Outline
Examples and Motivation
Associative λ-calculi
Chiastic λ-calculi
hhλ
→iiχL in action
Chiastic Lambda-Calculi
wren ng thornton
Cognitive Science & Computational Linguistics
Indiana University, Bloomington
NLCS, 28 June 2013
wren ng thornton (Indiana University)
Chiastic Lambda-Calculi
NLCS, 28 June 2013
0 / 34
Outline
Examples and Motivation
Associative λ-calculi
Chiastic λ-calculi
hhλ
→iiχL in action
Examples and Motivation
Associative λ-calculi
Chiastic λ-calculi
hhλ→iiχL in action
wren ng thornton (Indiana University)
Chiastic Lambda-Calculi
NLCS, 28 June 2013
1 / 34
Outline
Examples and Motivation
Associative λ-calculi
Chiastic λ-calculi
hhλ
→iiχL in action
Examples and Motivation
Associative λ-calculi
Chiastic λ-calculi
hhλ→iiχL in action
wren ng thornton (Indiana University)
Chiastic Lambda-Calculi
NLCS, 28 June 2013
1 / 34
Outline
Examples and Motivation
Associative λ-calculi
Chiastic λ-calculi
hhλ
→iiχL in action
Examples
Scrambling in Japanese
• Tarou -ga
hon -wo
book Acc
yon-da
read-Perf
Tarou -ga
book Acc
—
Nom
‘Taro read the book.’
yon-da
read-Perf
—
Nom
• hon -wo
Keyword arguments
• yonda(wo=‘hon’, ga=‘Tarou’)
• yonda(ga=‘Tarou’, wo=‘hon’)
Shorthands in category theory
• (FG )X = F (G X)
• (ηF )X = ηFX
• (F η)X = F (ηX )
wren ng thornton (Indiana University)
Chiastic Lambda-Calculi
NLCS, 28 June 2013
1 / 34
Outline
Examples and Motivation
Associative λ-calculi
Chiastic λ-calculi
hhλ
→iiχL in action
What do these have in common?
Juxtaposition is associative
f (g x) ≈ (f g ) x
Application is commutative
f xy ≈ f yx
Our Goal convert those “≈” into “=”
wren ng thornton (Indiana University)
Chiastic Lambda-Calculi
NLCS, 28 June 2013
2 / 34
Outline
Examples and Motivation
Associative λ-calculi
Chiastic λ-calculi
hhλ
→iiχL in action
Scrambling in Japanese
Many languages have “free word order”
• Tarou -ga
hon -wo
book Acc
yon-da
read-Perf
Tarou -ga
book Acc
—
Nom
‘Taro read the book.’
yon-da
read-Perf
—
Nom
• hon -wo
Both orders are normal and natural
• Both have the same propositional content
• Though, information structure may differ
•
wren ng thornton (Indiana University)
Chiastic Lambda-Calculi
NLCS, 28 June 2013
3 / 34
Outline
Examples and Motivation
Associative λ-calculi
Chiastic λ-calculi
hhλ
→iiχL in action
Arguments: Chomskian-style accounts
-ga
Tarou
...... .........
N
NPnom \N
J
NPnom
. . .-wo
.....
yonNPacc \N
................
J
NPacc
V \NPnom \NPacc
J
V \NPnom
.-da
....
J
V
S\V
J
S
.hon
...
N
-ga
.........
NPnom \N
J
yonNPnom
................
IT
V /(V \NPnom )
V \NPnom \NPacc
IBx
V \NPacc
.-da
....
J
V
S\V
J
S
Tarou
......
N
.hon
...
N
. . .-wo
.....
NPacc \N
J
NPacc
wren ng thornton (Indiana University)
Chiastic Lambda-Calculi
NLCS, 28 June 2013
4 / 34
Outline
Examples and Motivation
Associative λ-calculi
Chiastic λ-calculi
hhλ
→iiχL in action
Adjuncts: Radical neo-Davidsonian accounts
Tarou
......
N
-ga
.......
S/S\N
J
S/S
.hon
...
N
. .-wo
.....
S/S\N
J
S/S
S
S
.hon
...
N
. .-wo
.....
S/S\N
J
S/S
Tarou
......
N
-ga
.......
S/S\N
J
S/S
S
S
wren ng thornton (Indiana University)
Chiastic Lambda-Calculi
yon....
V
.-da
....
S\V
J
S
I
I
yon....
V
.-da
....
S\V
J
S
I
I
NLCS, 28 June 2013
5 / 34
Outline
Examples and Motivation
Associative λ-calculi
Chiastic λ-calculi
hhλ
→iiχL in action
Adjuncts: Radical neo-Davidsonian accounts
Tarou
......
N
.hon
...
N
-ga
.......
.....
.hon
. . . . .-wo
S/S\N
S/S\N
N
J
J
S/S
S/S
IB
S/S
S
-ga
Tarou
. .-wo
.....
...... .......
N
S/S\N
S/S\N
J
J
S/S
S/S
IB
S/S
S
wren ng thornton (Indiana University)
Chiastic Lambda-Calculi
yon....
V
.-da
....
S\V
J
S
I
yon....
V
.-da
....
S\V
J
S
I
NLCS, 28 June 2013
6 / 34
Outline
Examples and Motivation
Associative λ-calculi
Chiastic λ-calculi
hhλ
→iiχL in action
Arguments vs Adjuncts
Why prefer adjuncts?
• Avoids the need for T and Bx
(they’re dangerous together)
• Syntax matches morphology/prosody
• Same parse tree for different word orders
• Online and partial parsing is easy
wren ng thornton (Indiana University)
Chiastic Lambda-Calculi
(commutativity)
(associativity)
NLCS, 28 June 2013
7 / 34
Outline
Examples and Motivation
Associative λ-calculi
Chiastic λ-calculi
hhλ
→iiχL in action
Arguments vs Adjuncts
Why prefer adjuncts?
