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Outline Examples and Motivation Associative λ-calculi Chiastic λ-calculi 〈〈λ→〉〉χL in action

Chiastic Lambda-Calculi

wren gayle romano

Cognitive Science & Computational LinguisticsIndiana University, Bloomington

4 November 2014

wren gayle romano (Indiana University) Chiastic Lambda-Calculi 4 November 2014 0 / 34


Examples and Motivation

Associative λ-calculi

Chiastic λ-calculi

〈〈λ→〉〉χL in action



Examples

Scrambling in Japanese• Tarou

—-gaNom

honbook

-woAcc

yon-daread-Perf

• honbook

-woAcc

Tarou—

-gaNom

yon-daread-Perf

‘Taro read the book.’

Keyword arguments• yonda(wo=‘hon’, ga=‘Tarou’)• yonda(ga=‘Tarou’, wo=‘hon’)

Shorthands in category theory• FG where (FG )(X ) = F (GX)

• ηF where (ηF )(X ) = ηFX

• Fη where (Fη)(X ) = F (ηX

)



What do these have in common?

Juxtaposition is (essentially) associative

(f g) x ≈ f (g x)

Application is (essentially) commutative

f x y ≈ f y x

Our Goal convert those “≈” into “=”



Scrambling in Japanese

Many languages have “free word order”

• Tarou—

-gaNom

honbook

-woAcc

yon-daread-Perf

• honbook

-woAcc

Tarou—

-gaNom

yon-daread-Perf

‘Taro read the book.’

• Both orders are normal and natural

• Both have the same propositional content

• Though, information structure may differ



Arguments: Chomskian-style accounts

Tarou. . . . . .N

-ga. . . . . . . . .NPnom\N

JNPnom

hon. . . .N

-wo. . . . . . . .NPacc\N

JNPacc

yon-. . . . . . . . . . . . . . . .V \NPnom\NPacc

JV \NPnom

JV

-da. . . . .S\V

JS

hon. . . .N

-wo. . . . . . . .NPacc\N

JNPacc

Tarou. . . . . .N

-ga. . . . . . . . .NPnom\N

JNPnom

ITV /(V \NPnom)

yon-. . . . . . . . . . . . . . . .V \NPnom\NPacc

IBxV \NPacc

JV

-da. . . . .S\V

JS



Adjuncts: Radical neo-Davidsonian accounts

Tarou. . . . . .N

-ga. . . . . . .S/S\N

JS/S

hon. . . .N

-wo. . . . . . .S/S\N

JS/S

yon-. . . .V

-da. . . . .S\V

JSI

SI

S

hon. . . .N

-wo. . . . . . .S/S\N

JS/S

Tarou. . . . . .N

-ga. . . . . . .S/S\N

JS/S

yon-. . . .V

-da. . . . .S\V

JSI

SI

S



Adjuncts: Radical neo-Davidsonian accounts

Tarou. . . . . .N

-ga. . . . . . .S/S\N

JS/S

hon. . . .N

-wo. . . . . . .S/S\N

JS/S

IBS/S

yon-. . . .V

-da. . . . .S\V

JSI

S

hon. . . .N

-wo. . . . . . .S/S\N

JS/S

Tarou. . . . . .N

-ga. . . . . . .S/S\N

JS/S

IBS/S

yon-. . . .V

-da. . . . .S\V

JSI

S



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 (commutativity)

• Online and partial parsing is easy (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)





Chiastic λ-calculi




What are functions?

Traditional λ-calculi intentionally confuse two ideas

Procedures operations mapping values to valuesData values representing procedures

Category theory keeps them distinct

Morphisms functions as proceduresExponentials functions as data

For associativity, we must keep them distinct too

(λx. e) Unbracketed abstractions are procedures〈〈λx. e〉〉 Bracketed abstractions are values



Associative λ-calculi: 〈〈λ〉〉

Variables x , y, z ,. . .

Terms e, f, g ,. . . ::= x variables

| (λx. e) abstraction

| 〈〈e〉〉 bracketing

| e · f juxtaposition

Beta(λx. f ) · 〈〈e〉〉 {x 7→ e}f

Assoc(e · f ) · g ≡ e · (f · g)



What does juxtaposition mean?

