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
INTRODUCTION GARDNER/DE CARVALHO’S ITS M0 THE INFINITARY CALCULUS Λ001 TRUNCATION AND APPROXIMABILITY SEQUENCES AS INTERSECTION TYPES CONCLUSION
PLAN
INTRODUCTION
GARDNER/DE CARVALHO’S ITS M0
THE INFINITARY CALCULUS Λ001
TRUNCATION AND APPROXIMABILITY
SEQUENCES AS INTERSECTION TYPES
CONCLUSION
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)
.
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)
head variable head arguments
.
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)
head variable head arguments
I A term is head-normalizing (HN) if it can be reducedto a HNF (in a finite number of steps)
.
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)
head variable head arguments
I A term is head-normalizing (HN) if it can be reducedto a HNF (in a finite number of steps)
I I Normal Forms (NF): induction
t ::= λx1 . . . xp.x t1 . . . tq (p, q > 0)
.
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)
head variable head arguments
I A term is head-normalizing (HN) if it can be reducedto a HNF (in a finite number of steps)
I I Normal Forms (NF): induction
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)
.
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)
head variable head arguments
I A term is head-normalizing (HN) if it can be reducedto a HNF (in a finite number of steps)
I I Normal Forms (NF): induction
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.
.
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)
head variable head arguments
I A term is head-normalizing (HN) if it can be reducedto a HNF (in a finite number of steps)
I Coinductively, a term is hereditary head-normalizing(HHN) if it can be reduced to a HNF and all the headarguments are themselves HHN.
.
INTRODUCTION GARDNER/DE CARVALHO’S ITS M0 THE INFINITARY CALCULUS Λ001 TRUNCATION AND APPROXIMABILITY SEQUENCES AS INTERSECTION TYPES CONCLUSION
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?
INTRODUCTION GARDNER/DE CARVALHO’S ITS M0 THE INFINITARY CALCULUS Λ001 TRUNCATION AND APPROXIMABILITY SEQUENCES AS INTERSECTION TYPES CONCLUSION
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?
INTRODUCTION GARDNER/DE CARVALHO’S ITS M0 THE INFINITARY CALCULUS Λ001 TRUNCATION AND APPROXIMABILITY SEQUENCES AS INTERSECTION TYPES CONCLUSION
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?
INTRODUCTION GARDNER/DE CARVALHO’S ITS M0 THE INFINITARY CALCULUS Λ001 TRUNCATION AND APPROXIMABILITY SEQUENCES AS INTERSECTION TYPES CONCLUSION
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?
INTRODUCTION GARDNER/DE CARVALHO’S ITS M0 THE INFINITARY CALCULUS Λ001 TRUNCATION AND APPROXIMABILITY SEQUENCES AS INTERSECTION TYPES CONCLUSION
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.).
INTRODUCTION GARDNER/DE CARVALHO’S ITS M0 THE INFINITARY CALCULUS Λ001 TRUNCATION AND APPROXIMABILITY SEQUENCES AS INTERSECTION TYPES CONCLUSION
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.).
INTRODUCTION GARDNER/DE CARVALHO’S ITS M0 THE INFINITARY CALCULUS Λ001 TRUNCATION AND APPROXIMABILITY SEQUENCES AS INTERSECTION TYPES CONCLUSION
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.).
INTRODUCTION GARDNER/DE CARVALHO’S ITS M0 THE INFINITARY CALCULUS Λ001 TRUNCATION AND APPROXIMABILITY SEQUENCES AS INTERSECTION TYPES CONCLUSION
PLAN
INTRODUCTION
GARDNER/DE CARVALHO’S ITS M0
THE INFINITARY CALCULUS Λ001
TRUNCATION AND APPROXIMABILITY
SEQUENCES AS INTERSECTION TYPES
CONCLUSION
INTRODUCTION GARDNER/DE CARVALHO’S ITS M0 THE INFINITARY CALCULUS Λ001 TRUNCATION AND APPROXIMABILITY SEQUENCES AS INTERSECTION TYPES 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).
INTRODUCTION GARDNER/DE CARVALHO’S ITS M0 THE INFINITARY CALCULUS Λ001 TRUNCATION AND APPROXIMABILITY SEQUENCES AS INTERSECTION TYPES CONCLUSION
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
INTRODUCTION GARDNER/DE CARVALHO’S ITS M0 THE INFINITARY CALCULUS Λ001 TRUNCATION AND APPROXIMABILITY SEQUENCES AS INTERSECTION TYPES CONCLUSION
ALTERNATIVE PRESENTATION
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
INTRODUCTION GARDNER/DE CARVALHO’S ITS M0 THE INFINITARY CALCULUS Λ001 TRUNCATION AND APPROXIMABILITY SEQUENCES AS INTERSECTION TYPES CONCLUSION
ALTERNATIVE PRESENTATION
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
INTRODUCTION GARDNER/DE CARVALHO’S ITS M0 THE INFINITARY CALCULUS Λ001 TRUNCATION AND APPROXIMABILITY SEQUENCES AS INTERSECTION TYPES CONCLUSION
ALTERNATIVE PRESENTATION
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
INTRODUCTION GARDNER/DE CARVALHO’S ITS M0 THE INFINITARY CALCULUS Λ001 TRUNCATION AND APPROXIMABILITY SEQUENCES AS INTERSECTION TYPES CONCLUSION
ALTERNATIVE PRESENTATION
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
Where does this α come from?
INTRODUCTION GARDNER/DE CARVALHO’S ITS M0 THE INFINITARY CALCULUS Λ001 TRUNCATION AND APPROXIMABILITY SEQUENCES AS INTERSECTION TYPES CONCLUSION
ALTERNATIVE PRESENTATION
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
Where does this α come from?
From this axiom rule?
INTRODUCTION GARDNER/DE CARVALHO’S ITS M0 THE INFINITARY CALCULUS Λ001 TRUNCATION AND APPROXIMABILITY SEQUENCES AS INTERSECTION TYPES CONCLUSION
ALTERNATIVE PRESENTATION
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
Where does this α come from?
From this axiom rule? Or this one?
