Composing (Im)politeness in Dependent Type Semantics

Honoriﬁcation DTT DTS Solution Reference
Composing (Im)politeness
in Dependent Type Semantics
Daisuke Bekki
Ochanomizu University, Faculty of Core Research / CREST, Japan Science and
Technology Agency (JST) / National Institute of Advanced Industrial Science and
Technology (AIST) / National Institute of Informatics (NII)
Politeness workshop@LENLS12
Ochanomizu University, November 15th (Sun), 2015.
http://www.slideshare.net/kaleidotheater/
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Honoriﬁcation in Japanese
2 / 104

Puzzle 1: Morphology and composition
Honorifics indicate that the speaker (is behaving as if she) honors,
or feels some distance from, one of the arguments of a verb (or the
addressee):
(1) Sensei -ga
teacher-Nom
irassyat -ta.
come-Hon-Pst
Descriptive: ‘The teacher came.’ + Expressive: ‘The
speaker honors the teacher.’
(2) Sensei -ga
teacher-Nom
ringo-o
apple-Acc
mesiagat -ta.
eat.Hon-Pst
Descriptive: ‘The teacher ate an apple.’ + Expressive:
‘The speaker honors the teacher.’
These examples were suppletive honorifics: terms lexically specified
as honorific which completely replace nonhonorific forms.
3 / 104

It is also possible in Japanese to derive honorific forms via
systematic morphology from ordinary verbs; here, strategies for
honorification include use of the ‘passive’ form and the use of
‘pure’ honorific morphology.
(3) Sensei -ga
teacher-Nom
seito-o
student-Acc
home- rare -ta.
praise-Hon-Pst
Descriptive: ‘The teacher praised the student.’ +
Expressive: ‘The speaker honors the teacher.’
(4) Sensei -ga
teacher-Nom
seito-o
apple-Acc
o -home- ninat -ta.
Hon-praise-Hon-Pst
Descriptive: ‘The teacher praised the student.’ +
4 / 104

Japanese employs different morphology according to the
grammatical target of honorific meaning. One can find
subject-oriented honorifics, as in (5), and object honorifics, as in
(6).
(5) Sensei -ga
teacher-Nom
seito-o
student-Acc
o -tasuke- ninat -ta.
Hon-help-Hon-Pst
Descriptive: ‘The teacher helped the student.’ +
(6) Seito-ga
student-Nom
sensei -o
teacher-Acc
o -tasuke- si -ta.
Hon-help-Hon-Pst
Descriptive: ‘The student helped the teacher.’ +
As we will see, even simple examples like these can cause problems
for existing treatments of honorific composition.
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Following requirements should be satisfied:
Honorific contents require composition : Subject-oriented
and object-oriented honorifics
Honorific suffixes require higher-order composition : passive
suffix, honorific morphology ‘-o’, ‘-nasa(ru)’-suffix,
‘-su(ru)’-suffix, etc
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Puzzle 2: Honorific content projects
In 1970’s, honorification was considered as an instance of
syntactic agreement (Harada, 1976), which has been assumed
ever since (Gunji, 1987)(Siegel, 2000)(Boeckx and Niinuma,
2004)
The ’syntactic view’ is falsified by the following example
(Bekki et al., 2008):
(7) Dare -mo
nobody-Nom
o -mie- ni-nar -anakat-ta.
come-Hon-Neg-Pst
‘Nobody (in the contextually salient set) came-Hon.’
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Honorific content projects
(8) Sensei -ga
teacher-Nom
irassyat -ta
come-Hon-Pst
wakedehanai.
Neg
# Sensei -ga
teacher-Nom
ki- yagat -ta-nda.
come-AntiHon-Pst-Explanation
‘The teacher did not come-Hon. #The teacher
came-AntiHon.’
The honorific content projects over negation.
(9) Mosi
If
Sensei -ga
teacher-Nom
irassyat -tara
come-Hon-Pst
otya-o
tea-Acc
dasi-te
serve
kudasai.
please
# Sensei -ga
teacher-Nom
nomi- yagar -imasu.
drink-AntiHon-Nonpst
‘If the teacher comes-Hon, please serve a tea. #He will
drink-AntiHon it.’
The honorific content projects over implication. 8 / 104

Puzzle 2: Honorific content projects
Based on the projective feature of honorification, Bekki et al.
(2008) claimed that honorific contents are presuppositions.
However, it turns out that honorific contents CANNOT be
filtered by a presupposition filter. (McCready, 2010)
(10) Mosi
If
wareware-ga
we-NOM
Yamada-si-o
Mr.Yamada-ACC
sonkeisi-tei-ru-nara,
honor-Asp-NonPst-cond
Yamada -si-o
Mr.Yamada-ACC
go -syotai- suru -daroo.
invite-Hon-would
#Jissai-wa
actually
sonkeisi-tei-nai-ga.
honor-Asp-Neg-Contrastive
‘If we honor Mr.Yamada, we would invite-Hon Mr.Yamada.
#But actually we do not honor Mr.Yamada.’
(cf. ’If John is married, his wife will come.’) 9 / 104

Puzzule 1 + Puzzle 2 = ?
Honorific contents are expressive contents (not descriptive,
nor presuppositional)
What we need: a mechanism for composition of expressive
contents
Two-dimensional semantics? (Potts, 2005)(McCready,
2010)(Gutzmann, 2015)
Advantage: Projection, Immunity to presupposition filters, ...
Disadvantage: More than two-place honorific predicate
Disadvantage: The following case induces a binding problem!
(11) Dare -mo
nobody-Nom
come-Hon-Neg-Pst
‘Nobody (in the contextually salient set) came-Hon.’
∀x(Descriptive:¬came(x) ∧ Expressive:honor(sp, x))
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Dependent Type Theory
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What is Dependent Type Semantics?
Dependent Type Semantics (DTS; Bekki (2013, 2014)):
DTS is a discourse theory, an alternative framework to DRT,
DPL, continuation semantics, etc.
DTS is a compositional/lexicalized theory that serves as a
semantic component of most categorial grammars.
DTS is based on dependent type theory (Martin-L¨of,
1984)(Coquand and Huet, 1988), following the line of
Sundholm (1986), Ranta (1994), Cooper (2005), Mineshima
(2008) and Luo (2012).
DTS provides proof-theoretic semantics of natural language
(Dummett, 1975, 1976)(Prawitz, 1980)
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Three key concepts in DTS
1. Proof-theoretic semantics (vs. Model-theoretic semantics)
2. Curry-Howard Correspondence (between logic and type
theory)
3. Dependent types (vs. Simple types)
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A proof theory: Natural deduction rules
Implication Conjunction
Introduction
rules
Γ, A B
Γ A → B
(→I )
Γ A Γ B
Γ A ∧ B
(∧I )
Elimination
rules
Γ A → B Γ A
Γ B
(→E)
Γ A ∧ B
Γ A
(∧E)
Γ A ∧ B
Γ B
(∧E)
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Typing rules for simply-typed lambda calculus (STLC) with
binary products
Function type Product type
Type
construction
rules
Γ, x : A M : B
Γ λx.M
function
: A → B
(LAM )
Γ M : A Γ N : B
Γ (M, N)
pair
: A × B
(PROD)
Type
deconstruction
rules
Γ M : A → B Γ N : A
Γ MN : B
(APP)
Γ M : A × B
Γ π1(M) : A
(PROJ)
Γ M : A × B
Γ π2(M) : B
(PROJ)
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Typing rules for simply-typed lambda calculus (STLC) with
binary products
f : A → B, x : A f : A → B
(VAR)
f : A → B, x : A x : A
(VAR)
f : A → B, x : A fx : B
(APP)
f : A → B λx.fx : A → B
(LAM )
λf.λx.fx : (A → B) → (A → B)
(LAM )
The typing tree of a term is a record of rules used for typing.
The typing tree of a term (in STLC) can be recovered from
the structure of term. (cf. Milner (1978))
Fact 1
A term is an encoding of a typing tree.
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Curry-Howard Correspondence btw. function type and
implication
Typing rules in STLC Natural Deduction rules
Introduction
rules
Γ, x : A M : B
Γ λx.M
function
: A → B
(LAM )
Γ, A B
Γ A → B
(→I )
Elimination
rules
Γ M : A → B Γ N : A
Γ MN : B
(APP)
Γ A → B Γ A
Γ B
(→E)
Fact 2
Typing rules of STLC (almost exactly) correspond to natural
deduction rules in logic.
17 / 104

