Computational complexity of commutative grammars

Transcription

1 Computational complexity of commutative grammars Jérôme Kirman Sylvain Salvati Bordeaux I University - LaBRI INRIA October 1, 2013

2 Free word order Some languages allow free placement of several words or phrases: Fully non-configurational languages (Latin) Scrambling phenomenon (German) Case marking or pragmatics give information on syntactic roles Grammatically, all the possible word orders are equally valid, while the sentence s meaning and syntactic roles stay the same.

3 Scrambling in German subordinate clauses [BNR92] daß eine hiesige Firma meinem Onkel die Möbel vor drei Tagen ohne Voranmeldung zugestellt hat....that a local company to my uncle the furniture three days ago without warning has delivered....that a local company has delivered the furniture to my uncle three days ago without warning. All the possible orderings of the constituents (1,2,3,4,5) are possible Arguments of nested subordinate clauses can also be scrambled

4 Words modulo commutation There is a single sentence, which has many representations. One structure Many strings

5 Words modulo commutation There is a single sentence, which has many representations. One structure Many strings We need an object that represents a word modulo commutation of some of its letters/factors (sub-sequences) and a way to parse it

6 An algebra for words modulo commutation Words modulo commutation over an alphabet Σ will be constructed as terms of an algebra com(σ) Atomic terms are letters of the alphabet Σ and the empty word ε Terms are constructed by means of two operators: Concatenation of subterms (associative, as usual) Commutative combination of subterms (assoc + comm)

7 Terms and associated languages Ranked signature: com(σ) := {a 0 a Σ {ε 0 ; 2 ; 2

8 Terms and associated languages Ranked signature: com(σ) := {a 0 a Σ {ε 0 ; 2 ; 2 Equivalence relation for terms ( ): x y z x y z x y z x y z x y y x associativity associativity commutativity

9 Terms and associated languages Ranked signature: com(σ) := {a 0 a Σ {ε 0 ; 2 ; 2 Equivalence relation for terms ( ): x z x z y y z x y y z associativity associativity commutativity Language of a term: x L(t) := {yield(t ) t t x y y x

10 Examples of terms t 1 = a b c L(t 1 ) = {abc, acb

11 Examples of terms t 1 = t 2 = a a b c b c L(t 1 ) = {abc, acb L(t 2 ) = {abc, acb, bac, bca, cab, cba

12 Examples of terms t 1 = t 2 = t 3 = a a a b c b c ε b c L(t 1 ) = {abc, acb L(t 2 ) = {abc, acb, bac, bca, cab, cba L(t 3 ) = {abc, acb, bca, cba

13 Packed representation Using this algebra, we get objects to represent free word orders: 1 eine hiesige Firma a local company... daß... that 2 meinem Onkel to my uncle t = 3 die Möbel the furniture 4 zugestellt delivered 5 vor drei Tagen ohne Voranmeldung three days ago without warning hat has L(t) = daß zugestellt hat daß zugestellt hat daß zugestellt hat. daß zugestellt hat daß zugestellt hat. daß zugestellt hat daß zugestellt hat. daß zugestellt hat daß zugestellt hat 5! = 120 word orders

14 Towards commutative grammars t {w w L(t) sentence with free order set of representations

15 Towards commutative grammars t {w w L(t) sentence with free order set of representations Commutative grammars are defined as term grammars, with: L w (G) := L(t) t L(G)

16 Towards commutative grammars t {w w L(t) sentence with free order set of representations Commutative grammars are defined as term grammars, with: L w (G) := L(t) t L(G) L(G) = t 1 ; t 2 ;... L w (G) = L(t 1 ) L(t 2 )...

17 Commutative context-free grammars The first natural class of grammars we consider is CCFG: Regular term grammars on the terminal signature com(σ) CCFG generalize CFG (without the operator) G = N, com(σ), R, S The rules of R have the form N t, with t com(n Σ)

18 Example of CCFG G := {S, W, com({a, b, c, #), R, S S ; S ε ; W ; W ε R := S W # a b c W

19 Example of CCFG G := {S, W, com({a, b, c, #), R, S S ; S ε ; W ; W ε R := S W # a b c W L w (G) = {w 1 #... w n # n N w i a = w i b = w i c

