Towards a Montagovian Account of Dynamics Philippe de Groote LORIA - - PowerPoint PPT Presentation

Montagovian Dynamics

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Towards a Montagovian Account

• f Dynamics

Philippe de Groote LORIA & Inria-Lorraine

Montagovian Dynamics

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Introduction

An old problem: A man enters the room. He smiles. [[A man enters the room]] = ∃x.man(x) ∧ enters the room(x). x is bound. [[He smiles]] = smiles(x). x is free. How can we get from these: [[A man enters the room. He smiles]] = ∃x.man(x) ∧ enters the room(x) ∧ smiles(x). A well known solution: DRT.

• The reference markers of DRT act as existential quantifiers.
• Nevertheless, from a technical point of view, they must be considered

as free variables.

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Expressing propositions in context

“The key idea behind (...) Discourse Representation Theory is that each new sentence of a discourse is interpreted in the context provided by the sentences preceding it.” van Eijck and Kamp. Representing Discourse in Context. In Handbook of Logic and Language. Elsevier, 1997. We go two steps further:

• We will interpret a sentence according to both its left and right contexts.
• These two kinds of contexts will be abstracted over the meaning of the

sentences.

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Typing the left and the right contexts

Montague semantics is based on Church’s simple type theory, which provides a full hierarchy of functional types built upon two atomic types:

• ι, the type of individuals (a.k.a. entities).
• o, the type of propositions (a.k.a. truth values).

We add a third atomic type, γ, which stands for the type of the left contexts. What about the type of the right contexts?

• ↓

left context

• right context
• γ
• γ → o

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Semantic interpretation of the sentences

Let s be the syntactic category of sentences. Remember that we intend to abstract our notions of left and right contexts over the meaning of the sentences.

[[s]] = γ → (γ → o) → o

Composition of two sentence interpretations

[[S1. S2]] = λeφ. [[S1]] e (λe′. [[S2]] e′ φ)

Note that this operation is associative!

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Back to DRT and DRSs

Consider a DRS:

x1 . . . xn C1 . . . Cm

To such a structure, corresponds the following λ-term of type γ → γ → o → o:

λeφ. ∃x1 . . . xn. C1 ∧ · · · ∧ Cm ∧ φ e′

where e′ is a context made of e and of the variables x1, . . . , xn.

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Updating and accessing the context

John1 loves Mary2. He1 smiles at her2. nil : γ push : N → ι → γ → γ sel : N → γ → ι sel i (push j a l) = a if i = j sel i l

• therwise

[[John1 loves Mary2]] = λeφ. love j m ∧ φ (push 2 m (push 1 j e)) [[He1 smiles at her2]] = λeφ. smile (sel 1 e) (sel 2 e) ∧ φ e

Montagovian Dynamics

8 λeφ. [[John1 loves Mary2]] e (λe′. [[He1 smiles at her2]] e′ φ) = λeφ. (λeφ. love j m ∧ φ (push 2 m (push 1 j e))) e (λe′. [[He1 smiles at her2]] e′ φ) →β λeφ. (λφ. love j m ∧ φ (push 2 m (push 1 j e))) (λe′. [[He1 smiles at her2]] e′ φ) →β λeφ. love j m ∧ (λe′. [[He1 smiles at her2]] e′ φ) (push 2 m (push 1 j e)) →β λeφ. love j m ∧ [[He1 smiles at her2]] (push 2 m (push 1 j e)) φ = λeφ. love j m ∧ (λeφ. smile (sel 1 e) (sel 2 e) ∧ φ e) (push 2 m (push 1 j e)) φ →β λeφ. love j m ∧ (λφ. smile (sel 1 (push 2 m (push 1 j e))) (sel 2 (push 2 m (push 1 j e))) ∧ φ (push 2 m (push 1 j e))) φ →β λeφ. love j m ∧ smile (sel 1 (push 2 m (push 1 j e))) (sel 2 (push 2 m (push 1 j e))) ∧ φ (push 2 m (push 1 j e)) = λeφ. love j m ∧ smile j (sel 2 (push 2 m (push 1 j e))) ∧ φ (push 2 m (push 1 j e)) = λeφ. love j m ∧ smile j m ∧ φ (push 2 m (push 1 j e))

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Assigning a semantics to the lexical entries

[[s]] =

• [[n]]

= ι → o [[np]] = (ι → o) → o [[s]] =

• (1)

