Dependent Type Semantics and its Davidsonian extensions

Dependent Types Dynamics in NLS DTS Event Semantics Summary References
Dependent type semantics
and its Davidsonian extensions
Daisuke Bekki
Ochanomizu University, Faculty of Core Science
A talk at Laboratoire de linguistique formelle
15 Dec, 2022.
1 / 58

Dependent Type Semantics (DTS) (Bekki 2014;
Bekki and Mineshima 2017; Bekki 2021)
I A framework of natural language semantics
I Unified approach to general inferences and
anaphora/presupposition resolution in terms of tpe checking
and proof search
Main features:
1. Proof-theoretic semantics:
From truth-conditions (denotations, models) to verification
conditions (proofs, contexts)
2. Anaphora/Presuppositions: A proof-theoretic alternative to
Dynamic Semantics (DRT, DPL, etc.)
3. Compositionality: Syntax-semantics interface via categorial
grammars (e.g. CCG, TLG, ACG, etc)
4. Computation: Implementation, Applications to Natural
Language Processing
2 / 58

Dependent Types
3 / 58

Per Martin-Löf
Martin-Löf (1984) “Intuitionistic type theory”
4 / 58

What are Π-types
Π-type is a type of fibred functions.
Simple function space Fibred function space
5 / 58

What are Σ-types
Σ-type is a type of fibred products.
Simple product space Fibred product space
6 / 58

Notations
DTS notation Standard notation x 6∈ fv(B) x ∈ fv(B)
(x : A) → B (Πx : A)B A → B (∀x : A)B
(x : A) × B
or

x : A
B
(Σx : A)B A ∧ B (∃x : A)B
Scope of the variable in Π-types: (x : A) → B
Scope of the variable in Σ-types:

x : A
B

7 / 58

Π-type F/I/E rules
A : s1
x : A
i
.
.
.
.
B : s2
(x : A) → B : s2
(ΠF),i
where (s1, s2) ∈

(type, type),
(type, kind)

.
A : type
x : A
i
.
.
.
.
M : B
λx.M : (x : A) → B
(ΠI ),i
M : (x : A) → B N : A
MN : B[N/x]
(ΠE)
8 / 58

Σ-type F/I/E rules
A : type
x : A
i
.
.
.
.
B : type
(x : A) × B : type
(ΣF),i
M : A N : B[M/x]
(M, N) : (x : A) × B
(ΣI )
M : (x : A) × B
π1(M) : A
(ΣE)
M : (x : A) × B
π2(M) : B[π1(M)/x]
(ΣE)
9 / 58

Rules of DTS
Rules from Martin-Löf Type Theory
I Axioms and Structural rules
I Π-type (Dependent function type) [F/I/E]
I Σ-type (Dependent product type) [F/I/E]
I Intensional equality type [F/I/E]
I Disjoint union type [F/I/E]
I Enumeration type [F/I/E]
I Natural number type [F/I/E]
New rule in DTS
I @ (the ‘asperand’ operator)
I Anaphora and presupposition triggers
(linguistically speaking)
I Open proofs (logically speaking)
10 / 58

Conjunction, Implication, and Negation
Definition

A
B

def
≡ (x : A) × B where x /
∈ fv(B).
A → B
def
≡ (x : A) → B where x /
∈ fv(B).
¬A
def
≡ (x : A) → ⊥
11 / 58