INTRODUCTION GARDNER/DE CARVALHO’S ITS M0 THE INFINITARY CALCULUS Λ001 TRUNCATION AND APPROXIMABILITY SEQUENCES AS INTERSECTION TYPES CONCLUSION
SUBJECT REDUCTION PROPERTY FOR M0
If Π B Γ ` t : τ and t→ t′, then ∃Π′ B Γ ` t′ : τ
INTRODUCTION GARDNER/DE CARVALHO’S ITS M0 THE INFINITARY CALCULUS Λ001 TRUNCATION AND APPROXIMABILITY SEQUENCES AS INTERSECTION TYPES CONCLUSION
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
INTRODUCTION GARDNER/DE CARVALHO’S ITS M0 THE INFINITARY CALCULUS Λ001 TRUNCATION AND APPROXIMABILITY SEQUENCES AS INTERSECTION TYPES CONCLUSION
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
Axiom leavestyping x inside Πr
ax( x : [σi] ` x : )i∈Iσi ( )i ∈ IΠi
∆i ` s : σi
absΓ ` λx.r : [σi]i∈I → τ
appΓ +
∑i∈I
∆i ` (λx.r)s : τ
INTRODUCTION GARDNER/DE CARVALHO’S ITS M0 THE INFINITARY CALCULUS Λ001 TRUNCATION AND APPROXIMABILITY SEQUENCES AS INTERSECTION TYPES CONCLUSION
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
ax( x : [σi] ` x : )i∈Iσi ( )i ∈ IΠi
∆i ` s : σi
absΓ ` λx.r : [σi]i∈I → τ
appΓ +
∑i∈I
∆i ` (λx.r)s : τ
INTRODUCTION GARDNER/DE CARVALHO’S ITS M0 THE INFINITARY CALCULUS Λ001 TRUNCATION AND APPROXIMABILITY SEQUENCES AS INTERSECTION TYPES CONCLUSION
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
ax( x : [σi] ` x : )i∈Iσi “association”( )i ∈ IΠi
∆i ` s : σi
absΓ ` λx.r : [σi]i∈I → τ
appΓ +
∑i∈I
∆i ` (λx.r)s : τ
INTRODUCTION GARDNER/DE CARVALHO’S ITS M0 THE INFINITARY CALCULUS Λ001 TRUNCATION AND APPROXIMABILITY SEQUENCES AS INTERSECTION TYPES CONCLUSION
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
ax( x : [σi] ` x : )i∈Iσi “association”( )i ∈ IΠi
∆i ` s : σi
absΓ ` λx.r : [σi]i∈I → τ
appΓ +
∑i∈I
∆i ` (λx.r)s : τ
INTRODUCTION GARDNER/DE CARVALHO’S ITS M0 THE INFINITARY CALCULUS Λ001 TRUNCATION AND APPROXIMABILITY SEQUENCES AS INTERSECTION TYPES CONCLUSION
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+∑i∈I
∆i [s/x]
σi
( )i ∈ IΠi
∆i ` s :
INTRODUCTION GARDNER/DE CARVALHO’S ITS M0 THE INFINITARY CALCULUS Λ001 TRUNCATION AND APPROXIMABILITY SEQUENCES AS INTERSECTION TYPES CONCLUSION
SUBJECT REDUCTION PROPERTY FOR M0
If Π B Γ ` t : τ and t→ t′, then ∃Π′ B Γ ` t′ : τ
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 :
INTRODUCTION GARDNER/DE CARVALHO’S ITS M0 THE INFINITARY CALCULUS Λ001 TRUNCATION AND APPROXIMABILITY SEQUENCES AS INTERSECTION TYPES CONCLUSION
SUBJECT REDUCTION PROPERTY FOR M0
If Π B Γ ` t : τ and t→ t′, then ∃Π′ B Γ ` t′ : τ
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 :
INTRODUCTION GARDNER/DE CARVALHO’S ITS M0 THE INFINITARY CALCULUS Λ001 TRUNCATION AND APPROXIMABILITY SEQUENCES AS INTERSECTION TYPES CONCLUSION
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.
INTRODUCTION GARDNER/DE CARVALHO’S ITS M0 THE INFINITARY CALCULUS Λ001 TRUNCATION AND APPROXIMABILITY SEQUENCES AS INTERSECTION TYPES CONCLUSION
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.
INTRODUCTION GARDNER/DE CARVALHO’S ITS M0 THE INFINITARY CALCULUS Λ001 TRUNCATION AND APPROXIMABILITY SEQUENCES AS INTERSECTION TYPES CONCLUSION
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.
INTRODUCTION GARDNER/DE CARVALHO’S ITS M0 THE INFINITARY CALCULUS Λ001 TRUNCATION AND APPROXIMABILITY SEQUENCES AS INTERSECTION TYPES CONCLUSION
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.
INTRODUCTION GARDNER/DE CARVALHO’S ITS M0 THE INFINITARY CALCULUS Λ001 TRUNCATION AND APPROXIMABILITY SEQUENCES AS INTERSECTION TYPES CONCLUSION
PLAN
INTRODUCTION
GARDNER/DE CARVALHO’S ITS M0
THE INFINITARY CALCULUS Λ001
TRUNCATION AND APPROXIMABILITY
SEQUENCES AS INTERSECTION TYPES
CONCLUSION
INTRODUCTION GARDNER/DE CARVALHO’S ITS M0 THE INFINITARY CALCULUS Λ001 TRUNCATION AND APPROXIMABILITY SEQUENCES AS INTERSECTION TYPES 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
INTRODUCTION GARDNER/DE CARVALHO’S ITS M0 THE INFINITARY CALCULUS Λ001 TRUNCATION AND APPROXIMABILITY SEQUENCES AS INTERSECTION TYPES 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
INTRODUCTION GARDNER/DE CARVALHO’S ITS M0 THE INFINITARY CALCULUS Λ001 TRUNCATION AND APPROXIMABILITY SEQUENCES AS INTERSECTION TYPES 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
INTRODUCTION GARDNER/DE CARVALHO’S ITS M0 THE INFINITARY CALCULUS Λ001 TRUNCATION AND APPROXIMABILITY SEQUENCES AS INTERSECTION TYPES CONCLUSION
001-TERMS
Λ001: the set of∞-terms t s.t.:
b is an infinite branch of t⇒ ad(b) =∞.
INTRODUCTION GARDNER/DE CARVALHO’S ITS M0 THE INFINITARY CALCULUS Λ001 TRUNCATION AND APPROXIMABILITY SEQUENCES AS INTERSECTION TYPES CONCLUSION
001-TERMS
Λ001: the set of∞-terms t s.t.:
b is an infinite branch of t⇒ ad(b) =∞.
f ω := f (f (f (. . .)))
i.e. fω = f (fω) (fixpoint)
f
@f
@f
@f
@f
@
Infinite rightwardbranch
INTRODUCTION GARDNER/DE CARVALHO’S ITS M0 THE INFINITARY CALCULUS Λ001 TRUNCATION AND APPROXIMABILITY SEQUENCES AS INTERSECTION TYPES CONCLUSION
001-TERMS
Λ001: the set of∞-terms t s.t.:
b is an infinite branch of t⇒ ad(b) =∞.
