From Dependent Types to Natural Language Semantics

Dependent Types Anaphora DTS on Anaphora Presupposition DTS on Presupp. Summary
From Dependent Types
to Natural Language Semantics
Daisuke Bekki
Ochanomizu University
Faculty of Core Research
https://daisukebekki.github.io/
A plenary talk at Logic Colloquim 2024
25 June (Tue)
1 / 96

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 type checking
and proof search
Main features:
1. Proof-theoretic semantics:
From model theory (denotations and models) to proof theory
(proofs and 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. Implementation: Applications to Natural Language
Processing.
https://github.com/DaisukeBekki/lightblue/ 2 / 96

Dependent Types
3 / 96

Per Martin-Löf
Martin-Löf (1984) “Intuitionistic type theory”
4 / 96

What are Π-types
Π-type is a type of fibred functions.
Simple function space Fibred function space
5 / 96

What are Σ-types
Σ-type is a type of fibred products.
Simple product space Fibred product space
6 / 96

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 / 96

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),
(kind, 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 / 96

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 / 96

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 / 96

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 / 96

Anaphora in Natural Language
12 / 96

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 )))
13 / 96

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 .
14 / 96

Donkey anaphora: Geach (1962)
For the donkey sentences (9), a first-order formula (10), whose
truth condition is the same as those of (9), is a candidate of its
semantic representation (SR). (We only discuss its strong reading here. See Tanaka (2021)
for its weak reading.)
(9) a. Every farmer who owns [a donkey]1 beats it1.
b. If [a farmer]1 owns [a donkey]2 , he1 beats it2.
(10) ∀x(farmer(x) → ∀y (donkey(y) ∧ own(x, y) →
beat(x, y)))
But the translation from the sentence (9) to (10) is not
straightforward since i) the indefinite noun phrase a donkey is
translated into a universal quantifier in (10) instead of an
existential quantifier, and ii) the syntactic structure of (10) does
not corresponds to that of (9).
15 / 96

Donkey anaphora: Geach (1962)
(9) 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 (9) is, rather, the SR (11), in which the
indefinite noun phrase is translated into an existential
quantification.
(11) ∀x(farmer(x) ∧ ∃y(donkey(y) ∧ own(x, y)) →
beat(x, y ))
However, (11) does not represent the truth condition of (9)
correctly since the variable y in beat(x, y) fails to be bound by ∃.
Therefore, neither (10) nor (11) qualifies as the SR of (9).
16 / 96

Various approaches in discourse semantics
Dynamic Semantics
I Discourse Representation Theory (DRT): Kamp (1981),
Kamp and Reyle (1993)
I Dynamic Predicate Logic (DPL): Groenendijk and Stokhof
(1991)
I Dynamic Plural Predicate Logic (DPPL): van den Berg
(1996), Sudo (2012)
Type-theoretical Semantics
I Analysis of donkey anaphora: Sundholm (1986))
I Type Theoretical Grammar (TTG): Ranta (1994)
I Type Theory with Record (TTR): Cooper (2005)
I Modern Type Theory: Luo (1997, 1999, 2010, 2012), Asher
and Luo (2012), Chatzikyriakidis (2014)
I Dependent Type Semantics (DTS): Bekki (2014), Bekki and
Mineshima (2017)
17 / 96

Donkey anaphora: Sundholm (1986)
(9a) 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].
18 / 96

From TTG to DTS: Compositionality
Q: How could one get to these (dependently-typed)
representations from arbitrary sentences?
A: By lexicalization.
Q: But, how could we lexicalize context-dependent
words like pronouns?







u :







x : entity





farmer(x)



v :

y : entity
donkey(y)
#
own(x, π1v)






















→
beat(π1u, π1π1π2π2u )
19 / 96

From TTG to DTS: Compositionality
Q: How could one get to these (dependently-typed)
representations from arbitrary sentences?
A: By lexicalization.
Q: But, how could we lexicalize context-dependent
words like pronouns?
A: By using underspecified types.
Q: How could we retrieve a context for an underspecified
type?
A: By type checking.
20 / 96