20 Example of CCFG G := {S, W, com({a, b, c, #), R, S S ; S ε ; W ; W ε R := S W # a b c W L w (G) = {w 1 #... w n # n N w i a = w i b = w i c R = S x 1 S(x1) W (x2) ; W W (x1) x 2 # a b c x 1 S(ε) ; W (ε)

21 Commutative regular grammars CREG are a more constrained version of CCFG: Right-hand sides are restricted to right-branching terms with a single non-terminal at the end (rightmost leaf) CREG generalize right-linear (regular) word grammars A a b c B A typical CREG production

22 Commutative multiple regular grammar CMREG extend CCFG by allowing multiple terms: Productions have the form: A(t 1,..., t n ) B 1 (x 1,1,..., x 1,n1 ),..., B p (x p,1,..., x p,np ) Where t i com ( Σ { x i,j 1 i p 1 j n i ). Nonterminals are typed for consistency, with τ(s) = [o]. CMREG generalize multiple context free word grammars. A B(x 1,1, x 1,2 ) C(x 2,1, x 2,2 ) x 1,1 x 2,1 x 1,2 x 2,2 Example of a CMREG production.

23 Commutative macro grammars CMG are another generalization of CCFG: Nonterminals construct contexts (not just ground terms). Nonterminals are typed to enforce arities, with τ(s) = [o]. CMG generalize well-nested MCFGs. A x 2 x 1 [ ] B(x 1) C(x 2 ) a Example of a CMG production.

24 Commutative multiple context-free grammars CMCFG generalize the other classes: Nonterminals construct tuples of contexts. Intuitively the meet of CMG and CMREG. Nonterminals are also typed with τ(s) = [o]. x 1 A c [ ] c [ ], a x 3, b x 2 A(x 1, x 2, x 3 ) Example of a CMCFG production.

25 Outline of commutative grammar hierarchy MCFGwn CMG CMCFG CMREG MCFG CCFG CFG CREG RG Hierarchy of mildly context-sensitive commutative grammars (arrows denote inclusion)

26 Outline of commutative grammar hierarchy MCFGwn CMCFG?? CMG CMREG MCFG CCFG CFG CREG RG Hierarchy of mildly context-sensitive commutative grammars (arrows denote inclusion)

27 Two decisions problems for parsing We consider two classes of decision problems related to parsing: Universal membership problem Input A word w and a grammar G. Answer YES iff w L w (G). Membership problem for G Input A word w. Answer YES iff w L w (G). Solving both problems involves parsing w according to G; however, for membership, G is a fixed-size parameter (part of the constant).

28 Overview of results Class Membership Universal membership ND Full Terms O(1) NP-C CREG Nlogspace NP-C CCFG Logcfl-C NP-C CMG NP-C Pspace-C Exptime CMREG NP-C Pspace-C Exptime-C CMCFG NP-C Pspace-C Exptime-C

29 NP-completeness of universal term membership Class Membership Universal membership ND Full Terms O(1) NP-C CREG Nlogspace NP-C CCFG Logcfl-C NP-C CMG NP-C Pspace-C Exptime CMREG NP-C Pspace-C Exptime-C CMCFG NP-C Pspace-C Exptime-C

30 NP-hardness of UTM (1/2) 3-PARTproblem: Input A set S of 3m integers ( k 4 n i k ) 2 m n 1 n 2 n 3 n 4 n 5 n 6... n 3m 5 n 3m 4 n 3m 3 n 3m 2 n 3m 1 n 3m S k

31 NP-hardness of UTM (1/2) 3-PARTproblem: ( k Input A set S of 3m integers 4 n i k ) 2 Answer YES iff there is a partition S 1... S m of S that satisfies: n j = k nj Si S i m n 1 n 2 n 3 n 4 n 5 n 6... n 3m 5 n 3m 4 n 3m 3 S part. π n π(1) n π(2) n π(3) S 1 n π(4) n π(5) n π(6) S 2... n π(3m 5) n π(3m 4) n π(3m 3) S m 1 S n 3m 2 n 3m 1 n 3m n π(3m 2) n π(3m 1) n π(3m) S m k k

32 NP-hardness of UTM (2/2) t =... #... # {{ m a... a {{ n 1 a... a {{ n 2 a... a {{ n 3m 1 a... a {{ n 3m w = (a k #) m w = a. {{.. a # a. {{.. a #... a. {{.. a # k k {{ k m