• Avoids the need for T and Bx
(they’re dangerous together)
• Syntax matches morphology/prosody
• Same parse tree for different word orders
• Online and partial parsing is easy
(commutativity)
(associativity)
Only moves the problem from syntax to semantics!
• Also true of other CCG approaches to scrambling
wren ng thornton (Indiana University)
Chiastic Lambda-Calculi
NLCS, 28 June 2013
7 / 34
Outline
Examples and Motivation
Associative λ-calculi
Chiastic λ-calculi
hhλ
→iiχL in action
Arguments vs Adjuncts
Why prefer adjuncts?
• Avoids the need for T and Bx
(they’re dangerous together)
• Syntax matches morphology/prosody
• Same parse tree for different word orders
• Online and partial parsing is easy
(commutativity)
(associativity)
Only moves the problem from syntax to semantics!
• Also true of other CCG approaches to scrambling
Chiastic λ-calculi solve the problem
(in the semantics)
wren ng thornton (Indiana University)
Chiastic Lambda-Calculi
NLCS, 28 June 2013
7 / 34
Outline
Examples and Motivation
Associative λ-calculi
Chiastic λ-calculi
hhλ
→iiχL in action
Examples and Motivation
Associative λ-calculi
Chiastic λ-calculi
hhλ→iiχL in action
wren ng thornton (Indiana University)
Chiastic Lambda-Calculi
NLCS, 28 June 2013
8 / 34
Outline
Examples and Motivation
Associative λ-calculi
Chiastic λ-calculi
hhλ
→iiχL in action
What are functions?
Traditional λ-calculi intentionally confuse two ideas
Procedures operations mapping values to values
Data values representing procedures
Category theory keeps them distinct
Morphisms functions as procedures
Exponentials functions as data
For associativity, we must keep them distinct too
(λx. e) Unbracketed abstractions are procedures
hhλx. eii Bracketed abstractions are values
wren ng thornton (Indiana University)
Chiastic Lambda-Calculi
NLCS, 28 June 2013
8 / 34
Outline
Examples and Motivation
Associative λ-calculi
Chiastic λ-calculi
hhλ
→iiχL in action
Associative λ-calculi: hhλii
Variables
x, y, z,. . .
Terms
e, f, g,. . .
(λx. f ) · hheii
::=
x
variables
|
(λx. e)
abstraction
|
hheii
bracketing
|
e ·f
juxtaposition
{x 7→ e}f
(e · f ) · g ≡ e · (f · g )
wren ng thornton (Indiana University)
Chiastic Lambda-Calculi
Beta
Assoc
NLCS, 28 June 2013
9 / 34
Outline
Examples and Motivation
Associative λ-calculi
Chiastic λ-calculi
hhλ
→iiχL in action
What does juxtaposition mean?
Application (λx. e) · hhf ii
Composition (λx. e) · (λy. f )
(λx. e) · (λy. f ) · hhg ii ≡ (λx. e) · (λy. f ) · hhg ii
Tupling hhf ii · hhg ii
(λx.λy. e) · hhf ii · hhg ii ≡ (λx.λy. e) · hhf ii · hhg ii
wren ng thornton (Indiana University)
Chiastic Lambda-Calculi
NLCS, 28 June 2013
10 / 34
Outline
Examples and Motivation
Associative λ-calculi
Chiastic λ-calculi
hhλ
→iiχL in action
How powerful is it?
hhL ii is at least as powerful as L
• Every L -term has an evaluation-equivalent hhL ii-term
JxK = x
J(λx. e)K = (λx. JeK)
..
.
Je · f K = JeK · hhJf Kii
J(e)K = JeK
hhL ii can be more expressive than L
• hhλ
→ii has tuples, but they can’t be encoded in λ
→
• Then again, almost everything stronger than λ
→ has tuples
wren ng thornton (Indiana University)
Chiastic Lambda-Calculi
NLCS, 28 June 2013
11 / 34
Outline
Examples and Motivation
Associative λ-calculi
Chiastic λ-calculi
hhλ
→iiχL in action
Examples and Motivation
Associative λ-calculi
Chiastic λ-calculi
hhλ→iiχL in action
wren ng thornton (Indiana University)
Chiastic Lambda-Calculi
NLCS, 28 June 2013
12 / 34
Outline
Examples and Motivation
Associative λ-calculi
Chiastic λ-calculi
hhλ
→iiχL in action
Chiastic λ-calculi
The term level
Syntax — Two flavors of chiasmus
Equivalence
Reduction
Sanity check
The type level
Syntax
Equivalence
Reduction
Sanity check — Well-formed types
wren ng thornton (Indiana University)
Chiastic Lambda-Calculi
NLCS, 28 June 2013
12 / 34
Outline
Examples and Motivation
Associative λ-calculi
Chiastic λ-calculi
hhλ
→iiχL in action
Formalizing restricted commutativity
Actually we don’t want full commutativity
• Tarou -ga
kuruma -ga
—
Nom
car
Nom
‘Taro has a car.’
ar-u
have-Npst
% kuruma -ga
Tarou -ga
car
Nom
—
Nom
‘The car has a Taro.’
ar-u
have-Npst
• Let a dimension denote a class of non-commuting elements
• Elements along different dimensions don’t interfere with one another
wren ng thornton (Indiana University)
Chiastic Lambda-Calculi
NLCS, 28 June 2013
13 / 34
Outline
Examples and Motivation
Associative λ-calculi
Chiastic λ-calculi
hhλ
→iiχL in action
Chiastic λ-calculi: hhλiiχ
Variables
x, y, z,. . .
Dimensions
A, B, C ,. . .
Terms
e, f, g,. . .