Dependent Type Semantics (DTS)
21 / 96

Underspecified types
DTS = DTT + underspecified types
Definition (@-rule)
A : type M : A B[M/x] : type

x@A
B

: type
(@)
I @-rule states that the well-formedness of (x@A) × B
requires:
I A is a well-formed type
I the inhabitance of a proof (let it be M) of A, checking of
which launches a proof search
I B[M/x] is a well-formed type
This means that the truth of A is presupposed.
22 / 96

A model of natural language understanding via CCG
and DTS
A sentence . . . A sentence
⇓ ⇓
CCG Parsing . . . CCG Parsing
⇓ ⇓
An underspecified SRs in DTS . . . An underspecified SRs in DTS
⇓ ⇓
Discoruse Parsing
⇓
An underspecified discourse SRs in DTS
⇓
Type checking + Proof search
⇓
A proof diagram of the well-formedness of an SRs in DTT
⇓
Inference in DTT
23 / 96

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


24 / 96

Donkey anaphora: Syntactic Structure
Every
T /(T NP)/N
farmer
N
who
NN/(SNP)
owns
SNP/NP
a
T (T /NP)/N
donkey
N
T (T /NP)
SNP(SNP/NP)
∀E
SNP

NN

N

T /(T NP)

S/(SNP)
∀E
beat
SNP/NP
it
T (T /NP)
SNP(SNP/NP)
∀E
SNP

S

S̄
CC
25 / 96

Donkey anaphora: Semantic Composition (1/2)
farmer
farmer
who
λp.λn.λx.

n(x)
p(x)

owns
λy.λx.
own(x, y)
a
λn.λp.

 v :

y : entity
n(y)

p(π1v)


donkey
donkey
λp.

 v :

y : entity
donkey(y)

p(π1v)



λx.

 v :

y : entity
donkey(y)

own(x, π1v)



λn.λx.




n(x)
v :

y : entity
donkey(y)

own(x, π1v)





λx.




farmer(x)
v :

y : entity
donkey(y)

own(x, π1v)





26 / 96

Donkey anaphora: Semantic Composition (2/2)
Every
λn.λp.

u :

x : entity
n(x)

→ p(π1u)
farmer who owns a donkey
λx.




farmer(x)
v :

y : entity
donkey(y)

own(x, π1v)




λp.






u :






x : entity
farmer(x)
v :

y : entity
donkey(y)

own(x, π1v)












→ p(π1u)

beat
λy.λx.
beat(x, y)
it
λp.λx.



w@

z : entity
¬human(z)

p(π1w)x



λx.



w@

z : entity
¬human(z)

beat(x, π1w)










u :






x : entity
farmer(x)
v :

y : entity
donkey(y)

own(x, π1v)












→



w@

z : entity
¬human(z)

beat(π1u, π1w)




27 / 96

Donkey anaphora: Semantic Felicity Condition
(=Type Checking)
.
.
.
.






x : entity
farmer(x)
v :

y : entity
donkey(y)

own(x, π1v)






: type
.
.
.
.

z : entity
¬human(z)

: type
u :






x : entity
farmer(x)
v :

y : entity
donkey(y)

own(x, π1v)






1
.
.
.
.
? :

z : entity
¬human(z)

.
.
.
.
(beat(π1u, π1w))[?/w] : type



w@

z : entity
¬human(z)

beat(π1u, π1w)


 : type
(






u :






x : entity
farmer(x)
v :

y : entity
donkey(y)

own(x, π1v)












→



w@

z : entity
¬human(z)

beat(π1u, π1w)


 : type
(ΠI ),1
28 / 96

Donkey anaphora: Anaphora Resolusion (=Proof
Search)
u :






x : entity
farmer(x)
v :

y : entity
donkey(y)

own(x, π1v)






