Multiple Context-Free Grammars

Transcription

1 Ogden s Lemma, Multiple Context-Free Grammars, and the Control Language Hierarchy Makoto Kanazawa National Institute of Informatics and SOKENDAI Japan Multiple Context-Free Grammars Introduced by Seki, Matsumura, Fujii, and Kasami ( ) Independently by Vijay-Shanker,Weir, and Joshi (1987) Many equivalent models Arising from concerns in computational linguistics. CFGs are almost good enough for NL grammars, but not quite; a mild extension of CFGs is needed. Several criteria were put forward as to what constitutes a mild extension. Often thought to be an adequate formalization of mildly context-sensitive grammars (Joshi 1985) Which properties of CFGs are shared by/ generalize to MCFGs?

2 Multiple Context-Free Grammars A(α1,,αq) B1(x1,1,,x1,q 1 ),,Bn(xn,1,,xn,q n ) n 0, q, qi 1, αk (Σ { xi,j i [1,n], j [1,qi] })* each xi,j occurs exactly once in (α1,,αq) It s best to think of an MCFG as a kind of logic program. Each rule is a definite clause. Nonterminals are predicates on strings. q dim(a) (dimension of A) dim(s) 1 L(G) { w Σ* G S(w) } 4 S(x1#x) D(x1, x) D(x1y1, yx) E(x1,x), D(y1,y) E(ax1ā, āxa) D(x1,x) -MCFG -ary branching { w#w R w D1* } S(aaāāaā#āaāāaa) D(aaāāaā, āaāāaa) m-mcfg MCFG with nonterminal dimension not exceeding m 1-MCFG CFG Derivation tree for w proof of S(w) E(aaāā, āāaa) D(aā, āa) D(aā, āa) E(aā, āa) E(aā, āa) 5 derivation tree S(x1 xm) A(x1,,xm) A(ε,,ε) A(a1 x1 a,,am 1 xm am) A(x1,,xm) The languages of MCFGs form an infinite hierarchy. non-branching m-mcfg m-m (m 1)-M -M { a1 n a n am 1 n am n n 0 } Seki et al M 6

3 S(aaāāaā#āaāāaa) D(aaāāaā, āaāāaa) Pumping S(x1#x) D(x1, x) D(x1y1, yx) E(x1,x), D(y1,y) E(ax1ā, āxa) D(x1,x) Derivation trees of MCFGs are similar to those of CFGs. When the same nonterminal occurs twice on the same path of a derivation tree, E(aaāā, āāaa) D(aā, āa) D(aā, āa) E(aā, āa) E(aā, āa) 7 S(y1#y) D(y1, y) D(ax1āaā, āaāxa) E(ax1ā, āxa) D(aā, āa) D(x1, x) D(aā, āa) E(aā, āa) E(aā, āa) a n aā(āaā) n #(āaā) n āaa n L(G) 8 You can decompose the derivation tree into three parts, and the middle part can be iterated any number of times, including zero times. In the overall derivation tree, the variables x 1, x, y 1, y are instantiated by The number of iterated substrings (factors) larger than two. Iterative Properties L is k-iterative iff p z L( z p u1 uk+1 v1 vk( z u1v1 ukvkuk+1 v1 vk ε n 0(u1v1 n ukvk n uk+1 L)) For MCFGs, need to consider a generalized form of the condition of the puming lemma. Not straightforward; open question for a long time. L L is -iterative L m-m L is m-iterative? 9

4 Difficulty with Pumping The middle part of the derivation tree may look like this. A(v1 x1v, v xv4 ) A(v1x1v, vxv4) A(v1x1v, vxv4) A(x1, x) A(x1, x) 10 Difficulty with Pumping Or like this. A(v1 x1vxvvv4v, v4) A(v1x1vxv, v4) A(v1x1vxv, v4) A(x1, x) uneven pump A(x1, x) 11 S(x1#x#x) A(x1, x, x) A(ax1, y1cxc dyd x, yb) A(x1, x, x), A(x1, x, x) The pumping lemma fails for - MCFGs. A(a, ε, b) m-m -M -M not k-iterative for any k Kanazawa et al M 1