::=
x
variables
|
(λA x. e)
abstraction
|
hheiiA
bracketing
|
e ·f
juxtaposition
• Choose one
A 6= B
(λAx. λB y. e) ≡ (λB y. λAx. e)
Chi L
A 6= B
hheiiA · hhf iiB ≡ hhf iiB · hheiiA
wren ng thornton (Indiana University)
Chiastic Lambda-Calculi
Chi R
NLCS, 28 June 2013
14 / 34
Outline
Examples and Motivation
Associative λ-calculi
Chiastic λ-calculi
hhλ
→iiχL in action
Chiastic λ-calculi: hhλiiχ
Variables
x, y, z,. . .
Dimensions
A, B, C ,. . .
Terms
e, f, g,. . .
::=
x
variables
|
(λA x. e)
abstraction
|
hheiiA
bracketing
|
e ·f
juxtaposition
• For this talk, we’ll only consider Chi L
A 6= B
(λAx. λB y. e) ≡ (λB y. λAx. e)
Chi L
A 6= B
hheiiA · hhf iiB ≡ hhf iiB · hheiiA
wren ng thornton (Indiana University)
Chiastic Lambda-Calculi
Chi R
NLCS, 28 June 2013
14 / 34
Outline
Examples and Motivation
Associative λ-calculi
Chiastic λ-calculi
hhλ
→iiχL in action
Chi L vs Chi R
Should we accept terms like this?
(λAx. λB y. e) · (λC z. hhaiiA) · hhbiiB · hhciiC
Should we accept sentences like this?
• [sono hon -wo hHanako -gai Tarou -ga
kat-ta]
-to
that book Acc —
Nom —
Nom buy-Perf Comp
omot-te iru
think.Prog
‘Hanako thinks that Taro bought that book.’
wren ng thornton (Indiana University)
Chiastic Lambda-Calculi
NLCS, 28 June 2013
15 / 34
Outline
Examples and Motivation
Associative λ-calculi
Chiastic λ-calculi
hhλ
→iiχL in action
Term equivalence for hhλiiχL
e ≡ f
A 6= B
(λA x. λB y. e) ≡ (λB y. λA x. e)
e ≡ e0
(λA x. e) ≡ (λA x. e 0 )
e ≡ e
wren ng thornton (Indiana University)
e ≡ e0
hheiiA ≡ hhe 0 iiA
f ≡ e
e ≡ f
Chiastic Lambda-Calculi
e · (f · g ) ≡ (e · f ) · g
e ≡ e0
f ≡ f0
e · f ≡ e0 · f 0
e ≡ f
f ≡ g
e ≡ g
NLCS, 28 June 2013
16 / 34
Outline
Examples and Motivation
Associative λ-calculi
hhλ
→iiχL in action
Chiastic λ-calculi
Term reduction for hhλiiχL
e
e0
h ≡ (λA x. f )
h · hheiiA
{x 7→ e}f
e
hheiiA
e0
hhe 0 iiA
e ·f
e · (f · g )
wren ng thornton (Indiana University)
e
e ·f
h
h·g
e
(λA x. e)
e0
e0 · f
f ·g
(e · f ) · g
Chiastic Lambda-Calculi
e0
(λA x. e 0 )
f
e ·f
f0
e ·f 0
h
e ·h
NLCS, 28 June 2013
17 / 34
Outline
Examples and Motivation
Associative λ-calculi
Chiastic λ-calculi
hhλ
→iiχL in action
Do our terms make sense?
Theorem Term reduction is weak Church–Rosser.
Proof There are no critical pairs.
Corollary Term reduction is Church–Rosser.
Proof Supposing we can prove strong normalization,
then just use Newman’s lemma.
Conjecture Term reduction (for hhλ→iiχL ) is strongly normalizing.
Remark This is suspiciously difficult to prove.
wren ng thornton (Indiana University)
Chiastic Lambda-Calculi
NLCS, 28 June 2013
18 / 34
Outline
Examples and Motivation
Associative λ-calculi
Chiastic λ-calculi
hhλ
→iiχL in action
What are types?
The intrinsic view (à la Church)
• Types are manifest in terms
• Terms can have only one type
• Ill-typed terms “don’t exist”
The extrinsic view (à la Curry)
• Types characterize properties of terms
• Terms could have multiple types
• All terms exist, but we only care about the well-typed ones
wren ng thornton (Indiana University)
Chiastic Lambda-Calculi
NLCS, 28 June 2013
19 / 34
Outline
Examples and Motivation
Associative λ-calculi
Chiastic λ-calculi
hhλ
→iiχL in action
What are types?
The intrinsic view (à la Church)
• Types are manifest in terms
• Terms can have only one type
• Ill-typed terms “don’t exist”
The extrinsic view (à la Curry)
• Types characterize properties of terms
• Terms could have multiple types
• All terms exist, but we only care about the well-typed ones
Our view
• Types give abstract interpretations of terms
Γ ` e Bτ
τ ∗ τ0
∃e 0. e ∗ e 0 ∧ Γ ` e 0 B τ 0
wren ng thornton (Indiana University)
Chiastic Lambda-Calculi
NLCS, 28 June 2013
19 / 34
Outline
Examples and Motivation
Associative λ-calculi
hhλ
→iiχL in action
Chiastic λ-calculi
Simply-typed left-chiastic λ-calculus: hhλ→iiχL
Types
σ, τ , υ,. . .
