lipn.univ-paris13.frmazza/Elica/Slides/Elica161006-07-Vial.pdf · INTRODUCTION GARDNER/DE...

INTRODUCTION GARDNER/DE CARVALHO’S ITS M0 THE INFINITARY CALCULUS Λ001 TRUNCATION AND APPROXIMABILITY SEQUENCES AS INTERSECTION TYPES CONCLUSION

Infinitary Intersection Typesas Sequences

(A New Answer to Klop’s Problem)

Pierre VIALIRIF, Paris 7

Elica meeting, Bologna

October 7, 2016


PLAN

INTRODUCTION

GARDNER/DE CARVALHO’S ITS M0

THE INFINITARY CALCULUS Λ001

TRUNCATION AND APPROXIMABILITY

SEQUENCES AS INTERSECTION TYPES

CONCLUSION


HEREDITARY HEAD-NORMALIZATION

I I Head Normal Forms (HNF): terms t of the form:

λx1 . . . xp.x u1 . . . uq (p, q > 0)

.




λx1 . . . xp.x u1 . . . uq (p, q > 0)

head variable head arguments

.




λx1 . . . xp.x u1 . . . uq (p, q > 0)


I A term is head-normalizing (HN) if it can be reducedto a HNF (in a finite number of steps)

.




λx1 . . . xp.x u1 . . . uq (p, q > 0)



I I Normal Forms (NF): induction

t ::= λx1 . . . xp.x t1 . . . tq (p, q > 0)

.




λx1 . . . xp.x u1 . . . uq (p, q > 0)




t ::= λx1 . . . xp.x t1 . . . tq (p, q > 0)

I A term is weakly normalizing (WN) if it can be re-duced to a NF (in a finite number of steps)

.




λx1 . . . xp.x u1 . . . uq (p, q > 0)




t ::= λx1 . . . xp.x t1 . . . tq (p, q > 0)

I A term is weakly normalizing (WN) if it can be re-duced to a NF (in a finite number of steps)

I Inductively, a term is WN if it is HN and all the headarguments are themselves WN.

.




λx1 . . . xp.x u1 . . . uq (p, q > 0)



I Coinductively, a term is hereditary head-normalizing(HHN) if it can be reduced to a HNF and all the headarguments are themselves HHN.

.


KLOP’S PROBLEM

I The set of HN terms (resp. WN) terms have been staticallycharacterized by various intersection type assignement systems(ITS).

I Klop’s Problem [early 90s]: can the set of HHN terms can becharacterized by an ITS ?

I Tatsuta [07]: an inductive ITS cannot do it.

I Can a coinductive ITS characterize the set of HHN terms?


PURPOSES OF THIS TALK

I Present the key notions of truncations and approximability(meant to avoid irrelevant derivations).

I Understand why commutative intersection (here, Gardner/deCarvalho’s multiset intersection) is unfit to express those keynotions.

I Present the coinductive type assignment system S: intersectiontypes are sequences of types, instead of sets of types(idempotent intersection fw.) or multisets of types (regularnon-idempotent fw.).


PLAN

INTRODUCTION





CONCLUSION


TYPING RULES OF M0 (GARDNER/DE CARVALHO)Types (τ, σi): τ, σi := α ∈X | [σi]i∈I → τ .

Context (Γ,∆): assigns intersection types to variables.

axx : [τ ] ` x : τ

Γ, x : [σi]i∈I ` t : τabs

Γ ` λx.t : [σi]i∈I → τ

Γ ` t : [σi]i∈I → τ (∆i ` u : σi )i∈I

appΓ +

∑i∈I

∆i ` t(u) : τ

Remark

I Multiset equality: [σ, τ, σ] = [σ, σ, τ ] 6= [σ, τ ]

I Multiplicative rules: accumulation of typing information .

I Possibility to forget the argument (empty multiset).


ALTERNATIVE PRESENTATION

Standard presentation

axx : [[α, β, α]→ α] ` x : [α, β, α]→ α

axx : [α] ` x : α

axx : [β] ` x : β

axx : [α] ` x : α

appx : [α, β, α, [α, β, α]→ α] ` xx : α

abs` λx.xx : [α, β, α, [α, β, α]→ α]→ α

[α, β, α]→ α α

x

β

x

α

@

λx

λx.xx[α, β, α, [α, β, α]→ α]→ α



Alternative presentation

I Indicate the arity ofapplication rules.

