Dependent Types in Natural Language Semantics

Dependent Types in Natural Language Semantics
Daisuke Bekki†‡ §
†Ochanomizu University
‡CREST, Japan Science and Technology Agency
National Institute of Informatics
§National Institute of Advanced Industrial Science and Technology
4th July, 2015
The Second International Workshop on Linguistics of BA
at Future University Hakodate

1. Proof-theoretic turn in discourse
representation
“Contexts in DTS” Future University Hakodate, 4th July, 2015 Page 2/31

1.1. Donkey and E-type anaphora
Since 1980, the enterprise of dynamic semantics has pursued an alternative frame-
work to Montagovian semantics, which compensates for the gap between syntactic
structures of natural language sentences involving dynamic binding. The difficulty
of this pursuit implies that there is tension between dynamism and compositionality,
which have not yet been unified in a coherent semantic theory that accounts for both
aspects.
This tension has been the driving force behind dynamic semantics, and in fact
some theories have achieved partial success in unifying the two aspects. Thus, I
should clarify what I mean by dynamism and compositionality. Dynamic semantics
explores various empirical data concerning dynamic binding, whose nature is exem-
plified by the two paradigms of donkey sentences in (1) by Geach (1962) and E-type
anaphora in (2) by Evans (1980).
(1) a. Every farmer who owns [a donkey]i beats iti.
b. If [a farmer]i owns [a donkey]j , hei beats itj.
(2) [A man]i entered. Hei whistled.

As discussed elsewhere, (1a), for example, is problematic in terms of composition-
ality. Compositional semantic theory is such that it provides a way to calculate any
semantic representation of any target sentence from the semantic representations
of its parts. The structural analogue of (1a) (and (1b)), which allows us to give
a straightforward compositional analysis, is (3). However, it is not an appropriate
semantic representation for (1a) since variable y occurs as a free variable outside of
the scope of ∃y.
(3) ∀x(Farmer(x) ∧ ∃y(Donkey(y) ∧ Own(x, y)) → Beat(x, y))
In the same way, the structural analogue of (2) is (4), which is not an appropriate
representation for (2) since variable x in Whistle(x) is not bound by ∃x.
(4) ∃x(Man(x) ∧ Enter(x)) ∧ Whistle(x)

The ﬁrst-order representations for sentences (1) and (2) necessary in order to
correctly calculate their entailment relations are (5) and (6), respectively.
(5) ∀x(Farmer(x) → ∀y(Donkey(y) ∧ Own(x, y) → Beat(x, y)))
(6) ∃x(Man(x) ∧ Enter(x)∧Whistle(x))
These represent proper information that the sentences (1) and (2) contain, in
a sense that any proof system for ﬁrst-order predicate logic will prove that the
inferences in Example 1 and Example 2 are valid.
On the other hand, the structural similarity to the original sentences is lost in
(5) and (6), so their direct decomposition does not lead to the respective lexicalized
representations.

Example 1 (Donkey Syllogism).
Every farmer who owns [a donkey]i beats iti.
John is a farmer.
Bill is a donkey.
John owns Bill.
John beats Bill.
Example 2 (E-type Syllogisms).
[A man]i entered.
Hei whistled.
A man entered.
[A man]i entered.
Hei whistled.
A man whistled.

1.2. Proof-theoretic turn: Sundholm and Ranta
Sundholm (1986) noticed fairly early that dependent type theory provides se-
mantic representations for donkey sentences whose structures are parallel to their
syntactic structures in a diﬀerent way from DRT Kamp (1981), Kamp and Reyle
(1993), DPL Groenendijk and Stokhof (1991), and their successors.
(7) a. A man entered. He whistled.
⎡
⎢
⎢
⎣
u:
⎡
⎣
x:entity
man(x)
enter(x)
⎤
⎦
whistle(π1(u))
⎤
⎥
⎥
⎦
(8) a. Every farmer who owns a donkey beats it.
(x:entity) →
⎛
⎜
⎜
⎝u:
⎡
⎢
⎢
⎣
farmer(x)⎡
⎣
y:entity
donkey(y)
own(x, y)
⎤
⎦
⎤
⎥
⎥
⎦
⎞
⎟
⎟
⎠ → beat(x, π1π2(u))