Curry-Howard Correspondence btw. product type and
conjunction
Typing rules in STLC Natural Deduction rules
Introduction
rules
Γ M : A Γ N : B
Γ (M, N)
pair
: A × B
(PROD)
Γ A Γ B
Γ A ∧ B
(∧I )
Elimination
rules
Γ M : A × B
Γ π1(M) : A
(PROJ)
Γ A ∧ B
Γ A
(∧E)
Γ M : A × B
Γ π2(M) : B
(PROJ)
Γ A ∧ B
Γ B
(∧E)
Fact 2
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Curry-Howard Correspondence
Fact 1
A term is an encoding of a typing tree.
+
Fact 2
⇓
Fact 3
A term of type A is an encoding of a proof diagram of a
proposition A.
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conjunction
Fact 3
A term of type A is also an encoding of a proof diagram of a
proposition A (under the view that proposition is type). In
particular:
Functions encode proofs of →.
Pairs encode proofs of ∧.
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conjunction
The correspondence between the notions of logic and type theory :
Logic Type Theory
proposition type
proof term (or program)
axiom constant symbol
assumption variable
logical connective type constructor
implication functional type
conjunction product type
disjunction direct sum type
absurdity empty type
introduction constructor
elimination destructor
provability inhabitance
cut substitution
normalization reduction
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Three diﬀerent queries on Judgements
Γ M : A · · · Type check (returns yes/no)
Γ M : ? · · · Type inference (returns a type)
Γ ? : A · · · Proof search (returns a term)
Γ A true · · · There is a term M such that Γ M : A
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Simple function type vs. Dependent function type
Function type → in STLC Dependent function type (Π) in DTT
Introduction
rules
Γ, x : A M : B
Γ λx.M
function
: A → B
(LAM )
Γ A : s Γ, x : A M : B
Γ λx.M : (x:A) → B
(ΠI )
Elimination
rules
Γ M : A → B Γ N : A
Γ MN : B
(APP)
Γ M : (x:A) → B Γ N : A
Γ MN : B
(ΠE)
Scope: (x:A) → B
23 / 104

Simple product type vs. Dependent product type
Product type × in STLC Dependent product type (Σ) in DTT
Introduction
rules
Γ M : A Γ N : B
Γ (M, N)
pair
: A × B
(PROD) Γ M : A Γ N : B[M/x]
Γ (M, N) :
x:A
B
(ΣI )
Elimination
rules
Γ M : A × B
Γ π1(M) : A
(PROJ)
Γ M :
x:A
B
Γ π1(M) : A
(ΣE)
Γ M : A × B
Γ π2(M) : B
(PROJ)
Γ M :
x:A
B
Γ π2(M) : B[π1(M)/x]
(ΣE)
Scope:
x:A
B
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Dependent types and First-order logic
Dependent types Standard notation x ∈ fv (B) x ∈ fv (B)
(x:A) → B (Πx : A)B A → B (∀x : A)B
x:A
B
(Σx : A)B A ∧ B (∃x : A)B
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Examples
Types dependent on terms
List(n): lists of length n
append in DTT: List(m) × List(n) → List(m + n)
append in STLC: List × List → List
Linguist(x): proofs of x’s being a linguist.
Dependent product types
m:month
day(m)
: pairs of a month and its day
m:entity
linguist(m)
: pairs of an entity and its proof of being a
linguist (=“There is a linguist.”)
Dependent function types
u:
m:entity
linguist(m)
→ mortal(π1u): functions from pairs to
proofs of mortality (=“Every linguist is mortal.”)
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Dependent function type
Deﬁnition (Π-formation/introduction/elimination rules)
Γ A : s1 Γ, x : A B : s2
Γ (x:A) → B : s2
(ΠF) where s1, s2 ∈ {type, kind} .
Γ A : s Γ, x : A M : B
Γ λx.M : (x:A) → B
(ΠI ) where s ∈ {type, kind} , x /∈ fv (Γ) .
Γ M : (x:A) → B Γ N : A
Γ MN : B
(ΠE)
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Dependent function type
Deﬁnition (Negation)
¬A
def
≡ A → ⊥
Deﬁnition (¬-formation/introduction/elimination rules)
Γ A : type
Γ ¬A : type
(¬F)
Γ, x : A M : ⊥
Γ λx.M : ¬A
(¬I )
Γ M : ¬A Γ N : A
Γ MN : ⊥
(¬E)
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Dependent product type
Deﬁnition (Σ-formation/introduction/elimination rules)
DA
Γ A : s1 Γ, x : A B : s2
Γ
x:A
B
: s2
(ΣF) where s1, s2 ∈ {type, kind} .
Γ M : A Γ N : B[M/x]
Γ (M, N) :
x:A
B
(ΣI )
Γ M :
x:A
B
Γ π1(M) : A
(ΣE)
Γ M :
x:A
B
Γ π2(M) : B[π1(M)/x]
(ΣE)
29 / 104

Natural numbers
Deﬁnition (N-formation/introduction/elimination rules)
Γ N : type
(NF)
Γ 0 : N
(NI )
Γ n : N
Γ s(n) : N
(NI )
Γ P : N → type Γ e : P(0) Γ, n : N f : P(s(n))
Γ, n : N natrec(n, e, f) : P(n)
(NE)
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Intensional equality type
Definition (id-formation/introduction/elimination rules)
Γ A : s Γ M : A Γ N : A
Γ M =A N : type
(IdF)
Γ A : type Γ M : A
Γ reflA(M) : M =A M
(IdI )
Γ C : (x:A) → (y:A) → (x =A y) → type
Γ, x : A N : Cxx(reflA(x))
Γ, e : M1 =A M2 idpeel(e, N) : CM1M2e
(IdE)
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E-type anaphora: Ranta (1994)
(12) A man entered. He whistled.






v:



u:
x:entity
man(x)
enter(π1u)



whistle( π1π1v )