•b
•b0 (leaf)
I Start fromb ∈ supp(t)
I Move ↑ or↖I A leaf b0 must
be reached
INTRODUCTION GARDNER/DE CARVALHO’S ITS M0 THE INFINITARY CALCULUS Λ001 TRUNCATION AND APPROXIMABILITY SEQUENCES AS INTERSECTION TYPES CONCLUSION
001-TERMS
Λ001: the set of∞-terms t s.t.:
b is an infinite branch of t⇒ ad(b) =∞.
•b
•b0 (leaf)
I Start fromb ∈ supp(t)
I Move ↑ or↖
I A leaf b0 mustbe reached
INTRODUCTION GARDNER/DE CARVALHO’S ITS M0 THE INFINITARY CALCULUS Λ001 TRUNCATION AND APPROXIMABILITY SEQUENCES AS INTERSECTION TYPES CONCLUSION
001-TERMS
Λ001: the set of∞-terms t s.t.:
b is an infinite branch of t⇒ ad(b) =∞.
•b
•b0 (leaf)
I Start fromb ∈ supp(t)
I Move ↑ or↖
I A leaf b0 mustbe reached
INTRODUCTION GARDNER/DE CARVALHO’S ITS M0 THE INFINITARY CALCULUS Λ001 TRUNCATION AND APPROXIMABILITY SEQUENCES AS INTERSECTION TYPES CONCLUSION
001-TERMS
Λ001: the set of∞-terms t s.t.:
b is an infinite branch of t⇒ ad(b) =∞.
•b
•b0 (leaf)
I Start fromb ∈ supp(t)
I Move ↑ or↖
I A leaf b0 mustbe reached
INTRODUCTION GARDNER/DE CARVALHO’S ITS M0 THE INFINITARY CALCULUS Λ001 TRUNCATION AND APPROXIMABILITY SEQUENCES AS INTERSECTION TYPES CONCLUSION
001-TERMS
Λ001: the set of∞-terms t s.t.:
b is an infinite branch of t⇒ ad(b) =∞.
•b
•b0 (leaf)I Start from
b ∈ supp(t)
I Move ↑ or↖I A leaf b0 must
be reached
INTRODUCTION GARDNER/DE CARVALHO’S ITS M0 THE INFINITARY CALCULUS Λ001 TRUNCATION AND APPROXIMABILITY SEQUENCES AS INTERSECTION TYPES CONCLUSION
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) =∞.
INTRODUCTION GARDNER/DE CARVALHO’S ITS M0 THE INFINITARY CALCULUS Λ001 TRUNCATION AND APPROXIMABILITY SEQUENCES AS INTERSECTION TYPES CONCLUSION
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
INTRODUCTION GARDNER/DE CARVALHO’S ITS M0 THE INFINITARY CALCULUS Λ001 TRUNCATION AND APPROXIMABILITY SEQUENCES AS INTERSECTION TYPES CONCLUSION
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
f
@
INTRODUCTION GARDNER/DE CARVALHO’S ITS M0 THE INFINITARY CALCULUS Λ001 TRUNCATION AND APPROXIMABILITY SEQUENCES AS INTERSECTION TYPES CONCLUSION
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
f
@f
@
INTRODUCTION GARDNER/DE CARVALHO’S ITS M0 THE INFINITARY CALCULUS Λ001 TRUNCATION AND APPROXIMABILITY SEQUENCES AS INTERSECTION TYPES CONCLUSION
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
f
@f
@f
@
INTRODUCTION GARDNER/DE CARVALHO’S ITS M0 THE INFINITARY CALCULUS Λ001 TRUNCATION AND APPROXIMABILITY SEQUENCES AS INTERSECTION TYPES CONCLUSION
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
f
@f
@f
@f
@
INTRODUCTION GARDNER/DE CARVALHO’S ITS M0 THE INFINITARY CALCULUS Λ001 TRUNCATION AND APPROXIMABILITY SEQUENCES AS INTERSECTION TYPES CONCLUSION
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
@f
@f
@f
@
INTRODUCTION GARDNER/DE CARVALHO’S ITS M0 THE INFINITARY CALCULUS Λ001 TRUNCATION AND APPROXIMABILITY SEQUENCES AS INTERSECTION TYPES CONCLUSION
STRONG CONVERGENCE
Increaseof a.d.
t0
UnstablePart
INTRODUCTION GARDNER/DE CARVALHO’S ITS M0 THE INFINITARY CALCULUS Λ001 TRUNCATION AND APPROXIMABILITY SEQUENCES AS INTERSECTION TYPES CONCLUSION
STRONG CONVERGENCE
Increaseof a.d.
t1
UnstablePart
INTRODUCTION GARDNER/DE CARVALHO’S ITS M0 THE INFINITARY CALCULUS Λ001 TRUNCATION AND APPROXIMABILITY SEQUENCES AS INTERSECTION TYPES CONCLUSION
STRONG CONVERGENCE
Increaseof a.d.
t2
StabilizedPart
UnstablePart
INTRODUCTION GARDNER/DE CARVALHO’S ITS M0 THE INFINITARY CALCULUS Λ001 TRUNCATION AND APPROXIMABILITY SEQUENCES AS INTERSECTION TYPES CONCLUSION
STRONG CONVERGENCE
Increaseof a.d.
t3
StabilizedPart
UnstablePart
INTRODUCTION GARDNER/DE CARVALHO’S ITS M0 THE INFINITARY CALCULUS Λ001 TRUNCATION AND APPROXIMABILITY SEQUENCES AS INTERSECTION TYPES CONCLUSION
STRONG CONVERGENCE
Increaseof a.d.
t...
StabilizedPart
UnstablePart
INTRODUCTION GARDNER/DE CARVALHO’S ITS M0 THE INFINITARY CALCULUS Λ001 TRUNCATION AND APPROXIMABILITY SEQUENCES AS INTERSECTION TYPES CONCLUSION
STRONG CONVERGENCE
Increaseof a.d.
t50
StabilizedPart
UnstablePart
INTRODUCTION GARDNER/DE CARVALHO’S ITS M0 THE INFINITARY CALCULUS Λ001 TRUNCATION AND APPROXIMABILITY SEQUENCES AS INTERSECTION TYPES CONCLUSION
STRONG CONVERGENCE
Increaseof a.d.
t...