::=
T
primitive types
A
|
σ →τ
arrow types
|
hhτ iiA
bracketed types
|
σ·τ
juxtaposition
Γ ` e Bτ
`ctx Γ
Γ(x) ≡ τ
Γ ` x Bτ
Γ ` e Bτ
Γ ` hheiiA B hhτ iiA
wren ng thornton (Indiana University)
Γ, x : σ ` e B τ
A
Γ ` (λA x. e) B σ → τ
Γ ` e Bσ
Γ ` f Bτ
Γ ` e ·f B σ·τ
Chiastic Lambda-Calculi
NLCS, 28 June 2013
20 / 34
Outline
Examples and Motivation
Associative λ-calculi
Chiastic λ-calculi
hhλ
→iiχL in action
Type equivalence for hhλ
→iiχL
τ ≡ σ
A 6= B
A
B
B
A
σ→τ →υ ≡ τ →σ→υ
σ ≡ σ0
τ ≡ τ0
A
A
σ → τ ≡ σ0 → τ 0
τ ≡ τ
wren ng thornton (Indiana University)
τ ≡ τ0
hhτ iiA ≡ hhτ 0 iiA
σ ≡ τ
τ ≡ σ
Chiastic Lambda-Calculi
σ · (τ · υ) ≡ (σ · τ ) · υ
σ ≡ σ0
τ ≡ τ0
σ · τ ≡ σ0 · τ 0
σ ≡ τ
τ ≡ υ
σ ≡ υ
NLCS, 28 June 2013
21 / 34
Outline
Examples and Motivation
Associative λ-calculi
hhλ
→iiχL in action
Chiastic λ-calculi
Type reduction for hhλ
→iiχL
τ
τ0
A
ρ ≡ σ→τ
ρ · hhσiiA
τ
τ
hhτ iiA
τ0
hhτ 0 iiA
σ·τ
σ · (τ · υ)
wren ng thornton (Indiana University)
σ
A
σ→τ
σ
σ·τ
ρ
ρ·υ
σ0
τ
A
τ0
σ0 → τ
σ→τ
A
σ → τ0
σ0
σ0 · τ
τ
σ·τ
τ0
σ · τ0
τ ·υ
(σ · τ ) · υ
Chiastic Lambda-Calculi
A
ρ
τ ·ρ
NLCS, 28 June 2013
22 / 34
Outline
Examples and Motivation
Associative λ-calculi
Chiastic λ-calculi
hhλ
→iiχL in action
Do our types make sense?
Every type has a normal form
Theorem Type reduction for hhλ→iiχL is strongly normalizing
Proof
Theorem Type reduction for hhλ→iiχL is Church–Rosser
Proof
So we can define
Γ ` e B τ0
NF(τ0 ) ≡ τ
`type τ
Γ`e:τ
wren ng thornton (Indiana University)
Chiastic Lambda-Calculi
NLCS, 28 June 2013
23 / 34
Outline
Examples and Motivation
Associative λ-calculi
Chiastic λ-calculi
hhλ
→iiχL in action
Do our types make sense?
Every type has a normal form
But, what does unresolved type juxtaposition mean?
wren ng thornton (Indiana University)
Chiastic Lambda-Calculi
NLCS, 28 June 2013
24 / 34
Outline
Examples and Motivation
Associative λ-calculi
Chiastic λ-calculi
hhλ
→iiχL in action
Do our types make sense?
Every type has a normal form
But, what does unresolved type juxtaposition mean?
Good hhσiiA · hhτ iiB
wren ng thornton (Indiana University)
Chiastic Lambda-Calculi
NLCS, 28 June 2013
24 / 34
Outline
Examples and Motivation
Associative λ-calculi
Chiastic λ-calculi
hhλ
→iiχL in action
Do our types make sense?
Every type has a normal form
But, what does unresolved type juxtaposition mean?
Good hhσiiA · hhτ iiB
A
Bad (σ → hhτ iiB ) · hhυiiC
A
(σ → τ ) · hhυiiA
wren ng thornton (Indiana University)
where A 6= C
where σ ≡
6 υ
Chiastic Lambda-Calculi
NLCS, 28 June 2013
24 / 34
Outline
Examples and Motivation
Associative λ-calculi
Chiastic λ-calculi
hhλ
→iiχL in action
Do our types make sense?
Every type has a normal form
But, what does unresolved type juxtaposition mean?
Good hhσiiA · hhτ iiB
A
Bad (σ → hhτ iiB ) · hhυiiC
A
(σ → τ ) · hhυiiA
where A 6= C
where σ ≡
6 υ
B
Ugly hhσiiA · (τ → υ)
A
B
(ρ → σ) · (τ → υ)
wren ng thornton (Indiana University)
Chiastic Lambda-Calculi
NLCS, 28 June 2013
24 / 34
Outline
Examples and Motivation
Associative λ-calculi
Chiastic λ-calculi
hhλ
→iiχL in action
Do our types make sense?
Every type has a normal form
But, what does unresolved type juxtaposition mean?
Good hhσiiA · hhτ iiB
A
Bad (σ → hhτ iiB ) · hhυiiC
A
(σ → τ ) · hhυiiA
where A 6= C
where σ ≡
6 υ
B
Ugly hhσiiA · (τ → υ)
A
B
(ρ → σ) · (τ → υ)
• If `type τ doesn’t accept ugly terms,
then it doesn’t have the subterm property.
wren ng thornton (Indiana University)
Chiastic Lambda-Calculi
NLCS, 28 June 2013
24 / 34
Outline
Examples and Motivation
Associative λ-calculi
Chiastic λ-calculi
hhλ
→iiχL in action
Examples and Motivation
Associative λ-calculi
Chiastic λ-calculi
hhλ→iiχL in action
wren ng thornton (Indiana University)
Chiastic Lambda-Calculi
NLCS, 28 June 2013
25 / 34
Outline
Examples and Motivation
Associative λ-calculi
Chiastic λ-calculi
hhλ
→iiχL in action
Using hhλ
→iiχL to describe Japanese
Noun phrase scrambling
Tarou-ga hon-wo yonda
Hon-wo Tarou-ga yonda
Verbal morphology
“Paradoxical” behavior
Resolving the paradox
wren ng thornton (Indiana University)
Chiastic Lambda-Calculi
NLCS, 28 June 2013
25 / 34
Outline
Examples and Motivation
Associative λ-calculi
hhλ
→iiχL in action
Chiastic λ-calculi
Semantic analysis of Tarou-ga
. . .. .Tarou
. . .. . . .. . . .
