A Minimalist Grammar (MG) as Interpreted Regular Tree ......Nov 24, 2018 · Meaghan Fowlie and...

A Minimalist Grammar (MG)as Interpreted Regular Tree Grammar (IRTG)

Martin van Harmelen

Language Science and TechnologyComputational Linguistics & Phonetics (CoLi)

Universitat des Saarlandes

22-11-2018

Martin van Harmelen (LST) Fowlie and Koller (2017) 22-11-2018 1 / 39

Minimalist Tree vs. String Grammar

1. Minimalist Tree vs. String Grammar

Derived Tree vs. Derived String

〈make tortillas, =D V, ∅〉

move: Deleting vs. Storing

〈Maria will make tortillas, T, ∅〉

〈will make tortillas, +nom T, {nom 7→ 〈Maria, ε〉}〉

〈will make tortillas, =D +nom V, ∅〉〈Maria, D -nom, ∅〉

Minimalist String Grammar

2. Minimalist String Grammar

Objects

Derived string: 〈string , stack , storage〉storage: {feature 7→ 〈string , stack〉}

Example

〈Matt presented, T, ∅〉〈will make tortillas, +nom T, {nom 7→ 〈Maria, ε〉}〉

Derivation Tree

〈Matt presented, T, ∅〉move1

〈presented, +nom T, {nom 7→ 〈Matt, ε〉}〉merge1

〈ε, =V +nom T, ∅〉〈presented, V, {nom 7→ 〈Matt, ε〉}〉merge2

〈presented, =D V, ∅〉〈Matt, D -nom, ∅〉

Lexicon

〈ε, =V +nom T, ∅〉〈klingon, D, ∅〉〈Matt, D -nom, ∅〉〈Teresa, D -nom, ∅〉〈presented, =D V, ∅〉〈presented, =D =D V, ∅〉〈speaks, =D =D V, ∅〉

Grammar Rules: merge1

〈presented, +nom T, {nom 7→ 〈Matt, ε〉}〉merge1

〈ε, =V +nom T, ∅〉〈presented, V, {nom 7→ 〈Matt, ε〉}〉

If the feature X matchesand the feature-stack is emptythen concatenate strings and combine storages

merge1 (〈s, =Xα,S〉 , 〈t, X,T〉) = 〈s · t, α,S⊕ T〉

Grammar Rules: merge2

〈presented, V, {nom 7→ 〈Matt, ε〉}〉merge2

〈presented, =D V, ∅〉〈Matt, D -nom, ∅〉

If the feature X matchesand there are more features on the stack

(starting with -)then store string t and remaining features β under f

(for when you find +f)

merge2 (〈s, =Xα,S〉 , 〈t, X -fβ,T〉) = 〈s, α,S⊕ T⊕ {f 7→ 〈t, β〉}〉

Grammar Rules: move1

〈Matt presented, T, ∅〉move1

〈presented, +nom T, {nom 7→ 〈Matt, ε〉}〉

If the feature f exists in storageand the stored feature-stack is emptythen left-concatenate the stored string

move1 (〈s, +fα,S〉) = 〈t · s, α,S− f〉 if S(f) = 〈t, ε〉

Grammar Rules: move2

〈presented, T, {wh 7→ 〈Who, ε〉}〉move2

〈presented, +nom T, {nom 7→ 〈Who, -wh〉}〉

If the feature f exists in storageand there are more features on the stored stack

(starting with -)then re-store string t and remaining features γ under g

(for when you find +g)

move2 (〈s, +fα,S〉) = 〈s, α, (S− f)⊕ {g 7→ 〈t, γ〉}〉 if S(f) = 〈t, -gγ〉

Grammar Rules

merge(1|2): eat a =X, X feature-pair

move(1|2): eat a +f and stored (-)f

(merge|move)1: stick together strings

(merge|move)2: store strings (using features as index)

3. IRTG

Regular Tree Grammar (specifies allowed combinations)

Interpretation(s)

TreehomomorphismAlgebra

objects (things to be combined & result of evaluation)operations (different ways to combine)evaluation (specifies result of operations)

IRTG Algebra

3.1. Algebra

IRTG Algebra

Algebra Functionality

IRTG Algebra

Algebra Terminology

object 〈string , stack , {feature 7→ 〈string , stack〉 }〉operation ≈ (merge|move)(1|2)

term combinations of above (= tree)

