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1 Fifth Workshop on Lambda Calculus and Formal Grammar 2 Fifth - - PowerPoint PPT Presentation

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1 Fifth Workshop on Lambda Calculus and Formal Grammar 2 Fifth Workshop on Lambda Calculus and Formal Grammar A Type-Theoretic View of Dynamic Logic Philippe de Groote LORIA & Inria-Lorraine 3 Fifth Workshop on Lambda Calculus and

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Fifth Workshop on Lambda Calculus and Formal Grammar

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Fifth Workshop on Lambda Calculus and Formal Grammar

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A Type-Theoretic View

  • f Dynamic Logic

Philippe de Groote LORIA & Inria-Lorraine

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A Type-Theoretic Reconstruction of DRT

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A Type-Theoretic Reconstruction of DRT

Motivation:

  • to formalize DRT within Church’s simple theory of type (aka, Higher-Order Logic),

which will allow DRT and Montague semantics to rest on the same logical foundations.

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A Type-Theoretic Reconstruction of DRT

Motivation:

  • to formalize DRT within Church’s simple theory of type (aka, Higher-Order Logic),

which will allow DRT and Montague semantics to rest on the same logical foundations. Challenge:

  • to express dynamics using “static” primitives (in particular, to avoid the “destructive

assignment” problem, wich necessitates a LISP-like gensym operator).

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A Type-Theoretic Reconstruction of DRT

Motivation:

  • to formalize DRT within Church’s simple theory of type (aka, Higher-Order Logic),

which will allow DRT and Montague semantics to rest on the same logical foundations. Challenge:

  • to express dynamics using “static” primitives (in particular, to avoid the “destructive

assignment” problem, wich necessitates a LISP-like gensym operator). Proposed solution:

  • to interpret a sentence according to both its left and right contexts;
  • to abstract these two kinds of contexts over the meaning of the sentences.
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4 Typing the left and the right contexts

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4 Typing the left and the right contexts Montague semantics is based on Church’s simple type theory, which provides a full hierarchy

  • f 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).
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4 Typing the left and the right contexts Montague semantics is based on Church’s simple type theory, which provides a full hierarchy

  • f 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.

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4 Typing the left and the right contexts Montague semantics is based on Church’s simple type theory, which provides a full hierarchy

  • f 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?

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Fifth Workshop on Lambda Calculus and Formal Grammar

4 Typing the left and the right contexts Montague semantics is based on Church’s simple type theory, which provides a full hierarchy

  • f 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?

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Fifth Workshop on Lambda Calculus and Formal Grammar

4 Typing the left and the right contexts Montague semantics is based on Church’s simple type theory, which provides a full hierarchy

  • f 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?

  • ↓
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Fifth Workshop on Lambda Calculus and Formal Grammar

4 Typing the left and the right contexts Montague semantics is based on Church’s simple type theory, which provides a full hierarchy

  • f 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

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Fifth Workshop on Lambda Calculus and Formal Grammar

4 Typing the left and the right contexts Montague semantics is based on Church’s simple type theory, which provides a full hierarchy

  • f 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
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4 Typing the left and the right contexts Montague semantics is based on Church’s simple type theory, which provides a full hierarchy

  • f 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
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Fifth Workshop on Lambda Calculus and Formal Grammar

4 Typing the left and the right contexts Montague semantics is based on Church’s simple type theory, which provides a full hierarchy

  • f 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
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4 Typing the left and the right contexts Montague semantics is based on Church’s simple type theory, which provides a full hierarchy

  • f 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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5 Semantic interpretation of the sentences

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5 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.

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5 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

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5 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

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5 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)
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6 Semantic interpretation of the syntactic categories

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6 Semantic interpretation of the syntactic categories Montague’s interpretation s =

  • n

= ι → o np = (ι → o) → o

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6 Semantic interpretation of the syntactic categories Montague’s interpretation s =

  • n

= ι → o np = (ι → o) → o may be rephrased as follows: s =

  • (1)

n = ι →s (2) np = (ι →s) →s (3)

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6 Semantic interpretation of the syntactic categories Montague’s interpretation s =

  • n

= ι → o np = (ι → o) → o may be rephrased as follows: s =

  • (1)

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

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6 Semantic interpretation of the syntactic categories Montague’s interpretation s =

  • n

= ι → o np = (ι → o) → o may be rephrased as follows: 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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7 This interpretation results in handcrafted lexical semantics such as the following:

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7 This interpretation results in handcrafted lexical semantics such as the following: farmer = λxeφ. farmer x ∧ φ e donkey = λxeφ. donkey x ∧ φ e