1
π2u :




farmer(x)
v :

y : entity
donkey(y)

own(π1u, π1v)




(ΣE)
π2π2u :

 v :

y : entity
donkey(y)

own(π1u, π1v)


(ΣE)
π1π2π2u :

y : entity
donkey(y)
(ΣE)
π1π1π2π2u : entity
(ΣE)
u :






x : entity
farmer(x)
v :

y : entity
donkey(y)

own(x, π1v)






1
π2u :




farmer(x)
v :

y : entity
donkey(y)

own(π1u, π1v)




(ΣE)
π2π2u :

 v :

y : entity
donkey(y)

own(π1u, π1v)


(ΣE)
π1π2π2u :

y : entity
donkey(y)
(ΣE)
d :

u :

x : entity
donkey(x)

→ ¬human(π1u)
(CON )
d(π1π2π2u) : ¬human(π1π1π2π2u)
(ΠE)
(π1π1π2π2u, d(π1π2π2u)) :

z : entity
¬human(z)
(ΣI )
29 / 96

Donkey anaphora: Semantic Felicity Condition
(=Type Checking) continued
.
.
.
.






x : entity
farmer(x)
v :

y : entity
donkey(y)

own(x, π1v)






: type
u :






x : entity
farmer(x)
v :

y : entity
donkey(y)

own(x, π1v)






1
.
.
.
.
beat(π1u, π1π1π2π2u) : type






u :






x : entity
farmer(x)
v :

y : entity
donkey(y)

own(x, π1v)












→ beat(π1u, π1π1π2π2u) : type
(beat(π1u, π1w))[ (π1π1π2π2u, d(π1π2π2u)) /w] : type
β beat(π1u, π1π1π2π2u)
(ΠI ),1
30 / 96

Notes on donkey anaphora
I The same analysis applies to the implicational donkey
sentence:
(12) If [a farmer]1 owns [a donkey]2, he1 beats it2.
I This analysis only predicts the strong reading for donkey
sentences. For deriving both the strong readings and the weak
readings, we need to refine the semantic representations for
quantificational expressions: See Tanaka (2021).
I The refined analysis also explains why the anaphoric link to
the parametrized sum individual (Krifka, 1996) is allowed
(Tanaka, 2021).
(13) [Every farmer]1 who owns [a donkey]2 loves its2 tail.
But they1 beat it2.
31 / 96

Presupposition Joint work with Koji Mineshima
32 / 96

What is Presupposition? — Background content
(14) It is John who broke my iPhone.
Presupposition: Someone broke my iPhone.
I the background content
I its truth is usually taken for granted
Assertion: John was the one who did it.
I the foreground content
I the main point of an utterance
33 / 96

Two puzzules of Presupposition – (i) Projection
(14) It was John who broke my iPhone.
(15) Someone broke my iPhone. ((14) presupposes (15))
(15) projects out of all the embedded contexts in (16a–e).
(16) a. It wasn’t John who broke my iPhone. negation
b. Maybe it was John who broke my iPhone. modal
c. If it was John who broke my iPhone, then he has to fix
it. the antecedent of a conditional
d. Was it John who broke my iPhone? question
e. Suppose that it was John who broke my iPhone.
hypothetical assumption
34 / 96

The Case of Entailment
(17) John is an American pianist.
(18) John is American. ((17) entails (18))
(18) does not survive in the contexts (19a–e).
(19) a. John is not an American pianist. negation
b. Maybe John is an American pianist. modal
c. If John is an American pianist, he is skillful.
the antecedent of a conditional
d. Is John an American pianist? question
e. Suppose that John is an American pianist.
hypothetical assumption
35 / 96