Note:
x:A
B
is a type for pairs of A and B[x].
32 / 104

Donkey anaphora: Sundholm (1986)
(13) 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].
33 / 104

Accessibility: Ranta (1994)
(14) Every man entered. * He whistled.




v: u:
x:entity
man(x)
→ enter(π1u)
whistle( ? )




In this case, the pronoun CANNOT pick up the entity (=the man
who entered) from v, since v is a function. This explains the
following cases uniformly, since both implication and negation are
instances of dependent functional types:
(15) a. If John owns a car, it must be a Porsche. *It is red.
b. John did not buy a car. *It is a Porsche.
This accounts for accessibility, based on the structure of a proof.
34 / 104

Problem: Context retrieval
Ranta (1994), Krause (1995), Dávila-Pérez (1995), Krahmer
and Piwek (1999), Piwek and Krahmer (2000): Anaphora
resolution/presupposition binding as proof search of the form
Γ M : A
Context Γ consists of:
1. Knowledge of the hearer
2. Preceding discourse
3. Variables from c-commanding quantifiers
Problem of context retrieval for each anaphora/presupposition
trigger
Syntactic approaches: Ranta (1994)
Semantic approaches: Mineshima (2008, 2014), Bekki (2014)
Context retrieval by type checking
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Dependent Type Semantics
36 / 104

Underspecified DTT
UDTT = DTT + underspecified terms (@)
A map ‘@-elimination’ − : UDTT proof diagrams → DTT
terms
Definition (@-rule)
DA
Γ A : type Γ DA true
Γ @iA : A
(@)
A UDTT proof diagram contains information to specify the
antecedent of @s in the diagram
@-elimination replaces every occurrence of @ by a proof term
specified in the UDTT diagram
37 / 104

Organization of grammar
Sentences
⇓
CCG Parser
⇓
Underspeciﬁed SRs in UDTT
⇓
Discoruse relation (e.g. Progressive conjunction)
⇓
A underspeciﬁed SR of a discourse in UDTT
⇓
Type checking in UDTT (+ Proof search in a DTT fragment)
⇓
A proof diagram of the well-formedness of an SR in UDTT
⇓
@-elimination
⇓
A semantic representation in DTT
⇓
Inference in DTT
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Lexical items in DTS
PF CCG categories Semantic representations in DTS
if S/S/S λp.λq. (u:p) → q
everynom S/(SNP)/N λn.λp. u:
x:entity
nx
→ p(π1(u))
everyacc T (T /NP)/N λn.λp.λx. v:
y:entity
ny
→ p(π1(v))x
anom, somenom S/(SNP)/N λn.λp.

 u:
x:entity
nx
p(π1(u))


aacc, someacc T (T /NP)/N λn.λp.λx.

 v:
y:entity
ny
p(π1(v))x


farmer N λx.farmer(x)
donkey N λx.donkey(x)
who NN/(SNP) λp.λn.λx.(nx; px)
whom NN/(S/NP) λp.λn.λx.(nx; px)
owns SNP/NP λy.λx.own(x, y)
beats SNP/NP λy.λx.beat(x, y)
he NP π1@i
x:entity
male(x)
it NP π1@i
x:entity
¬human(x)
the NP/N λn.π1@i
x:entity
nx
39 / 104

Syntactic calculus/semantic composition
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)


>
40 / 104

Syntactic calculus/semantic composition
He
NP
: π1@
x:entity
man(x)
whistled
SNP
: λx.whistle(x)
S
: whistle π1@
x:entity
man(x)
<
41 / 104

Progressive conjunction: Ranta (1994)
Deﬁnition (Progressive conjunction)
M; N
def
≡
u:M
N
where u /∈ fv (N)
(12) A man entered. He whistled.


x:entity
man(x)
enter(x)

 ; whistle( π1@
x:entity
man(x)
)
underspeciﬁed term
=







u:


x:entity
man(x)
enter(x)


whistle( π1@
x:entity
man(x)
)







42 / 104

Presupposition projection as type checking
Felicity condition as Type Checking
K






u:


x:entity
man(x)
enter(x)


whistle @i
x:entity
man(x)






: type
⇓ Type checking in UDTT
K, u :


x:entity
man(x)
enter(x)

 @i
x:entity
man(x)
Type checking: reconstruction of context
Proof search: anaphora resolution/presupposition binding
43 / 104

Type checking in Underspeciﬁed DTT
D1
K

 u:
x:entity
man(x)
enter(π1u)

 : type
K, v :

 u:
x:entity
man(x)
enter(π1u)


whistle
: entity
→ type
(VAR)
D2
K, v :

 u:
x:entity
man(x)
enter(π1u)

 x:entity
man(x)
: type K, v :

 u:
x:entity
man(x)
enter(π1u)

 x:entity
man(x)
true
K, v :

 u:
x:entity
man(x)
enter(π1u)

 @i
x:entity
man(x)
:
x:entity
man(x)
(@)
K, v :

 u:
x:entity
man(x)
enter(π1u)

 π1@i
x:entity
man(x)
: entity
(ΣE)
K, v :

 u:
x:entity
man(x)
enter(π1u)

 whistle π1@i
x:entity
man(x)
: type
(→E)
K






v:

 u:
x:entity
man(x)
enter(π1u)


whistle π1@i
x:entity
man(x)






: type
(ΣF),2
44 / 104

Anaphora resolution is Proof search in DTT
D1
K

 u:
x:entity
man(x)
enter(π1u)

 : type
K, v :

 u:
x:entity
man(x)
enter(π1u)


whistle
: entity
→ type
(VAR)
D2
K, v :

 u:
x:entity
man(x)
enter(π1u)

 x:entity
man(x)
: type
K, v :

 u:
x:entity
man(x)
enter(π1u)

 v :

 u:
x:entity
man(x)
enter(π1u)


(VAR)
K, v :

 u:
x:entity
man(x)
enter(π1u)

 π1v :
x:entity
man(x)
(ΣE)
K, v :

 u:
x:entity
man(x)
enter(π1u)

 @i
x:entity
man(x)
:
x:entity
man(x)
(@)
K, v :

 u:
x:entity
man(x)
enter(π1u)

 π1 @i
x:entity
man(x)
: entity
(ΣE)
K, v :

 u:
x:entity
man(x)
enter(π1u)

 whistle π1 @i
x:entity
man(x)
: type
(→E)
K







v:

 u:
x:entity
man(x)
enter(π1u)


whistle π1 @i
x:entity
man(x)







: type
(ΣF),2
45 / 104

@-elimination
D1
K

 u:
x:entity
man(x)
enter(π1u)

 : type
K, v :

 u:
x:entity
man(x)
enter(π1u)


whistle
: entity
→ type
(VAR)
D2
K, v :

 u:
x:entity
man(x)
enter(π1u)

 x:entity
man(x)
: type
K, v :

 u:
x:entity
man(x)
enter(π1u)

 v :

 u:
x:entity
man(x)
enter(π1u)


(VAR)
K, v :

 u:
x:entity
man(x)
enter(π1u)

 π1v :
x:entity
man(x)
(ΣE)
K, v :

 u:
x:entity
man(x)
enter(π1u)

