Multiple Context-free Grammars
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1 Multiple Context-free Grammars Course 4: pumping properties Sylvain Salvati INRI Bordeaux Sud-Ouest ESSLLI 2011
2 The pumping Lemma for CFL Outline The pumping Lemma for CFL Weak pumping Lemma for MCFL No strong pumping Lemma for MCFL MCFL wn and mild context sensitivity
3 The pumping Lemma for CFL Ogden s pumping Lemma for CFL Ogden (1967)
4 The pumping Lemma for CFL The pumping Lemma s proof pump u x v y w u x y w The pump is one of the lowest xvy < K The derivation is minimal in size xy ɛ x v y
5 The pumping Lemma for CFL characterization of the recursive power of CFL Berstel (1979) Theorem Let L X be an algebraic language, and let π be a non-degenerated iterative pair in L. For any algebraic grammar generating L there exists an iterative pair π deduced from π and grammatical with respect to G.
6 Outline The pumping Lemma for CFL Weak pumping Lemma for MCFL No strong pumping Lemma for MCFL MCFL wn and mild context sensitivity
7 Generalization of pumbability: k-pumpability Definition (pumpability) language L is pumpable if there is K such that for all w such that w > K, w = uxvyw with: xy ɛ for all n N, ux n vy n w is in L Definition (k-pumpability) language L is k-pumpable if there is K such that for all w such that w > K, w = v 1 x 1 v 2 x 2...v k x k v k+1 with: x 1... x k ɛ for all n N, v 1 x n 1 v 2x n 2...v k x n k v k+1 pumpability = 2-pumpability k-pumpability k + 1-pumpability
8 Generalization of pumbability: k-pumpability Definition (pumpability) language L is pumpable if there is K such that for all w such that w > K, w = uxvyw with: xy ɛ for all n N, ux n vy n w is in L Definition (k-pumpability) language L is k-pumpable if there is K such that for all w such that w > K, w = v 1 x 1 v 2 x 2...v k x k v k+1 with: x 1... x k ɛ for all n N, v 1 x n 1 v 2x n 2...v k x n k v k+1 pumpability = 2-pumpability k-pumpability k + 1-pumpability
9 Generalization of pumbability: k-pumpability Definition (pumpability) language L is pumpable if there is K such that for all w such that w > K, w = uxvyw with: xy ɛ for all n N, ux n vy n w is in L Definition (k-pumpability) language L is k-pumpable if there is K such that for all w such that w > K, w = v 1 x 1 v 2 x 2...v k x k v k+1 with: x 1... x k ɛ for all n N, v 1 x n 1 v 2x n 2...v k x n k v k+1 pumpability = 2-pumpability k-pumpability k + 1-pumpability
10 Generalization of pumbability: k-pumpability Definition (pumpability) language L is pumpable if there is K such that for all w such that w > K, w = uxvyw with: xy ɛ for all n N, ux n vy n w is in L Definition (k-pumpability) language L is k-pumpable if there is K such that for all w such that w > K, w = v 1 x 1 v 2 x 2...v k x k v k+1 with: x 1... x k ɛ for all n N, v 1 x n 1 v 2x n 2...v k x n k v k+1 pumpability = 2-pumpability k-pumpability k + 1-pumpability
11 Iteration of the derivation tree (x 1 v 1 x 2, x 3 v 2 x 4 ) (x 1 x 1 v 1 x 2 x 2, x 3 x 3 v 2 x 4 x 4 ) (v 1, v 2 ) Even pump (v 1, v 2 )
12 Iteration of the derivation tree (x 1 v 1 x 2 v 2 x 3, x 4 ) (x 1 x 1 v 1 x 2 v 2 x 3 x 2 x 4 x 3, x 4 ) (v 1, v 2 ) Uneven pump (v 1, v 2 )
13 Even m-pump (x 1 v 1 y 1,..., x mv my m) Even m-pump (v 1,..., v m)
14 Even m-pump Definition rule of a m-mcfg: (s 1,..., s k ) B 1 (x 1 1,..., x 1 k 1 ),..., B n(x n 1,..., x n k n ) is m-proper in the i th premiss when k = k i = m and s j = w 1 x i j w 2. (y 1 x 1 y 2, x 2 ) B(x 1, x 2 ), C(y 1, y 2 ) (x 1 v 1 y 1,..., x mv my m) (v 1,..., v m) Even m-pump
15 Even m-pump Definition rule of a m-mcfg: (s 1,..., s k ) B 1 (x 1 1,..., x 1 k 1 ),..., B n(x n 1,..., x n k n ) is m-proper in the i th premiss when k = k i = m and s j = w 1 x i j w 2. (y 1 x 1 y 2, x 2 ) B(x 1, x 2 ), C(y 1, y 2 ) (x 1 v 1 y 1,..., x mv my m) only m-proper rules π 1 π k Even m-pump (v 1,..., v m)
16 Deviding the problem Strings that have a derivation tree containing an even m-pump are 2m-pumpable. The set of derivation trees that contain an even m-pump is recognizable. From a m-mcfg wn, G, we can construct two grammars G 1 and G 2, such that: L(G) = L(G1 ) L(G 2 ) the derivation trees of G 1 all contain an even m-pump no derivation tree of G 2 contains an even m-pump It remains to show that L(G 2 ) is 2m-pumpable.