The Case of Entailment
(17) John is an American pianist.
american(john) ∧ pianist(john)
(19) a. John is not an American pianist.
¬(american(j) ∧ pianist(j))
b. Maybe John is an American pianist.
3(american(j) ∧ pianist(j))
c. If John is an American pianist, he is skillful.
american(j) ∧ pianist(j) → skillful(j)
Standard semantics correctly predicts these patterns:
I (17) ` american(john)
I (19a) 6` american(john)
I (19b) 6` american(john)
I (19c) 6` american(john)
36 / 96

The Case of Presupposition
(14) It was John who broke my iPhone. SR1
(16) a. It wasn’t John who broke my iPhone. ¬SR1
b. Maybe it was John who broke my iPhone. 3SR1
c. If it was John who broke my iPhone, he has to fix it.
SR1 → · · ·
What SR accounts for the following inference patterns?
I SR1 ∃x(broke(x, my iphone))
I ¬SR1 ∃x(broke(x, my iphone))
I 3SR1 ∃x(broke(x, my iphone))
I SR1 →A ∃x(broke(x, my iphone))
Q: Can “ ” be defined as a standard consequence relation ”`”?
A: No. If that were the case, then ∃x(broke(x, my iphone)) was
a tautology (under the classical setting).
37 / 96

Two puzzules of Presupposition – (ii) Filteration
(20) presupposes that someone broke the window, but the
conditional in (21) does not inherit this presupposition.
(20) It was John who broke the window.
⇐
Someone broke the window
(21) If the window was broken, it was John who broke it.
6
⇐
Someone broke the window
Similarly for (22) and (23).
(22) The king of France is wise.
⇐
France has a king.
(23) If France has a king, the king of France is wise.
6
⇐
France has a king.
A presupposition is filtered when it occurs in certain contexts.
38 / 96

Presupposition triggers
(24) a. The elevator in this building is clean. Description
b. There is an elevator in this building.
(25) a. John’s sister is happy. Possessive
b. John has a sister.
(26) a. Bill regrets that he lied to Mary. Factive
b. Bill lied to Mary.
(27) a. John has stopped beating his wife. Aspectual
b. John has beaten his wife.
(28) a. Harry managed to find the book. Implicative
b. Finding the book required some effort.
39 / 96

Presupposition triggers
(29) a. Sam broke the window again today. Iterative
b. Sam broke the window before.
(30) a. It was Sam who broke the window. Cleft
b. Someone broke the window.
(31) a. What John broke was his typewriter. Pseudo-cleft
b. John broke something.
(32) a. [Pat]F is leaving, too. (Focus on Pat) Additive
b. Someone other than Pat is leaving.
For classical examples of presupposition triggers, see Levinson
(1983), Soames (1989), Geurts (1999), and Beaver (2001), among
others.
40 / 96

“Presupposition Is Anaphora”
hypothesis
41 / 96

“Presupposition Is Anaphora” hypothesis
There are striking parallels between anaphoric expressions and
presupposition triggers. (van der Sandt, 1992; Geurts, 1999)
Presupposition filtering:
(33) a. John has children and John’s children are wise.
b. If John has children, John’s children are wise.
(34) a. The window was broken and it was John who broke it.
b. If the window was broken, it was John who broke it.
Compare (33) and (34) with the paradigm examples of anaphora
resolution.
Anaphora resolution:
(35) a. John owns a donkey and he beats it.
b. If John owns a donkey, he beats it.
42 / 96

DTS on Projection
I The projection inferences of presupposition can be naturally
accounted for using the framework of DTS.
I Note that negation is defined to be an implication of the form
¬A ≡ A → ⊥.the formation rule for negation can be
derived as on the right:
I According to the formation rule (¬F ) for negation, the
proposition A and its negation ¬A have the same
presupposition.
Example:
It is not the case that the king of France is bald.
SR ¬



u@

x : entity
king(x, fr)

bold(π1u)



43 / 96

Projection: Syntax
It is not
the case that
S/S
the
S/(SNP)/N
king
N/PPof
of
PPof /NP
France
NP
PPof