StabilizedPart
UnstablePart
INTRODUCTION GARDNER/DE CARVALHO’S ITS M0 THE INFINITARY CALCULUS Λ001 TRUNCATION AND APPROXIMABILITY SEQUENCES AS INTERSECTION TYPES CONCLUSION
STRONG CONVERGENCE
Increaseof a.d.
t...
StabilizedPart
UnstablePart
INTRODUCTION GARDNER/DE CARVALHO’S ITS M0 THE INFINITARY CALCULUS Λ001 TRUNCATION AND APPROXIMABILITY SEQUENCES AS INTERSECTION TYPES CONCLUSION
STRONG CONVERGENCE
Increaseof a.d.
t1000
StabilizedPart
UnstablePart
INTRODUCTION GARDNER/DE CARVALHO’S ITS M0 THE INFINITARY CALCULUS Λ001 TRUNCATION AND APPROXIMABILITY SEQUENCES AS INTERSECTION TYPES CONCLUSION
STRONG CONVERGENCE
Increaseof a.d.
t...
StabilizedPart
UnstablePart
INTRODUCTION GARDNER/DE CARVALHO’S ITS M0 THE INFINITARY CALCULUS Λ001 TRUNCATION AND APPROXIMABILITY SEQUENCES AS INTERSECTION TYPES CONCLUSION
STRONG CONVERGENCE
Increaseof a.d.
t...
StabilizedPart
UnstablePart
INTRODUCTION GARDNER/DE CARVALHO’S ITS M0 THE INFINITARY CALCULUS Λ001 TRUNCATION AND APPROXIMABILITY SEQUENCES AS INTERSECTION TYPES CONCLUSION
STRONG CONVERGENCE
Increaseof a.d.
t...
StabilizedPart
UnstablePart
INTRODUCTION GARDNER/DE CARVALHO’S ITS M0 THE INFINITARY CALCULUS Λ001 TRUNCATION AND APPROXIMABILITY SEQUENCES AS INTERSECTION TYPES CONCLUSION
STRONG CONVERGENCE
Increaseof a.d.
t′
StabilizedPart
INTRODUCTION GARDNER/DE CARVALHO’S ITS M0 THE INFINITARY CALCULUS Λ001 TRUNCATION AND APPROXIMABILITY SEQUENCES AS INTERSECTION TYPES CONCLUSION
STRONG CONVERGENCE
Conclusion
A strongly converging reduction sequence (s.c.r.s) allows us todefine its limit.
INTRODUCTION GARDNER/DE CARVALHO’S ITS M0 THE INFINITARY CALCULUS Λ001 TRUNCATION AND APPROXIMABILITY SEQUENCES AS INTERSECTION TYPES CONCLUSION
STRONG CONVERGENCE
ConclusionA strongly converging reduction sequence (s.c.r.s) allows us todefine its limit.
INTRODUCTION GARDNER/DE CARVALHO’S ITS M0 THE INFINITARY CALCULUS Λ001 TRUNCATION AND APPROXIMABILITY SEQUENCES AS INTERSECTION TYPES CONCLUSION
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.
INTRODUCTION GARDNER/DE CARVALHO’S ITS M0 THE INFINITARY CALCULUS Λ001 TRUNCATION AND APPROXIMABILITY SEQUENCES AS INTERSECTION TYPES CONCLUSION
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.
INTRODUCTION GARDNER/DE CARVALHO’S ITS M0 THE INFINITARY CALCULUS Λ001 TRUNCATION AND APPROXIMABILITY SEQUENCES AS INTERSECTION TYPES CONCLUSION
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.
INTRODUCTION GARDNER/DE CARVALHO’S ITS M0 THE INFINITARY CALCULUS Λ001 TRUNCATION AND APPROXIMABILITY SEQUENCES AS INTERSECTION TYPES CONCLUSION
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.
INTRODUCTION GARDNER/DE CARVALHO’S ITS M0 THE INFINITARY CALCULUS Λ001 TRUNCATION AND APPROXIMABILITY SEQUENCES AS INTERSECTION TYPES CONCLUSION
PLAN
INTRODUCTION
GARDNER/DE CARVALHO’S ITS M0
THE INFINITARY CALCULUS Λ001
TRUNCATION AND APPROXIMABILITY
SEQUENCES AS INTERSECTION TYPES
CONCLUSION
INTRODUCTION GARDNER/DE CARVALHO’S ITS M0 THE INFINITARY CALCULUS Λ001 TRUNCATION AND APPROXIMABILITY SEQUENCES AS INTERSECTION TYPES 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
@
INTRODUCTION GARDNER/DE CARVALHO’S ITS M0 THE INFINITARY CALCULUS Λ001 TRUNCATION AND APPROXIMABILITY SEQUENCES AS INTERSECTION TYPES CONCLUSION
TRUNCATION (FIGURES)
We can use the same derivation frame Π∗1 to type f (. . .)