Dynamics in Natural Language
Semantics
12 / 58

A theory of anaphora
I Anaphora representable by a constant symbol:
I Deictic use:
(1) (Pointing at John)
He was born in Detroit.
bornIn( j , d)
I Coreference:
(2) John loves a girl who hates him .
∃x(girl(x) ∧ love( j , x) ∧ hate(x, j ))
I Anaphora representable by a variable
I Bound variable anaphora:
(3) Every boy loves his father.
∀x (boy(x) → love(x, fatherOf( x )))
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A theory of anaphora
I Anaphora not representable by FoL:
I E-type anaphora:
(4) A man entered into the park. He whistled.
I Donkey anaphora:
(5) Every farmer who owns a donkey beats it .
(6) If a farmer owns a donkey , he beats it .
I Anaphora not representable by FoL nor dynamic semantics:
I Syllogistic anaphora:
(7) Every girl received a present . Some girl opened it .
I Disjunctive antecedent:
(8) If Mary sees a horse or a pony , she waves to it .
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E-type anaphora: Evans (1980)
(9) [A man]1 entered. He1 whistled.
The first-order SR (10) represents the truth condition of (9), thus
is a candidate of the SR of (9).
(10) ∃x(man(x) ∧ enter(x) ∧ whistle(x))
But the syntactic structure of the SR (10) does not correspond to
that of (9), where consists of two independent sentences. The
sentential boundary of (9) prefers the first-order
representation (11).
(11) ∃x(man(x) ∧ enter(x)) ∧ whistle( x )
However, the truth condition of (11) is different from that of the
mini-discourse (9) since the variable x in whistle(x) is not bound
by ∃.
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Donkey anaphora: Geach (1962)
For the donkey sentences (12), a first-order formula (13), whose
truth condition is the same as those of (12), is a candidate of its
SR.
(12) a. Every farmer who owns [a donkey]1 beats it1.
b. If [a farmer]1 owns [a donkey]2 , he1 beats it2.
(13) ∀x(farmer(x) → ∀y (donkey(y) ∧ own(x, y) →
beat(x, y)))
But the translation from the sentence (12) to (13) is not
straightforward since i) the indefinite noun phrase a donkey is
translated into a universal quantifier in (13) instead of an
existential quantifier, and ii) the syntactic structure of (13) does
not corresponds to that of (12).
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Donkey anaphora: Geach (1962)
(12) a. Every farmer who owns [a donkey]1 beats it1 .
b. If [a farmer]1 owns [a donkey]2, he1 beats it2 .
The syntactic parallel of (12) is, rather, the SR (14), in which the
indefinite noun phrase is translated into an existential
quantification.
(14) ∀x(farmer(x) ∧ ∃y(donkey(y) ∧ own(x, y)) →
beat(x, y ))
However, (14) does not represent the truth condition of (12)
correctly since the variable y in beat(x, y) fails to be bound by ∃.
Therefore, neither (13) nor (14) qualifies as the SR of (12).
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E-type anaphora: Ranta (1994)
(9) A man entered. He whistled.







u :



x : entity

man(x)
enter(x)




whistle( π1(u) )







Note:

x : A
B

is a type for pairs of A and B[x].
18 / 58

Donkey anaphora: Sundholm (1986)
(12a) Every farmer who owns a donkey beats it .







u :







x : entity





farmer(x)



v :

y : entity
donkey(y)
#
own(x, π1v)






















→ beat(π1u, π1π1π2π2u
Note: (x : A) → B is a type for functions from A to B[x].
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From TTG to DTS: Compositionality
Q: How could one get to these (dependently-typed)
representations from arbitrary sentences?
A: By lexicalization.
Q: How could we lexicalize context-dependent words like
pronouns?







u :



x : entity

man(x)
enter(x)




whistle( π1(u) )







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Dependent Type Semantics (DTS)
21 / 58

From TTG to DTS: Compositionality
Q: How could one get to these (dependently-typed)
representations from arbitrary sentences?
A: By lexicalization.
Q: How could we lexicalize context-dependent words like
pronouns?
A: By using underspecified terms.
Q: But how could we retrieve a context for an
underspecified term?
A: By type checking.
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Underspecified terms
DTS = DTT + underspecified terms @A
Definition (@-rule)
A : typej A true
@iA : A
where j ∈ N
(@)
I @-rule states that the well-formedness of @A requires:
I A is a well-formed type.
I the inhabitance of A (namely, A is a presuppositional content)
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On parsing
We assume that an SR of a sentence is obtained by parsing and
semantic composition (assumed by one’s syntactic theory).
A sentence
⇓
Parsing
⇓
An underspecified SRs in DTS
24 / 58