N

S/(SNP)

is
SNP/(SNP)
bald
SNP
SNP

S

S

44 / 96

Projection: Semantic Composition
It is not
the case that
λp.¬p
the
λn.λp.



u@

x : entity
n(x)

p(π1u)



king
λz.λx.
kingOf(x, z)
of
id
France
fr
fr

λx.kingOf(x, fr)

λp.



u@

x : entity
kingOf(x, fr)

p(π1u)



is
id
bald
bald
bald




u@

x : entity
kingOf(x, fr)

bald(π1u)




¬



u@

x : entity
kingOf(x, fr)

bald(π1u)




45 / 96

Projection: Type checking
.
.
.
.

x : entity
kingOf(x, fr)

? :

x : entity
kingOf(x, fr)

¬bald(π1u)[ ? /u] : type



u@

x : entity
kingOf(x, fr)

¬bald(π1u)


 : type
(@)
⊥ : type
({}F)
¬



u@

x : entity
kingOf(x, fr)

¬bald(π1u)


 : type
(ΠF)
I In order for the sentence “The king of France is bald” to be
well-formed, the context Γ must be such that the following
type inhabits a proof (namely, there exists a king of France).
Γ `? :

x : entity
kingOf(x, fr)

46 / 96

DTS on Projection
I The same inference is triggered for the antecedent of a
conditional sentence like (36):
(36) If the king of France is wise, people will be happy.
.
.
.
.



u :

x : entity
kingOf(x, fr)

wise(π1u)


 : type
.
.
.
.
happy(people) : type



u :

x : entity
kingOf(x, fr)

wise(π1u)


 → happy(people) : type
(ΠF)
47 / 96

DTS on Filtering
I The present account can explain the filtering inference
without further stipulation.
I Take a look at the case of a conditional sentence:
(37) If France has a king, the king of France is wise.
SR

u :

x : entity
kingOf(x, fr)

→



v@

x : entity
kingOf(x, fr)

wise(π1v)



48 / 96

Filtering: Type checking
.
.
.
.

x : entity
kingOf(x, fr)

: type
.
.
.
.

x : entity
kingOf(x, fr)

: type
u :

x : entity
kingOf(x, fr)
1
.
.
.
.
? :

x : entity
kingOf(x, fr)

.
.
.
.
wise(π1v)[ ? /v]



v@

x : entity
kingOf(x, fr)

wise(π1v)



(@)

u :

x : entity
kingOf(x, fr)

→



v@

x : entity
kingOf(x, fr)

wise(π1v)


 : type
(ΠF),1
49 / 96

DTS on Filtering
.
.
.
.

x : entity
kingOf(x, fr)

: type
wise : entity → type
(CON )
u :

x : entity
kingOf(x, fr)
1
π1u : entity
(ΣE)
wise(π1u) : type
(ΠE)

u :

x : entity
kingOf(x, fr)

→ wise(π1u) : type
(ΠF),1
I Type checking algorithm returns a fully-specified semantic
representation.
I Presupposition filtering is performed via exactly the same
process as anaphora resolution.
50 / 96

Summary and History
51 / 96

A Unified, Compositional Theory of Projective
Meaning
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
52 / 96

A Unified, Compositional Theory of Projective
Meaning
I The background theory for DTS is an extention of DTT with
underspecified types and the @-rule .
I Lexical items of anaphoric expressions and presupposition
triggers are represented by using underspecified types.
I Context retrieval in DTS reduces to type checking .
I Anaphora resolution and presupposition binding in DTS
reduces to proof search .
I Type checker translates a proof diagram of DTS into a proof
diagram of DTT, by which an SR in DTT is obtained with all
anaphora resolved.
53 / 96

Natural language semantics via dependent types:
Early works
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)
54 / 96

Natural language semantics via dependent types:
The second generation
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 Dynamic Categorial Grammar: Martin and Pollard (2014)
I Dependent Type Semantics (DTS): Bekki (2014), Bekki
and Mineshima (2017)
55 / 96

