categorical methods in linguistics categorical... · 2014. 10. 21. · flow of ambiguity a noun...
TRANSCRIPT
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Categorical methods in linguistics
Robin Piedeleu
2014-10-16
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What does
share with
?
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Categories capture composition
B C
A
g
fg ○ f
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Reconciling syntax and semantics
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Formal, symbolic, logical
"I reject the contention that an important theoretical differenceexists between formal and natural languages."
- English as a Formal Language, Montague
Compositional but incomplete model of meaning.
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Distributional
"You shall know the word by the company it keeps"
- J.R. Firth
queen
plant
economy
monarch
law
growth
vegetable
Good model of meaning of individual words but not obviouslycompositional
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Compositional distributional model
▸ With categorical methods we wish to provide a general andabstract interface between these two approaches.
▸ A functorFSyntax Semantics
▸ What structure do we need to express basic syntactic andsemantic information? Category in which
▸ objects are grammatical types;▸ morphisms are ???.
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What structure?
▸ Monoidal: type of a sentence is the tensor of the types of thewords
My fake plants died ∶ Pr ⊗Adj ⊗N ⊗V i
▸ Closed: exponentials equipped with evaluation morphisms∗
NP ⊗ (NP ⇒ S)EvalNP,SÐÐÐÐ→ S
∗ satisfying the appropriate universal properties.
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What structure?
▸ Monoidal: type of a sentence is the tensor of the types of thewords
My fake plants died ∶ Pr ⊗Adj ⊗N ⊗V i
▸ Closed: exponentials equipped with evaluation morphisms∗
NP ⊗ (NP ⇒ S)EvalNP,SÐÐÐÐ→ S
∗ satisfying the appropriate universal properties.
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What structure?
▸ Monoidal: type of a sentence is the tensor of the types of thewords
My fake plants died ∶ Pr ⊗Adj ⊗N ⊗V i
▸ Closed: exponentials equipped with evaluation morphisms∗
(S ⇐ NP)⊗NPLaveNP,SÐÐÐÐ→ S
∗ satisfying the appropriate universal properties.
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A more convenient framework: compact closed categories
We get a diagrammatic calculus for free!
ηlA =
εlA =
A A
A AεrA =A A
AηrA =
A
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A more convenient framework: compact closed categories
We get a diagrammatic calculus for free!
A
A
= =A
A
A
and
A
A
A
= A
A
A
A=
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A more convenient framework: compact closed categories
We get a diagrammatic calculus for free!
A⊗ (Ar ⊗B)εrA⊗1BÐÐÐ→ B (B ⊗Al)⊗A
1B⊗εlAÐÐÐ→ A
A A
B
A
B
A
EvalA,B LaveA,B
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Reductions
▸ Morphisms in the internal language of a monoidal closedcategory.
▸ A reduction looks like
John likes MaryN r ⊗ S ⊗N lN N
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What structure?
▸ Monoidal: type of a sentence is the tensor of the types of thewords
My fake plants died ∶ Pr ⊗Adj ⊗N ⊗V i
▸ Closed: exponentials equipped with evaluation morphisms∗
NP ⊗ (NP ⇒ S)EvalNP,SÐÐÐÐ→ S
▸ Dagger: involutive, identity on object contravariant functor∗.To compare the proximity of meaning:
IloveÐÐ→ N r ⊗ S ⊗N l like†
ÐÐ→ I
∗satisfying coherence conditions with the compact closed structure.
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What structure?
▸ Monoidal: type of a sentence is the tensor of the types of thewords
My fake plants died ∶ Pr ⊗Adj ⊗N ⊗V i
▸ Closed: exponentials equipped with evaluation morphisms∗
NP ⊗ (NP ⇒ S)EvalNP,SÐÐÐÐ→ S
▸ Dagger: involutive, identity on object contravariant functor∗.To compare the proximity of meaning:
IloveÐÐ→ N r ⊗ S ⊗N l like†
ÐÐ→ I
∗satisfying coherence conditions with the compact closed structure.
![Page 17: Categorical methods in linguistics categorical... · 2014. 10. 21. · Flow of ambiguity a noun (adj noun a Morerefinedprocess: CP∗ Moral of the story Contextual features of natural](https://reader033.vdocuments.us/reader033/viewer/2022060705/6071c66af034ad5f60470617/html5/thumbnails/17.jpg)
Functorial interpretation
Strict monoidal functor from our syntactic category to our semanticcategory.
John likes Mary
N r ⊗ S ⊗N lN N
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Functorial interpretation
Strict monoidal functor from our syntactic category to our semanticcategory.
John likes Mary
N r ⊗ S ⊗N lN NJohn
SMary
likes
=
Examples:
GrFÐÐÐÐ→ FdHilb, ∑
ijk
⟨John∣vi ⟩sj⟨vk ∣Mary⟩
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Functorial interpretation
Strict monoidal functor from our syntactic category to our semanticcategory.
John likes Mary
N r ⊗ S ⊗N lN NJohn
SMary
likes
=
Examples:
GrFÐÐÐÐ→ FdHilb, ∑
ijk
⟨John∣vi ⟩sj⟨vk ∣Mary⟩
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Functorial interpretation
Strict monoidal functor from our syntactic category to our semanticcategory.
John likes Mary
N r ⊗ S ⊗N lN NJohn
SMary
likes
=
Examples:
GrFÐÐÐÐ→ CPM(FdHilb)
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Examples:
GrFÐÐÐÐ→ CPM(FdHilb)
⌜TrN,N(ρ(like) ○ (ρ(John)⊗ 1S ⊗ ρ(Mary)))⌝
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Disambiguation
Queen
Context determines meaning:
▸ "the queen overruled the decision of the prime minister"▸ "Queen captures the rook in a1"
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Toy example
Bank
Context determines meaning:
"river bank"
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Flow of ambiguity
adj
noun
↦adj noun
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Flow of ambiguity
a
noun
↦adj noun
a
More refined process: CP∗
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Flow of ambiguity
a
noun
↦adj noun
a
More refined process: CP∗
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Moral of the story
Contextual features of natural language can be builtin the wires.