UntypedSubterm
Γ1 = f : [[ ]→ α]
[ ]→ α
α
f
@
INTRODUCTION GARDNER/DE CARVALHO’S ITS M0 THE INFINITARY CALCULUS Λ001 TRUNCATION AND APPROXIMABILITY SEQUENCES AS INTERSECTION TYPES CONCLUSION
TRUNCATION (FIGURES)
Π11 B Γ1 ` f (∆f ∆f ) : α
UntypedSubterm
Γ1 = f : [[ ]→ α]
[ ]→ α
α
∆f ∆f
f
@
INTRODUCTION GARDNER/DE CARVALHO’S ITS M0 THE INFINITARY CALCULUS Λ001 TRUNCATION AND APPROXIMABILITY SEQUENCES AS INTERSECTION TYPES CONCLUSION
TRUNCATION (FIGURES)
Π21 B Γ1 ` f (f (∆f ∆f )) : α
UntypedSubterm
Γ1 = f : [[ ]→ α]
[ ]→ α
α
∆f ∆f
f
@f
@
INTRODUCTION GARDNER/DE CARVALHO’S ITS M0 THE INFINITARY CALCULUS Λ001 TRUNCATION AND APPROXIMABILITY SEQUENCES AS INTERSECTION TYPES CONCLUSION
TRUNCATION (FIGURES)
Π31 B Γ1 ` f 3(∆f ∆f ) : α
UntypedSubterm
Γ1 = f : [[ ]→ α]
[ ]→ α
α
∆f ∆f
f
@f
@f
@
INTRODUCTION GARDNER/DE CARVALHO’S ITS M0 THE INFINITARY CALCULUS Λ001 TRUNCATION AND APPROXIMABILITY SEQUENCES AS INTERSECTION TYPES CONCLUSION
TRUNCATION (FIGURES)
Π41 B Γ1 ` f 4(∆f ∆f ) : α
UntypedSubterm
Γ1 = f : [[ ]→ α]
[ ]→ α
α
∆f ∆f
f
@f
@f
@f
@
INTRODUCTION GARDNER/DE CARVALHO’S ITS M0 THE INFINITARY CALCULUS Λ001 TRUNCATION AND APPROXIMABILITY SEQUENCES AS INTERSECTION TYPES CONCLUSION
TRUNCATION (FIGURES)
Π′1 B Γ1 ` f ω : α
UntypedSubterm
Γ1 = f : [[ ]→ α]
[ ]→ α
α
f
@f
@f
@f
@f
@
INTRODUCTION GARDNER/DE CARVALHO’S ITS M0 THE INFINITARY CALCULUS Λ001 TRUNCATION AND APPROXIMABILITY SEQUENCES AS INTERSECTION TYPES CONCLUSION
TRUNCATION (FIGURES)
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
@
INTRODUCTION GARDNER/DE CARVALHO’S ITS M0 THE INFINITARY CALCULUS Λ001 TRUNCATION AND APPROXIMABILITY SEQUENCES AS INTERSECTION TYPES CONCLUSION
TRUNCATION (FIGURES)
Π22 B Γ2 ` f (f (∆f ∆f )) : α
UntypedSubterm
Γ2 = f : [[α]→ α] + [[ ]→ α]
[ ]→ α
α
[α]→ α
α
∆f ∆f
f
@f
@
INTRODUCTION GARDNER/DE CARVALHO’S ITS M0 THE INFINITARY CALCULUS Λ001 TRUNCATION AND APPROXIMABILITY SEQUENCES AS INTERSECTION TYPES CONCLUSION
TRUNCATION (FIGURES)
Π32 B Γ2 ` f 3(∆f ∆f ) : α
UntypedSubterm
Γ2 = f : [[α]→ α] + [[ ]→ α]
[ ]→ α
α
[α]→ α
α
∆f ∆f
f
@f
@f
@
INTRODUCTION GARDNER/DE CARVALHO’S ITS M0 THE INFINITARY CALCULUS Λ001 TRUNCATION AND APPROXIMABILITY SEQUENCES AS INTERSECTION TYPES CONCLUSION
TRUNCATION (FIGURES)
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
@
INTRODUCTION GARDNER/DE CARVALHO’S ITS M0 THE INFINITARY CALCULUS Λ001 TRUNCATION AND APPROXIMABILITY SEQUENCES AS INTERSECTION TYPES CONCLUSION
TRUNCATION (FIGURES)
Π′2 B Γ2 ` f ω : α
UntypedSubterm
Γ2 = f : [[α]→ α] + [[ ]→ α]
[ ]→ α
α
[α]→ α
α
f
@f
@f
@f
@f
@
INTRODUCTION GARDNER/DE CARVALHO’S ITS M0 THE INFINITARY CALCULUS Λ001 TRUNCATION AND APPROXIMABILITY SEQUENCES AS INTERSECTION TYPES CONCLUSION
TRUNCATION (FIGURES)
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
@
INTRODUCTION GARDNER/DE CARVALHO’S ITS M0 THE INFINITARY CALCULUS Λ001 TRUNCATION AND APPROXIMABILITY SEQUENCES AS INTERSECTION TYPES CONCLUSION
TRUNCATION (FIGURES)
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
@
INTRODUCTION GARDNER/DE CARVALHO’S ITS M0 THE INFINITARY CALCULUS Λ001 TRUNCATION AND APPROXIMABILITY SEQUENCES AS INTERSECTION TYPES CONCLUSION
TRUNCATION (FIGURES)
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
@
INTRODUCTION GARDNER/DE CARVALHO’S ITS M0 THE INFINITARY CALCULUS Λ001 TRUNCATION AND APPROXIMABILITY SEQUENCES AS INTERSECTION TYPES CONCLUSION
TRUNCATION (FIGURES)
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
@
INTRODUCTION GARDNER/DE CARVALHO’S ITS M0 THE INFINITARY CALCULUS Λ001 TRUNCATION AND APPROXIMABILITY SEQUENCES AS INTERSECTION TYPES CONCLUSION
TRUNCATION (FIGURES)
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
@
INTRODUCTION GARDNER/DE CARVALHO’S ITS M0 THE INFINITARY CALCULUS Λ001 TRUNCATION AND APPROXIMABILITY SEQUENCES AS INTERSECTION TYPES CONCLUSION
TRUNCATION (FIGURES)
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
@
INTRODUCTION GARDNER/DE CARVALHO’S ITS M0 THE INFINITARY CALCULUS Λ001 TRUNCATION AND APPROXIMABILITY SEQUENCES AS INTERSECTION TYPES CONCLUSION
TRUNCATION (FIGURES)
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
@
INTRODUCTION GARDNER/DE CARVALHO’S ITS M0 THE INFINITARY CALCULUS Λ001 TRUNCATION AND APPROXIMABILITY SEQUENCES AS INTERSECTION TYPES 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
α
f
@f
@f
@f
@f
@
INTRODUCTION GARDNER/DE CARVALHO’S ITS M0 THE INFINITARY CALCULUS Λ001 TRUNCATION AND APPROXIMABILITY SEQUENCES AS INTERSECTION TYPES CONCLUSION
TRUNCATION (FIGURES)
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
@
INTRODUCTION GARDNER/DE CARVALHO’S ITS M0 THE INFINITARY CALCULUS Λ001 TRUNCATION AND APPROXIMABILITY SEQUENCES AS INTERSECTION TYPES CONCLUSION
TRUNCATION (FIGURES)
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
@
INTRODUCTION GARDNER/DE CARVALHO’S ITS M0 THE INFINITARY CALCULUS Λ001 TRUNCATION AND APPROXIMABILITY SEQUENCES AS INTERSECTION TYPES CONCLUSION
TRUNCATION (FIGURES)
Π′ can be truncated into Π′4:
[ ]→ α
α
[α]→ α
α
[α]→ α
α
[α]→ α
α
f
@f
@f
@f
@f
@
INTRODUCTION GARDNER/DE CARVALHO’S ITS M0 THE INFINITARY CALCULUS Λ001 TRUNCATION AND APPROXIMABILITY SEQUENCES AS INTERSECTION TYPES CONCLUSION
TRUNCATION (FIGURES)
Π′ can be truncated into Π′3:
[ ]→ α
αα
[α]→ α
α
[α]→ α
α
[α]→ α
α
f
@f
@f
@f
@f
@
INTRODUCTION GARDNER/DE CARVALHO’S ITS M0 THE INFINITARY CALCULUS Λ001 TRUNCATION AND APPROXIMABILITY SEQUENCES AS INTERSECTION TYPES CONCLUSION
TRUNCATION (FIGURES)
Π′ can be truncated into Π′3:
[ ]→ α
α
[α]→ α
α
[α]→ α
α
f
@f
@f
@f
@f
@
INTRODUCTION GARDNER/DE CARVALHO’S ITS M0 THE INFINITARY CALCULUS Λ001 TRUNCATION AND APPROXIMABILITY SEQUENCES AS INTERSECTION TYPES CONCLUSION
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.