Application (λx. e) · 〈〈f 〉〉

Composition (λx. e) · (λy. f )((λx. e) · (λy. f )

)· 〈〈g〉〉 ≡ (λx. e) ·

((λy. f ) · 〈〈g〉〉

)Tupling 〈〈f 〉〉 · 〈〈g〉〉

(λx.λy. e) ·(〈〈f 〉〉 · 〈〈g〉〉

)≡

((λx.λy. e) · 〈〈f 〉〉

)· 〈〈g〉〉



How powerful is it?

〈〈L 〉〉 is at least as powerful as L• Every L -term has an evaluation-equivalent 〈〈L 〉〉-term

JxK = x

J(λx. e)K = (λx. JeK)

...

Je · f K = JeK · 〈〈Jf K〉〉J(e)K = JeK

〈〈L 〉〉 can be more expressive than L• 〈〈λ→〉〉 has tuples, but they can’t be encoded in λ→

• Then again, almost everything stronger than λ→ has tuples





Chiastic λ-calculi




Chiastic λ-calculi

The term level

Syntax — Two flavors of chiasmusEquivalenceReductionSanity check

The type level

SyntaxEquivalenceReductionSanity check — Well-formed types



Formalizing restricted commutativity

Actually we don’t want full commutativity

• Tarou—

-gaNom

kurumacar

-gaNom

ar-uhave-Npst

‘Taro has a car.’

% kurumacar

-gaNom

Tarou—

-gaNom

ar-uhave-Npst

‘The car has a Taro.’

• Let a dimension denote a class of non-commuting elements

• Elements along different dimensions don’t interfere with one another



Chiastic λ-calculi: 〈〈λ〉〉χVariables x , y, z ,. . .

Dimensions A, B, C ,. . .


| (λAx. e) abstraction

| 〈〈e〉〉A bracketing


• Choose one

For this talk, we’ll only consider Chi L

A 6= BChi L

(λAx. λBy. e) ≡ (λBy. λAx. e)

A 6= BChi R

〈〈e〉〉A · 〈〈f 〉〉B ≡ 〈〈f 〉〉B · 〈〈e〉〉A



Chiastic λ-calculi: 〈〈λ〉〉χVariables x , y, z ,. . .

Dimensions A, B, C ,. . .


| (λAx. e) abstraction

| 〈〈e〉〉A bracketing


• For this talk, we’ll only consider Chi L

A 6= BChi L

(λAx. λBy. e) ≡ (λBy. λAx. e)

A 6= BChi R

〈〈e〉〉A · 〈〈f 〉〉B ≡ 〈〈f 〉〉B · 〈〈e〉〉A



Chi L vs Chi R

Should we accept terms like this?

(λAx. λBy. e) · (λCz. 〈〈a〉〉A) · 〈〈b〉〉B · 〈〈c〉〉C

Should we accept sentences like this?• [sono

thathonbook

-woAcc

〈Hanako—

-ga〉Nom

Tarou—

-gaNom

kat-ta]buy-Perf

-toComp

omot-te iruthink.Prog

‘Hanako thinks that Taro bought that book.’



Term equivalence for 〈〈λ〉〉χL

e ≡ f

A 6= B

(λAx. λB y. e) ≡ (λB y. λAx. e) e · (f · g) ≡ (e · f ) · g

e ≡ e ′

(λAx. e) ≡ (λAx. e′)

e ≡ e ′

〈〈e〉〉A ≡ 〈〈e ′〉〉Ae ≡ e ′ f ≡ f ′

e · f ≡ e ′ · f ′

e ≡ ef ≡ e

e ≡ f

e ≡ f f ≡ ge ≡ g



Term reduction for 〈〈λ〉〉χL

e e ′

h ≡ (λAx. f )

h · 〈〈e〉〉A {x 7→ e}fe e ′

(λAx. e) (λAx. e′)

e e ′

〈〈e〉〉A 〈〈e ′〉〉Ae e ′

e · f e ′ · ff f ′

e · f e · f ′

e · f he · (f · g) h · g

f · g h

(e · f ) · g e · h



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 〈〈λ→〉〉χL) is strongly normalizing.