Lexical items in CCG-style
Surface form Syntactic category Semantic representation
every
(
T /(T NP)/N
T (T /NP)/N
λn.λp.λ~
x.

u :

x : entity
n(x)

→ p(π1u)~
x
a, some
(
T /(T NP)/N
T (T /NP)/N
λn.λp.λ~
x.

 u :

x : entity
n(x)

p(π1u)~
x


if
(
S/S/S
SS/S
λp.λq. (u : p) → q
who/whom
(
NN/(SNP )
NN/(S/NP )
λp.λn.λx.

nx
px

farmer N farmer
donkey N donkey
owns SNP/NP own
beats SNP/NP beat
he/him
(
T /(T NP) nom
T (T /NP) acc
λp.λ~
x.

 u@

x : entity
male(x)

p(π1u)~
x


it
(
T /(T NP) nom
T (T /NP) acc
λp.λ~
x.

 u@

x : entity
¬human(x)

p(π1u)~
x


the
(
T /(T NP)/N nom
T /(T NP)/N acc
λn.λp.λ~
x.

 u@

x : entity
n(x)

p(π1u)~
x


25 / 58

Lexical items in CCG-style (anaphoric expressions)
PF CCG categories Semantic representations in DTS
he NP π1 @

x : entity
male(x)
!
it NP π1 @

x : entity
¬human(x)
!
the NP/N λn.π1 @

x : entity
nx
!
26 / 58

E-type anaphora: Parsing
A
S/(SNP)/N
λn.λp.

 u :

x : entity
nx

p(π1u)


man
N
λx.man(x)
S/(SNP)
λp.

 u :

x : entity
man(x)

p(π1u)



entered
SNP
λx.enter(x)
S

 u :

x : entity
man(x)

enter(π1u)



27 / 58

E-type anaphora: Parsing
He
NP
π1 @

x : entity
male(x)

whistled
SNP
λx.whistle(x)
S
whistle π1 @

x : entity
male(x)
!

28 / 58

Progressive conjunction: Ranta (1994)
Definition (Progressive conjunction)
M; N
def
≡

u : M
N

where u /
∈ fv(N)
(9) [A man]1 entered. He 1 whistled.


x : entity

man(x)
enter(x)


 ; whistle(π1 @

x : entity
male(x)

)
=







u :


x : entity

man(x)
enter(x)



whistle(π1 @

x : entity
male(x)

)







29 / 58

E-type anahora: Type checking
.
.
.

 u :

x : entity
man(x)

enter(π1u)

 : type
whistle
: entity
→ type
(CON )
.
.
.

x : entity
male(x)

: type
v :

 u :

x : entity
man(x)

enter(π1u)


1
.
.
.
.

x : entity
male(x)

true
@

x : entity
male(x)

:

x : entity
male(x)
(@)
π1 @

x : entity
male(x)

: entity
(ΣE)
whistle π1 @

x : entity
male(x)
!
: type
(→E)







v :

 u :

x : entity
man(x)

enter(π1u)



x : entity
!







: type
(ΣF),1
30 / 58

E-type anaphora: Proof search
v :

 u :

x : entity
man(x)

enter(π1u)


1
π1v :

x : entity
man(x)
(ΣE)
π1π1v : entity
(ΣE)
v :

 u :

x : entity
man(x)

enter(π1u)


1
π1v :

x : entity
man(x)
(ΣE)
m :

u :

x : entity
man(x)

→ male(π1
m(π1v) : male(π1π1v)
(π1π1v, m(π1v)) :

x : entity
male(x)
(ΣE)
31 / 58

E-type anaphora: @-elimination
.
.
.
.

 u :

x : entity
man(x)

enter(π1u)

 : type
whistle : entity → type
v :

 u :

x : entity
man(x)

enter(π1u)


1
.
.
.
.
(π1π1v, m(π1v)) :

x : entity
male(x)

