showing P.L. holds doesn’t mean regularity ). If L is regular, then by P.L. ∃ n such that . . . Now let x = 0n 1n x ∈ L and x ≥ n, so by P.L. ∃u, v, w such that (1)–(4) hold. EXAMPLE 1
Prove that L = {0i 1i : i ≥ 0} is NOT regular. Proof. Show that P.L. doesn’t hold (note: showing P.L. holds doesn’t mean regularity ). If L is regular, then by P.L. ∃ n such that . . . Now let x = 0n 1n x ∈ L and x ≥ n, so by P.L. ∃u, v, w such that (1)–(4) hold. We show that ∀ u, v, w (1)–(4) don’t all hold. EXAMPLE 1
Prove that L = {0i 1i : i ≥ 0} is NOT regular. Proof. Show that P.L. doesn’t hold (note: showing P.L. holds doesn’t mean regularity ). If L is regular, then by P.L. ∃ n such that . . . Now let x = 0n 1n x ∈ L and x ≥ n, so by P.L. ∃u, v, w such that (1)–(4) hold. We show that ∀ u, v, w (1)–(4) don’t all hold. If (1), (2), (3) hold then x = 0n 1n = uvw with uv  ≤ n and v  ≥ 1. EXAMPLE 1
Prove that L = {0i 1i : i ≥ 0} is NOT regular. Proof. Show that P.L. doesn’t hold (note: showing P.L. holds doesn’t mean regularity ). If L is regular, then by P.L. ∃ n such that . . . Now let x = 0n 1n x ∈ L and x ≥ n, so by P.L. ∃u, v, w such that (1)–(4) hold. If (1), (2), (3) hold then x = 0n 1n = uvw with uv  ≤ n and v  ≥ 1. So, u = 0s , v = 0t , w = 0p 1n with We show that ∀ u, v, w (1)–(4) don’t all hold. EXAMPLE 1
Prove that L = {0i 1i : i ≥ 0} is NOT regular. Proof. Show that P.L. doesn’t hold (note: showing P.L. holds doesn’t mean regularity ). If L is regular, then by P.L. ∃ n such that . . . Now let x = 0n 1n x ∈ L and x ≥ n, so by P.L. ∃u, v, w such that (1)–(4) hold. If (1), (2), (3) hold then x = 0n 1n = uvw with uv  ≤ n and v  ≥ 1. So, u = 0s , v = 0t , w = 0p 1n with
s + t ≤ n, t ≥ 1, p ≥ 0, s + t + p = n. We show that ∀ u, v, w (1)–(4) don’t all hold. EXAMPLE 1
Prove that L = {0i 1i : i ≥ 0} is NOT regular. Proof. Show that P.L. doesn’t hold (note: showing P.L. holds doesn’t mean regularity ). If L is regular, then by P.L. ∃ n such that . . . Now let x = 0n 1n x ∈ L and x ≥ n, so by P.L. ∃u, v, w such that (1)–(4) hold. If (1), (2), (3) hold then x = 0n 1n = uvw with uv  ≤ n and v  ≥ 1. So, u = 0s , v = 0t , w = 0p 1n with We show that ∀ u, v, w (1)–(4) don’t all hold. But then (4) fails for i = 0: s + t ≤ n, t ≥ 1, p ≥ 0, s + t + p = n. EXAMPLE 1
Prove that L = {0i 1i : i ≥ 0} is NOT regular. Proof. Show that P.L. doesn’t hold (note: showing P.L. holds doesn’t mean regularity ). If L is regular, then by P.L. ∃ n such that . . . Now let x = 0n 1n x ∈ L and x ≥ n, so by P.L. ∃u, v, w such that (1)–(4) hold. If (1), (2), (3) hold then x = 0n 1n = uvw with uv  ≤ n and v  ≥ 1. So, u = 0s , v = 0t , w = 0p 1n with We show that ∀ u, v, w (1)–(4) don’t all hold. But then (4) fails for i = 0: s + t ≤ n, t ≥ 1, p ≥ 0, s + t + p = n. uv 0 w = uw = 0s 0p 1n = 0s+p 1n ∈ L, since s + p = n IN PICTURE
