— Tree Adjoining Grammars —

An Introduction to

Mildly Context-Sensitive Grammar Formalisms

— Tree Adjoining Grammars —

Gerhard J¨ager & Jens Michaelis Universit¨at Potsdam

{jaeger,michael}@ling.uni-potsdam.de

Mild context-sensitivity (Joshi 1985)

¥¥¥ a concept motivated by the intention of characterizing a narrow class of formal grammars which are “only slightly more powerful than CFGs,” and which nevertheless allow for descriptions of

natural languages in a linguistically significant way.

According to Joshi (1985, p. 225) a mildly context-sensitive language, L, has to fulfil three criteria, to be understood as a

“rough characterization.” Somewhat paraphrased, these are:

(1) the parsing problem for L is solvable in polynomial time, (2) L has the constant growth property, and

(3) there is a finite upper bound for L limiting the number of different instantiations of factorized cross-serial

dependencies occurring in a sentence of L.

Mild context-sensitivity (Joshi 1985)

¥¥¥ a concept motivated by the intention of characterizing a narrow class of formal grammars which are “only slightly more powerful than CFGs,” and which nevertheless allow for descriptions of

natural languages in a linguistically significant way.

According to Joshi (1985, p. 225) a mildly context-sensitive language, L, has to fulfil three criteria, to be understood as a

“rough characterization.” Somewhat paraphrased, these are:

(1) the parsing problem for L is solvable in polynomial time, (2) L has the constant growth property, and

(3) there is a finite upper bound for L limiting the number of different instantiations of factorized cross-serial

dependencies occurring in a sentence of L.

Mild context-sensitivity (Joshi 1985)

¥¥¥ a concept motivated by the intention of characterizing a narrow class of formal grammars which are “only slightly more powerful than CFGs,” and which nevertheless allow for descriptions of

natural languages in a linguistically significant way.

According to Joshi (1985, p. 225) a mildly context-sensitive language, L, has to fulfil three criteria, to be understood as a

“rough characterization.” Somewhat paraphrased, these are:

(1) the parsing problem for L is solvable in polynomial time, (2) L has the constant growth property, and

(3) there is a finite upper bound for L limiting the number of different instantiations of factorized cross-serial

dependencies occurring in a sentence of L.

Mild context-sensitivity

¥ A collection of mildly context-sensitive grammar (MCSG) formalisms is presented in Joshi et al. 1991:

¨ tree adjoining grammars (TAGs) (Joshi et al. 1975; Joshi 1985)

¨ (restricted) combinatory categorial grammars (CCGs) (as formalized e.g. in Weir & Joshi 1988 in accordance with the CCG-version developed in Steedman 1987, 1990)

¨ linear indexed grammars (LIGs) as they arise from Gazdar 1988

¨ head grammars (HGs) (Pollard 1984)

¨ multicomponent TAGs (MCTAGs) (Joshi 1987; Vijay-Shanker et al. 1987) as a generalization of TAGs

Mild context-sensitivity

¥ TAGs, CCGs, LIGs and HGs are weakly equivalent (see e.g. Vijay–Shanker & Weir 1994)

a , b

¥ MCTAGs and LCFRSs are weakly equivalent (Weir 1988)

c

a The weak equivalence to LIGs, CCGs and HGs holds for TAGs with local constraints (on tree adjoining) as formally introduced e.g. in Vijay–Shanker & Joshi 1985, following a suggestion in

Joshi et al. 1975, and capturing the intended use of local constraints (on adjoining) of the kind proposed in Joshi 1985. The class of TAGs with local constraints properly extends the strong as well as the weak generative capacity of the class of TAGs without such constraints.

b Note also that HGs as defined e.g. in Vijay–Shanker & Weir 1994 provide a modified version of HGs as originally defined in Pollard 1984. In terms of weak equivalence, HGs of this modified type

subsume HGs of the original type, and vice versa. Corresponding proofs can be found in Vijay–Shanker et al. 1986 and Seki et al. 1991, respectively.

c More precisely, MCTAGs in their set-local variant, i.e. MCTAGs which, during the course of a derivation, allow the members of a derived sequence of auxiliary trees to be (simultaneously) adjoined at distinct nodes to the members of a single elementary tree sequence (cf. Definition 2.7.1 in Weir 1988).