π1π1v : entity
(ΣE)
whistle( π1π1v ) : type
(ΠE)




v :

 u :

x : entity
man(x)

enter(π1u)


whistle( π1π1v )



 : type
(ΣF),1
32 / 58

@-elimination rules
Definition (@-elimination rules (excerpt))
u
w
v
DA
Γ ` A : s Γ ` M : JDAK
Γ ` ( @A ) : A
(@)
}

~ = M
u
w
w
v
DA
A : s1
x : A0
DB
B : s2
(x : A) → B : s2
(ΠF)
}

~ =
JDAK
A0 : s1
x : A0
JDBK
B0 : s2
(x : A0) → B0 : s2
(ΠF)
u
w
w
v
DA
A : s1
x : A0
DM
M : B
λx.M : (x : A) → B
(ΠI )
}

~ =
JDAK
A0 : s1
x : A0
JDM K
M0 : B0
λx.M0 : (x : A0) → B0
(ΠI )
u
w
v
DM
M : (x : A) → B
DN
N : A
MN : B0
(ΠE)
}

~ =
JDM K
M0 : (x : A0) → B0
JDN K
N0 : A0
M0N0 : B00
where B0[N0/x] β B00
(ΠE)
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The model of language understanding
A sentence . . . A sentence
⇓ ⇓
Parsing . . . Parsing
⇓ ⇓
An underspecified SRs in UDTT . . . An underspecified SRs in UDTT
⇓ ⇓
Discoruse relation (e.g. Progressive conjunction)
⇓
An underspecified discourse representation in UDTT
⇓
Type checking + Proof search in UDTT
⇓
A proof diagram of the well-formedness of an SR in UDTT
⇓
@-elimination
⇓
A proof diagram of the well-formedness of an SR in DTT
⇓
Inference in DTT
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Event Semantics
35 / 58

Davidsonian Extension
(15) John loves Mary.
(16) Classical DTS:
love(john, mary)
(17) Davidsonian DTS:

e : entity
love(e, john, mary)

(18) Neo-Davidsonian DTS:




e : Entity
love(e)
Ag(e) =entity john
Th(e) =entity mary)




36 / 58

Event Continuation: Champollion (2015),
Mineshima et al. (2015)
John
S/(SNP)
λp.p(john)
runs
SNP
λx λk


e : entity
run(e, x)
k (e)


slowly
SNP(SNP)
λp.λx.λk.p(x)

λe.

slowly(e)
k(e)

SNP
λx.λk.




e : entity
run(e, x)
slowly(e)
k(e)





S
λk.




e : entity
run(e, john)
slowly(e)
k(e)





U




e : entity
run(e, john)
slowly(e)




CL
37 / 58

Event Anaphora
(19) John ran slowly. Mary saw it.


e : entity
run(e, john)
slowly(e)

 ;



e : entity
see(e, mary, π1 @

e : entity
event(e)




=









u :


e : entity
run(e, john)
slowly(e)





e : entity
see(e, mary, π1 @

e : entity
event(e)













38 / 58

Event Anaphora
(20) John ran slowly. Mary saw it.


e : entity
run(e, john)
slowly(e)

 ;



e : entity
see(e, mary, π1 @

e : entity
event(e)




=






u :


e : entity
run(e, john)
slowly(e)



e : entity
see(e, mary, π1u )







Given that:
` (e : entity) → (x : entity) → run(e, x) → event(e)
39 / 58

Negative Events
Higginbotham (1983), Bernard and Champollion (2018):
(21) a. John saw Mary not leave.
b.
∃e








actual(e)∧
see(e)∧
exp(e) = John∧
∃e0


th(e) = e0∧
e0 ∈ Neg

λe00.
leave(e00)∧
ag(e00) = mary











cf. Axiom of Negation (Higginbotham 2000):
∃e ∈ Neg(P ).actual(e) ⇐
⇐
∀e0
∈ P.¬actual(e0
)
40 / 58

