Derivational order and ACGs

Transcription

1 Derivational order and ACGs Gregory M. Kobele 1 Jens Michaelis 2 1 University of Chicago, Chicago, USA 2 Bielefeld University, Bielefeld, Germany ACG@10, Bordeaux, December 7 9, 2011 Greg & Jens (Chicago & Bielefeld) Derivational order and ACGs ACG@10 1 / 51

2 What good are ACGs? ACGs provide a way to understand many ideas from a uniform perspective Architectures for grammar Syntactico-centrism vs Parallelism Grammar formalisms CFHGs CFG, LCFTG, MCFG DTWT MG Distributional learning We want to add to the list: Timing (SP) (SP, PdG, & CP) (MK) (PdG, SP) (SS) (SS) (RY) Greg & Jens (Chicago & Bielefeld) Derivational order and ACGs ACG@10 2 / 51

3 Chris Barker and a new approach to TAG semantics The basic idea of using cosubstitution to handle scope is that we can build the nuclear scope of a quantifier before the quantifier enters the derivation. Greg & Jens (Chicago & Bielefeld) Derivational order and ACGs ACG@10 3 / 51

4 Chris Barker and a new approach to TAG semantics The basic idea of using cosubstitution to handle scope is that we can build the nuclear scope of a quantifier before the quantifier enters the derivation. In other words, the nuclear scope of a quantifier is quite simply and quite literally its syntactic and semantic argument. Greg & Jens (Chicago & Bielefeld) Derivational order and ACGs ACG@10 3 / 51

5 Chris Barker and a new approach to TAG semantics The basic idea of using cosubstitution to handle scope is that we can build the nuclear scope of a quantifier before the quantifier enters the derivation. In other words, the nuclear scope of a quantifier is quite simply and quite literally its syntactic and semantic argument. Thus on the cosubstitution approach, quantifier scope ambiguity is a matter of timing: quantifiers that enter later in the derivation take wider scope. Greg & Jens (Chicago & Bielefeld) Derivational order and ACGs ACG@10 3 / 51

6 Chris Barker and a new approach to TAG semantics The basic idea of using cosubstitution to handle scope is that we can build the nuclear scope of a quantifier before the quantifier enters the derivation. In other words, the nuclear scope of a quantifier is quite simply and quite literally its syntactic and semantic argument. Thus on the cosubstitution approach, quantifier scope ambiguity is a matter of timing: quantifiers that enter later in the derivation take wider scope. In other words, co-tag has exactly the same weak and strong generative capacity as TAG. Greg & Jens (Chicago & Bielefeld) Derivational order and ACGs ACG@10 3 / 51

7 Actually, an old approach to semantics (and other things) Montague Grammar (Montague) [... ] ambiguity can arise even when there is no element of intensionality, simply because quantifying terms may be introduced in more than one order. Minimalist Grammars (Gärtner & Michaelis; Kobele) [... ] an expression may bind only those expressions that it c-commands when first merged Greg & Jens (Chicago & Bielefeld) Derivational order and ACGs ACG@10 4 / 51

8 What s behind this? The problem: Sometimes we have a semantic ambiguity without any obvious corresponding difference in the derivation tree. The idea: Enrich the derivation tree with information about order. (not only did I apply this rule, but I applied this rule before that one) Greg & Jens (Chicago & Bielefeld) Derivational order and ACGs ACG@10 5 / 51

9 A concrete example S A B A a B b S A a B b Example The representation of a CFG derivation as a tree equates sequences of rewriting steps which rewrite nonterminal instances the same way. (Leftmost, rightmost, etc) S AB ab ab S AB Ab ab Greg & Jens (Chicago & Bielefeld) Derivational order and ACGs ACG@10 6 / 51

10 What s to come The goals: Make sense of the idea of timing in terms of derivations Understand (in particular) co-substitution in TAGs in these terms If SGC/WGC unchanged, what is going on? Our main claims are that: Third order ACGs provide an elegant description of the derivations of context-free formalisms. Timing is not derivational order Greg & Jens (Chicago & Bielefeld) Derivational order and ACGs ACG@10 7 / 51

11 Outline 1 Everyone s favourite example (Montague) 2 3 rd order ACGs and context-free derivations 3 Cosubstitution in TAGs Greg & Jens (Chicago & Bielefeld) Derivational order and ACGs ACG@10 8 / 51

12 Frameworks which make use of late operations Montague Grammar Quantifying In Tree Adjoining Grammars co-substitution flexible composition Minimalist Grammars Late adjunction hypothetical reasoning Greg & Jens (Chicago & Bielefeld) Derivational order and ACGs 9 / 51

13 Montague Grammar (MG) MG has order sensitive rules in the form of Quantifying In (QI), namely, S14: Rules of quantification S14. If a 2 P T and f 2 P t, then F 10, n(a, f) 2 P t, where either (i) a does not have the form he k, and F 10, n(a, f) comes from f by replacing the first occurrence of he n or him n by a and all other occurrences of he n or < he = < him = him n by she : ; or her respectively, according as the gender of the : ; it it 8 9 < masc: = first B CN or B T in a is fem: : ;, or neuter (ii) a ˆ he k, and F 10, n(a, f) comes from f by replacing all occurrences of he n or him n by he k or him k respectively. S15. If a 2 P T and z 2 P CN, then F 10, n(a, z) 2 P CN. S16. If a 2 P T and d 2 P IV, then F 10, n(a, d) 2 P IV. Greg & Jens (Chicago & Bielefeld) Derivational order and ACGs ACG@10 10 / 51