5 Pumping Lemma for Subclasses Pumping possible for special cases. Well-nested MCFGs. m-mwn m-iterative m-m -Mwn 1-Mwn -M 1-M 4-iterative well-nested MCFGs Kanazawa Well-Nestedness Has a natural equivalent characterization: ycft sp { w#w R w D1* } S(x1#x) D(x1, x) D(x1y1, yx) E(x1,x), D(y1,y) { w#w w D1* } S(x1#x) D(x1, x) D(x1y1, xy) E(x1,x), D(y1,y) E(ax1ā, āxa) D(x1,x) well-nested E(ax1ā, axā) D(x1,x) non-well-nested { w#w w D1* } Mwn 14 Kanazawa and Salvati 010 Difficulty with Pumping A(v1 x1vxvvv4v, v4) Pumping not easy to prove even form well-nested MCFGs: this situation can still arise. A(v1x1vxv, v4) A(v1x1vxv, v4) A(x1, x) uneven pump A(x1, x) 15

6 S(x1x) A(x1, x) A(ax1bxc, d) A(x1, x) non-branching well-nested A(ε, ε) S(ε) S(abcd) S(aabcbdcd) S(aaabcbdcbdc,d) A(ε, ε) A(abc, d) A(ε, ε) A(aabcbdc,d) A(abc, d) A(ε, ε) A(aaabcbdcbdc,d) A(aabcbdc,d) A(abc, d) A(ε, ε) i0 i1 i i { ε } { a i 1 abc(bdc) i 1 d i 1 } A very simple example. The only choice you can make is the number of times you use the second rule. Actually -iterative, but no straightforward connection between the iterated substrings and parts of derivation trees. 16 Pumping Lemma for Subclasses m-iterative m-m m-mwn My proof of the pumping lemma for m-m wn and -M is not straightforward. -Mwn 1-Mwn -M 1-M 4-iterative Kanazawa 009, by grammar splitting and transformation What about Ogden s Lemma? 17 Ogden s Lemma for L p z L(at least p positions of z are marked u1uu v1v( z u1v1uvu (u1, v1, u each contain a marked position u, v, u each contain a marked position) v1uv contains no more than p marked positions n 0(u1v1 n uv n u L)) Ogden

7 There are various ways of generalizing Ogden s lemma suitable for MCFGs. At least this much should be implied. L has the weak Ogden property iff p z L(at least p positions of z are marked k 1 u1 uk+1 v1 vk( z u1v1 ukvkuk+1 i(vi contains a marked position) n 0(u1v1 n ukvk n uk+1 L)) 19 The Failure of Ogden s Lemma This is the first new result in this talk. m-mwn -Mwn -Mwn 1-Mwn m-iterative 6-iterative m-m -M 1-M 4-iterative The weak Ogden property fails for -Mwn and -M. 0 { a i1 b i0 $a i b i1 $a i b i $ $a in b in 1 n, i0,,in 0 } A language for which the weak Ogden property fails. -Mwn -Mwn -M 1

8 A(ε) A(bx1) A(x1) B(x1, ε) A(x1) B(ax1, bx) B(x1, x) C(x1, x, ε) B(x1, x) C(x1, ax, bx) C(x1, x, x) C(x1$x, x, ε) C(x1, x, x) D(x1$x, x) C(x1, x, x) D(x1, ax) D(x1, x) S(x1$x) D(x1, x) non-branching -MCFG A(ε) A(bx1) A(x1) B(x1, ε) A(x1) B(ax1, bx) B(x1, x) C(ε, ε) C(ax1, bx) C(x1, x) D(x1$y1x, y) B(x1, x), C(y1, y) D(x1$y1x, y) D(x1, x), C(y1, y) E(x1, x) D(x1, x, x) E(x1, ax) E(x1, x) S(x1$x) E(x1, x) -MCFG { a i1 b i0 $a i b i1 $a i b i $ $a in b in 1 n, i0,,in 0 } Mark the positions of $. { a i1 b i0 $a i b i1 $a i b i $ $a in b in 1 n, i0,,in 0 } is -iterative a$a b$a b $ $a p+1 b p Weir s (199) Control Language Hierarchy Subclasses of M that are known to have an Ogden property. Ck Ck k 1 -M Kanazawa and Salvati 007 C -Mwn C1 Generalization of Ogden s lemma Palis and Shende