17 Deviding the problem Strings that have a derivation tree containing an even m-pump are 2m-pumpable. The set of derivation trees that contain an even m-pump is recognizable. From a m-mcfg wn, G, we can construct two grammars G 1 and G 2, such that: L(G) = L(G1 ) L(G 2 ) the derivation trees of G 1 all contain an even m-pump no derivation tree of G 2 contains an even m-pump It remains to show that L(G 2 ) is 2m-pumpable.
18 Deviding the problem Strings that have a derivation tree containing an even m-pump are 2m-pumpable. The set of derivation trees that contain an even m-pump is recognizable. From a m-mcfg wn, G, we can construct two grammars G 1 and G 2, such that: L(G) = L(G1 ) L(G 2 ) the derivation trees of G 1 all contain an even m-pump no derivation tree of G 2 contains an even m-pump It remains to show that L(G 2 ) is 2m-pumpable.
19 Deviding the problem Strings that have a derivation tree containing an even m-pump are 2m-pumpable. The set of derivation trees that contain an even m-pump is recognizable. From a m-mcfg wn, G, we can construct two grammars G 1 and G 2, such that: L(G) = L(G1 ) L(G 2 ) the derivation trees of G 1 all contain an even m-pump no derivation tree of G 2 contains an even m-pump It remains to show that L(G 2 ) is 2m-pumpable.
20 Kanazawa s Lemma Kanazawa (2010) Lemma n m-mcfg wn whose derivation trees do not contain even m-pumps as an equivalent m 1-MCFG wn.
21 Proof of the Lemma Proof. eliminate m-proper rules by unfolding Make rules deriving non-terminal of arity m have no premise of arity m (relies on well-nestedness): (s 1,..., s m) B(x 1,..., x m), Γ (t 1,..., t m) D(y 1,..., y p), Γ 1 D(u 1,..., u p) B(x 1,..., x m), Γ 2 Unfold the rules of non-terminal of arity m The resulting grammar defines the same language and does not use non-terminals of arity m: it is an m 1-MCFG wn Thus by induction on m (the case where m = 1 is the CFL case), L(G 2 ) is 2(m 1)-pumpable and therefore m-pumpable.
22 Proof of the Lemma Proof. eliminate m-proper rules by unfolding Make rules deriving non-terminal of arity m have no premise of arity m (relies on well-nestedness): (s 1,..., s m) B(x 1,..., x m), Γ (t 1,..., t m) D(y 1,..., y p), Γ 1 D(u 1,..., u p) B(x 1,..., x m), Γ 2 Unfold the rules of non-terminal of arity m The resulting grammar defines the same language and does not use non-terminals of arity m: it is an m 1-MCFG wn Thus by induction on m (the case where m = 1 is the CFL case), L(G 2 ) is 2(m 1)-pumpable and therefore m-pumpable.
23 Proof of the Lemma Proof. eliminate m-proper rules by unfolding Make rules deriving non-terminal of arity m have no premise of arity m (relies on well-nestedness): (s 1,..., s m) B(x 1,..., x m), Γ (t 1,..., t m) D(y 1,..., y p), Γ 1 D(u 1,..., u p) B(x 1,..., x m), Γ 2 Unfold the rules of non-terminal of arity m The resulting grammar defines the same language and does not use non-terminals of arity m: it is an m 1-MCFG wn Thus by induction on m (the case where m = 1 is the CFL case), L(G 2 ) is 2(m 1)-pumpable and therefore m-pumpable.
24 Proof of the Lemma Proof. eliminate m-proper rules by unfolding Make rules deriving non-terminal of arity m have no premise of arity m (relies on well-nestedness): (s 1,..., s m) B(x 1,..., x m), Γ (t 1,..., t m) D(y 1,..., y p), Γ 1 D(u 1,..., u p) B(x 1,..., x m), Γ 2 Unfold the rules of non-terminal of arity m The resulting grammar defines the same language and does not use non-terminals of arity m: it is an m 1-MCFG wn Thus by induction on m (the case where m = 1 is the CFL case), L(G 2 ) is 2(m 1)-pumpable and therefore m-pumpable.