∃ u, v, w such that (1) x = uvw (2) uv  ≤ n (3) v  ≥ 1 (4) ∀ i ∈ N : uv i w ∈ L. IN PICTURE
∃ u, v, w such that (1) x = uvw (2) uv  ≤ n (3) v  ≥ 1 (4) ∀ i ∈ N : uv i w ∈ L. 00000 0 . . . 01111 . . . 1 ∈ L u v w IN PICTURE
∃ u, v, w such that (1) x = uvw (2) uv  ≤ n (3) v  ≥ 1
nonempty (4) ∀ i ∈ N : uv i w ∈ L. 00000 0 . . . 01111 . . . 1 ∈ L u v w IN PICTURE
∃ u, v, w such that (1) x = uvw (2) uv  ≤ n (3) v  ≥ 1
nonempty (4) ∀ i ∈ N : uv i w ∈ L. 00000 0 . . . 01111 . . . 1 ∈ L u v w If (1), (2), (3) hold then (4) fails: it is not the case that for all i, uv i w is in L. In particular, let i = 0. uw ∈ L. EXAMPLE 2
Prove that L = {0i : i is a prime} is NOT regular. EXAMPLE 2
Prove that L = {0i : i is a prime} is NOT regular.
Proof. We show that P.L. doesn’t hold. EXAMPLE 2
Prove that L = {0i : i is a prime} is NOT regular.
Proof. We show that P.L. doesn’t hold. If L is regular, then by P.L. ∃ n such that . . . EXAMPLE 2
Prove that L = {0i : i is a prime} is NOT regular.
Proof. We show that P.L. doesn’t hold. If L is regular, then by P.L. ∃ n such that . . . Now let x = 0m where m ≥ n + 2 is prime. EXAMPLE 2
Prove that L = {0i : i is a prime} is NOT regular.
Proof. We show that P.L. doesn’t hold. If L is regular, then by P.L. ∃ n such that . . . Now let x = 0m where m ≥ n + 2 is prime. x ∈ L and x ≥ n, so by P.L. ∃u, v, w such that (1)–(4) hold. EXAMPLE 2
Prove that L = {0i : i is a prime} is NOT regular.
Proof. We show that P.L. doesn’t hold. If L is regular, then by P.L. ∃ n such that . . . Now let x = 0m where m ≥ n + 2 is prime. x ∈ L and x ≥ n, so by P.L. ∃u, v, w such that (1)–(4) hold. We show that ∀ u, v, w (1)–(4) don’t all hold. EXAMPLE 2
Prove that L = {0i : i is a prime} is NOT regular.
Proof. We show that P.L. doesn’t hold. If L is regular, then by P.L. ∃ n such that . . . Now let x = 0m where m ≥ n + 2 is prime. x ∈ L and x ≥ n, so by P.L. ∃u, v, w such that (1)–(4) hold. We show that ∀ u, v, w (1)–(4) don’t all hold. If 0m is written as 0m = uvw, then 0m = 0u 0v 0w . EXAMPLE 2
Prove that L = {0i : i is a prime} is NOT regular.
Proof. We show that P.L. doesn’t hold. If L is regular, then by P.L. ∃ n such that . . . Now let x = 0m where m ≥ n + 2 is prime. x ∈ L and x ≥ n, so by P.L. ∃u, v, w such that (1)–(4) hold. We show that ∀ u, v, w (1)–(4) don’t all hold. If 0m is written as 0m = uvw, then 0m = 0u 0v 0w . If uv  ≤ n and v  ≥ 1, then consider i = v  + w: EXAMPLE 2
Prove that L = {0i : i is a prime} is NOT regular.
Proof. We show that P.L. doesn’t hold. If L is regular, then by P.L. ∃ n such that . . . Now let x = 0m where m ≥ n + 2 is prime. x ∈ L and x ≥ n, so by P.L. ∃u, v, w such that (1)–(4) hold. We show that ∀ u, v, w (1)–(4) don’t all hold. If 0m is written as 0m = uvw, then 0m = 0u 0v 0w . If uv  ≤ n and v  ≥ 1, then consider i = v  + w: uv w i =00 =0 v  v (v +w) w (v +1)(v +w) 0 ∈ L EXAMPLE 2
Prove that L = {0i : i is a prime} is NOT regular.