14 Montague Grammar (MG) MG has order sensitive rules in the form of Quantifying In (QI), namely, S14: QI has some complications we do not need to worry about right now: multiple occurrences of he n can be affected by rule F 10,n QI is order sensitive in the sense that if we have a derivation d = C[F 10,i (α, D[F 10,j (β, φ)])], then α takes semantic scope over β Greg & Jens (Chicago & Bielefeld) Derivational order and ACGs ACG@10 10 / 51

15 An Example - De Re vs De Dicto Readings John seeks a unicorn. De Re There is a particular unicorn that John is looking for. y[unicorn(y) & try(find(y))(j)] De Dicto John will be satisfied with any unicorn. try( y[unicorn(y) & find(y)])(j) x seeks y Can be represented in terms of operator scope: try(find(y))(x) Greg & Jens (Chicago & Bielefeld) Derivational order and ACGs ACG@10 11 / 51

16 Derivations for De Re vs De Dicto Readings John seeks a unicorn, 4 John seek a unicorn, 5 seek a unicorn, 2 John seeks a unicorn, 10, 0 unicorn a unicorn, 2 unicorn John seeks him 0, 4 John seek him 0, 5 seek he 0 Greg & Jens (Chicago & Bielefeld) Derivational order and ACGs ACG@10 12 / 51

17 An ACG implementation (following de Groote 2001) A faithful ACG implementation of the derivation structure of MG might look as follows: to each n-ary predicate (seek, love,... ) we associate a function symbol of type np n s to each common noun (boy, unicorn,... ) we associate a function symbol of type n to each determiner (some, every,... ) we associate a function symbol of type n d we have a function symbol QI of type ( np s ) ( d s ) This allows us to have abstract terms like the following: QI(λy np.qi(λx np.seek(x)(y))(some(unicorn)))(every(boy)) Greg & Jens (Chicago & Bielefeld) Derivational order and ACGs ACG@10 13 / 51

18 An ACG implementation The lexica are as follows: on the concrete phonology side, the symbol QI is interpreted as the (forwards application) combinator λxy.x(y) (and everything else is interpreted in the obvious way, given that all abstract types n, d, np and s are interpreted as the object type o o, the type of strings). on the concrete semantic side, the symbol QI is interpreted as the (backwards application) combinator λxy.y(x). Greg & Jens (Chicago & Bielefeld) Derivational order and ACGs ACG@10 14 / 51

19 An ACG implementation (alternative) An alternative implementation substitutes for the type d the type (np s) s, and eliminates the symbol QI. Then we have abstract terms like: every(boy)(λy np.some(unicorn)(λx np.seek(x)(y))) Under this alternative, the lexica are as follows: The concrete phonological interpretation of a determiner every is as the function λxf.f(every x) The concrete semantic interpretation of a determiner every is as the function λxf. (x)(f) Greg & Jens (Chicago & Bielefeld) Derivational order and ACGs ACG@10 15 / 51

20 Summary The ACG implementation does currently not deal with the ignored complexities of QI. The straightforward implementation takes the abstract terms to be pretty much isomorphic to the concrete semantic terms. The big picture (that it admittedly seems is not very clear in this presentation) is that: higher order constants mark scope positions, and apply to lambda abstracts which make the variables in lower positions accessible. In other words, timing requires of an expression that it simultaneously be connected to both an underlying (argument) position, and a surface (timing) position. Greg & Jens (Chicago & Bielefeld) Derivational order and ACGs ACG@10 16 / 51

21 Tree Adjoining Grammars Strongly equivalent to monadic linear CFTGs (Fujiyoshi & Kasai; Mönnich; Kepser & Rogers) Weakly equivalent to 2-MCFL wn (Vijay-Shanker et al.; Seki et al.; Kanazawa) A TAG consists of initial trees a finite set of trees with leaves labelled by either terminals or X, where X is a nonterminal symbol auxiliary trees as before, but with exactly one leaf node labelled by X, where X is the label of the root Greg & Jens (Chicago & Bielefeld) Derivational order and ACGs ACG@10 17 / 51

22 Tree Adjoining Grammars - Substitution Substitute (1st Order Substitution) DP + S = S John DP VP DP VP left John left unction: Interior nodes whose labels match Greg & Jens (Chicago & Bielefeld) Derivational order and ACGs ACG@10 18 / 51

23 Tree Adjoining Grammars - Adjunction Adjoin (2nd Order Substitution) S + VP = S DP VP quietly VP* DP VP John left John quietly VP left This is the familiar TAG story, simple and elegant Greg & Jens (Chicago & Bielefeld) Derivational order and ACGs ACG@10 19 / 51

24 Barker s Innovation: Unrestricted Derivation and Scope Fact Linear CFTGs derive the same tree languages under all (IO, OI, unrestricted) derivation modes. Barker: Γ scopes over iff Γ was substituted in after a tree containing was Greg & Jens (Chicago & Bielefeld) Derivational order and ACGs ACG@10 20 / 51

25 Linear CFTGs Definition (repeated) G = N, T, P, s a linear CFTG N the ranked alphabet of nonterminals T the ranked alphabet of terminals s the start symbol of rank 0 from N P the production rules each of which of the form a(x 1,..., x n ) t a N such that rank(a) = n x 1,..., x n variables of rank 0 t a linear tree over N T {x 1,..., x n } For expository reasons we assume that the start symbol s N does not appear on the righthand side of any rule Greg & Jens (Chicago & Bielefeld) Derivational order and ACGs ACG@10 21 / 51

26 Linear CFTGs as ACGs Composition as second-order substitution (de Groote and Pogodalla 2004) G = N, T, P, s a linear CFTG Greg & Jens (Chicago & Bielefeld) Derivational order and ACGs ACG@10 22 / 51