I Indicate the types given inaxiom leaves.

I Compute the type of theterm.

x

[α, β, α]→ α

x

α

x

β

x

α

@

λx

λx.xx

[α, β, α, [α, β, α]→ α]→ α







x

[α, β, α]→ α

x

α

x

β

x

α

@

λx

λx.xx[α, β, α, [α, β, α]→ α]→ α







x

[α, β, α]→ α

x

α

x

β

x

α

@

λx

λx.xx[α, β, α, [α, β, α]→ α]→ α

Where does this α come from?







x

[α, β, α]→ α

x

α

x

β

x

α

@

λx

λx.xx[α, β, α, [α, β, α]→ α]→ α


From this axiom rule?







x

[α, β, α]→ α

x

α

x

β

x

α

@

λx

λx.xx[α, β, α, [α, β, α]→ α]→ α


From this axiom rule? Or this one?


SUBJECT REDUCTION PROPERTY FOR M0

If Π B Γ ` t : τ and t→ t′, then ∃Π′ B Γ ` t′ : τ




(λx.r)s→ r[s/x]

Πr

Γ ` r : τ, x : [σi]i∈I

( )i ∈ IΠi

∆i ` s : σi

absΓ ` λx.r : [σi]i∈I → τ

appΓ +

∑i∈I

∆i ` (λx.r)s : τ




(λx.r)s→ r[s/x]

Πr

Γ ` r : τ, x : [σi]i∈I

Axiom leavestyping x inside Πr

ax( x : [σi] ` x : )i∈Iσi ( )i ∈ IΠi

∆i ` s : σi


appΓ +

∑i∈I

∆i ` (λx.r)s : τ




(λx.r)s→ r[s/x]

Πr

Γ ` r : τ, x : [σi]i∈I

ax( x : [σi] ` x : )i∈Iσi ( )i ∈ IΠi

∆i ` s : σi


appΓ +

∑i∈I

∆i ` (λx.r)s : τ




(λx.r)s→ r[s/x]

Πr

Γ ` r : τ, x : [σi]i∈I

ax( x : [σi] ` x : )i∈Iσi “association”( )i ∈ IΠi

∆i ` s : σi


appΓ +

∑i∈I

∆i ` (λx.r)s : τ




(λx.r)s→ r[s/x]

Πr

Γ ` r : τ+∑i∈I

∆i [s/x]

σi

( )i ∈ IΠi

∆i ` s :




Vocabulary:We say each association (between x-axiom leaves and arg-derivations)yields a derivation reduct Π′ typing r[s/x].

(λx.r)s→ r[s/x]

Πr

Γ ` r : τ+∑i∈I

∆i [s/x]

σi

( )i ∈ IΠi

∆i ` s :




Observation:If a type σ occurs several times in [σi]i∈I , there can be several associations,each one yielding a possibly different derivation reducts Π′.

(λx.r)s→ r[s/x]

Πr

Γ ` r : τ+∑i∈I

∆i [s/x]

σi

( )i ∈ IΠi

∆i ` s :


NORMALIZABILITY RESULTS

PropositionA term is HN iff it is typable in M0.

PropositionA term is WN iff it is typable in M0 by using an unforgetfuljudgment.

DefinitionA judgement Γ ` t : τ is unforgetful if there is no negativeoccurrence of [ ] in Γ and no positive occurrence of [ ] in τ .

I [ ] occurs negatively in [ ]→ τ

I If [ ] occurs negatively in σ2 then [ ] occurs positively in [σ1, σ2, σ3]→ τand so on.


PLAN

INTRODUCTION





CONCLUSION


∞-TERMS

Variable x

x

Abstraction λx.u

u

0

λx

Application u v

2

v

1

u

@

I Position: finite sequence in {0, 1, 2}∗, e.g. 0 · 0 · 2 · 1 · 2.

I Applicative Depth (a.d.): number of↗-edges e.g.

ad(1 · 2 · 2 · 0 · 2 · 1 · 2) = 4


001-TERMS

Λ001: the set of∞-terms t s.t.:

b is an infinite branch of t⇒ ad(b) =∞.