Semantic Analyses by DTS
I Generalized Quantifiers: Tanaka (2014)
I Honorification: Watanabe et al. (2014)
I Conventional Implicature: Bekki and McCready (2015),
Matsuoka et al. (2023)
I Factive Presuppositions: Tanaka et al. (2015)
I Dependent Plural Anaphora: Tanaka et al. (2017)
I Paycheck sentences: Tanaka et al. (2018)
I Coercion and Metaphor: Kinoshita et al. (2017, 2018)
I Questions: Watanabe et al. (2019), Funakura (2022)
I Comparision with DRT: Yana et al. (2019)
I The proviso problem: Yana et al. (2021)
I Weak Crossover: Bekki (2023)
56 / 96

Computational Aspects of DTS
I Type Checker for (the fragment of) DTS: Bekki and Sato
(2015)
I Development of an automated theorem prover (for the
fragment of) DTS: Daido and Bekki (2020)
I Integrating Deep Neural Network with DTS: Bekki et al.
(2023, 2022)
57 / 96

Thank you!
58 / 96

Type System Type Check/Inference Algorithm References
Appendix I: Type System in DTS
59 / 96

Axioms and Structural Rules
x : A
(VAR)
c : A
(CON )
where (c : A) ∈ σ.
type : kind
(typeF)
M : A N : B
M : A
(WK)
M : A
M : B
(CONV )
where A =β B.
60 / 96

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),
(kind, 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)
61 / 96

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)
62 / 96

Disjoint Union Type F/I/E rules
A : type B : type
A + B : type
(+F)
M : A
ι1(M) : A + B
(+I )
N : B
ι2(N) : A + B
(+I )
M : A + B P : (A + B) → type N1 : (x : A) → P (ι1(x)) N2 : (x : B) → P (ι2(x))
unpackP
M (N1, N2) : P (M)
(+E),i
63 / 96

Natural Number Type F/I/E rules
N : type
(NF)
0 : N
(NI )
n : N
s(n) : N
(NI )
n : N P : N → type e : P (0) f : (k : N) → P (k) → P (s(k))
natrecP
n (e, f) : P (n)
(NE)
64 / 96

Enumeration Type F/I/E rules
{a1, . . . , an} : type
({}F)
ai : {a1, . . . , an}
({}I )
M : {a1, . . . , an} P : {a1, . . . , an} → type N1 : P (a1) . . . Nn : P (an)
caseP
M (N1, . . . , Nn) : P (M)
({}E)
65 / 96

Intensional Equality Type F/I/E rules
A : type M : A N : A
M =A N : type
(=F)
M : A
reflA(M) : M =A M
(=I )
E : M =A N P : (x : A) → (y : A) → (x =A y) → type R : (x : A) → P xx(reflA(x))
idpeelP
E (R) : P MNE
(=E)
66 / 96

@-rule
A : type M : A B[M/x] : type

x@A
B

: type
(@)
67 / 96

Appendix II: Type Check/Inference
Algorithm
68 / 96

Inferable terms (1/2)
The collection of inferable terms (notation M↑ ) and the
collection of checkable terms (notation M↓) are simultaneously
defined by the following BNF notations, where v, v0 are values.
M↑ ::= x variable
| c constant symbol
| type the type of types
| (x : M↓) → M↓ dependent functional type
| M↑ M↓ functional application
| (x : M↓) × M↓ dependent sum type
| πi(M↑ ) projections
| M↓ + M↓ disjoint union types
| unpack
M↓
M↑
(M↓, M↓) unpack
69 / 96

Checkable terms
M↓ ::= M↑ inferable terms
| λx.M↓ lambda abstraction
| (M↓, M↓) pair
| ιi(M↓) injections
71 / 96