INTRODUCTION GARDNER/DE CARVALHO’S ITS M0 THE INFINITARY CALCULUS Λ001 TRUNCATION AND APPROXIMABILITY SEQUENCES AS INTERSECTION TYPES CONCLUSION
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.
INTRODUCTION GARDNER/DE CARVALHO’S ITS M0 THE INFINITARY CALCULUS Λ001 TRUNCATION AND APPROXIMABILITY SEQUENCES AS INTERSECTION TYPES CONCLUSION
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 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 .
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.
INTRODUCTION GARDNER/DE CARVALHO’S ITS M0 THE INFINITARY CALCULUS Λ001 TRUNCATION AND APPROXIMABILITY SEQUENCES AS INTERSECTION TYPES CONCLUSION
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 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 .
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.
INTRODUCTION GARDNER/DE CARVALHO’S ITS M0 THE INFINITARY CALCULUS Λ001 TRUNCATION AND APPROXIMABILITY SEQUENCES AS INTERSECTION TYPES CONCLUSION
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 unforgetfulderivation 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.
INTRODUCTION GARDNER/DE CARVALHO’S ITS M0 THE INFINITARY CALCULUS Λ001 TRUNCATION AND APPROXIMABILITY SEQUENCES AS INTERSECTION TYPES CONCLUSION
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.
INTRODUCTION GARDNER/DE CARVALHO’S ITS M0 THE INFINITARY CALCULUS Λ001 TRUNCATION AND APPROXIMABILITY SEQUENCES AS INTERSECTION TYPES CONCLUSION
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.
INTRODUCTION GARDNER/DE CARVALHO’S ITS M0 THE INFINITARY CALCULUS Λ001 TRUNCATION AND APPROXIMABILITY SEQUENCES AS INTERSECTION TYPES CONCLUSION
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.
INTRODUCTION GARDNER/DE CARVALHO’S ITS M0 THE INFINITARY CALCULUS Λ001 TRUNCATION AND APPROXIMABILITY SEQUENCES AS INTERSECTION TYPES CONCLUSION
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.
INTRODUCTION GARDNER/DE CARVALHO’S ITS M0 THE INFINITARY CALCULUS Λ001 TRUNCATION AND APPROXIMABILITY SEQUENCES AS INTERSECTION TYPES CONCLUSION
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.
INTRODUCTION GARDNER/DE CARVALHO’S ITS M0 THE INFINITARY CALCULUS Λ001 TRUNCATION AND APPROXIMABILITY SEQUENCES AS INTERSECTION TYPES CONCLUSION
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.
INTRODUCTION GARDNER/DE CARVALHO’S ITS M0 THE INFINITARY CALCULUS Λ001 TRUNCATION AND APPROXIMABILITY SEQUENCES AS INTERSECTION TYPES CONCLUSION
APPROXIMABILITY (FIGURE)
Π
INTRODUCTION GARDNER/DE CARVALHO’S ITS M0 THE INFINITARY CALCULUS Λ001 TRUNCATION AND APPROXIMABILITY SEQUENCES AS INTERSECTION TYPES CONCLUSION
APPROXIMABILITY (FIGURE)
ΠfΠ
INTRODUCTION GARDNER/DE CARVALHO’S ITS M0 THE INFINITARY CALCULUS Λ001 TRUNCATION AND APPROXIMABILITY SEQUENCES AS INTERSECTION TYPES CONCLUSION
APPROXIMABILITY (FIGURE)
ΠfΠ
•
•
••
•
•
•
•
•
INTRODUCTION GARDNER/DE CARVALHO’S ITS M0 THE INFINITARY CALCULUS Λ001 TRUNCATION AND APPROXIMABILITY SEQUENCES AS INTERSECTION TYPES CONCLUSION
APPROXIMABILITY (FIGURE)
Π
•
•
••
•
•
•
•
• fΠ
INTRODUCTION GARDNER/DE CARVALHO’S ITS M0 THE INFINITARY CALCULUS Λ001 TRUNCATION AND APPROXIMABILITY SEQUENCES AS INTERSECTION TYPES CONCLUSION
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.
INTRODUCTION GARDNER/DE CARVALHO’S ITS M0 THE INFINITARY CALCULUS Λ001 TRUNCATION AND APPROXIMABILITY SEQUENCES AS INTERSECTION TYPES CONCLUSION
NON-DETERMINISM AND TRUNCATION
(λx.r)s
fσ1· x#1
fσ2· x#2
fΠr
λx
@
fΠ1
fσ1
fΠ2
fσ2
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.
INTRODUCTION GARDNER/DE CARVALHO’S ITS M0 THE INFINITARY CALCULUS Λ001 TRUNCATION AND APPROXIMABILITY SEQUENCES AS INTERSECTION TYPES CONCLUSION
NON-DETERMINISM AND TRUNCATION
(λx.r)s
fσ1· x#1
fσ2· x#2
fΠr
λx
@
fΠ1
fσ2
fΠ2
fσ1
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.
INTRODUCTION GARDNER/DE CARVALHO’S ITS M0 THE INFINITARY CALCULUS Λ001 TRUNCATION AND APPROXIMABILITY SEQUENCES AS INTERSECTION TYPES CONCLUSION
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.
INTRODUCTION GARDNER/DE CARVALHO’S ITS M0 THE INFINITARY CALCULUS Λ001 TRUNCATION AND APPROXIMABILITY SEQUENCES AS INTERSECTION TYPES CONCLUSION
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.