 @i
x:entity
man(x)
:
x:entity
man(x)
(@)
K, v :

 u:
x:entity
man(x)
enter(π1u)

 π1 @i
x:entity
man(x)
: entity
(ΣE)
K, v :

 u:
x:entity
man(x)
enter(π1u)

 whistle π1 @i
x:entity
man(x)
: type
(→E)
K







v:

 u:
x:entity
man(x)
enter(π1u)


whistle π1 @i
x:entity
man(x)







: type
(ΣF),2
=




v:

 u:
x:entity
man(x)
enter(π1u)


whistle(π1π1v)




46 / 104

@-elimination
Deﬁnition (@-elimination rules (excerpt))
Γ A : s Γ M : DA
Γ (@iA) : A
(@) = M
DM
Γ M : (x:A) → B
DN
Γ N : A
Γ MN : B[N/x]
(ΠE)
= DM DN
47 / 104

Veriﬁcation of the theory
(16) a. [A man]i entered. Hei whistled. A man entered.
b. [A man]i entered. Hei whistled. A man whistled.
c. A man entered and whistled. [A man]i entered. Hei
whistled.
(17) a. K,




v:

 u:
x:entity
man(x)
enter(π1(u))


whistle(π1π1(v))





 u:
x:entity
man(x)
enter(π1(u))


b. K,




v:

 u:
x:entity
man(x)
enter(π1(u))


whistle(π1π1(v))





 u:
x:entity
man(x)
whistle(π1(u))


c. K,




u:
x:entity
man(x)
enter(π1(u))
whistle(π1(u))








v:

 u:
x:entity
man(x)
enter(π1(u))


whistle(π1π1(v))




48 / 104

Entailments Predicted
(17a) K,




v:

 u:
x:entity
man(x)
enter(π1(u))


whistle(π1π1(v))





 u:
x:entity
man(x)
enter(π1(u))


K, t :




v:

 u:
x:entity
man(x)
enter(π1(u))


whistle(π1(π1(v)))



 t :




v:

 u:
x:entity
man(x)
enter(π1(u))






(VAR)
K, t :




v:

 u:
x:entity
man(x)
enter(π1(u))





 π1(t) :

 u:
x:entity
man(x)
enter(π1(u))


(ΣE)
49 / 104

(17b) K,




v:

 u:
x:entity
man(x)
enter(π1(u))


whistle(π1π1(v))





 u:
x:entity
man(x)
whistle(π1(u))


K, t :




v:

 u:
x:entity
man(x)
enter(π1(u))


whistle(π1π1(v))



 t :




v:

 u:
x:entity
man(x)
enter(π1(u))


whistle(π1π1(v))




(VAR)
K, t :




v:

 u:
x:entity
man(x)
enter(π1(u))


whistle(π1π1(v))



 π1(t) :

 u:
x:entity
man(x)
enter(π1(u))


(ΣE)
K, t :




v:

 u:
x:entity
man(x)
enter(π1(u))


whistle(π1π1(v))



 π1π1(t) :
x:entity
man(x)
(ΣE)
K, t :




v:

 u:
x:entity
man(x)
enter(π1(u))


whistle(π1π1(v))



 t :




v:

 u:
x:entity
man(x)
enter(π1(u))


whistle(π1π1(v))




(VAR)
K, t :




v:

 u:
x:entity
man(x)
enter(π1(u))


whistle(π1π1(v))



 π2(t) : whistle(π1π1(v))[π1(t)/v]
=β whistle(π1π1π1(t))
=β whistle(π1(u))[π1π1(t)/u]
(ΣE)
K, t :




v:

 u:
x:entity
man(x)
enter(π1(u))


whistle(π1π1(v))



 (π1π1(t), π2(t)) :

 u:
x:entity
man(x)
whistle(π1(u))


(ΣI )
50 / 104

(17c) K,




u:
x:entity
man(x)
enter(π1(u))
whistle(π1(u))








v:

 u:
x:entity
man(x)
enter(π1(u))


whistle(π1π1(v))




K, t :




u:
x:entity
man(x)
enter(π1(u))
whistle(π1(u))



 t :




u:
x:entity
man(x)
enter(π1(u))
whistle(π1(u))




(VAR)
K, t :




u:
x:entity
man(x)
enter(π1(u))
whistle(π1(u))



 π1t :
x:entity
man(x)
(ΣE)
K, t :




u:
x:entity
man(x)
enter(π1(u))
whistle(π1(u))



 t :




u:
x:entity
man(x)
enter(π1(u))
whistle(π1(u))




(VAR)
K, t :




u:
x:entity
man(x)
enter(π1(u))
whistle(π1(u))



 π2t :
enter(π1π1t)
whistle(π1π1t)
(ΣE)
K, t :




u:
x:entity
man(x)
enter(π1(u))
whistle(π1(u))



 π1π2t : enter(π1π1t)
=β enter(π1(u))[π1t/u]
(ΣE)
K, t :




u:
x:entity
man(x)
enter(π1(u))
whistle(π1(u))



 (π1t, π1π2t) :

 u:
x:entity
man(x)
enter(π1(u))


(ΣI )
K, t :




u:
x:entity
man(x)
enter(π1(u))
whistle(π1(u))



 t :




u:
x:entity
man(x)
enter(π1(u))
whistle(π1(u))




(VAR)
K, t :




u:
x:entity
man(x)
enter(π1(u))
whistle(π1(u))



 π2t :
enter(π1π1t)
whistle(π1π1t)
(ΣE)
K, t :




u:
x:entity
man(x)
enter(π1(u))
whistle(π1(u))




π2π2t : whistle(π1π1t)
=β whistle(π1π1v)[(π1t, π1π2t)/v]
(ΣE)
K, t :




u:
x:entity
man(x)
enter(π1(u))
whistle(π1(u))








v:

 u:
x:entity
man(x)
enter(π1(u))


whistle(π1π1(v))




(ΣI )
51 / 104

Presupposition projection
The
NP/N
: λn.π1@i
x:entity
nx
king of France
N
: λx.KoF(x)
NP
: π1@i
x:entity
KoF(x)
<
S/(SNP)
: λp.p π1@i
x:entity
KoF(x)
>T
is not
S(S/(SNP))/(SNP)
: λp.λq.¬(qp)
bald
SNP
: λx.bald(x)
S(S/(SNP))
: λq.¬(q(λx.bald(x)))
>
S
: ¬bald π1@i
x:entity
KoF(x)
<
52 / 104

Presupposition projection
K bald : entity → type
(VAR)
K entity : type
(VAR)
K, x : entity KoF : entity → type
(VAR)
K, x : entity x : entity
(VAR)
K, x : entity KoF(x) : type
(→E)
K
x:entity
KoF(x)
: type
(ΣF)
K
x:entity
KoF(x)
true
K @
x:entity
KoF(x)
:
x:entity
KoF(x)
(@)
K π1@
x:entity
KoF(x)
: entity
(ΣE)
K bald π1@
x:entity
KoF(x)
: type
(→E)
K ¬bald π1@
x:entity
KoF(x)
: type
(¬F)
53 / 104