1.3. Key concepts
1. Proof-theoretic semantics (vs. Model-theoretic semantics)
2. Curry-Howard Correspondence (between logic type theory)
3. Dependent types (vs. Simple types)

2. Proof-theoretic Semantics,
Curry-Howard Correspondence,
and Dependent Types

2.1. Proof-theoretic vs. model-theoretic semantics
The ‘meaning’ of a given proposition φ:
The Proof-theoretic Semantics Model-theoretic Semantics
Provability, or {Γ | Γ φ} Truth-condition, or {(M, g) | φ M,g = 1}
Inference rules Semantic Values
(natural deduction): (classical logic):
A....
B
i
A → B
(→I ),i
A A → B
B
(→E)
A B
A ∧ B
(∧I )
A1 ∧ A2
Ai
(∧E),i=1,2
A → B M,g = 1
⇐⇒ A M,g = 0 or B M,g = 1
A ∧ B M,g = 1
⇐⇒ A M,g = 1 and B M,g = 1

2.2. Curry-Howard Correspondence
Propositional Logic Simply-Typed Lambda Calculus
A....
B
i
A → B
(→I ),i
x : A....
B
i
λx.M : A → B
(→I ),i
A A → B
B
(→E)
N : A M : A → B
MN : B
(→E)
A B
A ∧ B
(∧I )
M : A N : B
(M, N) : A × B
(∧I )
A1 ∧ A2
Ai
(∧E),i=1,2
(M1, M2) : A1 × A2
πi(M) : Ai
(×E),i=1,2

2.3. Curry-Howard Isomorphism
The correspondence between the notions of logic and type theory:
Logic Type Theory
proposition type
proof term
axiom constant symbol
assumption variable
provability inhabitance
cut substitution
normalization reduction
. . . . . .
• A term is an encoding of a proof of a type(=proposition)
• A proposition can be regarded as a collection of proofs.
• φ is true under Γ iﬀ φ is inhabited under Γ.

2.4. Dependent function/sum Types
• Dependent function type (x:A) → B is a generalized form of the function/implication
type A → B:
A → B
def
≡ (x:A) → B where x /∈ fv(B).
∀xB
def
≡ (x:entity) → B
• Dependent sum type
x:A
B
is a generalized form of the product/conjunction
type A × B:
A ∧ B
def
≡
x:A
B
where x /∈ fv(B).
∃xB
def
≡
x:entity
B

2.4. Dependent function/sum Types
Natural deduction rules for dependent types:
(x:A) → B : s
x : A....
M : B
i
λx : A.M : (x:A) → B
(ΠI ),i
M : (x:A) → B N : A
MN : B[N/x]
(ΠE)
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)

2.5. Dependent Types in Natural Language
Semantics
(9) Donkey anaphora: Sundholm (1986)
a. Every farmer who owns a donkey beats it.
b. (x:entity) →
⎛
⎜
⎜
⎝u:
⎡
⎢
⎢
⎣
farmer(x)⎡
⎣
y:entity
donkey(y)
own(x, y)
⎤
⎦
⎤
⎥
⎥
⎦
⎞
⎟
⎟
⎠ → beat(x, π1π2(u))
(10) E-type anaphora: Ranta (1994)
a. A man entered. He whistled.
b.
⎡
⎢
⎢
⎣
u:
⎡
⎣
x:entity
man(x)
enter(x)
⎤
⎦
whistle(π1(u))
⎤
⎥
⎥
⎦
Recall that (x:A) → B is a type for functions from A to B[x], and
x:A
B
is a
type for pairs of A and B[x].