INTRODUCTION GARDNER/DE CARVALHO’S ITS M0 THE INFINITARY CALCULUS Λ001 TRUNCATION AND APPROXIMABILITY SEQUENCES AS INTERSECTION TYPES CONCLUSION
NON-DETERMINISM AND TRUNCATION
(λx.r)s
fσ1· x#1
fσ2· x#2
fΠr
λx
@
fΠ1
fσ1
fΠ2
fσ2
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.
INTRODUCTION GARDNER/DE CARVALHO’S ITS M0 THE INFINITARY CALCULUS Λ001 TRUNCATION AND APPROXIMABILITY SEQUENCES AS INTERSECTION TYPES CONCLUSION
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.
INTRODUCTION GARDNER/DE CARVALHO’S ITS M0 THE INFINITARY CALCULUS Λ001 TRUNCATION AND APPROXIMABILITY SEQUENCES AS INTERSECTION TYPES CONCLUSION
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.
INTRODUCTION GARDNER/DE CARVALHO’S ITS M0 THE INFINITARY CALCULUS Λ001 TRUNCATION AND APPROXIMABILITY SEQUENCES AS INTERSECTION TYPES CONCLUSION
NON-DETERMINISM AND TRUNCATION
(λx.r)s
fσ1· x#1
fσ2· x#2
fΠr
λx
@
fΠ1
fσ1
fΠ2
fσ2
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.
INTRODUCTION GARDNER/DE CARVALHO’S ITS M0 THE INFINITARY CALCULUS Λ001 TRUNCATION AND APPROXIMABILITY SEQUENCES AS INTERSECTION TYPES CONCLUSION
PLAN
INTRODUCTION
GARDNER/DE CARVALHO’S ITS M0
THE INFINITARY CALCULUS Λ001
TRUNCATION AND APPROXIMABILITY
SEQUENCES AS INTERSECTION TYPES
CONCLUSION
INTRODUCTION GARDNER/DE CARVALHO’S ITS M0 THE INFINITARY CALCULUS Λ001 TRUNCATION AND APPROXIMABILITY SEQUENCES AS INTERSECTION TYPES 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
INTRODUCTION GARDNER/DE CARVALHO’S ITS M0 THE INFINITARY CALCULUS Λ001 TRUNCATION AND APPROXIMABILITY SEQUENCES AS INTERSECTION TYPES 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
INTRODUCTION GARDNER/DE CARVALHO’S ITS M0 THE INFINITARY CALCULUS Λ001 TRUNCATION AND APPROXIMABILITY SEQUENCES AS INTERSECTION TYPES 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
INTRODUCTION GARDNER/DE CARVALHO’S ITS M0 THE INFINITARY CALCULUS Λ001 TRUNCATION AND APPROXIMABILITY SEQUENCES AS INTERSECTION TYPES 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
INTRODUCTION GARDNER/DE CARVALHO’S ITS M0 THE INFINITARY CALCULUS Λ001 TRUNCATION AND APPROXIMABILITY SEQUENCES AS INTERSECTION TYPES 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
INTRODUCTION GARDNER/DE CARVALHO’S ITS M0 THE INFINITARY CALCULUS Λ001 TRUNCATION AND APPROXIMABILITY SEQUENCES AS INTERSECTION TYPES 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
INTRODUCTION GARDNER/DE CARVALHO’S ITS M0 THE INFINITARY CALCULUS Λ001 TRUNCATION AND APPROXIMABILITY SEQUENCES AS INTERSECTION TYPES 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): . . .→ T.
1
S S′ S
T
→root ε
INTRODUCTION GARDNER/DE CARVALHO’S ITS M0 THE INFINITARY CALCULUS Λ001 TRUNCATION AND APPROXIMABILITY SEQUENCES AS INTERSECTION TYPES 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): . . .→ T.
1
S S′ S
T
→root ε
Track 1 dedicated tothe⊕ side of arrow
INTRODUCTION GARDNER/DE CARVALHO’S ITS M0 THE INFINITARY CALCULUS Λ001 TRUNCATION AND APPROXIMABILITY SEQUENCES AS INTERSECTION TYPES 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): F→ T.
1
S S′ S
T
→root ε
Track 1 dedicated tothe⊕ side of arrow
INTRODUCTION GARDNER/DE CARVALHO’S ITS M0 THE INFINITARY CALCULUS Λ001 TRUNCATION AND APPROXIMABILITY SEQUENCES AS INTERSECTION TYPES 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): F→ T.
8 3 2 1
S S′ S T
→root ε
Track 1 dedicated tothe⊕ side of arrow
INTRODUCTION GARDNER/DE CARVALHO’S ITS M0 THE INFINITARY CALCULUS Λ001 TRUNCATION AND APPROXIMABILITY SEQUENCES AS INTERSECTION TYPES CONCLUSION
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.
INTRODUCTION GARDNER/DE CARVALHO’S ITS M0 THE INFINITARY CALCULUS Λ001 TRUNCATION AND APPROXIMABILITY SEQUENCES AS INTERSECTION TYPES CONCLUSION
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 : S′k )k∈K′ (Sk)k∈K = (S′k)k∈K′app
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.
INTRODUCTION GARDNER/DE CARVALHO’S ITS M0 THE INFINITARY CALCULUS Λ001 TRUNCATION AND APPROXIMABILITY SEQUENCES AS INTERSECTION TYPES CONCLUSION
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.
INTRODUCTION GARDNER/DE CARVALHO’S ITS M0 THE INFINITARY CALCULUS Λ001 TRUNCATION AND APPROXIMABILITY SEQUENCES AS INTERSECTION TYPES CONCLUSION
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.