N
-ga
.......................................
S/S\N
J
S/S
β
S/S
wren ng thornton (Indiana University)
Chiastic Lambda-Calculi
NLCS, 28 June 2013
26 / 34
Outline
Examples and Motivation
Associative λ-calculi
hhλ
→iiχL in action
Chiastic λ-calculi
Semantic analysis of Tarou-ga
. . .. .Tarou
. . .. . . .. . . .
hhTaro 0 iiN : N
-ga
.......................................
(λN n. λS s. hhs·hhniinom iiS ) : S/S\N
(λN n. λS s. hhs·hhniinom iiS ) · hhTaro 0 iiN : S/S
(λS s. hhs·hhTaro 0 iinom iiS ) : S/S
wren ng thornton (Indiana University)
Chiastic Lambda-Calculi
J
β
NLCS, 28 June 2013
26 / 34
Outline
Examples and Motivation
Associative λ-calculi
hhλ
→iiχL in action
Chiastic λ-calculi
Semantic analysis of Tarou-ga
. . .. .Tarou
. . .. . . .. . . .
hhTaro 0 iiN
-ga
.......................................
(λN n. λS s. hhs·hhniinom iiS )
(λN n. λS s. hhs·hhniinom iiS ) · hhTaro 0 iiN
(λS s. hhs·hhTaro 0 iinom iiS )
wren ng thornton (Indiana University)
Chiastic Lambda-Calculi
J
β
NLCS, 28 June 2013
26 / 34
Outline
Examples and Motivation
Associative λ-calculi
hhλ
→iiχL in action
Chiastic λ-calculi
Semantic analysis of Tarou-ga
. . .. .Tarou
. . .. . . .. . . .
hhTaro 0 iiN
-ga
.......................................
(λN n. λS s. hhs·hhniinom iiS )
(λN n. λS s. hhs·hhniinom iiS ) · hhTaro 0 iiN
(λS s. hhs·hhTaro 0 iinom iiS )
wren ng thornton (Indiana University)
Chiastic Lambda-Calculi
J
β
NLCS, 28 June 2013
26 / 34
Outline
Examples and Motivation
Associative λ-calculi
Chiastic λ-calculi
hhλ
→iiχL in action
Semantic analysis of Tarou-ga hon-wo yonda
yonda
hon-wo
0
hhλacc a. λnom n. n read 0 aiiS
(λS s. hhs·hhbook iiacc iiS )
(λS s. hhs·hhbook 0 iiacc iiS ) · hhλacc a. λnom n. n read 0 aiiS
Tarou-ga
hh(λacc a. λnom n. n read 0 a) · hhbook 0 iiacc iiS
(λS s. hhs·hhTaro 0 iinom iiS )
hhλnom n. n read 0 book 0 iiS
(λS s. hhs·hhTaro 0 iinom iiS ) · hhλnom n. n read 0 book 0 iiS
hh(λnom n. n read 0 book 0 ) · hhTaro 0 iinom iiS
hhTaro 0 read 0 book 0 iiS
wren ng thornton (Indiana University)
Chiastic Lambda-Calculi
I
β
β
I
β
β
NLCS, 28 June 2013
27 / 34
Outline
Examples and Motivation
Associative λ-calculi
Chiastic λ-calculi
hhλ
→iiχL in action
Semantic analysis of Tarou-ga hon-wo yonda
yonda
hon-wo
0
hhλacc a. λnom n. n read 0 aiiS
(λS s. hhs·hhbook iiacc iiS )
(λS s. hhs·hhbook 0 iiacc iiS ) · hhλacc a. λnom n. n read 0 aiiS
Tarou-ga
hh(λacc a. λnom n. n read 0 a) · hhbook 0 iiacc iiS
(λS s. hhs·hhTaro 0 iinom iiS )
hhλnom n. n read 0 book 0 iiS
(λS s. hhs·hhTaro 0 iinom iiS ) · hhλnom n. n read 0 book 0 iiS
hh(λnom n. n read 0 book 0 ) · hhTaro 0 iinom iiS
hhTaro 0 read 0 book 0 iiS
wren ng thornton (Indiana University)
Chiastic Lambda-Calculi
I
β
β
I
β
β
NLCS, 28 June 2013
27 / 34
Outline
Examples and Motivation
Associative λ-calculi
Chiastic λ-calculi
hhλ
→iiχL in action
Semantic analysis of Tarou-ga hon-wo yonda
yonda
hon-wo
0
hhλacc a. λnom n. n read 0 aiiS
(λS s. hhs·hhbook iiacc iiS )
(λS s. hhs·hhbook 0 iiacc iiS ) · hhλacc a. λnom n. n read 0 aiiS
Tarou-ga
hh(λacc a. λnom n. n read 0 a) · hhbook 0 iiacc iiS
(λS s. hhs·hhTaro 0 iinom iiS )
hhλnom n. n read 0 book 0 iiS
(λS s. hhs·hhTaro 0 iinom iiS ) · hhλnom n. n read 0 book 0 iiS
hh(λnom n. n read 0 book 0 ) · hhTaro 0 iinom iiS
hhTaro 0 read 0 book 0 iiS
wren ng thornton (Indiana University)
Chiastic Lambda-Calculi
I
β
β
I
β
β
NLCS, 28 June 2013
27 / 34
Outline
Examples and Motivation
Associative λ-calculi
Chiastic λ-calculi
hhλ
→iiχL in action
Semantic analysis of Tarou-ga hon-wo yonda
yonda
hon-wo
0
hhλacc a. λnom n. n read 0 aiiS
(λS s. hhs·hhbook iiacc iiS )
(λS s. hhs·hhbook 0 iiacc iiS ) · hhλacc a. λnom n. n read 0 aiiS
Tarou-ga
hh(λacc a. λnom n. n read 0 a) · hhbook 0 iiacc iiS
(λS s. hhs·hhTaro 0 iinom iiS )
hhλnom n. n read 0 book 0 iiS
(λS s. hhs·hhTaro 0 iinom iiS ) · hhλnom n. n read 0 book 0 iiS
hh(λnom n. n read 0 book 0 ) · hhTaro 0 iinom iiS
hhTaro 0 read 0 book 0 iiS
wren ng thornton (Indiana University)