Proof. We show that P.L. doesn’t hold. If L is regular, then by P.L. ∃ n such that . . . Now let x = 0m where m ≥ n + 2 is prime. x ∈ L and x ≥ n, so by P.L. ∃u, v, w such that (1)–(4) hold. We show that ∀ u, v, w (1)–(4) don’t all hold. If 0m is written as 0m = uvw, then 0m = 0u 0v 0w . If uv  ≤ n and v  ≥ 1, then consider i = v  + w: uv w i =00 =0 v  v (v +w) w (v +1)(v +w)
Both factors ≥ 2 0 ∈ L EXAMPLE 3
Prove that L = {yy : y ∈ {0, 1}∗ } is NOT regular. EXAMPLE 3
Prove that L = {yy : y ∈ {0, 1}∗ } is NOT regular.
Again we try to show that P.L. doesn’t hold. EXAMPLE 3
Prove that L = {yy : y ∈ {0, 1}∗ } is NOT regular.
Again we try to show that P.L. doesn’t hold. If L is regular, then by P.L. ∃ n such that . . . EXAMPLE 3
Prove that L = {yy : y ∈ {0, 1}∗ } is NOT regular.
Again we try to show that P.L. doesn’t hold. If L is regular, then by P.L. ∃ n such that . . . Let us consider x = 0n 0n ∈ L. Obviously x ≥ n. EXAMPLE 3
Prove that L = {yy : y ∈ {0, 1}∗ } is NOT regular.
Again we try to show that P.L. doesn’t hold. If L is regular, then by P.L. ∃ n such that . . . Let us consider x = 0n 0n ∈ L. Obviously x ≥ n. Can 0n 0n be written as 0n 0n = uvw such that uv  ≤ n v  ≥ 1 and that for all i: uv i w ∈ L? EXAMPLE 3
Prove that L = {yy : y ∈ {0, 1}∗ } is NOT regular.
Again we try to show that P.L. doesn’t hold. If L is regular, then by P.L. ∃ n such that . . . Let us consider x = 0n 0n ∈ L. Obviously x ≥ n. Can 0n 0n be written as 0n 0n = uvw such that uv  ≤ n v  ≥ 1 and that for all i: uv i w ∈ L?
YES! Let u = , v = 00, and w = 02n−2 . EXAMPLE 3
Prove that L = {yy : y ∈ {0, 1}∗ } is NOT regular.
Again we try to show that P.L. doesn’t hold. If L is regular, then by P.L. ∃ n such that . . . Let us consider x = 0n 0n ∈ L. Obviously x ≥ n. Can 0n 0n be written as 0n 0n = uvw such that uv  ≤ n v  ≥ 1 and that for all i: uv i w ∈ L?
YES! Let u = , v = 00, and w = 02n−2 . Then ∀i , uv i w is of the form 02k = 0k 0k . EXAMPLE 3
Prove that L = {yy : y ∈ {0, 1}∗ } is NOT regular.
Again we try to show that P.L. doesn’t hold. If L is regular, then by P.L. ∃ n such that . . . Let us consider x = 0n 0n ∈ L. Obviously x ≥ n. Can 0n 0n be written as 0n 0n = uvw such that uv  ≤ n v  ≥ 1 and that for all i: uv i w ∈ L?
YES! Let u = , v = 00, and w = 02n−2 . Then ∀i , uv i w is of the form 02k = 0k 0k . Does this mean that L is regular? EXAMPLE 3
Prove that L = {yy : y ∈ {0, 1}∗ } is NOT regular.
Again we try to show that P.L. doesn’t hold. If L is regular, then by P.L. ∃ n such that . . . Let us consider x = 0n 0n ∈ L. Obviously x ≥ n. Can 0n 0n be written as 0n 0n = uvw such that uv  ≤ n v  ≥ 1 and that for all i: uv i w ∈ L?