001-TERMS



f ω := f (f (f (. . .)))

i.e. fω = f (fω) (fixpoint)

f

@f

@f

@f

@f

@

Infinite rightwardbranch


001-TERMS



•b

•b0 (leaf)

I Start fromb ∈ supp(t)

I Move ↑ or↖I A leaf b0 must

be reached


001-TERMS



•b

•b0 (leaf)

I Start fromb ∈ supp(t)

I Move ↑ or↖

I A leaf b0 mustbe reached


001-TERMS



•b

•b0 (leaf)I Start from

b ∈ supp(t)

I Move ↑ or↖I A leaf b0 must

be reached


STRONG CONVERGENCE

DefinitionA reduction sequence t0

b0→ t1b1→ t2

b2→ . . .bn−1→ tn

bn→ . . . is stronglyconverging if it is of finite length or if limad(bn) =∞.


STRONG CONVERGENCE

∆f := λx.f (xx) ∆f ∆f : ”Curry”

∆f ∆f → f (∆f ∆f )→ f 2(∆f ∆f )→ f 3(∆f ∆f )→ f 4(∆f ∆f )→ . . .→∞ fω

∆f ∆f


STRONG CONVERGENCE



∆f ∆f

f

@


STRONG CONVERGENCE



∆f ∆f

f

@f

@


STRONG CONVERGENCE



∆f ∆f

f

@f

@f

@


STRONG CONVERGENCE



∆f ∆f

f

@f

@f

@f

@


STRONG CONVERGENCE



f

@f

@f

@f

@f

@


STRONG CONVERGENCE

Increaseof a.d.

t0

UnstablePart


STRONG CONVERGENCE

Increaseof a.d.

t1

UnstablePart


STRONG CONVERGENCE

Increaseof a.d.

t2

StabilizedPart

UnstablePart


STRONG CONVERGENCE

Increaseof a.d.

t3

StabilizedPart

UnstablePart


STRONG CONVERGENCE

Increaseof a.d.

t...

StabilizedPart

UnstablePart


STRONG CONVERGENCE

Increaseof a.d.

t50

StabilizedPart

UnstablePart


STRONG CONVERGENCE

Increaseof a.d.

t...

StabilizedPart

UnstablePart


STRONG CONVERGENCE

Increaseof a.d.

t1000

StabilizedPart

UnstablePart


STRONG CONVERGENCE

Increaseof a.d.

t...

StabilizedPart

UnstablePart


STRONG CONVERGENCE

Increaseof a.d.

t′

StabilizedPart


STRONG CONVERGENCE

Conclusion

A strongly converging reduction sequence (s.c.r.s) allows us todefine its limit.


STRONG CONVERGENCE

ConclusionA strongly converging reduction sequence (s.c.r.s) allows us todefine its limit.


INFINITARY NORMALIZATION

I The notions of redex and head-normalizability do not change.

I The NF of Λ001 are generated by the coinductive grammar:

t = λx1 . . . λxp.x t1 . . . tq (p, q > 0)

Definition (Infinitary WN)A 001-term is WN if it can be reduced to a NF through at least ones.c.r.s.

I Thus, a (finite) term is HHN iff it is 001-WN.


PLAN

INTRODUCTION





CONCLUSION


TRUNCATION (FIGURES)

Π′ B Γ ` f ω : α

Every Variableis Typed

Γ = f : [[α]→ α]ω (infinite multiplicity)

[α]→ α

α

[α]→ α

α

[α]→ α

α

[α]→ α

α

[α]→ α

α

f

@f

@f

@f

@f

@



We can use the same derivation frame Π∗1 to type f (. . .)

UntypedSubterm

Γ1 = f : [[ ]→ α]

[ ]→ α

α

f

@



Π11 B Γ1 ` f (∆f ∆f ) : α

UntypedSubterm

Γ1 = f : [[ ]→ α]

[ ]→ α

α

∆f ∆f

f

@



Π21 B Γ1 ` f (f (∆f ∆f )) : α

UntypedSubterm

Γ1 = f : [[ ]→ α]

[ ]→ α

α

∆f ∆f

f

@f

@



Π31 B Γ1 ` f 3(∆f ∆f ) : α

UntypedSubterm

Γ1 = f : [[ ]→ α]

[ ]→ α

α

∆f ∆f

f

@f

@f

@



Π41 B Γ1 ` f 4(∆f ∆f ) : α

UntypedSubterm

Γ1 = f : [[ ]→ α]

[ ]→ α

α

∆f ∆f

f

@f

@f

@f

@



Π′1 B Γ1 ` f ω : α

UntypedSubterm

Γ1 = f : [[ ]→ α]

[ ]→ α

α

f

@f

@f

@f

@f

@



We can use the same derivation frame Π∗2 to typef (f (. . .))