Chiastic Lambda-Calculi
I
β
β
I
β
β
NLCS, 28 June 2013
27 / 34
Outline
Examples and Motivation
Associative λ-calculi
Chiastic λ-calculi
hhλ
→iiχL in action
Semantic analysis of Tarou-ga hon-wo yonda
yonda
hon-wo
0
hhλacc a. λnom n. n read 0 aiiS
(λS s. hhs·hhbook iiacc iiS )
(λS s. hhs·hhbook 0 iiacc iiS ) · hhλacc a. λnom n. n read 0 aiiS
Tarou-ga
hh(λacc a. λnom n. n read 0 a) · hhbook 0 iiacc iiS
(λS s. hhs·hhTaro 0 iinom iiS )
hhλnom n. n read 0 book 0 iiS
(λS s. hhs·hhTaro 0 iinom iiS ) · hhλnom n. n read 0 book 0 iiS
hh(λnom n. n read 0 book 0 ) · hhTaro 0 iinom iiS
hhTaro 0 read 0 book 0 iiS
wren ng thornton (Indiana University)
Chiastic Lambda-Calculi
I
β
β
I
β
β
NLCS, 28 June 2013
27 / 34
Outline
Examples and Motivation
Associative λ-calculi
Chiastic λ-calculi
hhλ
→iiχL in action
Semantic analysis of Tarou-ga hon-wo yonda
yonda
hon-wo
0
hhλacc a. λnom n. n read 0 aiiS
(λS s. hhs·hhbook iiacc iiS )
(λS s. hhs·hhbook 0 iiacc iiS ) · hhλacc a. λnom n. n read 0 aiiS
Tarou-ga
hh(λacc a. λnom n. n read 0 a) · hhbook 0 iiacc iiS
(λS s. hhs·hhTaro 0 iinom iiS )
hhλnom n. n read 0 book 0 iiS
(λS s. hhs·hhTaro 0 iinom iiS ) · hhλnom n. n read 0 book 0 iiS
hh(λnom n. n read 0 book 0 ) · hhTaro 0 iinom iiS
hhTaro 0 read 0 book 0 iiS
wren ng thornton (Indiana University)
Chiastic Lambda-Calculi
I
β
β
I
β
β
NLCS, 28 June 2013
27 / 34
Outline
Examples and Motivation
Associative λ-calculi
Chiastic λ-calculi
hhλ
→iiχL in action
Semantic analysis of hon-wo Tarou-ga yonda
Tarou-ga
yonda
0
hhλacc a. λnom n. n read 0 aiiS
(λS s. hhs·hhTaro iinom iiS )
(λS s. hhs·hhTaro 0 iinom iiS ) · hhλacc a. λnom n. n read 0 aiiS
hon-wo
hh(λacc a. λnom n. n read 0 a) · hhTaro 0 iinom iiS
(λS s. hhs·hhbook 0 iiacc iiS )
hhλacc a. Taro 0 read 0 aiiS
(λS s. hhs·hhbook 0 iiacc iiS ) · hhλacc a. Taro 0 read 0 aiiS
hh(λacc a. Taro 0 read 0 a) · hhbook 0 iiacc iiS
hhTaro 0 read 0 book 0 iiS
wren ng thornton (Indiana University)
Chiastic Lambda-Calculi
I
β
β(χL )
I
β
β
NLCS, 28 June 2013
28 / 34
Outline
Examples and Motivation
Associative λ-calculi
Chiastic λ-calculi
hhλ
→iiχL in action
“Paradoxical” verbal morphology
Causative and passive verb forms
• tabe-ru
‘to eat’
• tabe-sase-ru ‘to cause to eat’
• tabe-rare-ru ‘to be made to eat’
“Paradoxical” behavior of causative and passive
Morpho-phonologically behaves as a single word
Semantically behaves as if involving complementation
• E.g., adverb scope ambiguity
But this “paradox” is due to traditional notions of constituency
• Kubota 2008 vs GB, LFG, HPSG
wren ng thornton (Indiana University)
Chiastic Lambda-Calculi
NLCS, 28 June 2013
29 / 34
Outline
Examples and Motivation
Associative λ-calculi
Chiastic λ-calculi
hhλ
→iiχL in action
General scheme for verbal morphology
Let the dimension E denote eventualities
Verbal roots have types of the general form
hh· · · → hhτ iiE iiV
Verbal inflections use “multicomposition”
(λV v. hh(λE e. f ) · v iiS )
• v can have any arity
• The semantic content f , has access to the whole eventuality e
• So if e is a compound eventuality, f can affect all or part of it
wren ng thornton (Indiana University)
Chiastic Lambda-Calculi
NLCS, 28 June 2013
30 / 34
Outline
Examples and Motivation
Associative λ-calculi
Chiastic λ-calculi
hhλ
→iiχL in action
Lexical entries for a few verbal inflections
Form = Semantics
Non-past
Perfect
-(r)u = λV v. hh(λE e. e ∧ Tense(e)=Npst) · v iiS
-ta
= λV v. hh(λE e. e ∧ Tense(e)=Perf) · v iiS
Causative
-(s)ase- = λV v. hhλnom n. λdat d. (λE e. hhe ∧ Cause(e)=niiE ) · v · hhdiinom iiV
Passive
-(r)are- = λV v. hhλnom n. λdat d. (λE e. hhe ∧ Exper(e)=niiE ) · v · hhdiinom iiV
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Chiastic Lambda-Calculi
NLCS, 28 June 2013
31 / 34
Outline
Examples and Motivation
Associative λ-calculi
hhλ
→iiχL in action
Chiastic λ-calculi
Semantic analysis for yonda
yon..............................
hhλacc a. λnom n. hhn reads 0 aiiE iiV
-da
.......................................