[[n]] = ι →[[s]] (2) [[np]] = (ι →[[s]]) →[[s]] (3) Replacing (1) with: [[s]] = γ → (γ → o) → o we obtain: [[n]] = ι → γ → (γ → o) → o [[np]] = (ι → γ → (γ → o) → o) → γ → (γ → o) → o

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Nouns

[[n]] = ι → γ → (γ → o) → o [[man]] = λxeφ. man x ∧ φ e [[woman]] = λxeφ. woman x ∧ φ e [[farmer]] = λxeφ. farmer x ∧ φ e [[donkey]] = λxeφ. donkey x ∧ φ e

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Noun phrases

[[np]] = (ι → γ → (γ → o) → o) → γ → (γ → o) → o [[Johni]] = λψeφ. ψ j e (λe. φ (push i j e)) [[Maryi]] = λψeφ. ψ m e (λe. φ (push i m e)) [[hei]] = λψeφ. ψ (sel i e) e φ [[heri]] = λψeφ. ψ (sel i e) e φ [[iti]] = λψeφ. ψ (sel i e) e φ

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Determiners

[[det]] = [[n]]→[[np]]

[[ai]] = λnψeφ. ∃x. n x e (λe. ψ x (push i x e) φ) [[everyi]] = λnψeφ. (∀x. ¬(n x e (λe. ¬(ψ x (push i x e) (λe. ⊤))))) ∧ φ e

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Transitive verbs

[[tv]] = [[np]]→[[np]]→[[s]] [[loves]] = λos. s (λx. o (λyeφ. love x y ∧ φ e)) [[owns]] = λos. s (λx. o (λyeφ. own x y ∧ φ e)) [[beats]] = λos. s (λx. o (λyeφ. beat x y ∧ φ e))

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Relative pronouns

[[rel]] = ([[np]]→[[s]])→[[n]]→[[n]] [[who]] = λrnxeφ. n x e (λe. r (λψ. ψ x) e φ)

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15 [[beats]] [[it2]] ([[every1]] ([[who]] ([[owns]] ([[a2]] [[donkey]])) [[farmer]])) [[a2]] [[donkey]] = (λnψeφ. ∃y. n y e (λe. ψ y (push 2 y e) φ)) [[donkey]] → →β λψeφ. ∃y. [[donkey]] y e (λe. ψ y (push 2 y e) φ) = λψeφ. ∃y. (λxeφ. donkey x ∧ φ e) y e (λe. ψ y (push 2 y e) φ) → →β λψeφ. ∃y. donkey y ∧ (λe. ψ y (push 2 y e) φ) e → →β λψeφ. ∃y. donkey y ∧ ψ y (push 2 y e) φ [[owns]] ([[a2]] [[donkey]]) = [[owns]] (λψeφ. ∃y. donkey y ∧ ψ y (push 2 y e) φ) = (λos. s (λx. o (λyeφ. own x y ∧ φ e))) (λψeφ. ∃y. donkey y ∧ ψ y (push 2 y e) φ) → →β λs. s (λx. (λψeφ. ∃y. donkey y ∧ ψ y (push 2 y e) φ) (λyeφ. own x y ∧ φ e)) → →β λs. s (λxeφ. ∃y. donkey y ∧ (λyeφ. own x y ∧ φ e) y (push 2 y e) φ) → →β λs. s (λxeφ. ∃y. donkey y ∧ own x y ∧ φ (push 2 y e))

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16 [[who]] ([[owns]] ([[a2]] [[donkey]])) = [[who]] (λs. s (λxeφ. ∃y. donkey y ∧ own x y ∧ φ (push 2 y e))) = (λrnxeφ. n x e (λe. r (λψ. ψ x) e φ)) (λs. s (λxeφ. ∃y. donkey y ∧ own x y ∧ φ (push 2 y e))) → →β λnxeφ. n x e (λe. (λs. s (λxeφ. ∃y. donkey y ∧ own x y ∧ φ (push 2 y e))) (λψ. ψ x) e φ) → →β λnxeφ. n x e (λe. (λψ. ψ x) (λxeφ. ∃y. donkey y ∧ own x y ∧ φ (push 2 y e)) e φ) → →β λnxeφ. n x e (λe. (λxeφ. ∃y. donkey y ∧ own x y ∧ φ (push 2 y e)) x e φ) → →β λnxeφ. n x e (λe. ∃y. donkey y ∧ own x y ∧ φ (push 2 y e)) [[who]] ([[owns]] ([[a2]] [[donkey]])) [[farmer]] = (λnxeφ. n x e (λe. ∃y. donkey y ∧ own x y ∧ φ (push 2 y e))) [[farmer]] → →β λxeφ. [[farmer]] x e (λe. ∃y. donkey y ∧ own x y ∧ φ (push 2 y e)) = λxeφ. (λxeφ. farmer x ∧ φ e) x e (λe. ∃y. donkey y ∧ own x y ∧ φ (push 2 y e)) → →β λxeφ. farmer x ∧ (λe. ∃y. donkey y ∧ own x y ∧ φ (push 2 y e)) e → →β λxeφ. farmer x ∧ (∃y. donkey y ∧ own x y ∧ φ (push 2 y e))