Presupposition binding
If
S/S/S
: λp.λq. (u:p) → q
John
NP
: j
has
SNP/NP
: λy.λx.have(x, y)
a
T (T /NP)/N
: λn.λp.λx.

 v:
y:entity
ny
p(π1(v))x


wife
N
: λx.wife(x)
T (T /NP)
: λp.λx.

 v:
y:entity
wife(y)
p(π1(v))x


>
SNP
: λx.

 v:
y:entity
wife(y)
have(x, π1(v))


<
S
:

 v:
y:entity
wife(y)
have(j, π1(v))


<
S/S
: λq.

u:

 v:
y:entity
wife(y)
have(j, π1(v))



 → q
>
54 / 104

If John has a wife, John’s wife is happy: Parsing
John
NP
: j
’s
NP/N NP
: λy.λn.π1@i


x:entity
nx
have(y, x)


NP/N
: λn.π1@i


x:entity
nx
have(j, x)


<
wife
N
: λx.wife(x)
NP
: π1@i


x:entity
wife(x)
have(j, x)


>
is
SNP/(SNP)
: id
happy
SNP
: λx.happy(x)
SNP
: λx.happy(x)
>
S
: happy

π1@i


x:entity
wife(x)
have(j, x)




<
55 / 104

If John has a wife, John’s wife is happy: Parsing
If John has a wife
S/S
: λq.

u:

 v:
y:entity
wife(y)
have(j, π1(v))



 → q
John’s wife is happy
S
: happy

π1@i


x:entity
wife(x)
have(j, x)




S
:

u:

 v:
y:entity
wife(y)
have(j, π1(v))



 → happy

π1@i


x:entity
wife(x)
have(j, x)




>
56 / 104

If John has a wife, John’s wife is happy: Type Cheking
D
 v:
y:entity
wife(y)
have(j, π1(v))

 : type
happy :
entity
→ type
(CON )
....

x:entity
wife(x)
have(j, x)

 : type
u :

 v:
y:entity
wife(y)
have(j, π1(v))


1
....

x:entity
wife(x)
have(j, x)

 true
@i


x:entity
wife(x)
have(j, x)

 :


x:entity
wife(x)
have(j, x)


(@)
π1@i


x:entity
wife(x)
have(j, x)

 : entity
(ΣE)
happy

π1@i


x:entity
wife(x)
have(j, x)



 : type
(→E)

u:

 v:
y:entity
wife(y)
have(j, π1(v))



 → happy

π1@i


x:entity
wife(x)
have(j, x)



 : type
(ΠF),1
57 / 104

If John has a wife, John’s wife is happy: Proof Search
u :

 v:
y:entity
wife(y)
have(j, π1v)


1
π1u :
y:entity
wife(y)
(ΣE)
π1π1u : entity
(ΣE)
wife(x)
have(j, x)
[π1π1u/x]
⇓
wife(π1π1u)
have(j, π1π1u)
u :

 v:
y:entity
wife(y)
have(j, π1v)


1
π1u :
y:entity
wife(y)
(ΣE)
wife(y)[π1π1u/y]
⇓ wife(π1π1u)
π2π1u : wife(π1π1u)
(ΣE)
u :

 v:
y:entity
wife(y)
have(j, π1v)


1
have(j, π1v)[π1u/v]
⇓ have(j, π1π1u)
π2u : have(j, π1π1u)
(ΣE)
(π2π1u, π2u) :
wife(π1π1u)
have(j, π1π1u)
(ΣI )
(π1π1u, (π2π1u, π2u)) :


x:entity
wife(x)
have(j, x)


(ΣI )
58 / 104

Solution
59 / 104

Subject honorification: lexical
Analysis of Japanese honorification based on DTS (Watanabe,
McCready and Bekki, 2014): No problem arises for more than
two-place honorific predicates. Descriptive and expressive contents
use the different channels: types and proofs. But still integrated in
a single-dimensional semantic representation.
Sensei-ga
T /(T NPga)
: λp.p(sensei)
ringo-o
T /(T NPo)
: λp.p(apple)
mesiagat-ta
Sv::5::r
term|attr
NPgaNPo
: λy.λx.λc.
eat(x, y)
CI(honor(sp, x))
Sv::5::r
term|attr
NPga
: λx.λc.
eat(x, apple)
CI(honor(sp, x))
>
Sv::5::r
term|attr
: λc.
eat(sensei, apple)
CI(honor(sp, sensei))
>
60 / 104

Subject honoriﬁcation: verb stem + passive form
homera
Sv::5::r
vo::r
NPgaNPo
: λy.λx.λc.praise(x, y)
re
Sv::1
stem
NPga$(Svo::r NPga$)
: λp.λy.λx.λc.
pyxc
CI(honor(sp, x))
Sv::1
stem
NPgaNPo
: λy.λx.λc.
praise(x, y)
CI(honor(sp, x))
< ta
S1
term|attr
+t
S1
euph::t
: id
Sv::1
term|attr
+t
NPgaNPo
: λy.λx.λc.
praise(x, y)
CI(honor(sp, x))
<
61 / 104

Subject honoriﬁcation: o-verb stem
o-
Sn::da|no
stem
+hon
NPga$/(Scont
+O
NPga$)
: λp.λy.λx.λc.
pyxc
CI(honor(sp, x))
home
Sv::5::r
cont
+O
NPgaNPo
: λy.λx.λc.praise(x, y)
Sn::da|no
stem
+hon
NPgaNPo
: λy.λx.λc.
praise(x, y)
CI(honor(sp, x))
> nina-ru
Sv::5::r
term|attr
Sn::da
stem
+hon
: id
Sv::5::r
term|attr
NPgaNPo
: λy.λx.λc.
praise(x, y)
CI(honor(sp, x))
<B2
62 / 104

Object honoriﬁcation
Taro-ga
T /(T NPga)
: λp.p(taro)
sensei-o
T /(T NPo)
: λp.p(sensei)
o-tasuke
Sv::S
stem
NPgaNPo
: λy.λx.λc.


help(x, y)
CI(
honor(sp, y)
honor(x, y)
)


suru
Sv::S
term|attr
Sv::S
stem
: id
Sv::S
term|attr
NPgaNPo
: λy.λx.λc.


help(x, y)
CI(
honor(sp, y)
honor(x, y)
)


<B2
Sv::S
term|attr
NPga
: λx.λc.


help(x, sensei)
CI(
honor(sp, sensei)
honor(x, sensei)
)


>
Sv::S
term|attr
: λc.


help(taro, sensei)
CI(
honor(sp, sensei)
honor(taro, sensei)
)


>
63 / 104

What is the CI-operator?
Defined by means of intentional equality type:
Definition
CIi(A)
def
≡ @iA =A @iA
Γ A : s
Anaphora resolution within A
....
Γ A : type
Proof search for A (=projection of A)
....
Γ M : A
Γ @iA : A
(@)
Γ @iA =A @iA : type
(IdF)
where fv (M) ⊆ fv (K) .
Not filtered by the linguistic antecedent!
64 / 104

Honoriﬁc contents project over negation
Sensei-ga
NP
: teacher
irassyat-ta
SNP
: λx.
come(x)
CI(honor(sp, x))
S
:
come(teacher)
CI(honor(sp, teacher))
<
n
wakedehanai
SS
: λp.¬p
S
: ¬
come(teacher)
<
65 / 104