27 Linear CFTGs as ACGs Composition as second-order substitution (de Groote and Pogodalla 2004) G = N, T, P, s a linear CFTG G G = Σ G, Σ T, L G, s associated 2nd order ACG Σ G = N, { p p P }, τ higher-order linear signature τ(p) = a 1 a m a p P with skeleton a, a 1 a m Greg & Jens (Chicago & Bielefeld) Derivational order and ACGs ACG@10 22 / 51

28 Linear CFTGs as ACGs Composition as second-order substitution (de Groote and Pogodalla 2004) G = N, T, P, s a linear CFTG G G = Σ G, Σ T, L G, s associated 2nd order ACG Σ G = N, { p p P }, τ higher-order linear signature τ(p) = a 1 a m a p P with skeleton a, a 1 a m For p = a(x 1,..., x n ) t P with skeleton a, a 1 a m by induction, linear λ-term p = λy 1... y m. λx 1... x n. t Greg & Jens (Chicago & Bielefeld) Derivational order and ACGs ACG@10 22 / 51

29 Linear CFTGs as ACGs Composition as second-order substitution (de Groote and Pogodalla 2004) G = N, T, P, s a linear CFTG G G = Σ G, Σ T, L G, s associated 2nd order ACG Σ G = N, { p p P }, τ higher-order linear signature τ(p) = a 1 a m a p P with skeleton a, a 1 a m For p = a(x 1,..., x n ) t P with skeleton a, a 1 a m by induction, linear λ-term p = λy 1... y m. λx 1... x n. t x i = x i f( ) = λx. ( f x ) f T f(t 1,..., t k ) = t t k a j( ) = y j a j from skeleton a j(t 1,..., t k ) = y j t 1 t k Greg & Jens (Chicago & Bielefeld) Derivational order and ACGs ACG@10 22 / 51

30 Linear CFTGs as ACGs Composition as second-order substitution (de Groote and Pogodalla 2004) G = N, T, P, s a linear CFTG G G = Σ G, Σ T, L G, s associated 2nd order ACG Σ G = N, { p p P }, τ higher-order linear signature τ(p) = a 1 a m a p P with skeleton a, a 1 a m For p = a(x 1,..., x n ) t P with skeleton a, a 1 a m by induction, linear λ-term p = λy 1... y m. λx 1... x n. t L G Σ G Σ T L G (a) = string rank(a) string for a N L G (p) = p for p P Greg & Jens (Chicago & Bielefeld) Derivational order and ACGs ACG@10 22 / 51

31 Linear CFTGs as ACGs Composition as second-order substitution (de Groote and Pogodalla 2004) G = N, T, P, s a linear CFTG G G = Σ G, Σ T, L G, s associated 2nd order ACG Σ G = N, { p p P }, τ higher-order linear signature τ(p) = a 1 a m a p P with skeleton a, a 1 a m For p = a(x 1,..., x n ) t P with skeleton a, a 1 a m by induction, linear λ-term p = λy 1... y m. λx 1... x n. t L G Σ G Σ T L G (a) = string rank(a) string for a N L G (p) = p for p P Σ T = { o }, T, τ higher-order linear signature τ(f) = string = o o f T Greg & Jens (Chicago & Bielefeld) Derivational order and ACGs ACG@10 22 / 51

32 Linear CFTGs as ACGs Composition as second-order substitution (de Groote and Pogodalla 2004) G = N, T, P, s a linear CFTG G G = Σ G, Σ T, L G, s associated 2nd order ACG Σ G = N, { p p P }, τ higher-order linear signature τ(p) = a 1 a m a Greg & Jens (Chicago & Bielefeld) Derivational order and ACGs ACG@10 23 / 51

33 Linear CFTGs as ACGs Composition as third-order substitution G = N, T, P, s a linear CFTG (including derivation order) G G = Σ G, Σ T, L G, s associated 2nd order ACG Σ G = N, { p p P }, τ higher-order linear signature τ(p) = a 1 a m a G G = Σ G, Σ T, L G, s 3rd order ACG resulting from G G Greg & Jens (Chicago & Bielefeld) Derivational order and ACGs ACG@10 24 / 51

34 Linear CFTGs as ACGs Composition as third-order substitution G = N, T, P, s a linear CFTG (including derivation order) G G = Σ G, Σ T, L G, s associated 2nd order ACG Σ G = N, { p p P }, τ higher-order linear signature τ(p) = a 1 a m a G G = Σ G, Σ T, L G, s 3rd order ACG resulting from G G Σ G = N, { p p P }, τ higher-order linear signature τ (p) = trans(τ(p)) = trans(a 1 a m a) trans(s) = s s the start symbol trans(a) = (a s) s a N {s} trans( α β ) = α trans( β ) Greg & Jens (Chicago & Bielefeld) Derivational order and ACGs ACG@10 24 / 51

35 Example grammar CFTG p 1 = S f( T (c), G(c) ) p 2 = T (x) f( T (x), G(c) ) p 3 = G(x) h(x) p 4 = T (x) G(x) 2nd order ACG p 1 T G S p 2 T G T p 3 G p 4 G T Greg & Jens (Chicago & Bielefeld) Derivational order and ACGs ACG@10 25 / 51

36 Example grammar CFTG p 1 = S f( T (c), G(c) ) p 2 = T (x) f( T (x), G(c) ) p 3 = G(x) h(x) p 4 = T (x) G(x) 2nd order ACG p 1 T G S p 2 T G T p 3 G p 4 G T 3rd order ACG p 1 T G S p 2 T G ( T S ) S p 3 ( G S ) S p 4 G ( T S ) S Greg & Jens (Chicago & Bielefeld) Derivational order and ACGs ACG@10 25 / 51