UntypedSubterm

Γ2 = f : [[α]→ α] + [[ ]→ α]

[ ]→ α

α

[α]→ α

α

f

@f

@



Π22 B Γ2 ` f (f (∆f ∆f )) : α

UntypedSubterm

Γ2 = f : [[α]→ α] + [[ ]→ α]

[ ]→ α

α

[α]→ α

α

∆f ∆f

f

@f

@



Π32 B Γ2 ` f 3(∆f ∆f ) : α

UntypedSubterm

Γ2 = f : [[α]→ α] + [[ ]→ α]

[ ]→ α

α

[α]→ α

α

∆f ∆f

f

@f

@f

@



Π42 B Γ2 ` f 4(∆f ∆f ) : α

UntypedSubterm

Γ2 = f : [[α]→ α] + [[ ]→ α]

[ ]→ α

α

[α]→ α

α

∆f ∆f

f

@f

@f

@f

@



Π′2 B Γ2 ` f ω : α

UntypedSubterm

Γ2 = f : [[α]→ α] + [[ ]→ α]

[ ]→ α

α

[α]→ α

α

f

@f

@f

@f

@f

@



We can use the same derivation frame Π∗3 to type f 3(. . .)

UntypedSubterm

Γ3 = f : [[α]→ α]2 + [[ ]→ α]

[ ]→ α

α

[α]→ α

α

[α]→ α

α

f

@f

@f

@



Π33 B Γ3 ` f 3(∆f ∆f ) : α

UntypedSubterm

Γ3 = f : [[α]→ α]2 + [[ ]→ α]

[ ]→ α

α

[α]→ α

α

[α]→ α

α

∆f ∆f

f

@f

@f

@



Π43 B Γ3 ` f 4(∆f ∆f ) : α

UntypedSubterm

Γ3 = f : [[α]→ α]2 + [[ ]→ α]

[ ]→ α

α

[α]→ α

α

[α]→ α

α

∆f ∆f

f

@f

@f

@f

@



Π′3 B Γ3 ` f ω : α

UntypedSubterm

Γ3 = f : [[α]→ α]2 + [[ ]→ α]

[ ]→ α

α

[α]→ α

α

[α]→ α

α

f

@f

@f

@f

@f

@



We can use the same derivation frame Π∗4 to type f 4(. . .)

UntypedSubterm

Γ4 = f : [[α]→ α]3 + [[ ]→ α]

[ ]→ α

α

[α]→ α

α

[α]→ α

α

[α]→ α

α

f

@f

@f

@f

@



Π44 B Γ4 ` f 4(∆f ∆f ) : α

UntypedSubterm

Γ4 = f : [[α]→ α]3 + [[ ]→ α]

[ ]→ α

α

[α]→ α

α

[α]→ α

α

[α]→ α

α

∆f ∆f

f

@f

@f

@f

@



Π′4 B Γ4 ` f ω : α

UntypedSubterm

Γ4 = f : [[α]→ α]3 + [[ ]→ α]

[ ]→ α

α

[α]→ α

α

[α]→ α

α

[α]→ α

α

f

@f

@f

@f

@f

@



Π′ B Γ ` f ω : α

Every Variableis Typed

Γ = f : [[α]→ α]ω (infinite multiplicity)

[α]→ α

α

[α]→ α[ ]→ α

α

[α]→ α

α

[α]→ α

α

[α]→ α

α

f

@f

@f

@f

@f

@



Π′ can be truncated into Π′4:[α]→ α

α

[α]→ α

α

[α]→ α

α

[α]→ α

α

[α]→ α

α

f

@f

@f

@f

@f

@



Π′ can be truncated into Π′4:

[ ]→ α

α

[α]→ α

α

[α]→ α

α

[α]→ α

α

f

@f

@f

@f

@f

@




[ ]→ α

αα

[α]→ α

α

[α]→ α

α

[α]→ α

α

f

@f

@f

@f

@f

@




[ ]→ α

α

[α]→ α

α

[α]→ α

α

f

@f

@f

@f

@f

@


INFINITARY SUBJECT EXPANSION

I How do we perform∞-subject expansion on Π′ (typing fω)?