Negative Events in DTS
(21a) John saw Mary not leave.
Proposal 1: ‘Neg’-event as a proof-term of a negative proposition




e : entity
ne : ¬

e0 : entity
leave(e0, mary)

see(John, e, ne)




I see becomes polymorphic
41 / 58

Negative Events in DTS
(21a) John saw Mary not leave.
Proposal 2: ‘Neg’-event as a universe






e : entity
ne : U
dec (ne) =type ¬

e0 : entity
leave(e0, mary)

see(John, e, ne)






I see becomes monomorphic
I sorttype-level equation is used
I Some postulate is necessary to imply the factivity
42 / 58

Summary
43 / 58

Compositional Theory of Anaphora
I DTS provides a unified analysis for (general) inferences and
anaphora resolusion mechanisms (at least) for:
I Deictic use and coreference
I Bound variable anaphora (BVA)
I E-type anaphora
I Donkey anaphora
I Bridging anaphora
I Syllogistic anaphora
I Disjunctive antecedents
44 / 58

Compositional Theory of Anaphora
I The background theory for DTS is an extention of DTT with
underspecified terms and the @-rule .
I Lexical items of anaphoric expressions and presupposition
triggers are represented by using underspecified terms.
I Context retrieval in DTS reduces to type checking .
I Anaphora resolution and presupposition binding in DTS
reduces to proof search .
I @-elimination translates a proof diagram of DTS into a proof
diagram of DTT, by which an SR in DTT is obtained with all
anaphora resolved.
45 / 58

Natural language semantics via dependent types:
Classics
I Donkey anaphora: Sundholm (1986)
I Translation from DRS to dependent type representations: Ahn
and Kolb (1990)
I Summation: Fox (1994a,b)
I Ranta’s TTG (Relative and Implicational Donkey Sentences,
Branching Quantifiers, Intensionality, Tense): Ranta (1994)
I Translation from Montague Grammar to dependent type
representations: Dávila-Pérez (1995)
I Presupposition Binding and Accommodation, Bridging:
Krahmer and Piwek (1999), Piwek and Krahmer (2000)
46 / 58

Natural language semantics via dependent types:
Recent frameworks
I Type Theory with Record (TTR): Cooper (2005)
I Modern Type Theory: Luo (1997, 1999, 2010, 2012), Asher
and Luo (2012), Chatzikyriakidis (2014)
I Semantics with Dependent Types: Grudzinska and
Zawadowski (2014; 2017)
I Dependent Type Semantics (DTS): Bekki (2014), Bekki
and Mineshima (2017)
I (Dynamic Categorial Grammar: Martin and Pollard (2014))
47 / 58

Semantic Analyses by DTS
I Generalized Quantifiers : Tanaka (2014)
I Honorification : Watanabe et al. (2014)
I Conventional Implicature : Bekki and McCready (2015)
I Factive Presuppositions : Tanaka et al. (2015)(2017)
I Dependent Plural Anaphora : Tanaka+(2017)
I Paycheck sentences : Tanaka+(2018) in NLCS2018
I Coercion and Metaphor : Kinoshita+(2018)
I Questions : Watanabe+(NLCS’19), Funakura (2022) in
LENLS19
I Comparision with DRT : Yana+(2019) in JoLLI
48 / 58

Semantic Analyses by DTS
I Development of an automated theorem prover for the
fragment of DTS: Daido and Bekki (2020) in LENLS17
I A Proof-theoretic Analysis of Weak Crossover : Bekki (2021)
in LENLS18
I The proviso problem from a proof-theoretic perspective:
Yana+(2021) in LACL2021
I Integrating Deep Neural Network with Dependent Type
Semantics: Bekki+(2021) in LACompLing2021,
Bekki+(2022) in NALOMA’22
49 / 58

Thank you!
50 / 58

Reference I
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Reference II
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Reference III
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Reference IV
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Reference V
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Reference VI
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Reference VII
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Reference VIII
Watanabe, N., E. McCready, and D. Bekki. (2014) “Japanese
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