Honoriﬁc contents project over negation
K come(teacher)
K honor(sp, teacher) : type K honor(sp, teacher) true
K CI(honor(sp, teacher))
(@)
K
come(teacher)
: type
(ΣF)
K ¬
come(teacher)
: type
(¬F)
66 / 104

Honoriﬁc contents are not ﬁltered
K honor(we, Y ) : type
K, u : honor(we, Y )
invite(we, Y ) : type
honor(we, Y ) : type
K, u : honor(we, Y ) honor(we, Y ) true
K, u : honor(we, Y ) CI(honor(we, Y )) : type
(IdF)
invite(we, Y )
CI(honor(we, Y ))
(ΣF)
K (u:honor(we, Y )) →
invite(we, Y )
CI(honor(we, Y ))
(ΠF)
Proof search for a proof term M such that fv (M) ⊆ fv (K)
→ a term of type honor(we, Y ) without u
67 / 104

Binding problem
(18) Dare -mo
nobody-Nom
come-Hon-Neg-Pst
‘Nobody (in the contextually salient set) came.’
∀x(Descriptive:¬came(x) ∧ Expressive:honor(sp, x))
68 / 104

Binding problem
K
x:entity
human(x)
: type
K, u :
x:entity
human(x)
come(π1u) : type
K, u :
x:entity
human(x)
honor(sp, π1u) : type K, u :
x:entity
human(x)
honor(sp, π1u) true
K, u :
x:entity
human(x)
CI(honor(sp, π1u)) : type
(IdF)
K, u :
x:entity
human(x)
come(π1u)
CI(honor(sp, π1u))
: type
(ΣF)
K, u :
x:entity
human(x)
¬
come(π1u)
CI(honor(sp, π1u))
: type
(¬F)
K u:
x:entity
human(x)
→ ¬
come(π1u)
CI(honor(sp, π1u))
: type
(ΠF)
69 / 104

Local accommodation
K
x:entity
human(x)
: type
K, u :
x:entity
human(x)
come(π1u) : type
K, u :
x:entity
human(x)
honor(sp, π1u) : type
K, u :
x:entity
human(x)
@i u:
x:entity
human(x)
→ honor(sp, π1u) u : honor(sp, x)
K, u :
x:entity
human(x)
CI(honor(sp, π1u)) : type
(IdF)
K, u :
x:entity
human(x)
come(π1u)
CI(honor(sp, π1u))
: type
(ΣF)
K, u :
x:entity
human(x)
¬
come(π1u)
CI(honor(sp, π1u))
: type
(¬F)
K u:
x:entity
human(x)
→ ¬
come(π1u)
CI(honor(sp, π1u))
: type
(ΠF)
A new variable of type u:
x:entity
human(x)
→ honor(sp, π1u)
will be added to K
“The speaker honors everyone”
This seems to be a bit strong, but if the restriction of the
universal quantiﬁcation is contextually speciﬁed, it gives a
correct information to be “locally-accommodated.”
70 / 104

Summary
Dependent type semantics: a proof-theoretic, compositional
alternative to DRT(s) and Dynamic Logic(s)
Two levels of information in single-dimensional
representations: types and terms, propositions and proofs,
formation rules and introduction/elimination rules
Gives a solution to compositionality of expressive contents,
including Japanese honoriﬁcation
In DTS, calculations of presupposition projection and binding
reduce to type checking and proof search.
71 / 104

Reference I
Bekki, D. (2013) “Dependent Type Semantics: An Introduction”,
In: the 2012 edition of the LIRa yearbook: a selection of papers.
University of Amsterdam.
Bekki, D. (2014) “Representing Anaphora with Dependent Types”,
In the Proceedings of N. Asher and S. V. Soloviev (eds.):
Logical Aspects of Computational Linguistics (8th international
conference, LACL2014, Toulouse, France, June 2014
Proceedings), LNCS 8535. Toulouse, pp.14–29, Springer,
Heiderburg.
Bekki, D., A. Kawazoe, K. Kataoka, and M. Saito. (2008)
“Semantics of Honoriﬁcation”, In the Proceedings of NLP2008.
University of Tokyo, pp.681–684.
72 / 104

Reference II
Boeckx, C. and F. Niinuma. (2004) “Conditions on Agreement in
Japanese”, Natural Language and Linguistic Theory 22,
pp.453–480.
Cooper, R. (2005) “Austinian truth, attitudes and type theory”,
Research on Language and Computation 3, pp.333–362.
Coquand, T. and G. Huet. (1988) “The Calculus of Constructions”,
Information and Computation 76(2-3), pp.95–120.
D´avila-P´erez, R. (1995) “Semantics and Parsing in Intuitionistic
Categorial Grammar”, Ph.d. thesis, University of Essex.
Dummett, M. (1975) “What is a Theory of Meaning?”, In: S.
Guttenplan (ed.): Mind and Language. Oxford, Oxford
University Press, pp.97–138.
73 / 104

Reference III
Dummett, M. (1976) “What is a Theory of Meaning? (II)”, In:
Evans and McDowell (eds.): Truth and Meaning. Oxford,
Oxford University Press, pp.67–137.
Gunji, T. (1987) Japanese Phrase Structure Grammar: A
Uniﬁcation-based Approach. Dordrecht, D. Reidel.
Gutzmann, D. (2015) Use-conditional meaning, Vol. 6 of Studies
in Semantics and Pragmatics, Oxford Studies in Semantics and
Pragmatics. Oxford, Oxford University Press.
Harada, S.-I. (1976) “Honoriﬁcs”, In: M. Shibatani (ed.): Syntax
and Semantics 5, Vol. 499-561. New York, Academic Press.
74 / 104

Reference IV
Krahmer, E. and P. Piwek. (1999) “Presupposition Projection as
Proof Construction”, In: H. Bunt and R. Muskens (eds.):
Computing Meanings: Current Issues in Computational
Semantics, Studies in Linguistics Philosophy Series. Dordrecht,
Kluwer Academic Publishers.
Krause, P. (1995) “Presupposition and Abduction in Type
Theory”, In the Proceedings of K. E., S. Manandhar, W. Nutt,
and J. Siekman (eds.): Edinburgh Conference on Computational
Logic and Natural Language Processing. Edinburgh: HCRC.
Luo, Z. (2012) “Formal Semantics in Modern Type Theories with
Coercive Subtyping”, Linguistics and Philosophy 35(6).
Martin-L¨of, P. (1984) Intuitionistic Type Theory, Vol. 17. Naples,
Italy: Bibliopolis. Sambin, Giovanni (ed.).
75 / 104

Reference V
McCready, E. (2010) “Varieties of Conventional Implicature”,
Semantics and Pragmatics 3(8), pp.1–57.
Milner, R. (1978) “A Theory of Type Polymorphism in
Programming”, Journal of Computer and System Science
(JCSS) 17, pp.348–374.
Mineshima, K. (2008) “A presuppositional analysis of deﬁnite
descriptions in proof theory”, In: K. Satoh, A. Inokuchi, K.
Nagao, and T. Kawamura (eds.): New Frontiers in Artiﬁcial
Intelligence: JSAI 2007 Conference and Workshops, Revised
Selected Papers, Lecture Notes in Computer Science Vol. 4914.
Springer-Verlag, pp.214–227.
Mineshima, K. (2014) “Presupposition and Descriptions: A
Proof-Theoretical Approach”, In: Aspects of Inference in
Natural Language. Keio University, p.135.
76 / 104