I Π′, that types fω , cannot be expanded (yet).I Π′n, that also types fω , cannot be expanded (yet).I But Πk

n, that types f k(∆f ∆f ), can be expanded.I Πk

n yields a derivation Πn typing ∆f ∆f (after k exp-steps).I We can build a “join” of the Πn, thus producing an infinite unforgetful

derivation Π typing ∆f ∆f .

I Derivation Π features a type γ coinductively defined by thefixpoint equation γ = [γ]ω → α.

I Type γ allows to type ∆∆. Need for a validity criterion.




I Π′, that types fω , cannot be expanded (yet).

I Π′n, that also types fω , cannot be expanded (yet).I But Πk









I Π′, that types fω , cannot be expanded (yet).I Π′n, that also types fω , cannot be expanded (yet).

I But Πkn, that types f k(∆f ∆f ), can be expanded.

I Πkn yields a derivation Πn typing ∆f ∆f (after k exp-steps).

I We can build a “join” of the Πn, thus producing an infinite unforgetfulderivation Π typing ∆f ∆f .







n, that types f k(∆f ∆f ), can be expanded.

I Πkn yields a derivation Πn typing ∆f ∆f (after k exp-steps).









n yields a derivation Πn typing ∆f ∆f (after k exp-steps).





APPROXIMABILITY (HEURISTIC)

I Informally, see a derivation Π as a set of symbols (type variablesα or→ that we found inside each jugdment of P).

I A (finite) approximation fΠ of a derivation Π is a finite subset ofsymbols of Π which is itself a derivation. We write fΠ 6 Π.

I A derivation Π is said to be approximable if for all finite subsetB of symbols of Π, there is an approximation fΠ 6 Π thatcontains B.


APPROXIMABILITY (FIGURE)

Π



ΠfΠ



ΠfΠ

•

•

••

•

•

•

•

•



Π

•

•

••

•

•

•

•

• fΠ


NON-DETERMINISM AND TRUNCATION

(λx.r)s

f

σ1· x#1

f

σ2· x#2

f

Πr

λx

@

f

Π1

f

σ1

f

Π2

f

σ2

Truncation possibily af-fects every type nestedinside Π.

Assume σ1 = σ2.

I Possible in Π:#1 7→ Π2,#2 7→ Π1

I If fσ1 6= fσ2, not in fP.

Assume σ1 6= σ2

I Not possible in Π:#1 7→ Π2,#2 7→ Π1

I If fσ1 = fσ2, possible in fP.



(λx.r)s

fσ1· x#1

fσ2· x#2

fΠr

λx

@

fΠ1

fσ1

fΠ2

fσ2


Assume σ1 = σ2.



Assume σ1 6= σ2





(λx.r)s

fσ1· x#1

fσ2· x#2

fΠr

λx

@

fΠ1

fσ2

fΠ2

fσ1


Assume σ1 = σ2.



Assume σ1 6= σ2





(λx.r)s

f

σ1· x#1

f

σ2· x#2

f

Πr

λx

@

f

Π1

f

σ1

f

Π2

f

σ2


Assume σ1 = σ2.



Assume σ1 6= σ2





(λx.r)s

f

σ1· x#1

f

σ2· x#2

f

Πr

λx

@

f

Π1

f

σ1

f

Π2

f

σ2


Assume σ1 = σ2.I Possible in Π:

#1 7→ Π2,#2 7→ Π1


Assume σ1 6= σ2





(λx.r)s

fσ1· x#1

fσ2· x#2

fΠr

λx

@

fΠ1

fσ1

fΠ2

fσ2


Assume σ1 = σ2.I Possible in Π:

#1 7→ Π2,#2 7→ Π1


Assume σ1 6= σ2





(λx.r)s

f

σ1· x#1

f

σ2· x#2

f

Πr

λx

@

f

Π1

f

σ1

f

Π2

f

σ2


Assume σ1 = σ2.