37 Example derivation p 1 p 4 p 3 p 3 left-to-right p 1 T G S (var, app) t 1 T p 1 t 1 G S (var, app) p 4 G (T S) S g 1 G, t 1 T p 1 t 1 g 1 S (var, app) (abs) g 2 G p 4 g 2 (T S) S g 1 G λt 1. p 1 t 1 g 1 T S (app) g 1 G, g 2 G p 4 g 2 (λt 1. p 1 t 1 g 1 ) S (abs) p 3 (G S) S g 2 G λg 1. p 4 g 2 (λt 1. p 1 t 1 g 1 ) G S (app) g 2 G p 3 (λg 1. p 4 g 2 (λt 1. p 1 t 1 g 1 )) S (abs) p 3 (G S) S λg 2. p 3 (λg 1. p 4 g 2 (λt 1. p 1 t 1 g 1 )) G S (app) p 3 (λg 2. p 3 (λg 1. p 4 g 2 (λt 1. p 1 t 1 g 1 ))) S Greg & Jens (Chicago & Bielefeld) Derivational order and ACGs ACG@10 26 / 51

38 Example derivation p 1 p 4 p 3 p 3 left-to-right p 1 T G S (var, app) t 1 T p 1 t 1 G S (var, app) p 4 G (T S) S g 1 G, t 1 T p 1 t 1 g 1 S (var, app) (abs) g 2 G p 4 g 2 (T S) S g 1 G λt 1. p 1 t 1 g 1 T S (app) g 1 G, g 2 G p 4 g 2 (λt 1. p 1 t 1 g 1 ) S (abs) p 3 (G S) S g 2 G λg 1. p 4 g 2 (λt 1. p 1 t 1 g 1 ) G S (app) g 2 G p 3 (λg 1. p 4 g 2 (λt 1. p 1 t 1 g 1 )) S p 3 (G S) S λg 2. p 3 (λg 1. p 4 g 2 (λt 1. p 1 t 1 g 1 )) G S (abs) p 3 (app) p 3 (λg 2. p 3 (λg 1. p 4 g 2 (λt 1. p 1 t 1 g 1 ))) S λ p 3 λ p 4 g 2 λ p 1 t 1 g 1 Greg & Jens (Chicago & Bielefeld) Derivational order and ACGs ACG@10 26 / 51

39 Example derivation p 1 p 3 p 4 p 3 right-to-left p 1 T G S (var, app) t 1 T p 1 t 1 G S (var, app) g 1 G, t 1 T p 1 t 1 g 1 S (abs) p 3 (G S) S t 1 T λg 1. p 1 t 1 g 1 G S (var, app) p 4 G (T S) S t 1 T p 3 (λg 1. p 1 t 1 g 1 ) S (var, app) (abs) g 2 G p 4 g 2 (T S) S λt 1. p 3 (λg 1. p 1 t 1 g 1 ) T S (app) g 2 G p 4 g 2 (λt 1. p 3 (λg 1. p 1 t 1 g 1 )) S (abs) p 3 (G S) S λg 2. p 4 g 2 (λt 1. p 3 (λg 1. p 1 t 1 g 1 )) G S (app) p 3 (λg 2. p 4 g 2 (λt 1. p 3 (λg 1. p 1 t 1 g 1 ))) S Greg & Jens (Chicago & Bielefeld) Derivational order and ACGs ACG@10 27 / 51

40 Example derivation p 1 p 3 p 4 p 3 right-to-left p 1 T G S (var, app) t 1 T p 1 t 1 G S (var, app) g 1 G, t 1 T p 1 t 1 g 1 S (abs) p 3 (G S) S t 1 T λg 1. p 1 t 1 g 1 G S (var, app) p 4 G (T S) S t 1 T p 3 (λg 1. p 1 t 1 g 1 ) S (var, app) (abs) g 2 G p 4 g 2 (T S) S λt 1. p 3 (λg 1. p 1 t 1 g 1 ) T S (app) g 2 G p 4 g 2 (λt 1. p 3 (λg 1. p 1 t 1 g 1 )) S p 3 (G S) S λg 2. p 4 g 2 (λt 1. p 3 (λg 1. p 1 t 1 g 1 )) G S (abs) p 3 (app) p 3 (λg 2. p 4 g 2 (λt 1. p 3 (λg 1. p 1 t 1 g 1 ))) S λ p 4 g 2 λ p 3 λ p 1 t 1 g 1 Greg & Jens (Chicago & Bielefeld) Derivational order and ACGs ACG@10 27 / 51

41 Back towards the object language The transformation g which identifies equivalent derivations: p p λ y 1... y m f. f ( p y 1 y m ) if p P is a non-lifted rule otherwise This actually maps a derivation with order information (qua third order lambda term) to a derivation tree, i.e. a representation which keeps track only of which rule was used to rewrite which nonterminal, and not the relative order in which the rules were used. Greg & Jens (Chicago & Bielefeld) Derivational order and ACGs ACG@10 28 / 51

42 Example The transformation g which identifies equivalent derivations: p p λ y 1... y m f. f ( p y 1 y m ) if p P is a non-lifted rule otherwise p 3 ( λg 2. p 3 ( λg 1. p 4 g 2 ( λt 1. p 1 t 1 g 1 ) ) ) left-to-right and p 3 ( λg 2. p 4 g 2 ( λt 1. p 3 ( λg 1. p 1 t 1 g 1 ) ) ) right-to-left both are identified as p 1 ( p 4 p 3 ) p 3 transformation under g Greg & Jens (Chicago & Bielefeld) Derivational order and ACGs ACG@10 29 / 51