Mild context-sensitivity

¥ TAGs, CCGs, LIGs and HGs are weakly equivalent (see e.g. Vijay–Shanker & Weir 1994) ^{a , b}

¥ MCTAGs and LCFRSs are weakly equivalent (Weir 1988) ^c

a The weak equivalence to LIGs, CCGs and HGs holds for TAGs with local constraints (on tree adjoining) as formally introduced e.g. in Vijay–Shanker & Joshi 1985, following a suggestion in

Joshi et al. 1975, and capturing the intended use of local constraints (on adjoining) of the kind proposed in Joshi 1985. The class of TAGs with local constraints properly extends the strong as well as the weak generative capacity of the class of TAGs without such constraints.

b Note also that HGs as defined e.g. in Vijay–Shanker & Weir 1994 provide a modified version of HGs as originally defined in Pollard 1984. In terms of weak equivalence, HGs of this modified type subsume HGs of the original type, and vice versa. Corresponding proofs can be found in

Vijay–Shanker et al. 1986 and Seki et al. 1991, respectively.

c More precisely, MCTAGs in their

Finite labeled trees

t = hN_t , /^∗_t , ≺_t , label_t i

¥¥¥ hN_t , /^∗_t , ≺_t i a finite ordered tree : N_t the finite set nodes

/^∗_t and ≺_t the binary relations of dominance and precedence on N_t , respectively

i.e., /^∗_t is the reflexive-transitive closure of /_t , the binary relation of immediate dominance on N_t

¥¥¥ label_t the labeling (function) , a function from N_t into a set of labels.

Finite labeled trees

t = hN_t , /^∗_t , ≺_t , label_t i

¥¥¥ h N_t , /^∗_t , ≺_t i a finite ordered tree : N_t the finite set nodes

/^∗_t and ≺_t the binary relations of dominance and precedence on N_t , respectively

i.e., /^∗_t is the reflexive-transitive closure of /_t , the binary relation of immediate dominance on N_t

¥¥¥ label_t the labeling (function) , a function from N_t into a set of labels.

Finite labeled trees

t = hN_t , /^∗_t , ≺_t , label_t i

¥¥¥ h N_t , /^∗_t , ≺_t i a finite ordered tree : N_t the finite set nodes

/^∗_t and ≺_t the binary relations of dominance and precedence on N_t , respectively

i.e., /^∗_t is the reflexive-transitive closure of /_t , the binary relation of immediate dominance on N_t

¥¥¥ label_t the labeling (function) , a function from N_t into a set of labels.

Finite labeled trees

t = hN_t , /^∗_t , ≺_t , label_t i

¥¥¥ h N_t , /^∗_t , ≺_t i a finite ordered tree : N_t the finite set nodes

/^∗_t and ≺_t the binary relations of dominance and precedence on N_t , respectively

i.e., /^∗_t is the reflexive-transitive closure of /_t , the binary relation of immediate dominance on N_t

¥¥¥ label_t the labeling (function) , a function from N_t into a set of labels.

Finite labeled trees

t = hN_t , /^∗_t , ≺_t , label_t i

¥¥¥ h N_t , /^∗_t , ≺_t i a finite ordered tree : N_t the finite set nodes

/^∗_t and ≺_t the binary relations of dominance and precedence on N_t , respectively

i.e., /^∗_t is the reflexive-transitive closure of /_t , the binary relation of immediate dominance on N_t

¥¥¥ label_t the labeling (function) , a function from N_t into a set of labels.

Finite labeled trees

t = hN_t , /^∗_t , ≺_t , label_t i

¥¥¥ h N_t , /^∗_t , ≺_t i a finite ordered tree : N_t the finite set nodes

/^∗_t and ≺_t the binary relations of dominance and precedence on N_t , respectively

i.e., /^∗_t is the reflexive-transitive closure of /_t , the binary relation of immediate dominance on N_t