Assume σ1 6= σ2





(λx.r)s

fσ1· x#1

fσ2· x#2

fΠr

λx

@

fΠ1

fσ1

fΠ2

fσ2


Assume σ1 = σ2.



Assume σ1 6= σ2




PLAN

INTRODUCTION





CONCLUSION


TYPES OF S

I Type (metavariable S,T): coinductive grammar

Sk,T ::= α ∈X | (Sk)k∈K → T

I Sequence Type:

I Intersection type replacing multiset types.I F = (Tk)k∈K where Tk types and K ⊂ N− {0, 1}.

I Example (Sequence Type): (Tk)k=2,3,8 with T2 = T8 = S and T3 = S′.

F = S S′ S8 3 2 argument tracks (2, 3, 8)

I Example (Arrow Type): .

1

S S′ S

T

→root ε


TYPES OF S


Sk,T ::= α ∈X | (Sk)k∈K → T

I Sequence Type:I Intersection type replacing multiset types.

I F = (Tk)k∈K where Tk types and K ⊂ N− {0, 1}.I Example (Sequence Type): (Tk)k=2,3,8 with T2 = T8 = S and T3 = S′.



1

S S′ S

T

→root ε


TYPES OF S


Sk,T ::= α ∈X | (Sk)k∈K → T

I Sequence Type:I Intersection type replacing multiset types.I F = (Tk)k∈K where Tk types and K ⊂ N− {0, 1}.




1

S S′ S

T

→root ε


TYPES OF S


Sk,T ::= α ∈X | (Sk)k∈K → T




I Example (Arrow Type): . . .→ T.

1

S S′ S

T

→root ε


TYPES OF S


Sk,T ::= α ∈X | (Sk)k∈K → T




I Example (Arrow Type): . . .→ T.

1

S S′ S

T

→root ε

Track 1 dedicated tothe⊕ side of arrow


TYPES OF S


Sk,T ::= α ∈X | (Sk)k∈K → T




I Example (Arrow Type): F→ T.

1

S S′ S

T

→root ε



TYPES OF S


Sk,T ::= α ∈X | (Sk)k∈K → T




I Example (Arrow Type): F→ T.

8 3 2 1

S S′ S T

→root ε



DERIVATIONS OF S

The set Deriv of rigid derivations is coinductively generated by:

axx : (T)k ` x : T

C ` t : T absC− x ` λx.t : C(x)→ T

C ` t : (Sk)k∈K → T (Dk ` u : Sk )k∈Kapp

C ∪⋃

k∈KDk ` t u : T

I For the app-rule: syntactic equality.

I Warning !If Rt(C) and the Rt(Dk) are not pairwise disjoint, contexts areincompatible.

I Parsing: premise of abs on tr. 0, left premise of app on tr. 1.


DERIVATIONS OF S


axx : (T)k ` x : T


C ` t : (Sk)k∈K → T (Dk ` u : S′k )k∈K′ (Sk)k∈K = (S′k)k∈K′app

C ∪⋃

k∈KDk ` t u : T





DERIVATIONS OF S


axx : (T)k ` x : T


C ` t : (Sk)k∈K → T (Dk ` u : Sk )k∈Kapp

C ∪⋃

k∈KDk ` t u : T





RIGID DERIVATION (EXAMPLE)