43 The object language The object language of the new ACG is gotten by composing the map g which turns a new abstract term into the old IO one, and the original lexicon. (Note that this simply reuses the original ACG s lexicon the object language of the new ACG is obtained by first translating it back into the old ACG, and then interpreting the old ACG as usual.) Greg & Jens (Chicago & Bielefeld) Derivational order and ACGs ACG@10 30 / 51

44 The object language We need to prove that the translation trans over types engenders an ACG whose abstract language exactly corresponds to the derivations of the original linear context-free tree grammar. One problem: the notion of derivation is not rich enough. We need to enrich it with information about which nonterminal is being rewritten. One way to do this is to redefine the notion of derivation: (T Σ P N) T Σ That is, the one step derivation relation is between a tree (a sentential form), a rule, a natural number, and another tree: C[A(t 1,..., t k )] p n C[ t [ t 1,..., t k ] ] if there are exactly n 1 nonterminal symbols to the left of the bullet in C[ ], and if there is some rule p = A(x 1,..., x k ) t. Greg & Jens (Chicago & Bielefeld) Derivational order and ACGs ACG@10 31 / 51

45 The object language Now we want to say that each type derivation of an abstract term (a term in the abstract language) can be put into correspondence with a unique derivation in the CFTG, and what properties this correspondence should have. We will be interested in canonical (β-reduced, but η-long) terms. That is, in terms where all argument positions are explicitly filled, but where all applications have been computed. Note that the abstract language is a subset of the following: L = δ Y 1 Y m ρ Y 1 Y m λy.l where δ is a rule rewriting the start symbol, and ρ is not. Greg & Jens (Chicago & Bielefeld) Derivational order and ACGs ACG@10 32 / 51

46 The object language To define a mapping from such terms to derivations in the underlying CFTG, we first need to be able to turn a term into a sentential form. We do this in the following way: For each nonterminal T in the CFTG, there is a type T in the ACG. We introduce a new zero-place constant T of type T for each nonterminal T. We define the lexicon L to be the extension of L which maps each T to L (T ) = λx 1... x rank(t ). T x 1 x rank(t ) For M Λ(Σ G ) of atomic type with F V (M) = {y 1,..., y m } we define M = L ((λy 1,..., y m. M) T 1 T m ), where y i T i. Now M is simply the sentential form which M represents. Greg & Jens (Chicago & Bielefeld) Derivational order and ACGs ACG@10 33 / 51

47 The object language We can define the translation from terms to derivations after introducing the following notation: Let λx. M denote the number of free variables which occur to the left of the first occurrence of x in M. Note, this definition treats the λ-term as a syntactic object it distinguishes between terms which are β,η-equivalent. This is why we are working with the unique representative of a β,η-equivalence class which is β-reduced and η-long. Formally, define for any set B, any a being a constant or variable, and any lambda terms M and N: str(b, a) = { ε if a B a otherwise str(b, λx. M) = str(b {x}, M) str(b, (MN)) = str(b, M) str(b, N) Then set λx. M = u, where str(c, M[x ]) = u w Greg & Jens (Chicago & Bielefeld) Derivational order and ACGs ACG@10 34 / 51

48 The object language The translation can be given as M = S δ 1 M if M = δ y 1 y m N ρ n+1 M if M = ρ y 1 y m λy.n and if n = λy.n Claim For every term M of the abstract language A(G G ), M is a derivation in G of M. Greg & Jens (Chicago & Bielefeld) Derivational order and ACGs ACG@10 35 / 51

49 Translating back to ACGs S δ 1 D 1 = D ( λ y. δ y ) φ ρ n+1 D ( M ) = D ( λ x u v. ρ x λ z. ( M u z v)) such that n = λ z. ( M u z v ) φ ( M ) = M Claim For every complete derivation D of t, D 1 = t. Moreover, 1 is the inverse of. Greg & Jens (Chicago & Bielefeld) Derivational order and ACGs ACG@10 36 / 51

50 Third order lambda terms really are nicer dequotienting of equivalent derivations implicit in the use of derivation trees but easy to use because substitution is managed by β-reduction canonical terms correspond exactly to derivations Greg & Jens (Chicago & Bielefeld) Derivational order and ACGs 37 / 51

51 Barker s Analysis (qua CFTG) Some person left. S s(dp, v) DP d(det, NP ) NP n Some person from Jamaica left. DP d(det, NP (NP )) NP (x) NP (np(x, pp(p, DP ))) NP (x) x DP name Greg & Jens (Chicago & Bielefeld) Derivational order and ACGs ACG@10 38 / 51

52 A Derivation Some person from every city left. S s(dp, v) s(d(det, NP (NP )), v) s(d(det, NP (n)), v) s(d(det, NP (np(n, pp(p, DP )))), v) s(d(det, np(n, pp(p, DP ))), v) s(d(det, np(n, pp(p, d(det, NP (NP ))))), v) s(d(det, np(n, pp(p, d(det, NP )))), v) s(d(det, np(n, pp(p, d(det, n)))), v) Greg & Jens (Chicago & Bielefeld) Derivational order and ACGs ACG@10 39 / 51