33 Outline of NP algorithm for UTM To check wether w L(t): Guess a term t (s.t. t t in com(σ) by assoc/comm of ) Check that yield(t ) = w Check that t t w = acb L(t)? t : t 2 : t : c a a a b b c c b t t yield(t ) = acb acb L(t)

34 NP-completeness of universal CCFG membership Class Membership Universal membership ND Full Terms O(1) NP-C CREG Nlogspace NP-C CCFG Logcfl-C NP-C CMG NP-C Pspace-C Exptime CMREG NP-C Pspace-C Exptime-C CMCFG NP-C Pspace-C Exptime-C

35 Outline of NP algorithm for CCFG Deciding wether w L w (G): Construct a compact grammar G s.t. L w (G ) = L w (G). Guess a derivation of a term t L(G ) with t w k. Check that the derivation is valid in Ptime. Check that w L(t) in NP. Why is there such a G? (t, ε) ε t (ε, t) ε t (t, ε) ε t (ε, t) ε t iff t = 1 or t = (t 1, t 2 ) Language-preserving rewriting system for terms.

36 Logcfl-completeness of CCFG membership Class Membership Universal membership ND Full Terms O(1) NP-C CREG Nlogspace NP-C CCFG Logcfl-C NP-C CMG NP-C Pspace-C Exptime CMREG NP-C Pspace-C Exptime-C CMCFG NP-C Pspace-C Exptime-C

37 Notations for the CCFG algorithm Grammars are compact and in normal form: ( ) [op] A B(x) C(y) x y A(a) Derivation items are unordered vectors of NT between two positions: v, i, j (1 i, j w ) ψ(g, N) is the precomputed (semilinear) set of bags of non-terminals that N can generate in any order according to G.

38 CCFG parsing algorithm A(a) a = w[i, j] constant 1 A, i, j 1 B, i, j 1 C, j, k A(x y) B(x) C(y) combine 1 A, i, k v 1, i, j v 2, j, k 0 < v 1 0 < v 2 v 1 + v 2 w comm. combine v 1 + v 2, i, k v, i, j v ψ(g, A) comm. reduction 1 A, i, j Logcfl recognition algorithm for CCFG.

39 NP-hardness of fixed grammars beyond CCFG Class Membership Universal membership ND Full Terms O(1) NP-C CREG Nlogspace NP-C CCFG Logcfl-C NP-C CMG NP-C Pspace-C Exptime CMREG NP-C Pspace-C Exptime-C CMCFG NP-C Pspace-C Exptime-C

40 Yet another reduction to 3-PART Based on 3-PART - triplets and their sum are derived in parallel.... t m a. {{.. a n 1,1 # a. {{.. a n 1,2 # a. {{.. a n 1,3 # b. {{.. b # n 1,1 +n 1,2 +n 1,3 w = a n 1 #... a n 3m #(b k #) m w = a. {{.. a #... a. {{.. a # b. {{.. b #... b. {{.. b # n 1 n 3m k {{ k m

41 NP-hard fixed CMREG x 5 S x 1 x 2 x 3 x 4 ( A, x 2, x 3, a x ( 1 A x 1,, x 3, a x ( 2 A x 1, x 2,, a x 3 A(x 1 ) S(x 2 ) ) A(x 1, x 2, x 3, x 4 ) b x 4 ) A(x 1, x 2, x 3, x 4 ) b x 4 ) A(x 1, x 2, x 3, x 4 ) b x 4 S(ε) A (#, #, #, #)

42 NP-hard fixed CMG S S(ε) x 1 # # # # x 2 A [ ] [ ] [ ] [ ] A(x 1) S(x 2 ) x 1 A [ ] [ ] A(x 1) a [ ] b [ ] x 1 A [ ] [ ] A(x 1) a [ ] b [ ] x 1 A [ ] [ ] A(x 1) a [ ] b [ ]

43 Overview of results Class Membership Universal membership ND Full Terms O(1) NP-C CREG Nlogspace NP-C CCFG Logcfl-C NP-C CMG NP-C Pspace-C Exptime CMREG NP-C Pspace-C Exptime-C CMCFG NP-C Pspace-C Exptime-C

44 Conclusion and perspectives In summary, we have provided: An algebraic representation of sentences with free order A hierarchy of generative grammars to construct such representations A study of the associated computational complexities Next we want to look into: Closure properties : rational cones, AFL,... Relation with UVG, dependency parsing,...