25 Proof of the Lemma Proof. eliminate m-proper rules by unfolding Make rules deriving non-terminal of arity m have no premise of arity m (relies on well-nestedness): (s 1,..., s m) B(x 1,..., x m), Γ (t 1,..., t m) D(y 1,..., y p), Γ 1 D(u 1,..., u p) B(x 1,..., x m), Γ 2 Unfold the rules of non-terminal of arity m The resulting grammar defines the same language and does not use non-terminals of arity m: it is an m 1-MCFG wn Thus by induction on m (the case where m = 1 is the CFL case), L(G 2 ) is 2(m 1)-pumpable and therefore m-pumpable.
26 Proof of the Lemma Proof. eliminate m-proper rules by unfolding Make rules deriving non-terminal of arity m have no premise of arity m (relies on well-nestedness): (s 1,..., s m) B(x 1,..., x m), Γ (t 1,..., t m) D(y 1,..., y p), Γ 1 D(u 1,..., u p) B(x 1,..., x m), Γ 2 Unfold the rules of non-terminal of arity m The resulting grammar defines the same language and does not use non-terminals of arity m: it is an m 1-MCFG wn Thus by induction on m (the case where m = 1 is the CFL case), L(G 2 ) is 2(m 1)-pumpable and therefore m-pumpable.
27 Kanazawa s Theorems Kanazawa (2010) Theorem m-mcfl wn are 2m-pumpable. Theorem 2-MCFL are 4-pumpable.
28 Weak pumping Lemma for MCFL Outline The pumping Lemma for CFL Weak pumping Lemma for MCFL No strong pumping Lemma for MCFL MCFL wn and mild context sensitivity
29 Weak pumping Lemma for MCFL Seki et al. pumping lemma Seki, Matsumura, Fujii, Kasami Theorem In a infinite m-mcfl L, there is a string w that is 2m-pumpable in L.
30 Weak pumping Lemma for MCFL Proof of the Theorem Take an ordered grammar.
31 Weak pumping Lemma for MCFL Proof of the Theorem Take an ordered grammar. Iterate the pump until the variables are fixed. (x 1,..., x k )
32 Weak pumping Lemma for MCFL Proof of the Theorem Take an ordered grammar. Iterate the pump until the variables are fixed. Now iterating the pump results in an iteration on the string. (x 1,..., x k )
33 Weak pumping Lemma for MCFL n example The pump: (u 1 x 1 u 2 x 2 u 3, u 4 ) One iteration: The variables are fixed. Second iteration: (u 1 u 1 x 1 u 2 x 2 u 3 u 2 u 4 u 3, u 4 ) (u 1 u 1 u 1 x 1 u 2 x 2 u 3 u 2 u 4 u 3 u 2 u 4 u 3, u 4 )
34 Weak pumping Lemma for MCFL n example The pump: (u 1 x 1 u 2 x 2 u 3, u 4 ) One iteration: The variables are fixed. Second iteration: (u 1 u 1 x 1 u 2 x 2 u 3 u 2 u 4 u 3, u 4 ) (u 1 u 1 u 1 x 1 u 2 x 2 u 3 u 2 u 4 u 3 u 2 u 4 u 3, u 4 )
35 Weak pumping Lemma for MCFL n example The pump: (u 1 x 1 u 2 x 2 u 3, u 4 ) One iteration: The variables are fixed. Second iteration: (u 1 u 1 x 1 u 2 x 2 u 3 u 2 u 4 u 3, u 4 ) (u 1 u 1 u 1 x 1 u 2 x 2 u 3 u 2 u 4 u 3 u 2 u 4 u 3, u 4 )
36 Weak pumping Lemma for MCFL n example The pump: (u 1 x 1 u 2 x 2 u 3, u 4 ) One iteration: The variables are fixed. Second iteration: (u 1 u 1 x 1 u 2 x 2 u 3 u 2 u 4 u 3, u 4 ) (u 1 u 1 u 1 x 1 u 2 x 2 u 3 u 2 u 4 u 3 u 2 u 4 u 3, u 4 )
37 Weak pumping Lemma for MCFL n example The pump: (u 1 x 1 u 2 x 2 u 3, u 4 ) One iteration: The variables are fixed. Second iteration: (u 1 u 1 x 1 u 2 x 2 u 3 u 2 u 4 u 3, u 4 ) (u 1 u 1 u 1 x 1 u 2 x 2 u 3 u 2 u 4 u 3 u 2 u 4 u 3, u 4 )
38 No strong pumping Lemma for MCFL Outline The pumping Lemma for CFL Weak pumping Lemma for MCFL No strong pumping Lemma for MCFL MCFL wn and mild context sensitivity