Type Checking/Inference Algorithm
Type inference of a UDTT is a transformation of a UDTT
judgment to a set of DTT proof diagrams, recursively defined by
the following set of rules.
Definition (Inferable Terms: Structural Rules)
JΓ, x : A, ∆ ` x : ?K =
(
x : A0
.
.
.
.
A0 : type
∈ JΓ ` A : typeK
)
JΓ ` c : ?K =

c : A
(CON ) σ ` c : A

JΓ ` type : ?K =

Γ ` type : kind
(typeF)

72 / 96

Definition (Π-types)
JΓ ` (x : A) → B : ?K =









Γ
DA
A0 : type
Γ, x : A0
DB
B0 : type
(x : A0) → B0 : type
(ΠF)
DA ∈ JΓ ` A : typeK
DB ∈ JΓ, x : A0 ` B : typeK









JΓ ` λx.M : (x : A) → BK =









Γ
DA
A0 : type
Γ, x : A0
DM
M0 : B0
λx.M0 : (x : A0) → B0
(ΠI )
DM ∈ JΓ, x : A0 ` M : BK









JΓ ` MN : ?K =













Γ
DM
M0 : (x : A) → B
Γ
DN
N0 : A
M0N0 : B[N0/x]
(ΠE)
M0N0 : B0
(CONV )
DM ∈ JΓ ` M : ?K
DN ∈ JΓ ` N : AK
B[N0/x] β B0













73 / 96

Definition (Σ-types)
s
Γ `

x : A
B

: ?
{
=











Γ
DA
A0 : type
Γ, x : A0
DB
B0 : type

x : A0
B0

: type
(ΣF)
DB ∈ JΓ, x : A0 ` B : typeK











s
Γ ` (M, N) :

x : A
B
{
=









Γ
DM
M0 : A0
Γ
DN
N0 : B0
(M0, N0) :

x : A0
B0
(ΣI )
DM ∈ JΓ ` M : AK
B[M0/x] β B0
DN ∈ JΓ ` N : B0K









JΓ ` π1(M) : ?K =













Γ
DM
M0 :

x : A
B

π1(M0) : A
(ΣE)
DM ∈ JΓ ` M : ?K













JΓ ` π2(M) : ?K =

















Γ
DM
M0 :

x : A
B

π2(M0) : B[π1M0/x]
(ΣE)
π2(M0) : B0
(CONV )
DM ∈ JΓ ` M : ?K
B[π1M0/x] β B0

















74 / 96

Definition (Disjoint Union types)
JΓ ` A + B : ?K =







Γ
DA
A0 : type
Γ
DB
B0 : type
A0 + B0 : type
(+F)
DB ∈ JΓ ` B : typeK







JΓ ` ι1(M) : A + BK =







Γ
DM
M0 : A0
ι1(M0) : A0 + B
(+I )
DM ∈ JΓ ` M : AK







JΓ ` ι2(N) : A + BK =







Γ
DN
N0 : B0
ι2(N0) : A + B0
(+I )
DN ∈ JΓ ` N : BK







q
Γ ` unpackP
L (M, N) : ?
y
=













Γ
DL
L0 : A + B
Γ
DP
P 0 : (A + B) → type
Γ
DM
M0 : (x : A) → P 0(ι1(x))
Γ
DN
N0 : (x : B) → P 0(ι2(x))
unpackP
L0 (M0, N0) : P 0(L0)
(+E),i
unpackP B
L0 (M0, N0) : PL
(CONV )
DL ∈ JΓ ` L : ?K
DP ∈ JΓ ` P : (A + B) → typeK
P 0(L0) β PL
P 0(ι1(x)) β PM
P 0(ι2(x)) β PN
DM ∈ JΓ ` M : (x : A) → PM K
DN ∈ JΓ ` N : (x : B) → PN K



















75 / 96

Definition (Enumeration types)
JΓ ` {a1, . . . , an} : ?K =

{a1, . . . , an} : type
({}F)