Remark This is suspiciously difficult to prove.



What are types?

The intrinsic view (a la Church)• Types are manifest in terms• Terms can have only one type• Ill-typed terms “don’t exist”

The extrinsic view (a 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 B τ τ ∗ τ ′

∃e ′. e ∗ e ′ ∧ Γ ` e ′ B τ ′



Simply-typed left-chiastic λ-calculus: 〈〈λ→〉〉χL

Types σ, τ , υ,. . . ::= T primitive types

| σA→ τ arrow types

| 〈〈τ〉〉A bracketed types

| σ · τ juxtaposition

Γ ` e B τ

`ctx Γ Γ(x) ≡ τ

Γ ` x B τΓ, x :σ ` e B τ

Γ ` (λAx. e) B σA→ τ

Γ ` e B τΓ ` 〈〈e〉〉A B 〈〈τ〉〉A

Γ ` e B σ Γ ` f B τΓ ` e ·f B σ·τ



Type equivalence for 〈〈λ→〉〉χL

τ ≡ σ

A 6= B

σA→ τ

B→ υ ≡ τB→ σ

A→ υ σ · (τ · υ) ≡ (σ · τ) · υ

σ ≡ σ′ τ ≡ τ ′

σA→ τ ≡ σ′

A→ τ ′

τ ≡ τ ′

〈〈τ〉〉A ≡ 〈〈τ ′〉〉Aσ ≡ σ′ τ ≡ τ ′

σ · τ ≡ σ′ · τ ′

τ ≡ τσ ≡ ττ ≡ σ

σ ≡ τ τ ≡ υσ ≡ υ



Type reduction for 〈〈λ→〉〉χL

τ τ ′

ρ ≡ σA→ τ

ρ · 〈〈σ〉〉A τσ σ′

σA→ τ σ′

A→ τ

τ τ ′

σA→ τ σ

A→ τ ′

τ τ ′

〈〈τ〉〉A 〈〈τ ′〉〉Aσ σ′

σ · τ σ′ · ττ τ ′

σ · τ σ · τ ′

σ · τ ρ

σ · (τ · υ) ρ · υτ · υ ρ

(σ · τ) · υ τ · ρ



Do our types make sense?

Every type has a normal form

Theorem Type reduction for 〈〈λ→〉〉χL is strongly normalizing Proof

Theorem Type reduction for 〈〈λ→〉〉χL is Church–Rosser Proof

So we can define

Γ ` e B τ0 NF(τ0) ≡ τ `type τΓ ` e : τ





But, what does unresolved type juxtaposition mean?

Good 〈〈σ〉〉A · 〈〈τ〉〉BBad (σ

A→ 〈〈τ〉〉B) · 〈〈υ〉〉C where A 6= C

(σA→ τ) · 〈〈υ〉〉A where σ 6≡ υ

Ugly 〈〈σ〉〉A · (τB→ υ)

(ρA→ σ) · (τ B→ υ)

• If `type τ doesn’t accept ugly terms,then it doesn’t have the subterm property.






Good 〈〈σ〉〉A · 〈〈τ〉〉B

Bad (σA→ 〈〈τ〉〉B) · 〈〈υ〉〉C where A 6= C



(ρA→ σ) · (τ B→ υ)







Good 〈〈σ〉〉A · 〈〈τ〉〉BBad (σ

A→ 〈〈τ〉〉B) · 〈〈υ〉〉C where A 6= C



(ρA→ σ) · (τ B→ υ)






Chiastic λ-calculi




Using 〈〈λ→〉〉χL to describe Japanese

Noun phrase scrambling

Tarou-ga hon-wo yondaHon-wo Tarou-ga yonda

Verbal morphology

“Paradoxical” behaviorResolving the paradox



Semantic analysis of Tarou-ga

Tarou. . . . . . . . . . . . . . . .