Objects specified by a tree adjoining grammar

V_N a set of nonterminals V_N a set of nonterminals

V_T a set of terminals V_T a set of terminals

A

B D ↓ a

C

A ^FFF D

E ↓ B

b ε

t = hN_t , /^∗_t , ≺_t , label_t i

label_t : NonLeaves _t → V_N

Leaves _t → V_N {↓ ,^F } ∪ V_T ∪ {ε}

Objects specified by a tree adjoining grammar

V_N a set of nonterminals

V_N a set of nonterminals V_T a set of terminals

V_T a set of terminals A

B D ↓ a

C

A ^FFF D

E ↓ B

b ε

t = hN_t , /^∗_t , ≺_t , label_t i label_t : NonLeaves _t → V_N

Leaves _t → V_N {↓ ,^F } ∪ V_T ∪ {ε}

Objects specified by a tree adjoining grammar

V_N a set of nonterminals

V_N a set of nonterminals V_T a set of terminals

V_T a set of terminals

A B

D ↓ a

C A ^FFF

D E ↓

B

b ε

t = hN_t , /^∗_t , ≺_t , label_t i

label_t : NonLeaves_t → V_N Leaves _t → V_N {↓ ,^F } ∪ V_T ∪ {ε}

Objects specified by a tree adjoining grammar

V_N a set of nonterminals

V_N a set of nonterminals V_T a set of terminals

V_T a set of terminals

A

B D

↓ a

C A

FFF

D E

↓

B

b ε

t = hN_t , /^∗_t , ≺_t , label_t i

label_t : NonLeaves_t → V_N Leaves → V {↓ ,^F } ∪ V_T ∪ {ε}

Objects specified by a tree adjoining grammar

V_N a set of nonterminals

V_N a set of nonterminals V_T a set of terminals

V_T a set of terminals

A

B D↓

a

C

A^FFF D

E↓ B

b ε

t = hN_t , /^∗_t , ≺_t , label_t i

label_t : NonLeaves_t → V_N

Leaves_t → V_N {↓ ,^F }

∪ V_T ∪ {ε}

Objects specified by a tree adjoining grammar

V_N a set of nonterminals

V_N a set of nonterminals V_T a set of terminals

V_T a set of terminals

A

B D↓ a

C

A^FFF D

E↓ B

b ε

t = hN_t , /^∗_t , ≺_t , label_t i

label_t : NonLeaves_t → V_N

Leaves → V {↓ ,^F } ∪ V

∪ {ε}

Objects specified by a tree adjoining grammar

V_N a set of nonterminals

V_N a set of nonterminals V_T a set of terminals

V_T a set of terminals

A

B D↓ a

C

A^FFF D

E↓ B

b ε

t = hN_t , /^∗_t , ≺_t , label_t i

label_t : NonLeaves_t → V_N

Leaves_t → V_N {↓ ,^F } ∪ V_T ∪ {ε}

Deriving (labeled) trees by a tree adjoining grammar

¥¥¥ Labeled trees can be derived from others by applying the

structure building operators, namely, substitution and adjoining.

¥¥¥ Derivations start from initial trees.

A

w

A ∈ V_N

w ∈ Strings( V_N{↓} ∪ V_T )

¥¥¥ W.r.t. adjoining, auxiliary trees are of central importance.

A

w₁ A^FFF w₂ w₁ A^FFF w₂ w₁ A^FFF w₂

A ∈ V_N

w₁ , w₂ ∈ Strings( V_N{↓} ∪ V_T ) foot node spine

Deriving (labeled) trees by a tree adjoining grammar

¥¥¥ Labeled trees can be derived from others by applying the

structure building operators, namely, substitution and adjoining.

¥¥¥ Derivations start from initial trees.

A

w

A ∈ V_N

w ∈ Strings( V_N{↓} ∪ V_T )

¥¥¥ W.r.t. adjoining, auxiliary trees are of central importance.

A

w₁ A^FFF w₂ w₁ A^FFF w₂ w₁ A^FFF w₂

A ∈ V_N

w₁ , w₂ ∈ Strings( V_N{↓} ∪ V_T ) foot node spine

Deriving (labeled) trees by a tree adjoining grammar

¥¥¥ Labeled trees can be derived from others by applying the

structure building operators, namely, substitution and adjoining.

¥¥¥ Derivations start from initial trees.

A

w

A ∈ V_N

w ∈ Strings( V_N{↓} ∪ V_T )

¥¥¥ W.r.t. adjoining, auxiliary trees are of central importance.

A w₁ A^FFF w₂

w₁ A^FFF w₂

A ∈ V_N foot node spine

Deriving (labeled) trees by a tree adjoining grammar

¥¥¥ Labeled trees can be derived from others by applying the

structure building operators, namely, substitution and adjoining.

¥¥¥ Derivations start from initial trees.

A

w

A ∈ V_N

w ∈ Strings( V_N{↓} ∪ V_T )

¥¥¥ W.r.t. adjoining, auxiliary trees are of central importance.