53 A Derivation Some person from every city left. S s(dp, v) s(d(det, NP (NP )), v) s(d(det, NP (n)), v) s(d(det, NP (np(n, pp(p, DP )))), v) s(d(det, np(n, pp(p, DP ))), v) s(d(det, np(n, pp(p, d(det, NP (NP ))))), v) s(d(det, np(n, pp(p, d(det, NP )))), v) s(d(det, np(n, pp(p, d(det, n)))), v) Greg & Jens (Chicago & Bielefeld) Derivational order and ACGs ACG@10 39 / 51

54 A Derivation Some person from every city left. S s(dp, v) s(d(det, NP (NP )), v) s(d(det, NP (n)), v) s(d(det, NP (np(n, pp(p, DP )))), v) s(d(det, np(n, pp(p, DP ))), v) s(d(det, np(n, pp(p, d(det, NP (NP ))))), v) s(d(det, np(n, pp(p, d(det, NP )))), v) s(d(det, np(n, pp(p, d(det, n)))), v) Greg & Jens (Chicago & Bielefeld) Derivational order and ACGs ACG@10 39 / 51

55 A Derivation Some person from every city left. S s(dp, v) s(d(det, NP (NP )), v) s(d(det, NP (n)), v) s(d(det, NP (np(n, pp(p, DP )))), v) s(d(det, np(n, pp(p, DP ))), v) s(d(det, np(n, pp(p, d(det, NP (NP ))))), v) s(d(det, np(n, pp(p, d(det, NP )))), v) s(d(det, np(n, pp(p, d(det, n)))), v) Greg & Jens (Chicago & Bielefeld) Derivational order and ACGs ACG@10 39 / 51

56 A Derivation Some person from every city left. S s(dp, v) s(d(det, NP (NP )), v) s(d(det, NP (n)), v) s(d(det, NP (np(n, pp(p, DP )))), v) s(d(det, np(n, pp(p, DP ))), v) s(d(det, np(n, pp(p, d(det, NP (NP ))))), v) s(d(det, np(n, pp(p, d(det, NP )))), v) s(d(det, np(n, pp(p, d(det, n)))), v) Greg & Jens (Chicago & Bielefeld) Derivational order and ACGs ACG@10 39 / 51

57 A Derivation Some person from every city left. S s(dp, v) s(d(det, NP (NP )), v) s(d(det, NP (n)), v) s(d(det, NP (np(n, pp(p, DP )))), v) s(d(det, np(n, pp(p, DP ))), v) s(d(det, np(n, pp(p, d(det, NP (NP ))))), v) s(d(det, np(n, pp(p, d(det, NP )))), v) s(d(det, np(n, pp(p, d(det, n)))), v) Greg & Jens (Chicago & Bielefeld) Derivational order and ACGs ACG@10 39 / 51

58 A Derivation Some person from every city left. S s(dp, v) s(d(det, NP (NP )), v) s(d(det, NP (n)), v) s(d(det, NP (np(n, pp(p, DP )))), v) s(d(det, np(n, pp(p, DP ))), v) s(d(det, np(n, pp(p, d(det, NP (NP ))))), v) s(d(det, np(n, pp(p, d(det, NP )))), v) s(d(det, np(n, pp(p, d(det, n)))), v) Greg & Jens (Chicago & Bielefeld) Derivational order and ACGs ACG@10 39 / 51

59 A Derivation Some person from every city left. S s(dp, v) s(d(det, NP (NP )), v) s(d(det, NP (n)), v) s(d(det, NP (np(n, pp(p, DP )))), v) s(d(det, np(n, pp(p, DP ))), v) s(d(det, np(n, pp(p, d(det, NP (NP ))))), v) s(d(det, np(n, pp(p, d(det, NP )))), v) s(d(det, np(n, pp(p, d(det, n)))), v) Greg & Jens (Chicago & Bielefeld) Derivational order and ACGs ACG@10 39 / 51

60 The abstract language of Barker s TAG Here are the rules, and their second-order types: S s(dp, v) (ρ S ) DP d(det, NP (NP )) (ρ DP ) NP n (ρ N ) NP (x) NP (np(x, pp(p, DP ))) (ρ NP a ) NP (x) x (ρ NP b ) Greg & Jens (Chicago & Bielefeld) Derivational order and ACGs ACG@10 40 / 51

61 The abstract language of Barker s TAG Here are the rules, and their second-order types: S s(dp, v) DP S (ρ S ) DP d(det, NP (NP )) (ρ DP ) NP n (ρ N ) NP (x) NP (np(x, pp(p, DP ))) (ρ NP a ) NP (x) x (ρ NP b ) Greg & Jens (Chicago & Bielefeld) Derivational order and ACGs ACG@10 40 / 51

62 The abstract language of Barker s TAG Here are the rules, and their second-order types: S s(dp, v) DP S (ρ S ) DP d(det, NP (NP )) NP NP DP (ρ DP ) NP n (ρ N ) NP (x) NP (np(x, pp(p, DP ))) (ρ NP a ) NP (x) x (ρ NP b ) Greg & Jens (Chicago & Bielefeld) Derivational order and ACGs ACG@10 40 / 51

63 The abstract language of Barker s TAG Here are the rules, and their second-order types: S s(dp, v) DP S (ρ S ) DP d(det, NP (NP )) NP NP DP (ρ DP ) NP n NP (ρ N ) NP (x) NP (np(x, pp(p, DP ))) (ρ NP a ) NP (x) x (ρ NP b ) Greg & Jens (Chicago & Bielefeld) Derivational order and ACGs ACG@10 40 / 51

64 The abstract language of Barker s TAG Here are the rules, and their second-order types: S s(dp, v) DP S (ρ S ) DP d(det, NP (NP )) NP NP DP (ρ DP ) NP n NP (ρ N ) NP (x) NP (np(x, pp(p, DP ))) NP DP NP (ρ NP a ) NP (x) x (ρ NP b ) Greg & Jens (Chicago & Bielefeld) Derivational order and ACGs ACG@10 40 / 51