9 G: Control Grammars 1: S asās : S bsb S : S ε CFG with child selection K { 1 n n n 0 } control set a a S 1 S b S b S b S b S ε ε ε 1 S ā ā S 1 S ε a S ā S b S b S ε ε ε Languages in each level of the control language hierarchy are given by control grammars. L(G, K) D* ({ a n b n n 1} b {ā,b }*)* 5 Control Language Hierarchy C1 Ck+1 { L(G, K) K Ck } 6 Ogden s Lemma for C k L Ck It is quite straightforward to prove an Ogden s lemma for C k. p z L(at least p positions of z are marked u1 uk+1 v1 vk( z u1v1 ukvkuk+1 i(ui, vi, ui+1 each contain a marked position) vk 1uk 1vk 1+1 contains no more than p marked positions n 0(u1v1 n uv n ukvk n uk+1 L)) 7 Palis and Shende 1995 k 1 gives Ogden s (1968) original lemma.

10 Proper MCFGs A(α1,,αq) Approach this existing result from the MCFG formalism. Sufficient condition for an Ogden property. dimension q αi contains xi A(x1,,xq) 8 S(x1x) A(x1, x) A(ax1c, d) B(x1) B(x1bx) A(x1, x) A(ε, ε) proper -MCFG Slight variation of an earlier example. S(aabcbdcd) A(ax1bxc,d) A(aabcbdc,d) dimension < B(x1bx) B(abcbd) B(ax1cbd) A(x1, x) A(abc, d) A(ax1c, d) dimension 1 B(b) B(x1) A(ε, ε) 9 Ogden s Lemma for m-m prop Constrain all of v 1,,v m L m-mprop p z L(at least p positions of z are marked u1 um+1 v1 vm( z u1v1 umvmum+1 i(ui, vi, ui+1 each contain a marked position) v1uv,vu4v4,,vm 1umvm together contain no more than p marked positions n 0(u1v1 n uv n umvm n um+1 L)) m 1 gives Ogden s (1968) original lemma. 0

11 C k k 1 -M prop The earlier result is subsumed by the present result. G: 1: S asās : S bsb S : S ε l(1) a, r(1) ās l() b, r() b S l() ε, r() ε homomorphisms K { 1 n n n 0 } K K(x1) E(x1) E(1x1) E(x1) E(ε) m-mcfg (1,R), l, r (K) { l(w)r(w R ) w K} homomorphic replication Kʹ(x1x) K(x1, x) K(x1l(), r()x) E(x1, x) E(l(1)x1l(), r()xr(1)) E(x1, x) E(ε, ε) 1 m-mcfg 1 S Generates strings with nonterminals. a 1 S a S b S b S b S b S ε ε ε ā S ā S ε 1 a S ā S b S b S ε ε ε l(11)r(11) aabbb Sb SāSāS l()r() ε l(1)r(1) abb SāS (1,R), l, r (K) { l(w)r(w R ) w K} Kʹ(x1x) K(x1, x) K(x1l(), r()x) E(x1, x) E(l(1)x1l(), r()xr(1)) E(x1, x) E(ε, ε) Kʹ(x1x) K(x1, x) K(x1, x) E(x1, x) E(ax1b, b SxāS) E(x1, x) E(ε, ε) a a S 1 S b S b S b S b S ε ε ε 1 S ā ā S 1 S ε a S ā S b S b S ε ε ε (1,R), l, r (K) { l(w)r(w R ) w K} L(G, K) Kʹ(x1x) K(x1, x) K(x1, x) E(x1, x) E(ax1b, b SxāS) E(x1, x) E(ε, ε) Kʹ(x1x) K(x1, x) K(x1, x) E(x1, x) E(ax1b, b yxāz) E(x1, x), Kʹ(y), Kʹ(z) E(ε, ε) S Kʹ iterated substitution Σ* The construction doubles the dimension, preserves properness. m-mcfg

12 Ck Ck k 1 -Mprop m-mprop The different requirement shows the properness of the inclusion. C -Mwn C1 -Mprop 1-Mprop p z u1v1uvuvu4v4u5 p 4 Summary Pumping doesn t imply Ogden: There is no Ogdenlike theorem for -Mwn -M There is a natural Ogden s lemma for m-mprop Covers Weir s control language hieararchy