JΓ ` ai : ?K =

ai : {a1, . . . , an}
({}I )

q
Γ ` caseP
M (N1, . . . , Nn) : ?
y
=













Γ
DM
M : {a1, . . . , an}
Γ
DP
P : {a1, . . . , an} → type
Γ
DP1
N1 : P1
N1 : P (a1)
(CONV )
. . .
Γ
DPn
Nn : Pn
Nn : P (an)
(CONV )
caseP
M (N1, . . . , Nn) : P (M)
({}E)
caseP
M (N1, . . . , Nn) : PM
(CONV )
DM ∈ JΓ ` M : ?K
DP ∈ JΓ ` P : {a1, . . . , an} → typeK
P (a1) β P1
DP1 ∈ JΓ ` N1 : P1K
. . .
P (an) β Pn
DPn ∈ JΓ ` Nn : PnK
P (M) β PM

















76 / 96

Definition (Intensional Equality types)
JΓ ` M =A N : ?K =







Γ
DA
A0 : type
Γ
DM
M0 : A0
Γ
DN
N0 : A0
M0 =A0 N0 : type
(=F)
DM ∈ JΓ ` M : A0K
DN ∈ JΓ ` N : A0K







JΓ ` reflA(M) : ?K =







Γ
DA
A0 : type
Γ
DM
M0 : A0
reflA0 (M0) : M0 =A0 M0
(=I )
DM ∈ JΓ ` M : A0K







q
Γ ` idpeelP
E (R) : ?
y
=









Γ
DE
E0 : M =A N
Γ
DP
P 0 : (x : A) → (y : A) → (x =A y) → type
Γ
DR
R0 : (x : A) → P 0xx(reflA(x))
idpeelP 0
E0 (R0) : P 0MNE0
(=E)
idpeelP 0
E0 (R0) : P 00
(CONV )
DE ∈ JΓ ` E : ?K
DP ∈ JΓ ` P : (x : A) → (y : A) → (x =A y) → typeK
DR ∈ JΓ ` R : (x : A) → P 0xx(reflA(x))K
P MNE β P 00







77 / 96

Definition (Natural Number types)
JΓ ` N : ?K =

N : type
(NF)

JΓ ` 0 : ?K =

0 : N
(NI )

JΓ ` s(n) : ?K =







Γ
Dn
n : N
s(n) : N
(NI )
Dn ∈ JΓ ` n : NK







q
Γ ` natrecP
n (e, f) : ?
y
=













Γ
Dn
n0 : N P : N → type e : P (0) f : (k : N) → P (k) → P (s(k))
natrecP
n (e, f) : P (n)
(NE)
natrecP
n (e, f) : Pn
(CONV )
Dn ∈ JΓ ` n : NK
P (n) β Pn

78 / 96

Definition (Underspecified types)
s
Γ `

x@A
B

: ?
{
=





















Γ
DB
B00 : type
.
.
.
.
A0 : type
∈ JΓ ` A : typeK
.
.
.
.
M : A0
∈ JΓ ` ? : A0K
B[M/x] β B0
DB ∈ JΓ ` B0 : typeK





















79 / 96

Definition (Checkable Terms)
JΓ ` M : AK =



Γ
DM
M : A0
DM ∈ JΓ ` M : ?K
A =β A0



where M inferable.
80 / 96

β-reduction Rules
M β λx.v v[N/x] β v0
MN β v0
(β)
M β (v1, v2)
πiM β vi
(β)
x β x
(VAR=)
c β c
(CON =)
type β type
(type=)
kind β kind
(kind=)
81 / 96

β-reduction Rules
M β v N β v0
(x : M) → N β (x : v) → v0
(Π =)
M β v
λx.M β λx.v
(λ=)
M β v N β v0

x : M
N

β

x : v
v0
(Σ=) M β v N β v0
(M, N) β (v, v0)
(PAIR=)
π
() β ()
(()=)
β
(=)
⊥ β ⊥
(⊥=)
82 / 96

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