A w₁ A^FFF w₂

w₁ A^FFF w₂ w₁ A^FFF w₂

A ∈ V_N

w₁ , w₂ ∈ Strings( V_N{↓} ∪ V_T ) foot node

spine

– p.7

Deriving (labeled) trees by a tree adjoining grammar

¥¥¥ Labeled trees can be derived from others by applying the

structure building operators, namely, substitution and adjoining.

¥¥¥ Derivations start from initial trees.

A

w

A ∈ V_N

w ∈ Strings( V_N{↓} ∪ V_T )

¥¥¥ W.r.t. adjoining, auxiliary trees are of central importance.

A w₁ A^FFF w₂

w₁ A^FFF w₂

A ∈ V_N foot node

Structure building operators

substitution : Trees(V) × Trees(V) −→^part 2Trees(V)

hφ , ψi ∈ Domain(substitution) :⇐⇒

¥ φ has a leaf labeled A↓ for some A ∈ V_N

¥ ψ’s root is labeled A

Structure building operators

substitution : Trees(V) × Trees(V) −→^part 2Trees(V)

A↓

φ A

ψ

Ã

φ⁰

A ψ

Structure building operators

substitution : Trees(V) × Trees(V) −→^part 2Trees(V)

A↓

φ A

ψ

Ã

φ⁰

A ψ

substitution

S

NP↓ VP

V saw

NP↓

NP D↓ N

man

Ã

NP D↓ N

man

VP V

saw

NP↓

substitution

S

NP↓ VP

V saw

NP↓

NP D↓ N

man

Ã

NP D↓ N

man

VP V

saw

NP↓

substitution

S

NP↓ VP

V saw

NP↓

NP D↓ N

man

Ã

NP D↓ N

man

VP V

saw

NP↓

substitution

S

NP↓ VP

V saw

NP↓

NP D↓ N

man

Ã

NP D↓ N

man

VP V

saw

NP↓

Structure building operators

adjoining : Trees(V) × Trees(V) −→^part 2Trees(V)

hφ , ψi ∈ Domain(adjoining) :⇐⇒

¥ φ has a node labeled A for some A ∈ V_N

¥ ψ’s root is labeled A and ψ has a leaf labeled A^F

Structure building operators

adjoining : Trees(V) × Trees(V) −→^part 2Trees(V)

A

φ A ψ

A^FFF

Ã

A

A

Structure building operators

adjoining : Trees(V) × Trees(V) −→^part 2Trees(V)

A

φ A ψ

A^FFF

Ã

A

A

Structure building operators

adjoining : Trees(V) × Trees(V) −→^part 2Trees(V)

A

φ A ψ

A^FFF

Ã

A

A

adjoining

NP D↓ N

man

N Adj

tall

N^FFF

Ã

D↓ N

Adj tall

N man

adjoining

NP D↓ N

man

N Adj

tall

N^FFF

Ã

D↓ N

Adj tall

N man

adjoining

NP D↓ N

man

N Adj

tall

N^FFF

Ã

D↓ N

Adj tall

N man

adjoining

S

NP↓ VP

V saw

NP↓

S Adv yesterday

S^FFF

Ã

Adv yesterday

S

NP↓ VP

V saw

NP↓

adjoining

S

NP↓ VP

V saw

NP↓

S

Adv yesterday

S^FFF

Ã

Adv yesterday

S

NP↓ VP

V saw

NP↓

adjoining

S

NP↓ VP

V saw

NP↓

S

Adv yesterday

S^FFF

Ã

Adv yesterday

S

NP↓ VP

V saw

NP↓

– p.18

Tree adjoining grammars (TAGs)

G = h V_N , V_T , T_Ini , T_Aux , S i

¥¥¥ V_N a set of nonterminals

¥¥¥ V_T a set of terminals

¥¥¥ T_Ini a finite set of initial trees

¥¥¥ T_Aux a finite set of auxiliary trees

¥¥¥ S ∈ V_N a distinguished nonterminal ( the start symbol )

Tree adjoining grammars (TAGs)

G = h V_N , V_T , T_Ini , T_Aux , S i

¥¥¥ V_N a set of nonterminals

¥¥¥ V_T a set of terminals

¥¥¥ T_Ini a finite set of initial trees

¥¥¥ T_Aux a finite set of auxiliary trees

¥¥¥ S ∈ V_N a distinguished nonterminal ( the start symbol )

