ESSLLI’08: wsclsi 1 · ESSLLI’08: wsclsi 6 Typing the left and the right contexts Montague...

ESSLLI’08: wsclsi 1

Typing

Binding &

Anaphora

Dynamic Contexts as λµ-Terms

Philippe de Groote

Inria Nancy - Grand Est

Outline

• A type-theoretic reconstruction of DRT.

Outline

• A type-theoretic view of dynamic logic.

Outline

• A type-theoretic view of dynamic logic.

• Contexts as λµ-terms.

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.

Motivation:

Challenge:

• to express dynamics using “static” primitives (in particular, to avoid the “destructiveassignment” problem, wich necessitates a LISP-like gensym operator).

Motivation:

Challenge:

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

Typing the left and the right contexts

Montague semantics is based on Church’s simple type theory, which provides a full hierarchyof 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︷︸︸︷

•↓left context︷︸︸︷ right context︷︸︸︷

︸︷︷︸γ

︸︷︷︸γ︸︷︷︸

︸︷︷︸γ → o

Semantic interpretation of the sentences

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

JsK = γ → (γ → o)→ o

Composition of two sentence interpretations

JsK = γ → (γ → o)→ o

Composition of two sentence interpretations

JS1. S2K = λeφ. JS1K e (λe′. JS2K e′ φ)

Semantic interpretation of the syntactic categories

Montague’s interpretation

JsK = oJnK = ι→ o

JnpK = (ι→ o)→ o

may be rephrased as follows:

JsK = o (1)JnK = ι→JsK (2)

JnpK = (ι→JsK)→JsK (3)

Replacing (1) with:

JsK = γ → (γ → o)→ o

Replacing (1) with:

JsK = γ → (γ → o)→ o

we obtain:

JnK = ι→ γ → (γ → o)→ oJnpK = (ι→ γ → (γ → o)→ o)→ γ → (γ → o)→ o

This interpretation results in handcrafted lexical semantics such as the following:

JfarmerK = λxeφ. farmerx ∧ φ eJdonkeyK = λxeφ.donkey x ∧ φ e

JownsK = λos. s (λx. o (λyeφ.ownx y ∧ φ e))JbeatsK = λos. s (λx. o (λyeφ.beat x y ∧ φ e))JwhoK = λrnxeφ. n x e (λe. r (λψ. ψ x) e φ)

JaK = λnψeφ.∃x. n x e (λe. ψ x (x::e)φ)JeveryK = λnψeφ. (∀x.¬(nx e (λe.¬(ψ x (x::e) (λe.>))))) ∧ φ e

JitK = λψeφ. ψ (sel e) e φ

JfarmerK = λxeφ. farmerx ∧ φ eJdonkeyK = λxeφ.donkey x ∧ φ e

JownsK = λos. s (λx. o (λyeφ.ownx y ∧ φ e))JbeatsK = λos. s (λx. o (λyeφ.beat x y ∧ φ e))JwhoK = λrnxeφ. n x e (λe. r (λψ. ψ x) e φ)

JaK = λnψeφ.∃x. n x e (λe. ψ x (x::e)φ)JeveryK = λnψeφ. (∀x.¬(nx e (λe.¬(ψ x (x::e) (λe.>))))) ∧ φ e

JitK = λψeφ. ψ (sel e) e φ

...which might seem a little bit involved.

Questions:

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

Questions:

• can we find any “modular” presentation of the approach?

Questions:

• can we find any “modular” presentation of the approach?

• is there some dynamic logic hidden in the approach?

A Type-Theoretic View of 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 consistsin the conjunction of its sentences);

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

We share with DRT the two following assumptions:

• discourse composition is mainly conjunctive (roughly speaking, a discourse consistsin 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 (definedas primitives). The other connectives will be obtained using negation (a third primitive)and de Morgan’s laws.

Formal Framework

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

Formal Framework

FIRST-ORDER LOGIC

> : o (truth)

¬ : o→ o (negation)

∧ : o→ o→ o (conjunction)

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

Formal Framework

FIRST-ORDER LOGIC

> : o (truth)

¬ : o→ o (negation)

∧ : o→ o→ o (conjunction)

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

DYNAMIC PRIMITIVES

:: : ι→ γ → γ (context updating)

sel : γ → ι (choice operator)

Conjunction

Conjunction is nothing but sentence composition. We therefore define:

A uB , λeφ.A e (λe.B e φ)

Conjunction

Existential quantification

Conjunction

Existential quantification introduces “reference markers”. It is therefore responsible forcontext updating:

Σx. P x , λeφ.∃x. P x (x::e)φ

Conjunction

Negation

Conjunction

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:

∼ A , λeφ.¬ (Ae (λe.>)) ∧ φ e

Implication and Universal Quantification

These are defined using de Morgan’s laws:

A A B , ∼(A u ∼B)

Πx. P x , ∼Σx.∼(P x)

A A B , ∼(A u ∼B)

Embedding of first-order logic into dynamic logic

A A B , ∼(A u ∼B)

R t1 . . . tn = λeφ.R t1 . . . tn ∧ φ e¬A = ∼A

A ∧B = A uB∃x.A = Σx.A

A A B , ∼(A u ∼B)

R t1 . . . tn = λeφ.R t1 . . . tn ∧ φ e¬A = ∼A

A ∧B = A uB∃x.A = Σx.A

This embedding is such that, for every term e of type γ:

A ≡ Ae (λe.>)

Donkey Sentence Revisited

Montague-like semantic interpretation:

JfarmerK = farmerJdonkeyK = donkey

JownsK = λOS. S (λx.O (λy.ownx y))JbeatsK = λOS. S (λx.O (λy.beatx y))JwhoK = λRQx.Qx ∧R (λP. P x)

JaK = λPQ.∃x. P x ∧QxJeveryK = λPQ.∀x. P x ⊃ Qx

JitK = ???

Montague-like semantic interpretation:

JfarmerK = farmerJdonkeyK = donkey

JownsK = λOS. S (λx.O (λy.ownx y))JbeatsK = λOS. S (λx.O (λy.beatx y))JwhoK = λRQx.Qx ∧R (λP. P x)

JaK = λPQ.∃x. P x ∧QxJeveryK = λPQ.∀x. P x ⊃ Qx

JitK = ???

Dynamic interpretation:

JfarmerK = farmer

JdonkeyK = donkeyJownsK = λOS. S (λx.O (λy.ownx y))

JbeatsK = λOS. S (λx.O (λy.beatx y))JwhoK = λRQx.Qx uR (λP. P x)

JaK = λPQ.Σx. P x uQxJeveryK = λPQ.Πx. P x A Qx

JitK = λPeφ. P (sel e) e φ

With the dynamic interpretation we have that:

JbeatsK JitK (JeveryK (JwhoK (JownsK (JaK JdonkeyK)) JfarmerK))

β-reduces to the following term (modulo de Morgan’s laws):

λeφ. (∀x. farmerx ⊃ (∀y.donkey y ⊃ (ownx y ⊃ beatx (sel (x::y::e))))) ∧ φ e

β-reduces to the following term (modulo de Morgan’s laws):

λeφ. (∀x. farmerx ⊃ (∀y.donkey y ⊃ (ownx y ⊃ beatx (sel (x::y::e))))) ∧ φ e

that is, assuming that sel is a “perfect” anaphora resolution operator:

λeφ. (∀x. farmerx ⊃ (∀y.donkey y ⊃ (ownx y ⊃ beatx y))) ∧ φ e

Observations

• Dynamic propositions are usually seen as contexts transformers.

Observations

• Here, we transform contexts (γ) into “type-raised” contexts ((γ → o)→ o).

Observations

• Interpreting (γ → o)→ o as a double negation, we may identify it with γ.

Observations

• To this end, we must use a calculus whose type system corresponds to classical logic.

Observations

• To this end, we must use a calculus whose type system corresponds to classical logic.

• The λµ-calculus is such a calculus.

Call by Value λµ-calculus

syntax

The set of λµ-terms is built upon two disjoint sets of variables (namely, λ-variables and µ)according to the following rules:

syntax

The set of λµ-terms is built upon two disjoint sets of variables (namely, λ-variables and µ)according to the following rules:

The set of values is defined as follows:

V ::= c V . . . V | x | λx. T

Intended operational meaning

(µa. . . . a (t1) . . . a (t2) . . . a (tn) . . .) u

(µa. . . . a (t1) . . . a (t2) . . . a (tn) . . .) u︸︷︷︸α→β

︸︷︷︸α

(µa. . . . a (t1) . . . a (t2) . . . a (tn) . . .) u︸︷︷︸α→β

︸︷︷︸α

α→β︷︸︸︷ α→β︷︸︸︷ α→β︷︸︸︷

(µa. . . . a (t1) . . . a (t2) . . . a (tn) . . .) u︸︷︷︸α→β

︸︷︷︸α

α→β︷︸︸︷ α→β︷︸︸︷ α→β︷︸︸︷

→ (µa. . . . a (t1 u) . . . a (t2 u) . . . a (tn u) . . .)

(µa. . . . a (t1) . . . a (t2) . . . a (tn) . . .) u︸︷︷︸α→β

︸︷︷︸α

α→β︷︸︸︷ α→β︷︸︸︷ α→β︷︸︸︷

→ (µa. . . . a (t1 u) . . . a (t2 u) . . . a (tn u) . . .)