IRTG Algebra

Algebra Evaluation: merge

merge1 (〈s,S〉 , 〈t,T〉) = 〈s · t, α,S⊕ T〉

merge2f (〈s,S〉 , 〈t,T〉) = 〈s,S⊕ T⊕ {f 7→ t}〉

If the feature X matchesand there are more features on the stack

(starting with -)then store string t and remaining features β under f

IRTG Algebra

Algebra Evaluation: move

move1f (〈s,S〉) = 〈t · s, α,S− f〉 where S(f) = t

move2f-g (〈s,S〉) = 〈s, (S− f)⊕ {g 7→ t}〉 where S(f) = t

IRTG Regular Tree Grammar

3.2. Regular Tree Grammar

Regular Tree Grammar Functionality

RTG Terminology

non-terminal 〈string , stack , {feature 7→ 〈string , stack〉 }〉

How-To

1 MG lexicon 7→strip strings RTG lexical rules

2 applications of (merge|move)(1|2) 7→ RTG non-lexical rules

The Grammar

lexicon from minimalist grammar

〈ε, =V +nom T, ∅〉〈klingon, D, ∅〉〈Matt, D -nom, ∅〉〈Teresa, D -nom, ∅〉〈presented, =D V, ∅〉〈presented, =D =D V, ∅〉〈speaks, =D =D V, ∅〉

lexical rules of regular tree grammar

〈=V +nom T, ∅〉 → lexcase

〈D, ∅〉 → lexklingon

〈D -nom, ∅〉 → lexMatt

〈D -nom, ∅〉 → lexTeresa

〈=D V, ∅〉 → lexpresented

〈=D =D V, ∅〉 → lexpresented2

〈=D =D V, ∅〉 → lexspeaks

Lexical Rules (condensed)

1 〈=V +nom T, ∅〉 → lexcase2 〈D, ∅〉 → lexklingon3 〈D -nom, ∅〉 → lexMatt |lexTeresa4 〈=D V, ∅〉 → lexpresented5 〈=D =D V, ∅〉 → lexpresented2|lexspeaks

Non-Lexical Rules: merge1

〈V, ∅〉 → mg1.1 (〈=D V, ∅〉 , 〈D, ∅〉)〈=D V, ∅〉 → mg1.2 (〈=D =D V, ∅〉 , 〈D, ∅〉)〈V, {nom 7→ ε}〉 → mg1.3 (〈=D V, {nom 7→ ε}〉 , 〈D, ∅〉)〈+nom T, ∅〉 → mg1.4 (〈=V +nom T, ∅〉 , 〈V, ∅〉)〈+nom T, {nom 7→ ε}〉 → mg1.5 (〈=V +nom T, ∅〉 , 〈V, {nom 7→ ε}〉)

Non-Lexical Rules: merge2

〈V, {nom 7→ ε}〉 → mg2.1 (〈=D V, ∅〉 , 〈D -nom, ∅〉)〈=D V, {nom 7→ ε}〉 → mg2.2 (〈=D =D V, ∅〉 , 〈D -nom, ∅〉)

If the feature X matchesand there are more features on the stackthen store string t and remaining features β under f

Non-Lexical Rules: move1

〈T, ∅〉!→ mv1.1 (〈+nom T, {nom 7→ ε}〉)

Non-Lexical Rules: move2

Example

lexicon from minimalist grammar

〈〈Who, D -nom -wh〉, ∅〉〈〈ε, =T +wh C〉, ∅〉

rules of regular tree grammar

〈V, {nom 7→ -wh}〉 → mg2.3 (〈=D V, ∅〉 , 〈D -nom -wh, ∅〉). . .

〈T, {wh 7→ ε}〉 → mv2.1 (〈+nom T, {nom 7→ -wh}〉)

IRTG Arrows

3.3. Arrows

IRTG Arrows

Derivation Tree

lexcase mg2.1

lexpresented lexMatt

IRTG Arrows

Treehomomorphism

mv1.1 7→ move1nom(x1)

mg1.5 7→ merge1(x1, x2)

mg2.1 7→ merge2nom(x1, x2)

lexcase 7→ 〈ε, ∅〉lexpresented 7→ 〈presented, ∅〉

lexMatt 7→ 〈Matt, ∅〉. . .

IRTG Arrows

Term of Algebra

move1nom

merge1

〈ε, ∅〉 merge2nom

〈presented, ∅〉〈Matt, ∅〉

IRTG Arrows

Evaluation

outcome of operations

Example

merge2nom

〈presented, ∅〉〈Matt, ∅〉〈presented, {nom 7→ Matt}〉

IRTG Arrows

Object of Algebra

〈Matt presented, ∅〉

IRTG Arrows

Conclusion

RTG decides allowed combinations

Algebra decides outcome of combinations

IRTG Arrows

Exercise

Say we would add the following two items to our Minimalist Grammar:

〈〈Who, D -nom -wh〉, ∅〉〈〈ε, =T +wh C〉, ∅〉

Do the following:

Write down the extra RTG rules needed

Write down the full homomorphism

Show how the following sentence would be derived:

Who presented Klingon?

IRTG Arrows

References

Meaghan Fowlie and Alexander Koller. Parsing minimalist languages withinterpreted regular tree grammars. In Proceedings of the 13thInternational Workshop on Tree Adjoining Grammars and RelatedFormalisms (TAG), pages 11–20. Association for ComputationalLinguistics, September 2017. URLhttps://aclanthology.info/papers/W17-6202/w17-6202.

A Minimalist Grammar (MG) as Interpreted Regular Tree ......Nov 24, 2018 · Meaghan Fowlie and...

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