x

[α, β, α]→ α

x

α

(

α

)5

x

β

(

β

)8

x

α

(

α

)2

@

λx

λx.xx[α, β, α, [α, β, α]→ α]→ α

(8 · β, 5 · α, 4 · (8 · α, 5 · α, 3 · α), 2 · α)→ α



x x

α

(

α

)5

x x

@

λx

λx.xx

(8 · β, 5 · α, 4 · (8 · α, 5 · α, 3 · α), 2 · α)→ α



0

1

x x

α

(

α

)5

x x

@

λx

λx.xx

(8 · β, 5 · α, 4 · (8 · α, 5 · α, 3 · α), 2 · α)→ α



0

13 5 8

x x

α

(

α

)5

x x

@

λx

λx.xx

(8 · β, 5 · α, 4 · (8 · α, 5 · α, 3 · α), 2 · α)→ α



0

13 5 8

x

(

(8 · α, 5 · β, 3 · α)→ α

)4

x

α(

α

)5

x

β(

β

)8

x

α(

α

)2

@

λx

λx.xx

(8 · β, 5 · α, 4 · (8 · α, 5 · α, 3 · α), 2 · α)→ α



0

13 5 8

x

((8 · α, 5 · β, 3 · α)→ α)4

x

α

(α)5

x

β

(β)8

x

α

(α)2

@

λx

λx.xx

(8 · β, 5 · α, 4 · (8 · α, 5 · α, 3 · α), 2 · α)→ α



0

13 5 8

x

((8 · α, 5 · β, 3 · α)→ α)4

x

α

(α)5

x

β

(β)8

x

α

(α)2

@

λx

λx.xx(8 · β, 5 · α, 4 · (8 · α, 5 · α, 3 · α), 2 · α)→ α


MAIN FEATURES

I Subject reduction is deterministic:

I Assume P types (λx.r)s. If there is an axiom rule typing x on track 5(#5-ax), by typing constraint, there will also be an argument derivation P5

typing s on track 5, concluded by exactly the same type S5I During reduction, #5-ax will be replaced by P5, even if there are other Pk

concluded by S = S5

I Trackability: every symbol used inside P can be pointed atunivocuously. Notion of biposition.


MAIN FEATURES



typing s on track 5, concluded by exactly the same type S5

I During reduction, #5-ax will be replaced by P5, even if there are other Pk

concluded by S = S5



MAIN FEATURES



typing s on track 5, concluded by exactly the same type S5I During reduction, #5-ax will be replaced by P5, even if there are other Pk

concluded by S = S5



BISUPPORT OF A DERIVATION

P A := supp(P) ⊂ N∗




C ` t : T•

pos. a




C ` t : T•

pos. a

Right biposition (a, c)

Right of `I c ∈ supp(T)

I (Tk)k∈K := C(x)I k ∈ K and c ∈ supp(Tk).




C ` t : T•

pos. a




C ` t : T•

pos. a

Left Biposition (a, x, k · c)Left of `

I x variable.

I (Tk)k∈K := C(x)I k ∈ K and c ∈ supp(Tk).




C ` t : T•

pos. a


I x variable.I (Tk)k∈K := C(x)

I k ∈ K and c ∈ supp(Tk).




C ` t : T•

pos. a


I x variable.I (Tk)k∈K := C(x)I k ∈ K and c ∈ supp(Tk).




C ` t : T•

pos. a

Bisupport of P: the set of (right or left) bipositions


APPROXIMABILITY

I Every symbol inside a rigid derivation P has a biposition (aposition pointing inside a type nested in a judgment of P).

I A finite part B of P is finite subset of bisupp(P).

I A finite approximation of P is a (finite) derivation induced by Pon a finite part of P.

I A rigid derivation P is said to be approximable if for all finitepart B of P, there is a finite approximation fP 6 P s.t. fP containsB.


CHARACTERIZATION OF INFINITARY WN

TheoremA 001-term t is WN iff t is unforgetfully typable by means of anapproximable derivation.

Argument 1: If a term is typable by an approximable derivation, thenit is head normalizing. Unforgetfulness makes HN hereditary.Argument 2: Subject reduction holds for s.c.r.s. (with or withoutapproximability condition).Argument 3: Every NF can be typed by quantitative unforgetfulderivations and every quantitative derivation typing a NF isapproximable.Argument 4: Subject expansion property holds for s.c.r.s. (assumingapproximability only).




Argument 1: If a term is typable by an approximable derivation, thenit is head normalizing. Unforgetfulness makes HN hereditary.

Argument 2: Subject reduction holds for s.c.r.s. (with or withoutapproximability condition).Argument 3: Every NF can be typed by quantitative unforgetfulderivations and every quantitative derivation typing a NF isapproximable.Argument 4: Subject expansion property holds for s.c.r.s. (assumingapproximability only).




Argument 1: If a term is typable by an approximable derivation, thenit is head normalizing. Unforgetfulness makes HN hereditary.Argument 2: Subject reduction holds for s.c.r.s. (with or withoutapproximability condition).