Symmetrically:

f (µa. . . . a (t1) . . . a (t2) . . . a (tn) . . .) → (µa. . . . a (f t1) . . . a (f t2) . . . a (f tn) . . .)

Typing rules

Γ, x : α; ∆⊥ − x : α

Γ, x : α; ∆⊥ − t : β

Γ; ∆⊥ − λx. t : α→ β

Γ; ∆⊥ − t : α→ β Γ; ∆⊥ − u : α

Γ; ∆⊥ − t u : β

Typing rules

Γ, x : α; ∆⊥ − x : α

Γ, x : α; ∆⊥ − t : β

Γ; ∆⊥ − λx. t : α→ β

Γ; ∆⊥ − t : α→ β Γ; ∆⊥ − u : α

Γ; ∆⊥ − t u : β

Γ; ∆⊥, a : α⊥ − t : ⊥

Γ; ∆⊥ − µa. t : α

Γ; ∆⊥, a : α⊥ − t : α

Γ; ∆⊥, a : α⊥ − a (t) : ⊥

Typing rules

Γ, x : α; ∆⊥ − x : α

Γ, x : α; ∆⊥ − t : β

Γ; ∆⊥ − λx. t : α→ β

Γ; ∆⊥ − t : α→ β Γ; ∆⊥ − u : α

Γ; ∆⊥ − t u : β

Γ; ∆⊥, a : α⊥ − t : ⊥

Γ; ∆⊥ − µa. t : α

Γ; ∆⊥, a : α⊥ − t : α

Γ; ∆⊥, a : α⊥ − a (t) : ⊥

Γ; ∆⊥ − t : ⊥

Γ; ∆⊥ − 〈t〉 : ⊥

Reduction rules

βV -reduction:

(λx. t) v → t[x:=v] (v is a value)

Reduction rules

βV -reduction:

µ-reduction:

(µa. t)u→ µb. t[a ( ):=b ( u)]

Reduction rules

βV -reduction:

µ-reduction:

(µa. t)u→ µb. t[a ( ):=b ( u)]

µV -reduction:

v (µa. t)→ µb. t[a ( ):=b (v )] (v is a value)

Reduction rules

βV -reduction:

µ-reduction:

(µa. t)u→ µb. t[a ( ):=b ( u)]

µV -reduction:

Renaming:

a (µb. t)→ t[b:=a]

Reduction rules

βV -reduction:

µ-reduction:

(µa. t)u→ µb. t[a ( ):=b ( u)]

µV -reduction:

Renaming:

a (µb. t)→ t[b:=a]

ηµ-reduction:

µa. a (t)→ t (a does not occur in t)

Reduction rules

βV -reduction:

µ-reduction:

(µa. t)u→ µb. t[a ( ):=b ( u)]

µV -reduction:

Renaming:

a (µb. t)→ t[b:=a]

ηµ-reduction:

Resetµ:

〈µa. t〉 → 〈t[a ( ):= ]〉

Reduction rules

βV -reduction:

µ-reduction:

(µa. t)u→ µb. t[a ( ):=b ( u)]

µV -reduction:

Renaming:

a (µb. t)→ t[b:=a]

ηµ-reduction:

Resetµ:

〈µa. t〉 → 〈t[a ( ):= ]〉

ResetV :

〈v〉 → v (v is a value)

Representig Contexts as λµ-Terms

Assumptions

• We take ⊥ to be o;

Assumptions

• The type of dynamic proposition is defined as Ω = γ → γ

Assumptions

Dynamic interpretation of atomic propositions

Assumptions

R t1 . . . tn = λe. µc.R t1 . . . tn ∧ c (e)

Assumptions

R t1 . . . tn = λe. µc.R t1 . . . tn ∧ c (e)

Indeed:

e : γ; c : γo − R t1 . . . tn : o

e : γ; c : γo − e : γ

e : γ; c : γo − c (e) : o

e : γ; c : γo − R t1 . . . tn ∧ c (e) : o

e : γ; − µc.R t1 . . . tn ∧ c (e) : γ

− λe. µc.R t1 . . . tn ∧ c (e) : γ → γ

Composition of two dynamic propositions

λe.B (Ae)

Example:

λe.B (Ae)

Example:

λe. p2 (p1 e)

λe.B (Ae)

Example:

λe. p2 (p1 e) = λe. (λe2. µc2. p2 ∧ c2 (e2)) (p2 e)

λe.B (Ae)

Example:

= λe. (λe2. µc2. p2 ∧ c2 (e2)) ((λe1. µc1. p1 ∧ c1 (e1)) e)

λe.B (Ae)

Example:

→ λe. (λe2. µc2. p2 ∧ c2 (e2)) (µc1. p1 ∧ c1 (e))

λe.B (Ae)

Example:

→ λe. (λe2. µc2. p2 ∧ c2 (e2)) (µc1. p1 ∧ c1 (e))

→ λe. µc1. p1 ∧ c1 ((λe2. µc2. p2 ∧ c2 (e2)) e)

λe.B (Ae)

Example:

→ λe. (λe2. µc2. p2 ∧ c2 (e2)) (µc1. p1 ∧ c1 (e))

→ λe. µc1. p1 ∧ c1 ((λe2. µc2. p2 ∧ c2 (e2)) e)

→ λe. µc1. p1 ∧ c1 (µc2. p2 ∧ c2 (e))

λe.B (Ae)

Example:

→ λe. (λe2. µc2. p2 ∧ c2 (e2)) (µc1. p1 ∧ c1 (e))

→ λe. µc1. p1 ∧ c1 ((λe2. µc2. p2 ∧ c2 (e2)) e)

→ λe. µc1. p1 ∧ c1 (µc2. p2 ∧ c2 (e))

→ λe. µc1. p1 ∧ (p2 ∧ c1 (e))

Reading a dynamic proposition

READ eA = 〈(λx.>) (Ae)〉

Example:

READ e p

Example:

READ e p = 〈(λx.>) (p e)〉

Example:

READ e p = 〈(λx.>) (p e)〉= 〈(λx.>) ((λe. µc. p ∧ c (e)) e)〉

Example:

READ e p = 〈(λx.>) (p e)〉= 〈(λx.>) ((λe. µc. p ∧ c (e)) e)〉→ 〈(λx.>) (µc. p ∧ c (e))〉

Example:

READ e p = 〈(λx.>) (p e)〉= 〈(λx.>) ((λe. µc. p ∧ c (e)) e)〉→ 〈(λx.>) (µc. p ∧ c (e))〉→ 〈µc. p ∧ c ((λx.>) e)〉

Example:

READ e p = 〈(λx.>) (p e)〉= 〈(λx.>) ((λe. µc. p ∧ c (e)) e)〉→ 〈(λx.>) (µc. p ∧ c (e))〉→ 〈µc. p ∧ c ((λx.>) e)〉→ 〈µc. p ∧ c (>)〉

Example:

READ e p = 〈(λx.>) (p e)〉= 〈(λx.>) ((λe. µc. p ∧ c (e)) e)〉→ 〈(λx.>) (µc. p ∧ c (e))〉→ 〈µc. p ∧ c ((λx.>) e)〉→ 〈µc. p ∧ c (>)〉→ p ∧ >

Example:

READ e p = 〈(λx.>) (p e)〉= 〈(λx.>) ((λe. µc. p ∧ c (e)) e)〉→ 〈(λx.>) (µc. p ∧ c (e))〉→ 〈µc. p ∧ c ((λx.>) e)〉→ 〈µc. p ∧ c (>)〉→ p ∧ >≡ p

Dynamic Logic Revisited

Conjunction

Conjunction amounts to functional composition.

A uB , λe.B (Ae)

Conjunction

A uB , λe.B (Ae)

Conjunction

A uB , λe.B (Ae)

Existential quantification introduces “reference markers” by updating the context:

Σx. P x , λe. µc.∃x. c (P x (x::e))

Conjunction

A uB , λe.B (Ae)

Negation

Conjunction

A uB , λe.B (Ae)

Negation

The scope of the negation must be restricted to the current proposition and, consequently,“reset” the continuation:

∼ A , λe. µc.¬(READ eA) ∧ c (e)

They could be defined using de Morgan’s laws. Alternative (simpler) definitions are asfollows:

A A B , λe. µc.¬〈(λe.¬(READ eB)) (Ae)〉 ∧ c (e)

Πx. P x , λe. µc. (∀x.READ (x::e) (P x)) ∧ c (e)

Donkey Sentence Again

The dynamic lexical semantics is kept unchanged:

JfarmerK = farmer

JitK = λPe. P (sel e) e

JfarmerK = farmer

Then, we have that:

JfarmerK = farmer

Then, we have that:

reduces to the following term :

λe. µc. (∀x. farmerx ⊃ (∀y.donkey y ⊃ (ownx y ⊃ beat x (sel (x::y::e))))) ∧ c (e)

Conclusions

• The λµ-calculus allows propositions (o) and contexts (γ) to be mixed in a same term.

Conclusions

• To this end, the use of the “reset” operator is central.

Conclusions

• The reset operator we have used is rather “cheap”.

Conclusions

• The reset operator we have used is rather “cheap”.

• More powerful versions (w.r.t. typing) should allow other dynamic phenomena (e.g.definite clauses, focus, presuppositions) to be handled similarly.

ESSLLI’08: wsclsi 1 · ESSLLI’08: wsclsi 6 Typing the left and the right contexts Montague...

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