Tree adjoining grammars (TAGs)

G = h V_N , V_T , T_Ini , T_Aux , S i

¥¥¥ V_N a set of nonterminals

¥¥¥ V_T a set of terminals

¥¥¥ T_Ini a finite set of initial trees

¥¥¥ T_Aux a finite set of auxiliary trees

¥¥¥ S ∈ V_N a distinguished nonterminal ( the start symbol )

Tree adjoining grammars (TAGs)

G = h V_N , V_T , T_Ini , T_Aux , S i

¥¥¥ V_N a set of nonterminals

¥¥¥ V_T a set of terminals

¥¥¥ T_Ini a finite set of initial trees

¥¥¥ T_Aux a finite set of auxiliary trees

¥¥¥ S ∈ V_N a distinguished nonterminal ( the start symbol )

Tree adjoining grammars (TAGs)

G = h V_N , V_T , T_Ini , T_Aux , S i

¥¥¥ V_N a set of nonterminals

¥¥¥ V_T a set of terminals

¥¥¥ T_Ini a finite set of initial trees

¥¥¥ T_Aux a finite set of auxiliary trees

¥¥¥ S ∈ V_N a distinguished nonterminal ( the start symbol )

Tree adjoining grammars (TAGs)

G = h V_N , V_T , T_Ini , T_Aux , S i

¥¥¥ V_N a set of nonterminals

¥¥¥ V_T a set of terminals

¥¥¥ T_Ini a finite set of initial trees

¥¥¥ T_Aux a finite set of auxiliary trees

¥¥¥ S ∈ V_N a distinguished nonterminal ( the start symbol )

Elementary trees: initial vs. auxiliary

t ∈ T_Ini t is a finite labeled tree h N_t , /^∗_t , ≺_t , label_t i such that A

w

A ∈ V_N

w ∈ Strings( V_N{↓} ∪ V_T )

t ∈ T_Aux t is a finite labeled tree hN_t , /^∗_t , ≺_t , label_t i such that A

w₁ A^FFF w₂ w₁ A^FFF w₂ w₁ A^FFF w₂

A ∈ V_N

w₁ , w₂ ∈ Strings( V_N{↓} ∪ V_T ) foot node spine

Elementary trees: initial vs. auxiliary

t ∈ T_Ini t is a finite labeled tree h N_t , /^∗_t , ≺_t , label_t i such that A

w

A ∈ V_N

w ∈ Strings( V_N{↓} ∪ V_T ) t ∈ T_Aux t is a finite labeled tree hN_t , /^∗_t , ≺_t , label_t i such that

A

w₁ A^FFF w₂ w₁ A^FFF w₂ w₁ A^FFF w₂

A ∈ V_N

w₁ , w₂ ∈ Strings( V_N{↓} ∪ V_T ) foot node spine

Elementary trees: initial vs. auxiliary

t ∈ T_Ini t is a finite labeled tree h N_t , /^∗_t , ≺_t , label_t i such that A

w

A ∈ V_N

w ∈ Strings( V_N{↓} ∪ V_T )

t ∈ T_Aux t is a finite labeled tree h N_t , /^∗_t , ≺_t , label_t i such that A

w₁ A^FFF w₂ w₁ A^FFF w₂

A ∈ V_N foot node spine

Elementary trees: examples

(β_yest) S Adv yesterday

S^FFF

(α_a) D a

(α_man) NP D↓ N

man

(α_saw) S

NP↓ VP

V saw

NP↓

(α_Mary) NP N Mary

Tree adjoining languages

Closure(G) , the closure of a TAG G = h V_N , V_T , T_Ini , T_Aux , S i, is the closure of T_Ini ∪ T_Aux under finitely many applications

of substitution and adjoining . t ∈ Closure(G) is complete :⇐⇒

t’s root-label is S and yield(t) ∈ Strings(V_T) .