  • wns = λos. s (λx. o (λyeφ. own x y ∧ φ e))

beats = λos. s (λx. o (λyeφ. beat x y ∧ φ e)) who = λrnxeφ. n x e (λe. r (λψ. ψ x) e φ) a = λnψeφ. ∃x. n x e (λe. ψ x (x::e) φ) every = λnψeφ. (∀x. ¬(n x e (λe. ¬(ψ x (x::e) (λe. ⊤))))) ∧ φ e it = λψeφ. ψ (sel e) e φ

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7 This interpretation results in handcrafted lexical semantics such as the following: farmer = λxeφ. farmer x ∧ φ e donkey = λxeφ. donkey x ∧ φ e

  • wns = λos. s (λx. o (λyeφ. own x y ∧ φ e))

beats = λos. s (λx. o (λyeφ. beat x y ∧ φ e)) who = λrnxeφ. n x e (λe. r (λψ. ψ x) e φ) a = λnψeφ. ∃x. n x e (λe. ψ x (x::e) φ) every = λnψeφ. (∀x. ¬(n x e (λe. ¬(ψ x (x::e) (λe. ⊤))))) ∧ φ e it = λψeφ. ψ (sel e) e φ ...which might seem a little bit involved.

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8 Questions:

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8 Questions:

  • is there a systematic way of obtaining the new lexical semantics from Montague’s?
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8 Questions:

  • is there a systematic way of obtaining the new lexical semantics from Montague’s?
  • can we find any “modular” presentation of the approach?
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8 Questions:

  • is there a systematic way of obtaining the new lexical semantics from Montague’s?
  • can we find any “modular” presentation of the approach?
  • is there some dynamic logic hidden in the approach?
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A Dynamic Logic

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A Dynamic Logic

Let Ω γ → (γ → o) → o. We intend to design a logic acting on propositions of type Ω

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A Dynamic Logic

Let Ω γ → (γ → o) → o. We intend to design a logic acting on propositions of type Ω We share with DRT the two following assumptions:

  • discourse composition is mainly conjunctive (roughly speaking, a discourse consists

in the conjunction of its sentences);

  • the main form of quantification is existential (it introduces referential markers).
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A Dynamic Logic

Let Ω γ → (γ → o) → o. We intend to design a logic acting on propositions of type Ω We share with DRT the two following assumptions:

  • discourse composition is mainly conjunctive (roughly speaking, a discourse consists

in the conjunction of its sentences);

  • the main form of quantification is existential (it introduces referential markers).

Consequently, our logic will be based on conjunction and existential quantification (defined as primitives). The other connectives will be obtained using negation (a third primitive) and de Morgan’s laws.

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10 Formal Framework We consider a simply-typed λ-calculus, the terms of which are built upon asignature in- cluding the following constants:

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10 Formal Framework We consider a simply-typed λ-calculus, the terms of which are built upon asignature in- cluding the following constants: FIRST-ORDER LOGIC ⊤ :

  • (truth)

¬ :

  • → o

(negation) ∧ :

  • → o → o

(conjunction) ∃ : (ι → o) → o (existential quantification)

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10 Formal Framework We consider a simply-typed λ-calculus, the terms of which are built upon asignature in- cluding the following constants: FIRST-ORDER LOGIC ⊤ :

  • (truth)

¬ :

  • → o

(negation) ∧ :

  • → o → o

(conjunction) ∃ : (ι → o) → o (existential quantification) DYNAMIC PRIMITIVES :: : ι → γ → γ (context updating) sel : γ → ι (choice operator)

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11 Conjunction

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11 Conjunction Conjunction is nothing but sentence composition. We therefore define: A ⊓ B λeφ. A e (λe. B e φ)

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11 Conjunction Conjunction is nothing but sentence composition. We therefore define: A ⊓ B λeφ. A e (λe. B e φ) Existential quantification

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11 Conjunction Conjunction is nothing but sentence composition. We therefore define: A ⊓ B λeφ. A e (λe. B e φ) Existential quantification Existential quantification introduces “reference markers”. It is therefore responsible for context updating: Σx. P x λeφ. ∃x. P x (x::e) φ

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11 Conjunction Conjunction is nothing but sentence composition. We therefore define: A ⊓ B λeφ. A e (λe. B e φ) Existential quantification Existential quantification introduces “reference markers”. It is therefore responsible for context updating: Σx. P x λeφ. ∃x. P x (x::e) φ Negation