Reference VI
Piwek, P. and E. Krahmer. (2000) “Presuppositions in Context:
Constructing Bridges”, In: P. Bonzon, M. Cavalcanti, and R.
Nossum (eds.): Formal Aspects of Context, Applied Logic
Series. Dordrecht, Kluwer Academic Publishers.
Potts, C. (2005) The Logic of Conventional Implicatures. Oxford
University Press.
Prawitz, D. (1980) “Intuitionistic Logic: A Philosophical
Challenge”, In: G. von Wright (ed.): Logics and Philosophy.
The Hague, Martinus Nijhoﬀ.
Ranta, A. (1994) Type-Theoretical Grammar. Oxford University
Press.
77 / 104

Reference VII
Siegel, M. (2000) “Japanese Honoriﬁcation in an HPSG
Framework”, In the Proceedings of PACLIC 14 : 14th Paciﬁc
Asia Conference on Language, Information and Computation.
Waseda University International Conference Center, Tokyo,
Japan, pp.289–300, PACLIC 14 Organizing Committee.
Sundholm, G. (1986) “Proof theory and meaning”, In: D. Gabbay
and F. Guenthner (eds.): Handbook of Philosophical Logic, Vol.
III. Reidel, Kluwer, pp.471–506.
78 / 104

PTS DTT formal presentation Underspeciﬁed DTT
Proof-theoretic Semantics
79 / 104

Model-theoretic vs. proof-theoretic semantics
Model-theoretic semantics (of mathematics and natural language)
Primitive notions: truth , model
Interpretation is a function: formula × model → truth
Validity of inference is defined as a derived notion.
Meaning of a sentence: its truth-condition
Proof-theoretic semantics (of mathematics and natural language)
Primitive notion:
formation / introduction / elimination rules
Provability is a binary relation between a list of formula and
formula
Truth is defined as a derived notion.
Meaning of a sentence: its verification condition
80 / 104

Model-theoretic vs. proof-theoretic semantics
In classical and intuitionistic ﬁrst-order logic, there are many pairs
of model theory and proof theory that satisfy the sound and
complete relation.
Γ A ⇐⇒ Γ A
In other words,
For every model M, if Γ M = 1 then A M = 1
⇐⇒ There is a proof from Γ to A
A pair of a model theory and a proof theory which is sound and
complete with each other do not make any diﬀerence with respect
to inferences. Given that (almost all) the data in semantics are
entailment relations between sentences, the two approaches are not
distinguishable as natural sciences.
81 / 104

A proof theory: General elimination rules
Implication Conjunction
Introduction
rules
Γ, A B
Γ A → B
(→I )
Γ A Γ B
Γ A ∧ B
(∧I )
Elimination
rules
Γ A → B Γ A
Γ B
(→E)
Γ A ∧ B
Γ A
(∧E)
Γ A ∧
Γ B
General
elimination
rules
Γ A B, Γ C
Γ, A → B C
(→GE)
Γ, A, B C
Γ, A ∧ B C
(∧GE)
82 / 104

General elimination rules
General elimination rules:
are equivalent to elimination rules
ca be automatically obtained from a set of introduction rules
ensure that the introduction rules are exhaustive
Thus
Introduction rules carry a primary meaning of the type:
veriﬁcational meaning
Elimination rules carry a secondary meaning of the type:
pragmatic meaning
83 / 104

Verification conditions: Example
(19) John jogs and Mary runs.
Verification of (19) requires the verifications of “John jogs” and
“Mary runs”.
(20) If x s a natural number, then either x is odd or even.
Verification of (20) requires the verification of “either x is odd or
even”, assuming the verification of “x s a natural number”.
84 / 104

DTT formal presentation
85 / 104

Type Theory
A deﬁnition of a type theory consists of following items:
1. Alphabets
2. Preterms
3. Free variables
4. Substitution
5. Reduction
6. Typing/inference rules
6.1 Formation rule(s)
6.2 Introduction rule(s)
6.3 Elimination rule(s)
86 / 104

Preterms
Definition (Preterms)
A collection of preterms (notation: Λ) for an alphabet (Var, Con)
is recursively defined by the following BNF grammar (where
x ∈ Var and c ∈ Con).
Λ ::= x | c | type | kind
| (x:Λ) → Λ | λx.Λ | ΛΛ
|
x:Λ
Λ
| (Λ, Λ) | π1(Λ) | π2(Λ)
| ⊥ | | ()
| Λ =Λ Λ | reflΛ(Λ) | idpeel(Λ, Λ)
Var
def
≡ {x, y, z, u, v, w, . . .}
Con
def
≡ {entity, man, donkey, enter, own, beats, . . .}
87 / 104

Free variables and substitution
fv (x)
def
≡ {x}
fv (c)
def
≡ ∅
fv (type)
def
≡ ∅
fv (kind)
def
≡ ∅
fv ((x:A) → B)
def
≡ fv (A) ∪ (fv (B) − {x})
fv (λx.M)
def
≡ fv (M) − {x}
fv (MN)
def
≡ fv (M) ∪ fv (N)
fv
x:A
B
def
≡ fv (A) ∪ (fv (B) − {x})
fv ((M, N))
def
≡ fv (M) ∪ fv (N)
fv (πi(M))
def
≡ fv (M)
fv (⊥)
def
≡ ∅
fv ( )
def
≡ ∅
fv (())
def
≡ ∅
fv (M =A N)
def
≡ fv (A) ∪ fv (M) ∪ fv (N)
fv (reflA(M))
def
≡ fv (A) ∪ fv (M)
fv (idpeel(M, N))
def
≡ fv (M) ∪ fv (N)
x[L/x]
def
≡ L
y[L/x]
def
≡ y
c[L/x]
def
≡ c
type[L/x]
def
≡ type
kind[L/x]
def
≡ kind
((x:M) → N)[L/x]
def
≡ (x:M[L/x]) → N
((y:M) → N)[L/x]
def
≡ (y:M[L/x]) → (N[L/x])
where x /∈ fv (N) ∨ y /∈ fv (L) .
(λx.M)[L/x]
def
≡ λx.M
(λy.M)[L/x]
def
≡ λy.M[L/x]
where x /∈ fv (N) ∨ y /∈ fv (L) .
(MN)[L/x]
def
≡ (M[L/x])(N[L/x])
x:M
N
[L/x]
def
≡
x:M[L/x]
N
y:M
N
[L/x]
def
≡
y:M[L/x]
(N[L/x])
where x /∈ fv (N) ∨ y /∈ fv (L) .
((M, N))[L/x]
def
≡ (M[L/x], N[L/x])
(π1(M))[L/x]
def
≡ π1(M[L/x])
(π2(M))[L/x]
def
≡ π2(M[L/x])
⊥[L/x]
def
≡ ()
[L/x]
def
≡ ()
()[L/x]
def
≡ ()
(M =A N)[L/x]
def
≡ M[L/x] =A[L/x] N[L/x]
(reflA(M))[L/x]
def
≡ reflA[L/x](M[L/x])
(idpeel(M, N))[L/x]
def
≡ idpeel(M[L/x], N[L/x])
88 / 104