39 No strong pumping Lemma for MCFL mythic pumping Lemma Radzinski (1991) I has then been assumed to hold: Groenink 1997, Kracht 2003
40 No strong pumping Lemma for MCFL Constructing a counter-example We must take a m-mcfl so that m > 2 It must not contain proper rules H(x 2 ) G(x 1, x 2, x 3 ) G(ax 1, y 1 cx 2 cdy 2 dx 3, y 3 b) G(x 1, x 2, x 3 ) G(y 1, y 2, y 3 ) G(a, ɛ, b)
41 No strong pumping Lemma for MCFL The language and binary trees H(x 2 ) G(x 1, x 2, x 3 ) G(ax 1, y 1 cx 2 cdy 2 dx 3, y 3 b) G(x 1, x 2, x 3 ) G(y 1, y 2, y 3 ) G(a, ɛ, b) Let ϕ be a morphism such that ϕ(a) = ϕ(b) = ɛ and leaves the other letter unchanged. Lemma ϕ(l(h)) is the CFL described by the grammar V cv cdv d ɛ The language L(V ) represent binary trees. The language L(H) can be seen as: a l cw 1 cdw 2 db k k l w 1 w 2
42 No strong pumping Lemma for MCFL The language and binary trees H(x 2 ) G(x 1, x 2, x 3 ) G(ax 1, y 1 cx 2 cdy 2 dx 3, y 3 b) G(x 1, x 2, x 3 ) G(y 1, y 2, y 3 ) G(a, ɛ, b) Let ϕ be a morphism such that ϕ(a) = ϕ(b) = ɛ and leaves the other letter unchanged. Lemma ϕ(l(h)) is the CFL described by the grammar V cv cdv d ɛ The language L(V ) represent binary trees. The language L(H) can be seen as: a l cw 1 cdw 2 db k k l w 1 w 2
43 No strong pumping Lemma for MCFL The language and binary trees H(x 2 ) G(x 1, x 2, x 3 ) G(ax 1, y 1 cx 2 cdy 2 dx 3, y 3 b) G(x 1, x 2, x 3 ) G(y 1, y 2, y 3 ) G(a, ɛ, b) Let ϕ be a morphism such that ϕ(a) = ϕ(b) = ɛ and leaves the other letter unchanged. Lemma ϕ(l(h)) is the CFL described by the grammar V cv cdv d ɛ The language L(V ) represent binary trees. The language L(H) can be seen as: a l cw 1 cdw 2 db k k l w 1 w 2
44 No strong pumping Lemma for MCFL 3-MCFL are not finitely pumpable Kanazawa, Kobele, S., Yoshinaka 2011 Theorem L(H) is not k-pumpable for any k Proof. Every string representing complete binary tree is not k-pumpable in L(H) for any k.
45 MCFL wn and mild context sensitivity Outline The pumping Lemma for CFL Weak pumping Lemma for MCFL No strong pumping Lemma for MCFL MCFL wn and mild context sensitivity
46 MCFL wn and mild context sensitivity MCFL wn and cross serial dependencies Joshi s informal notion I(x1y1,y2x2) J(x1,x2), K (y1,y2) I(x1y1,x2y2) J(x1,x2), K (y1,y2) (x1z1, z2x2y1, y2y3x3) B(x1, x2, x3) C(y1, y2, y3) D(z1, z2) (z1x1, y1x2z2y2x3, y3) B(x1, x2, x3) C(y1, y2, y3) D(z1, z2)
47 MCFL wn and mild context sensitivity MCFL wn and cross serial dependencies Joshi, Vijay Shanker, Weir (1991) Conjecture: MIX is not a MCFL wn
48 MCFL wn and mild context sensitivity MCFL wn and cross serial dependencies Groenink (PhD dissertation): finite pumpability k-mcfl wn are 2k-pumpable.
49 MCFL wn and mild context sensitivity MCFL wn and cross serial dependencies Kallmeyer (course Düsseldorf): finite copying For every k N, {w k w T } is a k-mcfl wn.
50 MCFL wn and mild context sensitivity MCFL wn and constant growth property Vijay Shanker, Weir, Joshi (1987) Theorem The language defined by an MCFG wn is semilinear.
51 MCFL wn and mild context sensitivity MCFL wn Polynomial parsing Membership Universal Membership m-mcfg LOGCFL-complete NP-complete (when m 2) m-mcfg wn LOGCFL-complete P-complete MCFG LOGCFL-complete PSPCE-complete/ EXPTIME-complete MCFG wn LOGCFL-complete PSPCE-complete/??
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