λV v. hh(λE e. e ∧ Tense(e) = Perf) · v iiS
(λV v. hh(λE e. e ∧ Tense(e) = Perf) · v iiS ) · hhλacc a. λnom n. hhn reads 0 aiiE iiV
hh(λE e. e ∧ Tense(e) = Perf) · (λacc a. λnom n. hhn reads 0 aiiE )iiS
hhλacc a. λnom n. (λE e. e ∧ Tense(e) = Perf) · hhn reads 0 aiiE iiS
hhλacc a. λnom n. hhn reads 0 a ∧ Tense(n reads 0 a) = PerfiiE iiS
wren ng thornton (Indiana University)
Chiastic Lambda-Calculi
J
β
η
β
NLCS, 28 June 2013
32 / 34
Outline
Examples and Motivation
Associative λ-calculi
hhλ
→iiχL in action
Chiastic λ-calculi
Semantic analysis for yonda
yon..............................
hhλacc a. λnom n. hhn reads 0 aiiE iiV
-da
.......................................
λV v. hh(λE e. e ∧ Tense(e) = Perf) · v iiS
(λV v. hh(λE e. e ∧ Tense(e) = Perf) · v iiS ) · hhλacc a. λnom n. hhn reads 0 aiiE iiV
hh(λE e. e ∧ Tense(e) = Perf) · (λacc a. λnom n. hhn reads 0 aiiE )iiS
hhλacc a. λnom n. (λE e. e ∧ Tense(e) = Perf) · hhn reads 0 aiiE iiS
hhλacc a. λnom n. hhn reads 0 a ∧ Tense(n reads 0 a) = PerfiiE iiS
wren ng thornton (Indiana University)
Chiastic Lambda-Calculi
J
β
η
β
NLCS, 28 June 2013
32 / 34
Outline
Examples and Motivation
Associative λ-calculi
hhλ
→iiχL in action
Chiastic λ-calculi
Semantic analysis for yonda
yon..............................
hhλacc a. λnom n. hhn reads 0 aiiE iiV
-da
.......................................
λV v. hh(λE e. e ∧ Tense(e) = Perf) · v iiS
(λV v. hh(λE e. e ∧ Tense(e) = Perf) · v iiS ) · hhλacc a. λnom n. hhn reads 0 aiiE iiV
hh(λE e. e ∧ Tense(e) = Perf) · (λacc a. λnom n. hhn reads 0 aiiE )iiS
hhλacc a. λnom n. (λE e. e ∧ Tense(e) = Perf) · hhn reads 0 aiiE iiS
hhλacc a. λnom n. hhn reads 0 a ∧ Tense(n reads 0 a) = PerfiiE iiS
wren ng thornton (Indiana University)
Chiastic Lambda-Calculi
J
β
η
β
NLCS, 28 June 2013
32 / 34
Outline
Examples and Motivation
Associative λ-calculi
hhλ
→iiχL in action
Chiastic λ-calculi
Semantic analysis for yonda
yon..............................
hhλacc a. λnom n. hhn reads 0 aiiE iiV
-da
.......................................
λV v. hh(λE e. e ∧ Tense(e) = Perf) · v iiS
(λV v. hh(λE e. e ∧ Tense(e) = Perf) · v iiS ) · hhλacc a. λnom n. hhn reads 0 aiiE iiV
hh(λE e. e ∧ Tense(e) = Perf) · (λacc a. λnom n. hhn reads 0 aiiE )iiS
hhλacc a. λnom n. (λE e. e ∧ Tense(e) = Perf) · hhn reads 0 aiiE iiS
hhλacc a. λnom n. hhn reads 0 a ∧ Tense(n reads 0 a) = PerfiiE iiS
wren ng thornton (Indiana University)
Chiastic Lambda-Calculi
J
β
η
β
NLCS, 28 June 2013
32 / 34
Outline
Examples and Motivation
Associative λ-calculi
hhλ
→iiχL in action
Chiastic λ-calculi
Semantic analysis for yonda
yon..............................
hhλacc a. λnom n. hhn reads 0 aiiE iiV
-da
.......................................
λV v. hh(λE e. e ∧ Tense(e) = Perf) · v iiS
(λV v. hh(λE e. e ∧ Tense(e) = Perf) · v iiS ) · hhλacc a. λnom n. hhn reads 0 aiiE iiV
hh(λE e. e ∧ Tense(e) = Perf) · (λacc a. λnom n. hhn reads 0 aiiE )iiS
hhλacc a. λnom n. (λE e. e ∧ Tense(e) = Perf) · hhn reads 0 aiiE iiS
hhλacc a. λnom n. hhn reads 0 a ∧ Tense(n reads 0 a) = PerfiiE iiS
J
β
η
β
The η is a lie!
wren ng thornton (Indiana University)
Chiastic Lambda-Calculi
NLCS, 28 June 2013
32 / 34
Outline
Examples and Motivation
Associative λ-calculi
Chiastic λ-calculi
hhλ
→iiχL in action
Conclusion
Associative λ-calculi
• Justifies shorthands in category theory
Chiastic λ-calculi (namely hhλ→iiχL )
• Captures linguistic phenomena
• Type reduction is CR and SN
• Term reduction is WCR
Current work
• Is term reduction SN?