Reduction
Deﬁnition (β-reduction in DTT)
x ⇓ x type ⇓ type kind ⇓ kind
A ⇓ A B ⇓ B
(x:A) → B ⇓ (x:A ) → B
M ⇓ M
λx.M ⇓ λx.M
M ⇓ λx.M M [N/x] ⇓ M
MN ⇓ M
M ⇓ n N ⇓ N
MN ⇓ nN
A ⇓ A B ⇓ B
x:A
B
⇓
x:A
B
M ⇓ M N ⇓ N
(M, N) ⇓ (M , N )
M ⇓ (M1, M2)
πiM ⇓ Mi
M ⇓ n
πiM ⇓ πin
⊥ ⇓ ⊥ ⇓ () ⇓ ()
89 / 104

Reduction
Deﬁnition (Values and neutral terms of DTT)
v ::= n | type | kind | (x:v) → v | λx.v |
x:v
v
| (v, v) | ⊥ | | ()
n ::= x | c | nv | πin
Theorem
If M is typable, then any M such that M ⇓ M is a value, and
the calculation of M terminates.
Proof.
By induction on the structure of M. In particular, if MN is
typable then M reduces to either the form λx.v or a neutral term;
if πiM is typable then M reduces to either the form (v, v) or a
neutral term.
90 / 104

Basic rules
Deﬁnition (Basic rules)
Γ type : kind
(typeF)
Γ M : A ∆ N : B
Γ M : A
(WK)
Γ M : A A =β B
Γ M : B
(CONV )
91 / 104

Underspeciﬁed dependent type theory
92 / 104

What is UDTT?
UDTT = DTT + underspecified terms (notation @iA, for i ∈ N)
Definition (Preterms of UDTT)
Let DTTpreterms be a collection of preterms in DTT. The
collection of preterms of UDTT (notation Λ) is defined by the
following BNF grammar (where i ∈ N):
Λ ::= DTTpreterms | @iΛ
Example (Anaphoric expressions/presupposition triggers)
hei = π1@i
x:entity
male(x)
thej = λn.π1@j
x:entity
nx
93 / 104

Typing rules of UDTT
The @-rule is the formation rule of underspeciﬁed terms.
Deﬁnition (@-rule)
DA
Γ A : type Γ DA true
Γ @iA : A
(@)
This is understood as the set of following instructions, when called
as a type checking of @iA : A.
1. Check if A : type, let its diagram be DA.
2. Eliminate @ in A if any, resulting in DA .
3. Then search for a proof term of type DA .
94 / 104

Deﬁnition (type-formation rule)
Γ A : s
Γ, x : A, ∆ x : A
(VAR)
where A is a DTT term.
Γ type : kind
(typeF)
95 / 104

Deﬁnition (Π-formation/introduction/elimination rules)
For any s1, s2 ∈ {type, kind}:
DA
Γ A : s1 Γ, x : DA B : s2
Γ (x:A) → B : s2
(ΠF)
DA
Γ A : s1 Γ, x : DA M : B
Γ λx.M : (x:A) → B
(ΠI )
Γ M : (x:A) → B Γ N : A
Γ MN : B[M/x]
(ΠE) where A is a DTT term.
96 / 104

Deﬁnition (Σ-formation/introduction/elimination rules)
DA
Γ A : s1 Γ, x : DA B : s2
Γ
x:A
B
: s2
(ΣF)
DM
Γ M : A Γ N : B[ DM /x]
Γ (M, N) :
x:A
B
(ΣI )
Γ M :
x:A
B
Γ π1(M) : A
(ΣE)
Γ M :
x:A
B
Γ π2(M) : B[π1M/x]
(ΣE)
97 / 104

@-elimination −
@-elimination − is a map from a proof diagram of UDTT
to a term of DTT (=a term without @)
Type checking diagram of UDTT may contain occurrences of
underspeciﬁed terms (@).
The @-rule for a term of the form @iA contains a proof of
type A. Let the proof term be M. Then speciﬁcation replaces
the occurrence of @ with such M, resulting in a DTT
diagram which contains no @.
....
A : s
....
M : A
@iA : A
(@)
....
[· · · @iA · · · ]
= [· · · M · · · ]
98 / 104

@-elimination −
Deﬁnition (@-elimination for @-rule)
Γ A : s Γ M : DA
Γ (@iA) : A
(@) = M
Deﬁnition (@-elimination for type-rule)
Γ A : s
Γ, x : A, ∆ x : A
(VAR) = x
Γ type : kind
(typeF)
= type
99 / 104

@-elimination −
Deﬁnition (@-elimination for Π-rules)
DA
Γ A : s1
DB
Γ, x : DA B : s2
Γ (x:A) → B : s2
(ΠF)
= (x: DA ) → DB
DA
Γ A : s1
DM
Γ, x : DA M : B
λx.M : (x:A) → B
(ΠI )
= λx. DM
DM
Γ M : (x:A) → B
DN
Γ N : A
Γ MN : B[N/x]
(ΠE)
= DM DN
100 / 104

@-elimination −
Deﬁnition (@-elimination for Σ-rules)
DA
Γ A : s1
DB
Γ, x : DA B : s2
Γ
x:A
B
: s2
(ΣF) =
x: DA
DB
DM
M : A
DN
N : B[ DM /x]
(M, N) :
x:A
B
(ΣI ) = ( DM , DN )
DM
Γ M :
x:A
B
Γ π1(M) : A
(ΣE)
= π1( DM )
DM
Γ M :
x:A
B
Γ π2(M) : B[π1(M)/x]
(ΣE)
= π2( DM )
101 / 104

@-elimination −
Definition (Free variables and substitution)
fv (@iA)
def
≡ fv (A)
(@iA)[M/x]
def
≡ @i(A[M/x])
Definition (Inferable and checkable terms of UDTT)
Λ↑ ::= x | type | kind
| (x:Λ↓) → Λ↓ | Λ↑Λ↓
|
x:Λ↓
Λ↓
| πiΛ↑
| ⊥ | | ()
| Λ =Λ Λ | reflΛ(Λ) | idpeel(Λ, Λ)
| @iΛ↑
Λ↓ ::= Λ↑ | λx.Λ↓ | (Λ↓, Λ↓) 102 / 104

@-elimination −
Definition (Values and neutral terms of UDTT)
Values of UDTT is defined in the same way as DTT, except that a
term of the form @iv is a neutral term.
Definition (β-reduction in UDTT)
β-reduction relation ⇓ in UDTT is defined as that of DTT
extended by the following rule:
A ⇓ A
@iA ⇓ @iA
(@⇓)
103 / 104

@-elimination −
Theorem
If M is a checkable term of UDTT that is typable, and M ⇓ M ,
then M is a value.
Proof.
By induction on the structure of M. Interesting cases are when M
is either functional application or projections.
104 / 104

Composing (Im)politeness in Dependent Type Semantics

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