〈〈Taro ′〉〉N :

N

-ga. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

(λN n. λS s. 〈〈s·〈〈n〉〉nom〉〉S) :

S/S\NJ

(λN n. λS s. 〈〈s·〈〈n〉〉nom〉〉S) · 〈〈Taro ′〉〉N :

S/Sβ

(λS s. 〈〈s·〈〈Taro ′〉〉nom〉〉S) :

S/S




Tarou. . . . . . . . . . . . . . . .〈〈Taro ′〉〉N : N

-ga. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .(λN n. λS s. 〈〈s·〈〈n〉〉nom〉〉S) : S/S\N

J(λN n. λS s. 〈〈s·〈〈n〉〉nom〉〉S) · 〈〈Taro ′〉〉N : S/S

β

(λS s. 〈〈s·〈〈Taro ′〉〉nom〉〉S) : S/S




Tarou. . . . . . . . . . . . . . . .〈〈Taro ′〉〉N

: N

-ga. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .(λN n. λS s. 〈〈s·〈〈n〉〉nom〉〉S)

: S/S\N

J(λN n. λS s. 〈〈s·〈〈n〉〉nom〉〉S) · 〈〈Taro ′〉〉N

: S/S

β

(λS s. 〈〈s·〈〈Taro ′〉〉nom〉〉S)

: S/S



Semantic analysis of Tarou-ga hon-wo yonda

Tarou-ga

(λS s. 〈〈s·〈〈Taro′〉〉nom〉〉S)

hon-wo

(λS s. 〈〈s·〈〈book ′〉〉acc〉〉S)

yonda

〈〈λacc a. λnom n. n read ′ a〉〉SI

(λS s. 〈〈s·〈〈book ′〉〉acc〉〉S) · 〈〈λacc a. λnom n. n read ′ a〉〉Sβ

〈〈(λacc a. λnom n. n read ′ a) · 〈〈book ′〉〉acc〉〉Sβ

〈〈λnom n. n read ′ book ′〉〉SI

(λS s. 〈〈s·〈〈Taro′〉〉nom〉〉S) · 〈〈λnom n. n read ′ book ′〉〉Sβ

〈〈(λnom n. n read ′ book ′) · 〈〈Taro′〉〉nom〉〉Sβ

〈〈Taro′ read ′ book ′〉〉S



Semantic analysis of hon-wo Tarou-ga yonda

hon-wo

(λS s. 〈〈s·〈〈book ′〉〉acc〉〉S)

Tarou-ga

(λS s. 〈〈s·〈〈Taro′〉〉nom〉〉S)

yonda

〈〈λacc a. λnom n. n read ′ a〉〉SI

(λS s. 〈〈s·〈〈Taro′〉〉nom〉〉S) · 〈〈λacc a. λnom n. n read ′ a〉〉Sβ

〈〈(λacc a. λnom n. n read ′ a) · 〈〈Taro′〉〉nom〉〉Sβ(χL)

〈〈λacc a. Taro′ read ′ a〉〉SI

(λS s. 〈〈s·〈〈book ′〉〉acc〉〉S) · 〈〈λacc a. Taro′ read ′ a〉〉Sβ

〈〈(λacc a. Taro′ read ′ a) · 〈〈book ′〉〉acc〉〉Sβ

〈〈Taro′ read ′ book ′〉〉S



“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 wordSemantically 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



General scheme for verbal morphology

Let the dimension E denote eventualities

Verbal roots have types of the general form

〈〈· · · → 〈〈τ〉〉E 〉〉V

Verbal inflections use “multicomposition”

(λVv. 〈〈(λEe. f ) · v〉〉S)

• 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



Lexical entries for a few verbal inflections

Form = Semantics

Non-past -(r)u = λVv. 〈〈(λE e. e ∧Tense(e)=Npst) · v〉〉SPerfect -ta = λVv. 〈〈(λE e. e ∧Tense(e)=Perf) · v〉〉SCausative -(s)ase- = λVv. 〈〈λnom n. λdat d. (λE e. 〈〈e ∧Cause(e)=n〉〉E ) · v · 〈〈d〉〉nom〉〉VPassive -(r)are- = λVv. 〈〈λnom n. λdat d. (λE e. 〈〈e ∧ Exper(e)=n〉〉E ) · v · 〈〈d〉〉nom〉〉V



