1 Montagovian Dynamics Towards a Montagovian Account of Dynamics Philippe de Groote LORIA & Inria-Lorraine
2 Montagovian Dynamics 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.
3 Montagovian Dynamics 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.
4 Montagovian Dynamics 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 � �� � o
5 Montagovian Dynamics 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 [[ S 1 . S 2 ]] = λeφ. [[ S 1 ]] e ( λe ′ . [[ S 2 ]] e ′ φ ) Note that this operation is associative!
6 Montagovian Dynamics Back to DRT and DRSs Consider a DRS: x 1 . . . x n C 1 . . . C m To such a structure, corresponds the following λ -term of type γ → γ → o → o : λeφ. ∃ x 1 . . . x n . C 1 ∧ · · · ∧ C m ∧ φ e ′ where e ′ is a context made of e and of the variables x 1 , . . . , x n .
7 Montagovian Dynamics Updating and accessing the context John 1 loves Mary 2 . He 1 smiles at her 2 . : γ nil : N → ι → γ → γ push : N → γ → ι sel � a if i = j sel i ( push j a l ) = otherwise sel i l [[ John 1 loves Mary 2 ]] = λeφ. love j m ∧ φ ( push 2 m ( push 1 j e )) [[ He 1 smiles at her 2 ]] = λeφ. smile ( sel 1 e ) ( sel 2 e ) ∧ φ e
8 Montagovian Dynamics λeφ. [[ John 1 loves Mary 2 ]] e ( λe ′ . [[ He 1 smiles at her 2 ]] e ′ φ ) λeφ. ( λeφ. love j m ∧ φ ( push 2 m ( push 1 j e ))) e ( λe ′ . [[ He 1 smiles at her 2 ]] e ′ φ ) = → β λeφ. ( λφ. love j m ∧ φ ( push 2 m ( push 1 j e ))) ( λe ′ . [[ He 1 smiles at her 2 ]] e ′ φ ) → β λeφ. love j m ∧ ( λe ′ . [[ He 1 smiles at her 2 ]] e ′ φ ) ( push 2 m ( push 1 j e )) → β λeφ. love j m ∧ [[ He 1 smiles at her 2 ]] ( 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 ))
9 Montagovian Dynamics Assigning a semantics to the lexical entries [[ s ]] = o [[ n ]] = ι → o [[ np ]] = ( ι → o ) → o [[ s ]] = o (1) [[ n ]] = ι → [[ s ]] (2) = ( ι → [[ s ]] ) → [[ s ]] (3) [[ np ]] 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
10 Montagovian Dynamics 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
11 Montagovian Dynamics Noun phrases [[ np ]] = ( ι → γ → ( γ → o ) → o ) → γ → ( γ → o ) → o [[ John i ]] = λψeφ. ψ j e ( λe. φ ( push i j e )) [[ Mary i ]] = λψeφ. ψ m e ( λe. φ ( push i m e )) [[ he i ]] = λψeφ. ψ ( sel i e ) e φ [[ her i ]] = λψeφ. ψ ( sel i e ) e φ [[ it i ]] = λψeφ. ψ ( sel i e ) e φ
12 Montagovian Dynamics Determiners [[ det ]] = [[ n ]] → [[ np ]] [[ a i ]] = λnψeφ. ∃ x. n x e ( λe. ψ x ( push i x e ) φ ) [[ every i ]] = λnψeφ. ( ∀ x. ¬ ( n x e ( λe. ¬ ( ψ x ( push i x e ) ( λe. ⊤ ))))) ∧ φ e
13 Montagovian Dynamics 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 ))
14 Montagovian Dynamics Relative pronouns [[ rel ]] = ( [[ np ]] → [[ s ]] ) → [[ n ]] → [[ n ]] [[ who ]] = λrnxeφ. n x e ( λe. r ( λψ. ψ x ) e φ )
15 Montagovian Dynamics [[ beats ]] [[ it 2 ]] ( [[ every 1 ]] ( [[ who ]] ( [[ owns ]] ( [[ a 2 ]] [[ donkey ]] )) [[ farmer ]] )) [[ a 2 ]] [[ 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 ]] ( [[ a 2 ]] [[ 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 )) →
16 Montagovian Dynamics [[ who ]] ( [[ owns ]] ( [[ a 2 ]] [[ 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 ]] ( [[ a 2 ]] [[ 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 ))
17 Montagovian Dynamics [[ every 1 ]] ( [[ who ]] ( [[ owns ]] ( [[ a 2 ]] [[ donkey ]] )) [[ farmer ]] ) [[ every 1 ]] ( 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 ]] [[ it 2 ]] = ( λos. s ( λx. o ( λyeφ. beat x y ∧ φ e ))) [[ it 2 ]] → β λs. s ( λx. [[ it 2 ]] ( 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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