YES! Let u = , v = 00, and w = 02n−2 . Then ∀i , uv i w is of the form 02k = 0k 0k . Does this mean that L is regular?
P.L., we only need to exhibit some x that cannot be “pumped” (and x ≥ n). NO. We have chosen a bad string x. To show that L fails the EXAMPLE 3, 2ND ATTEMPT
Prove that L = {yy : y ∈ {0, 1}∗ } is NOT regular. EXAMPLE 3, 2ND ATTEMPT
Prove that L = {yy : y ∈ {0, 1}∗ } is NOT regular.
Given n from the P.L., let x = (01)n (01)n . Obviously x ∈ L and x ≥ n. EXAMPLE 3, 2ND ATTEMPT
Prove that L = {yy : y ∈ {0, 1}∗ } is NOT regular.
Given n from the P.L., let x = (01)n (01)n . Obviously x ∈ L and x ≥ n. Q: Can x be “pumped” for some choice of u, v, w with uv  ≤ n and v  ≥ 1? EXAMPLE 3, 2ND ATTEMPT
Prove that L = {yy : y ∈ {0, 1}∗ } is NOT regular.
Given n from the P.L., let x = (01)n (01)n . Obviously x ∈ L and x ≥ n. Q: Can x be “pumped” for some choice of u, v, w with uv  ≤ n and v  ≥ 1?
A: Yes! Take u = , v = 0101, w = (01)2n−2 . EXAMPLE 3, 2ND ATTEMPT
Prove that L = {yy : y ∈ {0, 1}∗ } is NOT regular.
Given n from the P.L., let x = (01)n (01)n . Obviously x ∈ L and x ≥ n. Q: Can x be “pumped” for some choice of u, v, w with uv  ≤ n and v  ≥ 1?
A: Yes! Take u = , v = 0101, w = (01)2n−2 . Another bad choice of x! EXAMPLE 3, 3RD ATTEMPT
Prove that L = {yy : y ∈ {0, 1}∗ } is NOT regular. EXAMPLE 3, 3RD ATTEMPT
Given n from the P.L., let x = 0n 10n 1. Again x ∈ L and x ≥ n. Prove that L = {yy : y ∈ {0, 1}∗ } is NOT regular. EXAMPLE 3, 3RD ATTEMPT
Given n from the P.L., let x = 0n 10n 1. Again x ∈ L and x ≥ n. Prove that L = {yy : y ∈ {0, 1}∗ } is NOT regular. ∀u, v, w such that 0n 10n 1 = uvw and uv  ≤ n and v  ≥ 1: EXAMPLE 3, 3RD ATTEMPT
Given n from the P.L., let x = 0n 10n 1. Again x ∈ L and x ≥ n. Prove that L = {yy : y ∈ {0, 1}∗ } is NOT regular. ∀u, v, w such that 0n 10n 1 = uvw and uv  ≤ n and v  ≥ 1: must have uv contained in the ﬁrst group of 0n . Thus consider EXAMPLE 3, 3RD ATTEMPT
Given n from the P.L., let x = 0n 10n 1. Again x ∈ L and x ≥ n. Prove that L = {yy : y ∈ {0, 1}∗ } is NOT regular. ∀u, v, w such that 0n 10n 1 = uvw and uv  ≤ n and v  ≥ 1: must have uv contained in the ﬁrst group of 0n . Thus consider uv w = 0
0 n−v  10 1.
n EXAMPLE 3, 3RD ATTEMPT
Given n from the P.L., let x = 0n 10n 1. Again x ∈ L and x ≥ n. Prove that L = {yy : y ∈ {0, 1}∗ } is NOT regular. ∀u, v, w such that 0n 10n 1 = uvw and uv  ≤ n and v  ≥ 1: must have uv contained in the ﬁrst group of 0n . Thus consider uv w = 0
0 n−v  10 1.
n Since v  is at least 1, this is clearly not of the form yy . ...
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This note was uploaded on 04/11/2010 for the course CSC CSC236 taught by Professor Farzanazadeh during the Spring '10 term at University of Toronto.
 Spring '10
 FarzanAzadeh

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