Semantic analysis for yonda

yon-. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .〈〈λacc a. λnom n. 〈〈n reads ′ a〉〉E 〉〉V

-da. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .λV v. 〈〈(λE e. e ∧Tense(e) =Perf) · v〉〉S

J(λV v. 〈〈(λE e. e ∧Tense(e) =Perf) · v〉〉S) · 〈〈λacc a. λnom n. 〈〈n reads ′ a〉〉E 〉〉V

β〈〈(λE e. e ∧Tense(e) =Perf) · (λacc a. λnom n. 〈〈n reads ′ a〉〉E )〉〉S

η

〈〈λacc a. λnom n. (λE e. e ∧Tense(e) =Perf) · 〈〈n reads ′ a〉〉E 〉〉Sβ

〈〈λacc a. λnom n. 〈〈n reads ′ a ∧ Tense(n reads ′ a) =Perf〉〉E 〉〉S



Semantic analysis for yonda

yon-. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .〈〈λacc a. λnom n. 〈〈n reads ′ a〉〉E 〉〉V

-da. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .λV v. 〈〈(λE e. e ∧Tense(e) =Perf) · v〉〉S

J(λV v. 〈〈(λE e. e ∧Tense(e) =Perf) · v〉〉S) · 〈〈λacc a. λnom n. 〈〈n reads ′ a〉〉E 〉〉V

β〈〈(λE e. e ∧Tense(e) =Perf) · (λacc a. λnom n. 〈〈n reads ′ a〉〉E )〉〉S

η

〈〈λacc a. λnom n. (λE e. e ∧Tense(e) =Perf) · 〈〈n reads ′ a〉〉E 〉〉Sβ

〈〈λacc a. λnom n. 〈〈n reads ′ a ∧ Tense(n reads ′ a) =Perf〉〉E 〉〉S

The η is a lie!



Conclusion

Associative λ-calculi• Justifies shorthands in category theory

Chiastic λ-calculi (namely 〈〈λ→〉〉χL)• 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?



∼fin.


Type reduction is SN Type reduction is CR

Type reduction is strongly normalizing

Type reduction is Church–Rosser



Type reduction for 〈〈λ→〉〉χL is strongly normalizing

Definition The “length” of a type is the number of constructors

length(T ) = 1

length(σA→ τ) = 1 + length(σ) + length(τ)

length(〈〈τ〉〉A) = 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 ′. τ τ ′ ⇒ length(τ) > length(τ ′)

Back



Type reduction is strongly normalizing

Type reduction is Church–Rosser



Type reduction for 〈〈λ→〉〉χL is Church–Rosser

Lemma Type reduction commutes with type equivalence; i.e.,τ

σ τ ′

σ′*

Theorem Type reduction is weak Church–Rosser.

Proof There are no critical pairs. Use the key lemma to resolvepotential conflicts between β and itself.

Corollary Type reduction is Church–Rosser

Proof By Newman’s Lemma.

Back



Case 2

ρ · 〈〈σ〉〉A

τ ρ · 〈〈σ′〉〉A

ρ′ · 〈〈σ′〉〉A

β(e) (ρ · 〈〈−〉〉A)(s)

(− · 〈〈σ′〉〉A)(r)β(e′)

ρ ρ′

τA→ σ τ

A→ σ′

r

e

(τA→ −)(s)

e′



Case 3a

ρ · 〈〈σ〉〉A

τ ρ′ · 〈〈σ〉〉A

ρ′ · 〈〈σ′〉〉A

β(e) (− · 〈〈σ〉〉A)(r)

(ρ′ · 〈〈−〉〉A)(s)β(e′)

ρ ρ′

τA→ σ τ

A→ σ′

r

e

(τA→ −)(s)

e′



Case 3b

ρ · 〈〈σ〉〉A

τ ρ′ · 〈〈σ〉〉A

τ ′

β(e) (− · 〈〈σ〉〉A)(r)

β(e′)t

ρ ρ′

τA→ σ τ ′

A→ σ

r

e

(− A→ σ)(t)

e′


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