• Can we describe Γ ` e : τ more directly?
• What about η?
wren ng thornton (Indiana University)
Chiastic Lambda-Calculi
NLCS, 28 June 2013
33 / 34
Outline
Examples and Motivation
Associative λ-calculi
Chiastic λ-calculi
hhλ
→iiχL in action
∼fin.
wren ng thornton (Indiana University)
Chiastic Lambda-Calculi
NLCS, 28 June 2013
34 / 34
Visualizing dimensions
Type reduction is SN
Type reduction is CR
More “paradoxical” verbal morphology
Visualizing dimensions
Type reduction is strongly normalizing
Type reduction is Church–Rosser
More “paradoxical” verbal morphology
wren ng thornton (Indiana University)
Chiastic Lambda-Calculi
NLCS, 28 June 2013
35 / 34
Visualizing dimensions
Type reduction is SN
Type reduction is CR
More “paradoxical” verbal morphology
Visualizing dimensions
wren ng thornton (Indiana University)
Chiastic Lambda-Calculi
NLCS, 28 June 2013
35 / 34
Visualizing dimensions
Type reduction is SN
Type reduction is CR
More “paradoxical” verbal morphology
Visualizing dimensions
Type reduction is strongly normalizing
Type reduction is Church–Rosser
More “paradoxical” verbal morphology
wren ng thornton (Indiana University)
Chiastic Lambda-Calculi
NLCS, 28 June 2013
36 / 34
Visualizing dimensions
Type reduction is SN
Type reduction is CR
More “paradoxical” verbal morphology
Type reduction for hhλ→iiχL is strongly normalizing
Definition The “length” of a type is the number of constructors
length(T ) = 1
A
length(σ → τ ) = 1 + length(σ) + length(τ )
length(hhτ iiA ) = 1 + length(τ )
length(σ · τ ) = 1 + length(σ) + length(τ )
Lemma Equivalent types have equal length.
Theorem Type reduction diminishes length; i.e.,
∀τ, τ 0. τ
τ 0 ⇒ length(τ ) > length(τ 0 )
Back
wren ng thornton (Indiana University)
Chiastic Lambda-Calculi
NLCS, 28 June 2013
36 / 34
Visualizing dimensions
Type reduction is SN
Type reduction is CR
More “paradoxical” verbal morphology
Visualizing dimensions
Type reduction is strongly normalizing
Type reduction is Church–Rosser
More “paradoxical” verbal morphology
wren ng thornton (Indiana University)
Chiastic Lambda-Calculi
NLCS, 28 June 2013
37 / 34
Visualizing dimensions
Type reduction is SN
Type reduction is CR
More “paradoxical” verbal morphology
Type reduction for hhλ
→iiχL is Church–Rosser
Lemma Type reduction commutes with type equivalence; i.e.,
τ
τ0
σ
* σ0
Theorem Type reduction is weak Church–Rosser.
Proof There are no critical pairs. Use the key lemma to resolve
potential conflicts between β and itself.
Corollary Type reduction is Church–Rosser
Proof By Newman’s Lemma.
Back
wren ng thornton (Indiana University)
Chiastic Lambda-Calculi
NLCS, 28 June 2013
37 / 34
Visualizing dimensions
Type reduction is SN
Type reduction is CR
More “paradoxical” verbal morphology
Case 2
ρ · hhσiiA
(ρ · hh−iiA )(s)
β(e)
ρ · hhσ 0 iiA
τ
(− · hhσ 0 iiA )(r )
β(e 0 )
ρ0
e0
e
A
τ →σ
ρ0 · hhσ 0 iiA
wren ng thornton (Indiana University)
r
ρ
A
A
τ → σ0
(τ → −)(s)
Chiastic Lambda-Calculi
NLCS, 28 June 2013
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Visualizing dimensions
Type reduction is SN
Type reduction is CR
More “paradoxical” verbal morphology
Case 3a
ρ · hhσiiA
(− · hhσiiA )(r )
β(e)
ρ0 · hhσiiA
τ
(ρ0 · hh−iiA )(s)
β(e 0 )
ρ0
e0
e
A
τ →σ
ρ0 · hhσ 0 iiA
wren ng thornton (Indiana University)
r
ρ
A
A
τ → σ0
(τ → −)(s)
Chiastic Lambda-Calculi
NLCS, 28 June 2013
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Visualizing dimensions
Type reduction is SN
Type reduction is CR
More “paradoxical” verbal morphology
Case 3b
ρ · hhσiiA
(− · hhσiiA )(r )
β(e)
ρ0 · hhσiiA
τ
β(e 0 )
t
ρ0
e0
e
A
τ →σ
τ0
wren ng thornton (Indiana University)
r
ρ
A
A
τ0 → σ
(− → σ)(t)
Chiastic Lambda-Calculi
NLCS, 28 June 2013
40 / 34
Visualizing dimensions
Type reduction is SN
Type reduction is CR
More “paradoxical” verbal morphology
Visualizing dimensions
Type reduction is strongly normalizing
Type reduction is Church–Rosser
More “paradoxical” verbal morphology
wren ng thornton (Indiana University)
Chiastic Lambda-Calculi
NLCS, 28 June 2013
41 / 34
Visualizing dimensions
Type reduction is SN
Type reduction is CR
More “paradoxical” verbal morphology
More “paradoxical” verbal morphology
-i
-te
to
Interclausal scrambling
3
3
7
Adverb between V1 and V2
7
7
3
Argument cluster coordination involving V1
7
7
3
Postposing of ‘VP’ headed by V1
7
7
3
Clefting of ‘VP’ headed by V1
7
7
3
Coordination of ‘VP’ headed by V1
7
3
3
Focus particle between V1 and V2
7
3
3
Reduplication of V2 alone
7
3
3
(Kubota 2008)
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