The tree and string language generated by G

T(G) = { t | t ∈ Closure(G) and complete } L(G) = { yield(t) | t ∈ T(G) }

Tree adjoining languages

Closure(G) , the closure of a TAG G = h V_N , V_T , T_Ini , T_Aux , S i, is the closure of T_Ini ∪ T_Aux under finitely many applications

of substitution and adjoining .

t ∈ Closure(G) is complete :⇐⇒

t’s root-label is S and yield(t) ∈ Strings(V_T) . The tree and string language generated by G

T(G) = { t | t ∈ Closure(G) and complete } L(G) = { yield(t) | t ∈ T(G) }

Tree adjoining languages

Closure(G) , the closure of a TAG G = h V_N , V_T , T_Ini , T_Aux , S i, is the closure of T_Ini ∪ T_Aux under finitely many applications

of substitution and adjoining .

t ∈ Closure(G) is complete :⇐⇒

t’s root-label is S and yield(t) ∈ Strings(V_T) . The tree and string language generated by G

Linguistic applications

¥ Wh-movement

¥ Verbclusters

Linguistic applications: elementary trees for ‘likes’

(α₁) S

NP↓ VP

V likes

NP↓

(α₂) S

NP(wh)↓_i S

NP↓ VP

V likes

NP_i ε

Linguistic applications: sample elementary trees

(β₁) S

NP↓ VP

V think

S^FFF

(β₂) S

V does

S^FFF

(α₃) NP(wh) who

(α₄) NP Harry

(α₅) NP Bill

Linguistic applications: sample derivation

(α₂) S

NP(wh)↓_i S

NP↓ VP

V likes

NP_i ε

(γ₁) S

NP(wh)_i who

S NP

Harry

VP V

likes

NP_i ε

(α₃) NP(wh) (α₄) NP

Linguistic applications: sample derivation

(α₂) S

NP(wh)↓_i S

NP↓ VP

V likes

NP_i ε

(γ₁) S

NP(wh)_i who

S NP

Harry

VP V

likes

NP_i ε

(α₃) NP(wh) who

(α₄) NP Harry

Linguistic applications: sample derivation

(β₁) S

NP↓ VP

V think

S^FFF

(β₂) S V does

S^FFF

(α₅) NP Bill (γ₂) S

V does

S NP

Bill

VP V

think

S^FFF

Linguistic applications: sample derivation

(β₁) S

NP↓ VP

V think

S^FFF

(β₂) S V does

S^FFF

(α₅) NP Bill

(γ₂) S V

does

S NP

Bill

VP V

think

S^FFF

Linguistic applications: sample derivation

(γ₁) S

NP(wh)_i who

S

NP Harry

VP V

likes

NP_i ε

(γ₂) S

V does

S NP

Bill

VP V

think

S^FFF

Linguistic applications: sample derivation

(γ3) S

NP(wh)i who

S

V does

S

NP Bill

VP

V think

S

NP Harry

VP V

likes

NPi ε

Linguistic applications: “verb clusters”

(α₇) S

S NP Jan

VP ε₃

V₃

zwemmen

(β₄) S

S NP

Piet

VP S^FFF ε₂

V₂ leren

(β₅) S

S NP

Marie

VP S^FFF ε₁

V₁ helpen

(β₆) S

NP Karel

VP S^FFF V₁

laat

Linguistic applications: “verb clusters”

(α₇) S

S NP Jan

VP ε₃

V₃

zwemmen

(β₄) S

S NP

Piet

VP S^FFF ε₂

V₂ leren

(β₅) S

S NP

Marie

VP S^FFF ε₁

V₁ helpen

(β₆) S

NP Karel

VP S^FFF V₁

laat

Linguistic applications: “verb clusters”

(α₇) S

S NP Jan

VP ε₃

V₃

zwemmen

(β₄) S

S NP

Piet

VP S^FFF ε₂

V₂ leren

(β₅) S

S V₁

(β₆) S

NP Karel

VP S^FFF V₁

laat

Linguistic applications: “verb clusters”

(α₇) S

S NP Jan

VP ε₃

V₃

zwemmen

(β₄) S

S NP

Piet

VP S^FFF ε₂

V₂ leren

(β₅) S

S NP

Marie

VP S^FFF ε₁

V₁ helpen

(β₆) S

NP Karel

VP S^FFF V₁

laat

– p.30

‘dat Karel Marie Piet Jan laat helpen leren zwemmen’

(α₇) _S

S NP Jan

VP

ε3

V3

zwemmen

(β₄) _S

S NP

Piet

VP S^FFF ε2

V2 leren

(β₅) S

S NP

Marie

VP S^FFF ε₁

V₁ helpen

(β₆) S

NP Karel

VP S^FFF V₁

laat

‘dat Karel Marie Piet Jan laat helpen leren zwemmen’