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11 Conjunction Conjunction is nothing but sentence composition. We therefore define: A ⊓ B λeφ. A e (λe. B e φ) Existential quantification Existential quantification introduces “reference markers”. It is therefore responsible for context updating: Σx. P x λeφ. ∃x. P x (x::e) φ Negation We do not want the continuation of the discourse to fall into the scope of the negation. Consequently, negation must be defined as follows:

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12 ∼ A λeφ. ¬ (A e (λe. ⊤)) ∧ φ e

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13 Implication and Universal Quantification

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13 Implication and Universal Quantification These are defined using de Morgan’s laws: A ⊐ B ∼(A ⊓ ∼B) Πx. P x ∼Σx. ∼(P x)

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13 Implication and Universal Quantification These are defined using de Morgan’s laws: A ⊐ B ∼(A ⊓ ∼B) Πx. P x ∼Σx. ∼(P x) Embedding of first-order logic into dynamic logic

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13 Implication and Universal Quantification These are defined using de Morgan’s laws: A ⊐ B ∼(A ⊓ ∼B) Πx. P x ∼Σx. ∼(P x) Embedding of first-order logic into dynamic logic R t1 . . . tn = λeφ. R t1 . . . tn ∧ φ e ¬A = ∼A A ∧ B = A ⊓ B ∃x. A = Σx. A

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13 Implication and Universal Quantification These are defined using de Morgan’s laws: A ⊐ B ∼(A ⊓ ∼B) Πx. P x ∼Σx. ∼(P x) Embedding of first-order logic into dynamic logic R t1 . . . tn = λeφ. R t1 . . . tn ∧ φ e ¬A = ∼A A ∧ B = A ⊓ B ∃x. A = Σx. A This embedding is such that, for every term e of type γ: A ≡ A e (λe. ⊤)

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Donkey Sentence Revisited

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Donkey Sentence Revisited

Montague-like semantic interpretation: farmer = farmer donkey = donkey

  • wns = λOS. S (λx. O (λy. own x y))

beats = λOS. S (λx. O (λy. beat x y)) who = λRQx. Q x ∧ R (λP. P x) a = λPQ. ∃x. P x ∧ Q x every = λPQ. ∀x. P x ⊃ Q x it = ???

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Donkey Sentence Revisited

Montague-like semantic interpretation: farmer = farmer donkey = donkey

  • wns = λOS. S (λx. O (λy. own x y))

beats = λOS. S (λx. O (λy. beat x y)) who = λRQx. Q x ∧ R (λP. P x) a = λPQ. ∃x. P x ∧ Q x every = λPQ. ∀x. P x ⊃ Q x it = ???

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15 Dynamic interpretation: farmer = farmer donkey = donkey

  • wns = λOS. S (λx. O (λy. own x y))

beats = λOS. S (λx. O (λy. beat x y)) who = λRQx. Q x ⊓ R (λP. P x) a = λPQ. Σx. P x ⊓ Q x every = λPQ. Πx. P x ⊐ Q x it = λPeφ. P (sel e) e φ

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16 With the dynamic interpretation we have that: beats it (every (who (owns (a donkey)) farmer))

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16 With the dynamic interpretation we have that: beats it (every (who (owns (a donkey)) farmer)) β-reduces to the following term (modulo de Morgan’s laws): λeφ. (∀x. farmer x ⊃ (∀y. donkey y ⊃ (own x y ⊃ beat x (sel (x::y::e))))) ∧ φ e

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16 With the dynamic interpretation we have that: beats it (every (who (owns (a donkey)) farmer)) β-reduces to the following term (modulo de Morgan’s laws): λeφ. (∀x. farmer x ⊃ (∀y. donkey y ⊃ (own x y ⊃ beat x (sel (x::y::e))))) ∧ φ e that is, assuming that sel is a “perfect” anaphora resolution operator: λeφ. (∀x. farmer x ⊃ (∀y. donkey y ⊃ (own x y ⊃ beat x y))) ∧ φ e

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The Higher-Order Case

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The Higher-Order Case

Define type “dynamization” as follows: Dι = ι Do = Ω D(α → β) = Dα → Dβ

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The Higher-Order Case

Define type “dynamization” as follows: Dι = ι Do = Ω D(α → β) = Dα → Dβ Then, define term “dynamization” as follows: Dt = λx1 . . . xn. t (Rnilx1) . . . (Rnilxn) at type α1 → . . . αn → ι Dt = λx1 . . . xneφ. t (Rex1) . . . (Rexn) ∧ (φ e) at type α1 → . . . αn → o Ret = λx1 . . . xn. t (Dx1) . . . (Dxn) at type D(α1 → . . . αn → ι) Ret = λx1 . . . xn. t (Dx1) . . . (Dxn) e (λe. ⊤) at type D(α1 → . . . αn → o)