Argument 3: Every NF can be typed by quantitative unforgetfulderivations and every quantitative derivation typing a NF isapproximable.Argument 4: Subject expansion property holds for s.c.r.s. (assumingapproximability only).




Argument 1: If a term is typable by an approximable derivation, thenit is head normalizing. Unforgetfulness makes HN hereditary.Argument 2: Subject reduction holds for s.c.r.s. (with or withoutapproximability condition).Argument 3: Every NF can be typed by quantitative unforgetfulderivations and every quantitative derivation typing a NF isapproximable.

Argument 4: Subject expansion property holds for s.c.r.s. (assumingapproximability only).




Argument 1: If a term is typable by an approximable derivation, thenit is head normalizing. Unforgetfulness makes HN hereditary.Argument 2: Subject reduction holds for s.c.r.s. (with or withoutapproximability condition).Argument 3: Every NF can be typed by quantitative unforgetfulderivations and every quantitative derivation typing a NF isapproximable.Argument 4: Subject expansion property holds for s.c.r.s. (assumingapproximability only).


PLAN

INTRODUCTION





CONCLUSION


FUTURE WORK?

I Representation Theorem: every M -derivation is the collapse ofa S-derivation (already done, HOR 2016).

I Can we reformulate approximability ?

I Can infinitary Strong Normalization be characterized ?

I Is every term typable in S (without approximability) ?Yes ! S is completely unsound (difficult because of relevance).

I Categorical Adaptation of this framework (ongoing work with D.Mazza and L. Pellisier).


QUESTIONS

Thank you for your attention !


EXPANDED DERIVATIONS

Πn =

[α]→ α([ ]→ α if n = 1)

γn−1 γn−2 γ1

γn

γn−1 γ1

@

λx

n−1

@

f @n−2

x x x

Ψn−1 Ψ1

I Ψn is the left subderivation.I γ1 = [ ]→ α and γn = [γi]16i6n−1 → α.I Πn is the n-expanded of Πn

n, the (n + 1)-expanded of Πn+1n ,..., the

∞-expanded of Π′n.


EXPANDED DERIVATIONS

Π =

[α]→ α

γ γ γ

γ

γ γ

@

λx

ω

@

f @ω

x x x

Ψ Ψ

I Ψ is the left subderivation.I γ = [γ]ω → α.

I Πn is the n-expanded of Πnn, the (n + 1)-expanded of Πn+1

n ,..., the∞-expanded of Π′n.


SYSTEM M

I The relation ≡ (between types or seq. types) is definedcoinductively:

I α ≡ α.I (Sk)k∈K → T ≡ (S′k)k∈K′ → T′ if (Sk)k∈K ≡ (S′k)k′∈K′ and T ≡ T′.I (Sk)k∈K ≡ (S′k)k∈K′ if there is a bijection ρ : K→ K′ s.t. ∀k ∈ K, Sk ≡ S′

σ(k).

I We set TypesM = Types / ≡.The set of multiset types is STypes / ≡.

I To obtain System M , take the rules of M0 coinductively (withthose types and multiset types).


SYSTEM M


I α ≡ α.

I (Sk)k∈K → T ≡ (S′k)k∈K′ → T′ if (Sk)k∈K ≡ (S′k)k′∈K′ and T ≡ T′.I (Sk)k∈K ≡ (S′k)k∈K′ if there is a bijection ρ : K→ K′ s.t. ∀k ∈ K, Sk ≡ S′

σ(k).




SYSTEM M


I α ≡ α.I (Sk)k∈K → T ≡ (S′k)k∈K′ → T′ if (Sk)k∈K ≡ (S′k)k′∈K′ and T ≡ T′.

I (Sk)k∈K ≡ (S′k)k∈K′ if there is a bijection ρ : K→ K′ s.t. ∀k ∈ K, Sk ≡ S′σ(k).




SYSTEM M


I α ≡ α.I (Sk)k∈K → T ≡ (S′k)k∈K′ → T′ if (Sk)k∈K ≡ (S′k)k′∈K′ and T ≡ T′.I (Sk)k∈K ≡ (S′k)k∈K′ if there is a bijection ρ : K→ K′ s.t. ∀k ∈ K, Sk ≡ S′

σ(k).



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