(α₇) _S

S NP Jan

VP

ε3

V3

zwemmen

(β₄) _S

S NP

Piet

VP S^FFF ε2

V2 leren

(β₅) S

S NP

Marie

VP S^FFF ε₁

V₁ helpen

(β₆) S

NP Karel

VP S^FFF V₁

laat

‘dat Karel Marie Piet Jan laat helpen leren zwemmen’

(α₇) _S

S NP Jan

VP

ε3

V3

zwemmen

(β₄) _S

S NP

Piet

VP S^FFF ε2

V2 leren

(β₅) S

S NP

Marie

VP S^FFF ε₁

V₁ helpen

(β₆) S

NP Karel

VP S^FFF V₁

laat

‘dat Karel Marie Piet Jan laat helpen leren zwemmen’

(α₇) _S

S

NP Jan

VP

ε3

V3

zwemmen

(β₄) _S

S NP

Piet

VP S^FFF ε2

V2 leren

(γ₄) S

S

S

NP Piet

VP S^FFF

NP Jan

VP ε3

ε2

V2

leren

V3

zwemmen

(β₆) S

NP Karel

VP S^FFF V₁

laat

‘dat Karel Marie Piet Jan laat helpen leren zwemmen’

(γ₄) _S

S

S

NP Piet

VP S

NP VP

ε2

V2 leren

V3

zwemmen

(β₅) S

S NP

Marie

VP S^FFF ε₁

V₁ helpen

(β₆) S

NP Karel

VP S^FFF V₁

laat

‘dat Karel Marie Piet Jan laat helpen leren zwemmen’

(γ₄) _S

S

S

NP Piet

VP

S NP Jan

VP

ε3

ε2

V2 leren

V3

zwemmen

(β₅) _S

S

NP Marie

VP S^FFF ε1

V1 helpen

(β₆) S

NP Karel

VP S^FFF V₁

laat

‘dat Karel Marie Piet Jan laat helpen leren zwemmen’

(γ₄) _S

S

S

NP Piet

VP

S

NP VP

ε2

V2 leren

V3

zwemmen

(β₅) _S

S

NP Marie

VP S^FFF ε1

V1 helpen

(β₆) S

NP Karel

VP S^FFF V₁

laat

‘dat Karel Marie Piet Jan laat helpen leren zwemmen’

(γ₄) _S

S

S

NP Piet

VP

S NP Jan

VP

ε3

ε2

V2 leren

V3

zwemmen

(β₅) _S

S

NP Marie

VP S^FFF ε1

V1 helpen

(β₆) S

NP Karel

VP S^FFF V₁

laat

‘dat Karel Marie Piet Jan laat helpen leren zwemmen’

(γ₅) ^S

S

S

S

NP Marie

VP S

NP Piet

VP

S ε2

ε1 V1 helpen

V2 leren

V3 zwemmen

(β₆) _S

NP Karel

VP S^FFF V1

laat

‘dat Karel Marie Piet Jan laat helpen leren zwemmen’

(γ₅) ^S

S

S

S

NP Marie

VP S

NP Piet

VP S

NP Jan

VP ε3

ε2 ε1

V1 helpen

V2 leren

V3 zwemmen

(β₆) ^S

NP Karel

VP S^FFF V1

laat

– p.37

‘dat Karel Marie Piet Jan laat helpen leren zwemmen’

(γ₅) ^S

S

S

S

NP Marie

VP S

NP Piet

VP

S ε2

ε1 V1 helpen

V2 leren

V3 zwemmen

(β₆) ^S

NP Karel

VP S^FFF V1

laat

‘dat Karel Marie Piet Jan laat helpen leren zwemmen’

(γ₅) ^S

S

S

S

NP Marie

VP S

NP Piet

VP S

NP Jan

VP ε3

ε2 ε1

V1 helpen

V2 leren

V3 zwemmen

(β₆) ^S

NP Karel

VP S^FFF V1

laat

– p.38

‘dat Karel Piet Jan laat leren zwemmen’

(α₇) _S

S NP Jan

VP

ε3

V3

zwemmen

(β₄) _S

S NP

Piet

VP S^FFF ε2

V2 leren

(β₆) S

NP Karel

VP S^FFF V₁

laat

‘dat Karel Piet Jan laat leren zwemmen’