Montagovian Dynamics

17 [[every1]] ([[who]] ([[owns]] ([[a2]] [[donkey]])) [[farmer]]) = [[every1]] (λxeφ. farmer x ∧ (∃y. donkey y ∧ own x y ∧ φ (push 2 y e))) = (λnψeφ. (∀x. ¬(n x e (λe. ¬(ψ x (push 1 x e) (λe. ⊤))))) ∧ φ e) (λxeφ. farmer x ∧ (∃y. donkey y ∧ own x y ∧ φ (push 2 y e))) → →β λψeφ. (∀x. ¬( (λxeφ. farmer x ∧ (∃y. donkey y ∧ own x y ∧ φ (push 2 y e))) x e (λe. ¬(ψ x (push 1 x e) (λe. ⊤))))) ∧ φ e → →β λψeφ. (∀x. ¬(farmer x ∧ (∃y. donkey y ∧ own x y ∧ (λe. ¬(ψ x (push 1 x e) (λe. ⊤))) (push 2 y e)))) ∧ φ e → →β λψeφ. (∀x. ¬(farmer x ∧ (∃y. donkey y ∧ own x y ∧ ¬(ψ x (push 1 x (push 2 y e)) (λe. ⊤))))) ∧ φ e [[beats]] [[it2]] = (λos. s (λx. o (λyeφ. beat x y ∧ φ e))) [[it2]] → →β λs. s (λx. [[it2]] (λyeφ. beat x y ∧ φ e)) = λs. s (λx. (λψeφ. ψ (sel 2 e) e φ) (λyeφ. beat x y ∧ φ e)) → →β λs. s (λxeφ. (λyeφ. beat x y ∧ φ e) (sel 2 e) e φ) → →β λs. s (λxeφ. beat x (sel 2 e) ∧ φ e)

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18 [[beats]] [[it2]] ([[every1]] ([[who]] ([[owns]] ([[a2]] [[donkey]])) [[farmer]])) = (λs. s (λxeφ. beat x (sel 2 e) ∧ φ e)) ([[every1]] ([[who]] ([[owns]] ([[a2]] [[donkey]])) [[farmer]])) → →β [[every1]] ([[who]] ([[owns]] ([[a2]] [[donkey]])) [[farmer]]) (λxeφ. beat x (sel 2 e) ∧ φ e) = (λψeφ. (∀x. ¬(farmer x ∧ (∃y. donkey y ∧ own x y ∧ ¬(ψ x (push 1 x (push 2 y e)) (λe. ⊤))))) ∧ φ e) (λxeφ. beat x (sel 2 e) ∧ φ e) → →β λeφ. (∀x. ¬(farmer x ∧ (∃y. donkey y ∧ own x y ∧ ¬((λxeφ. beat x (sel 2 e) ∧ φ e) x (push 1 x (push 2 y e)) (λe. ⊤))))) ∧ φ e → →β λeφ. (∀x. ¬(farmer x ∧ (∃y. donkey y ∧ own x y ∧ ¬(beat x (sel 2 (push 1 x (push 2 y e))) ∧ (λe. ⊤) (push 1 x (push 2 y e)))))) ∧ φ e → →β λeφ. (∀x. ¬(farmer x ∧ (∃y. donkey y ∧ own x y ∧ ¬(beat x (sel 2 (push 1 x (push 2 y e))) ∧ ⊤)))) ∧ φ e = λeφ. (∀x. ¬(farmer x ∧ (∃y. donkey y ∧ own x y ∧ ¬(beat x y ∧ ⊤)))) ∧ φ e ≡ λeφ. (∀x. farmer x ⊃ (∀y. (donkey y ∧ own x y) ⊃ beat x y)) ∧ φ e