65 The abstract language of Barker s TAG Here are the rules, and their second-order types: S s(dp, v) DP S (ρ S ) DP d(det, NP (NP )) NP NP DP (ρ DP ) NP n NP (ρ N ) NP (x) NP (np(x, pp(p, DP ))) NP DP NP (ρ NP a ) NP (x) x NP (ρ NP b ) Greg & Jens (Chicago & Bielefeld) Derivational order and ACGs ACG@10 40 / 51

66 Lifting Barker s TAG And here the types lifted over S: ρ S DP S ρ DP NP NP DP ρ N NP ρ NP a NP DP NP ρ NP b NP Greg & Jens (Chicago & Bielefeld) Derivational order and ACGs ACG@10 41 / 51

67 Lifting Barker s TAG And here the types lifted over S: ρ S DP S ρ DP NP NP DP ρ N NP ρ NP a NP DP NP ρ NP b NP Greg & Jens (Chicago & Bielefeld) Derivational order and ACGs ACG@10 41 / 51

68 Lifting Barker s TAG And here the types lifted over S: ρ S DP S ρ DP NP NP DP ρ N NP ρ NP a NP DP NP ρ NP b NP Greg & Jens (Chicago & Bielefeld) Derivational order and ACGs ACG@10 41 / 51

69 Lifting Barker s TAG And here the types lifted over S: ρ S DP S ρ DP NP NP (DP S) S ρ N NP ρ NP a NP DP NP ρ NP b NP Greg & Jens (Chicago & Bielefeld) Derivational order and ACGs ACG@10 41 / 51

70 Lifting Barker s TAG And here the types lifted over S: ρ S DP S ρ DP NP NP (DP S) S ρ N NP ρ NP a NP DP NP ρ NP b NP Greg & Jens (Chicago & Bielefeld) Derivational order and ACGs ACG@10 41 / 51

71 Lifting Barker s TAG And here the types lifted over S: ρ S DP S ρ DP NP NP (DP S) S ρ N (NP S) S ρ NP a NP DP NP ρ NP b NP Greg & Jens (Chicago & Bielefeld) Derivational order and ACGs ACG@10 41 / 51

72 Lifting Barker s TAG And here the types lifted over S: ρ S DP S ρ DP NP NP (DP S) S ρ N (NP S) S ρ NP a NP DP NP ρ NP b NP Greg & Jens (Chicago & Bielefeld) Derivational order and ACGs ACG@10 41 / 51

73 Lifting Barker s TAG And here the types lifted over S: ρ S DP S ρ DP NP NP (DP S) S ρ N (NP S) S ρ NP a NP DP (NP S) S ρ NP b NP Greg & Jens (Chicago & Bielefeld) Derivational order and ACGs ACG@10 41 / 51

74 Lifting Barker s TAG And here the types lifted over S: ρ S DP S ρ DP NP NP (DP S) S ρ N (NP S) S ρ NP a NP DP (NP S) S ρ NP b NP Greg & Jens (Chicago & Bielefeld) Derivational order and ACGs ACG@10 41 / 51

75 Lifting Barker s TAG And here the types lifted over S: ρ S DP S ρ DP NP NP (DP S) S ρ N (NP S) S ρ NP a NP DP (NP S) S ρ NP b (NP S) S Greg & Jens (Chicago & Bielefeld) Derivational order and ACGs ACG@10 41 / 51

76 Derivations Some person left basic (2nd order): ρ S ( ρ DP ( ρ NP b ρ NP ) ) lifted (3rd order): ρ N ( λx N. ρ NP b ( λx NP. ρ DP ( x NP x N λx D. ρ S x D ) ) ) Some person from every city left basic: ρ S ( ρ DP ( ρ NP a ( ρ NP b ρ DP ( ρ NP b ρ NP ) ) ρ NP ) ) lifted p.ρ NP b (λx np.ρ DP (x np x np (λx d.ρ NP b (λx np.ρ NP a (x np x d (λx np.ρ N (λx np.ρ DP (x np x np (λx d.ρ S x d ))) Greg & Jens (Chicago & Bielefeld) Derivational order and ACGs ACG@10 42 / 51

77 The Problem of Inverse Linking Observation: derivation order does not permit surface scope in: every person from a city left. Greg & Jens (Chicago & Bielefeld) Derivational order and ACGs ACG@10 43 / 51

78 The Problem of Inverse Linking Observation: derivation order does not permit surface scope in: every person from a city left. The problem: the most deeply embedded quantifier cannot be introduced until after its argument position is present, which is introduced only once its containing DP is present Greg & Jens (Chicago & Bielefeld) Derivational order and ACGs ACG@10 43 / 51

79 The Problem of Inverse Linking Observation: derivation order does not permit surface scope in: every person from a city left. The problem: the most deeply embedded quantifier cannot be introduced until after its argument position is present, which is introduced only once its containing DP is present Example Every derivation begins: S s(dp, v) 1 = λx D.ρ S (x D ) Greg & Jens (Chicago & Bielefeld) Derivational order and ACGs ACG@10 43 / 51

80 The Problem of Inverse Linking Observation: derivation order does not permit surface scope in: every person from a city left. The problem: the most deeply embedded quantifier cannot be introduced until after its argument position is present, which is introduced only once its containing DP is present Example Every derivation begins: S s(dp, v) s(d(det, NP (NP )), v) 1 = λx NP x N. ρ DP ( x NP x N λx D.ρ S (x D ) ) Greg & Jens (Chicago & Bielefeld) Derivational order and ACGs ACG@10 43 / 51