(α₇) _S

S NP Jan

VP

ε3

V3

zwemmen

(β₄) _S

S NP

Piet

VP S^FFF ε2

V2 leren

(β₆) S

NP Karel

VP S^FFF V₁

laat

‘dat Karel Piet Jan laat leren zwemmen’

(α₇) _S

S NP Jan

VP

ε3

V3

zwemmen

(β₄) _S

S NP

Piet

VP S^FFF ε2

V2 leren

(β₆) S

NP Karel

VP S^FFF V₁

laat

‘dat Karel Piet Jan laat leren zwemmen’

(α₇) _S

S

NP Jan

VP

ε3

V3

zwemmen

(β₄) _S

S NP

Piet

VP S^FFF ε2

V2 leren

(γ₄) S

S

S

NP Piet

VP S^FFF

NP Jan

VP ε3

ε2

V2

leren

V3

zwemmen

(β₆) S

NP Karel

VP S^FFF V₁

laat

‘dat Karel Piet Jan laat helpen leren zwemmen’

(γ₄) _S

S

S

NP Piet

VP S

NP VP

ε2

V2 leren

V3

zwemmen

(β₆) S

NP Karel

VP S^FFF V₁

laat

‘dat Karel Piet Jan laat helpen leren zwemmen’

(γ₄) _S

S

S

NP Piet

VP

S NP Jan

VP

ε3

ε2

V2 leren

V3

zwemmen

(β₆) _S

NP Karel

VP S^FFF V1

laat

‘dat Karel Piet Jan laat helpen leren zwemmen’

(γ₄) _S

S

S

NP Piet

VP

S

NP VP

ε2

V2 leren

V3

zwemmen

(β₆) _S

NP Karel

VP S^FFF V1

laat

‘dat Karel Piet Jan laat helpen leren zwemmen’

(γ₄) _S

S

S

NP Piet

VP

S NP Jan

VP

ε3

ε2

V2 leren

V3

zwemmen

(β₆) _S

NP Karel

VP S^FFF V1

laat

‘dat Karel Piet Jan laat leren zwemmen’

(γ₆) _S

S

S

NP Karel

VP

S

NP Piet

VP

S ε2

V1

laat

V2 leren

V3 zwemmen

(Kroch & Santorini 1991)

‘dat Karel Piet Jan laat leren zwemmen’

(γ₆) _S

S

S

NP Karel

VP

S

NP Piet

VP S

NP Jan

VP ε3

ε2

V1

laat

V2 leren

V3 zwemmen

(Kroch & Santorini 1991)

Some formal properties

¥ TAGs (even TIGs) strongly lexicalize CFGs

¥ Lexicalized tree adjoining grammars (LTAGs):

each elementary tree has at least one lexical anchor

Some formal properties

¥ TAGs (even TIGs) strongly lexicalize CFGs

¥ Lexicalized tree adjoining grammars (LTAGs):

each elementary tree has at least one lexical anchor

Some formal properties

¥¥¥ CFL $$$ T A L $$$ CSL But so far, T A L is not an AFL, because in particular, it

is not closed under intersection with regular languages.

¥¥¥ Consider the TAG G_ex1 whose elementary trees are the following : (α_ex1) S

ε

(β_ex1) S

a S

b S^FFF c

L(G_ex1) ∩ {a ^k b ^l c ^m | k , l , m ≥ 0} = {a ⁿ b ⁿ c ⁿ | n ≥ 0}

Some formal properties

¥¥¥ CFL $$$ T A L $$$ CSL But

so far, T A L is not an AFL, because in particular, it is not closed under intersection with regular languages.

¥¥¥ Consider the TAG G_ex1 whose elementary trees are the following : (α_ex1) S

ε

(β_ex1) S

a S

b S^FFF c

L(G_ex1) ∩ {a ^k b ^l c ^m | k , l , m ≥ 0} = {a ⁿ b ⁿ c ⁿ | n ≥ 0}

Some formal properties

¥¥¥ CFL $$$ T A L $$$ CSL

But so far, T A L is not an AFL, because in particular, it is not closed under intersection with regular languages.

¥¥¥ Consider the TAG G_ex1 whose elementary trees are the following : (α_ex1) S

ε

(β_ex1) S

a S

b S^FFF c

L(G_ex1) ∩ {a ^k b ^l c ^m | k , l , m ≥ 0} = {a ⁿ b ⁿ c ⁿ | n ≥ 0}