2.6. Previous Works
Subsequently, the following three approaches have been proposed to obtain Sund-
holmian representations:
1. Ahn and Kolb (1990) provides a set of translation rules from DRS to Dependent
Type representations
2. D´avila-P´erez (1995) presented a reformulation of Montague Grammar Mon-
tague (1973) in terms of MLTT.
3. Ranta (1994) proposed a generative theory of grammar based on MLTT, known
as Type-Theoretical Grammar (TTG).

2.6. Previous Works
• Relative and Implicational Donkey Sentences, Branching Quantiﬁers, Inten-
sionality, Tense: Ranta (1994)
• Summation: Fox (1994a,b)
• Presupposition Binding and Accommodation, Bridging: Krahmer and Piwek
(1999), Piwek and Krahmer (2000)
• Coercion: Luo (1997, 1999, 2010, 2012b), Asher and Luo (2012)
• Adverbs: Chatzikyriakidis (2014)
• New frameworks: Cooper (2005), Luo (2012a), Bekki (2014), Martin and Pol-
lard (2014)

2.6. Previous Works
Recent works in our group:
• Generalized Quantiﬁers: Tanaka et al. (2013), Tanaka (2014)
• Type checking/inference in DTS and its implementation: Bekki and Sato
(2015)
• Modal Subordination: Tanaka et al. (2014)
• Factive Presupposition: Tanaka et al. (2015)
• Conventional Implicature: Bekki and McCready (2014), Watanabe et al. (2014)
• Double-Negated Antecedents: Bekki (2013)

A. Appendix

A.1. Dependent Types in Mathematics and Logic
• The notion of dependent types originates from:
– Martin-Löf Type Theory (MLTT) (Martin-Löf (1975, 1984)), which was
proposed as a foundation of constructive mathematics.
– Calculus of Constructions (CoC) (Coquand and Huet (1988)), which was
proposed as a foundation of functional programming and mathematical
proofs.
• Lately, fragments of MLTT and CoC have been integrated into a general the-
ory of the λ-cube (Barendregt (1992)) and Pure Type Systems (PTS) (Berardi
(1990); Barendregt (1991)) with other important type theories, such as Gi-
rard’s F (Girard et al. (1989)).
• Calculus of Inductive Constructions (CoIC) (=CoC with inductive types) is
known as an underlying language of proof assistants Coq (Bertot and Castéran
(2004)) and Agda (Nordström et al. (1990), Bove and Dybjer (2008)).

A.1. Dependent Types in Mathematics and Logic
The term “Dependent Type Theory” usually covers:
• λP is an extention of Simple Type Theory (λ→) with dependent functional
type Π. (cf. Barendregt Cube) λω λC
λ2

λP2

λω λPω
λ→

λP

• Martin-L¨of/Constructive/Intuitionistic/Modern Type Theory is an extention
of λP with dependent sum type Σ, (dependent) record type, Equational type,
and Natural Numbers, Inductive types, etc.

A.2. What is NOT Dependent Type Theory
• Type-Theoretic Semantics (ex. Montague (1973), Gallin (1975)), where a
proposition is a term of type t, while in Dependent Type Theory, a propo-
sition is a type (=a collection of proofs).
• Dependent Type Theory is proof-theoretic, but it has a denotational semantics
(cf. ﬁbred category theory: Jacobs (1999)) and other types of semantics (cf.
game-theoretic semantics) as well.

References
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Ahn, R. and H.-P. Kolb. (1990) “Discourse Representation meets Constructive Math-
ematics”, In: L. Kalman and L. Polos (eds.): Papers from the Second Symposium
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Bekki, D. (2013) “A Type-theoretic Approach to Double Negation Elimination in
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References
Bekki, D. (2014) “Representing Anaphora with Dependent Types”, In the Proceed-
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References
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References
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Dependent Types in Natural Language Semantics

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