81 Constituency in derivations A timing based approach to quantifier scope simply cannot derive surface scope relations between embedded scope-takers: the scope relations predicted are always inverted. ρ S ρ DP ρ NP a ρdp ρnp ρ NPb ρ NP b ρ NP The possible derivations for t are given by h(t), where h(σ(t 1,..., t n ) = σ {h(t 1 ),..., h(t n )} ρ S (ρ DP (ρ NP (ρ NP a (ρ NP b (ρ DP (ρ NP b ρ NP )))))) Greg & Jens (Chicago & Bielefeld) Derivational order and ACGs ACG@10 44 / 51

82 Timing /= Derivation Barker wants scope relations such that ρ DP < ρ DP There is no derivation which provides this No system which derives this can be expressed in terms of derivations Whatever people mean by timing it is not describable in terms of rewriting order Greg & Jens (Chicago & Bielefeld) Derivational order and ACGs 45 / 51

83 Back to Barker We simply lifted all of our types because we wanted to look at every possible derivation order Barker only allows for cosubstitution of certain initial trees (the DPs) this corresponds to us only lifting types ending in DP the resulting signature has nothing to do with derivations Greg & Jens (Chicago & Bielefeld) Derivational order and ACGs 46 / 51

84 Selectively Lifting Barker s TAG Only types ending in DP : ρ S DP S ρ DP NP NP DP ρ N NP ρ NP a NP DP NP ρ NP b NP Greg & Jens (Chicago & Bielefeld) Derivational order and ACGs ACG@10 47 / 51

85 Selectively Lifting Barker s TAG Only types ending in DP : ρ S DP S ρ DP NP NP DP ρ N NP ρ NP a NP DP NP ρ NP b NP Greg & Jens (Chicago & Bielefeld) Derivational order and ACGs ACG@10 47 / 51

86 Selectively Lifting Barker s TAG Only types ending in DP : ρ S DP S ρ DP NP NP DP ρ N NP ρ NP a NP DP NP ρ NP b NP Greg & Jens (Chicago & Bielefeld) Derivational order and ACGs ACG@10 47 / 51

87 Selectively Lifting Barker s TAG Only types ending in DP : ρ S DP S ρ DP NP NP (DP S) S ρ N NP ρ NP a NP DP NP ρ NP b NP Greg & Jens (Chicago & Bielefeld) Derivational order and ACGs ACG@10 47 / 51

88 Selectively Lifting Barker s TAG Only types ending in DP : ρ S DP S ρ DP NP NP (DP S) S ρ N NP ρ NP a NP DP NP ρ NP b NP Greg & Jens (Chicago & Bielefeld) Derivational order and ACGs ACG@10 47 / 51

89 Selectively Lifting Barker s TAG Only types ending in DP : ρ S DP S ρ DP NP NP (DP S) S ρ N NP ρ NP a NP DP NP ρ NP b NP Greg & Jens (Chicago & Bielefeld) Derivational order and ACGs ACG@10 47 / 51

90 Selectively Lifting Barker s TAG Only types ending in DP : ρ S DP S ρ DP NP NP (DP S) S ρ N NP ρ NP a NP DP NP ρ NP b NP Greg & Jens (Chicago & Bielefeld) Derivational order and ACGs ACG@10 47 / 51

91 Selectively Lifting Barker s TAG Only types ending in DP : ρ S DP S ρ DP NP NP (DP S) S ρ N NP ρ NP a NP DP NP ρ NP b NP Greg & Jens (Chicago & Bielefeld) Derivational order and ACGs ACG@10 47 / 51

92 Selectively Lifting Barker s TAG Only types ending in DP : ρ S DP S ρ DP NP NP (DP S) S ρ N NP ρ NP a NP DP NP ρ NP b NP Greg & Jens (Chicago & Bielefeld) Derivational order and ACGs ACG@10 47 / 51

93 Selectively Lifting Barker s TAG Only types ending in DP : ρ S DP S ρ DP NP NP (DP S) S ρ N NP ρ NP a NP DP NP ρ NP b NP Greg & Jens (Chicago & Bielefeld) Derivational order and ACGs ACG@10 47 / 51

94 We still don t get the surface reading! but this time its easy to fix: Barker s solution allow a version of rule ρ NP a to directly select for a continuized DP : ρ NP a = NP ((DP S) S) NP A lexical solution introduce new atomic type XP, and duplicate rules as appropriate: XP d(det, NP (NP )) (ρ XP ) NP (x) NP (np(x, pp(p, XP ))) (ρ NP c ) lexically interpret L(XP ) = L((DP S) S) Greg & Jens (Chicago & Bielefeld) Derivational order and ACGs ACG@10 48 / 51

95 Conclusions Cotags leave the realm of context-free grammar formalisms WGC and SGC preservation NOT because IO and OI coincide on linear CFTGs instead, because spellout involves going back through the 2 nd order derivation term Complexity heuristics in ACG differ from those of TAG: simpler to deal with in-situ readings by making lexicon higher order than the abstract language... (??) Greg & Jens (Chicago & Bielefeld) Derivational order and ACGs ACG@10 49 / 51

96 Conclusions Timing intuitive and oft-occuring not related to rewriting strategies (we may have been the only ones to have thought this... ) best seen as moving from trees to (3 rd order) terms Greg & Jens (Chicago & Bielefeld) Derivational order and ACGs ACG@10 50 / 51

97 Thank you! Greg & Jens (Chicago & Bielefeld) Derivational order and ACGs 51 / 51