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18 Finally, define λ-term translation as follows: x = x ∧ = ⊓ ∃ = Σ ¬ = ∼ k = Dk for the other constants λx. t = λx. t t u = t u

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18 Finally, define λ-term translation as follows: x = x ∧ = ⊓ ∃ = Σ ¬ = ∼ k = Dk for the other constants λx. t = λx. t t u = t u Then, for every closed term t of type o, and every context e, we have that: t e (λe. ⊤) ≡ t

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19 Comparison with existing works

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19 Comparison with existing works Most existing works on dynamics (DRT, Muskens’, Groenendijk & Stokhof’s) interpret dynamic propositions as binary relations on states (a.k.a., assignments or environments).

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19 Comparison with existing works Most existing works on dynamics (DRT, Muskens’, Groenendijk & Stokhof’s) interpret dynamic propositions as binary relations on states (a.k.a., assignments or environments). In our setting, these would be terms of type: γ → γ → o,

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19 Comparison with existing works Most existing works on dynamics (DRT, Muskens’, Groenendijk & Stokhof’s) interpret dynamic propositions as binary relations on states (a.k.a., assignments or environments). In our setting, these would be terms of type: γ → γ → o, and the semantics of Groenendijk & Stokhof’s DPL would be rephrased as follows: Ad λgh. h=g ∧ A (atomic proposition) (¬P)d λgh. h=g ∧ ¬(∃k. Pd h k) (negation) (P ∧ Q)d λgh. ∃k. Pd g k ∧ Qd k h (conjunction) (∃x. P)d λgh. ∃k. k[x]g ∧ Pd k h (existential)

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20

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20 There exists a canonical embedding ⌈·⌉ from γ → γ → o into γ → (γ → o) → o: ⌈R⌉ λeφ. ∃e′. φ e′ ∧ R e e′

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20 There exists a canonical embedding ⌈·⌉ from γ → γ → o into γ → (γ → o) → o: ⌈R⌉ λeφ. ∃e′. φ e′ ∧ R e e′ Then, we have: ⌈Ad⌉ ≡ A ⌈(¬P)d⌉ ≡ ∼ ⌈P d⌉ ⌈(P ∧ Q)d⌉ ≡ ⌈P d⌉⊓⌈Qd⌉

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20 There exists a canonical embedding ⌈·⌉ from γ → γ → o into γ → (γ → o) → o: ⌈R⌉ λeφ. ∃e′. φ e′ ∧ R e e′ Then, we have: ⌈Ad⌉ ≡ A ⌈(¬P)d⌉ ≡ ∼ ⌈P d⌉ ⌈(P ∧ Q)d⌉ ≡ ⌈P d⌉⊓⌈Qd⌉ As for the existential quantifier: ⌈(∃x. P)d⌉ = λeφ. ∃e′. φ e′ ∧ ∃k. k[x]e ∧ ⌈Pd⌉ k e′ Σx. ⌈Pd⌉ = λeφ. ∃e′. φ e′ ∧ ∃x. ⌈Pd⌉ (x :: e) e′

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Conclusions

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Conclusions

What do we gain?

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Conclusions

What do we gain?

  • No destructive assignment.
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Conclusions

What do we gain?

  • No destructive assignment.
  • Parametric in γ.
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Conclusions

What do we gain?

  • No destructive assignment.
  • Parametric in γ.
  • Relatively independent of the underlying logic:
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Conclusions

What do we gain?

  • No destructive assignment.
  • Parametric in γ.
  • Relatively independent of the underlying logic:

– generalizes to H.O.L.;

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Conclusions

What do we gain?

  • No destructive assignment.
  • Parametric in γ.
  • Relatively independent of the underlying logic:

– generalizes to H.O.L.; – an intuitionistic version could be worked out.

slide-81
SLIDE 81

Fifth Workshop on Lambda Calculus and Formal Grammar

21

Conclusions

What do we gain?

  • No destructive assignment.
  • Parametric in γ.
  • Relatively independent of the underlying logic:

– generalizes to H.O.L.; – an intuitionistic version could be worked out.

  • Relies on well-established existing theories (proof-theory and model theory for free).
slide-82
SLIDE 82

Fifth Workshop on Lambda Calculus and Formal Grammar

21

Conclusions

What do we gain?

  • No destructive assignment.
  • Parametric in γ.
  • Relatively independent of the underlying logic:

– generalizes to H.O.L.; – an intuitionistic version could be worked out.

  • Relies on well-established existing theories (proof-theory and model theory for free).
  • Deduction is replaced by computation.