INTRODUCTION GARDNER/DE CARVALHO’S ITS M0 THE INFINITARY CALCULUS Λ001 TRUNCATION AND APPROXIMABILITY SEQUENCES AS INTERSECTION TYPES CONCLUSION
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
INTRODUCTION GARDNER/DE CARVALHO’S ITS M0 THE INFINITARY CALCULUS Λ001 TRUNCATION AND APPROXIMABILITY SEQUENCES AS INTERSECTION TYPES CONCLUSION
RIGID DERIVATION (EXAMPLE)
x x
α
(
α
)5
x x
@
λx
λx.xx
(8 · β, 5 · α, 4 · (8 · α, 5 · α, 3 · α), 2 · α)→ α
INTRODUCTION GARDNER/DE CARVALHO’S ITS M0 THE INFINITARY CALCULUS Λ001 TRUNCATION AND APPROXIMABILITY SEQUENCES AS INTERSECTION TYPES CONCLUSION
RIGID DERIVATION (EXAMPLE)
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
INTRODUCTION GARDNER/DE CARVALHO’S ITS M0 THE INFINITARY CALCULUS Λ001 TRUNCATION AND APPROXIMABILITY SEQUENCES AS INTERSECTION TYPES CONCLUSION
RIGID DERIVATION (EXAMPLE)
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
INTRODUCTION GARDNER/DE CARVALHO’S ITS M0 THE INFINITARY CALCULUS Λ001 TRUNCATION AND APPROXIMABILITY SEQUENCES AS INTERSECTION TYPES CONCLUSION
RIGID DERIVATION (EXAMPLE)
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
INTRODUCTION GARDNER/DE CARVALHO’S ITS M0 THE INFINITARY CALCULUS Λ001 TRUNCATION AND APPROXIMABILITY SEQUENCES AS INTERSECTION TYPES CONCLUSION
RIGID DERIVATION (EXAMPLE)
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
INTRODUCTION GARDNER/DE CARVALHO’S ITS M0 THE INFINITARY CALCULUS Λ001 TRUNCATION AND APPROXIMABILITY SEQUENCES AS INTERSECTION TYPES CONCLUSION
RIGID DERIVATION (EXAMPLE)
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
INTRODUCTION GARDNER/DE CARVALHO’S ITS M0 THE INFINITARY CALCULUS Λ001 TRUNCATION AND APPROXIMABILITY SEQUENCES AS INTERSECTION TYPES CONCLUSION
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.
INTRODUCTION GARDNER/DE CARVALHO’S ITS M0 THE INFINITARY CALCULUS Λ001 TRUNCATION AND APPROXIMABILITY SEQUENCES AS INTERSECTION TYPES CONCLUSION
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 S5
I 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.
INTRODUCTION GARDNER/DE CARVALHO’S ITS M0 THE INFINITARY CALCULUS Λ001 TRUNCATION AND APPROXIMABILITY SEQUENCES AS INTERSECTION TYPES CONCLUSION
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.
INTRODUCTION GARDNER/DE CARVALHO’S ITS M0 THE INFINITARY CALCULUS Λ001 TRUNCATION AND APPROXIMABILITY SEQUENCES AS INTERSECTION TYPES CONCLUSION
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.
INTRODUCTION GARDNER/DE CARVALHO’S ITS M0 THE INFINITARY CALCULUS Λ001 TRUNCATION AND APPROXIMABILITY SEQUENCES AS INTERSECTION TYPES CONCLUSION
BISUPPORT OF A DERIVATION
P A := supp(P) ⊂ N∗
INTRODUCTION GARDNER/DE CARVALHO’S ITS M0 THE INFINITARY CALCULUS Λ001 TRUNCATION AND APPROXIMABILITY SEQUENCES AS INTERSECTION TYPES CONCLUSION
BISUPPORT OF A DERIVATION
P A := supp(P) ⊂ N∗
C ` t : T•
pos. a
INTRODUCTION GARDNER/DE CARVALHO’S ITS M0 THE INFINITARY CALCULUS Λ001 TRUNCATION AND APPROXIMABILITY SEQUENCES AS INTERSECTION TYPES CONCLUSION
BISUPPORT OF A DERIVATION
P A := supp(P) ⊂ N∗
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).
INTRODUCTION GARDNER/DE CARVALHO’S ITS M0 THE INFINITARY CALCULUS Λ001 TRUNCATION AND APPROXIMABILITY SEQUENCES AS INTERSECTION TYPES CONCLUSION
BISUPPORT OF A DERIVATION
P A := supp(P) ⊂ N∗
C ` t : T•
pos. a
INTRODUCTION GARDNER/DE CARVALHO’S ITS M0 THE INFINITARY CALCULUS Λ001 TRUNCATION AND APPROXIMABILITY SEQUENCES AS INTERSECTION TYPES CONCLUSION
BISUPPORT OF A DERIVATION
P A := supp(P) ⊂ N∗
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).
INTRODUCTION GARDNER/DE CARVALHO’S ITS M0 THE INFINITARY CALCULUS Λ001 TRUNCATION AND APPROXIMABILITY SEQUENCES AS INTERSECTION TYPES CONCLUSION
BISUPPORT OF A DERIVATION
P A := supp(P) ⊂ N∗
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).
INTRODUCTION GARDNER/DE CARVALHO’S ITS M0 THE INFINITARY CALCULUS Λ001 TRUNCATION AND APPROXIMABILITY SEQUENCES AS INTERSECTION TYPES CONCLUSION
BISUPPORT OF A DERIVATION
P A := supp(P) ⊂ N∗
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).
INTRODUCTION GARDNER/DE CARVALHO’S ITS M0 THE INFINITARY CALCULUS Λ001 TRUNCATION AND APPROXIMABILITY SEQUENCES AS INTERSECTION TYPES CONCLUSION
BISUPPORT OF A DERIVATION
P A := supp(P) ⊂ N∗
C ` t : T•
pos. a
Bisupport of P: the set of (right or left) bipositions
INTRODUCTION GARDNER/DE CARVALHO’S ITS M0 THE INFINITARY CALCULUS Λ001 TRUNCATION AND APPROXIMABILITY SEQUENCES AS INTERSECTION TYPES CONCLUSION
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.
INTRODUCTION GARDNER/DE CARVALHO’S ITS M0 THE INFINITARY CALCULUS Λ001 TRUNCATION AND APPROXIMABILITY SEQUENCES AS INTERSECTION TYPES CONCLUSION
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.
INTRODUCTION GARDNER/DE CARVALHO’S ITS M0 THE INFINITARY CALCULUS Λ001 TRUNCATION AND APPROXIMABILITY SEQUENCES AS INTERSECTION TYPES CONCLUSION
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.
INTRODUCTION GARDNER/DE CARVALHO’S ITS M0 THE INFINITARY CALCULUS Λ001 TRUNCATION AND APPROXIMABILITY SEQUENCES AS INTERSECTION TYPES CONCLUSION
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.
INTRODUCTION GARDNER/DE CARVALHO’S ITS M0 THE INFINITARY CALCULUS Λ001 TRUNCATION AND APPROXIMABILITY SEQUENCES AS INTERSECTION TYPES CONCLUSION
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).
INTRODUCTION GARDNER/DE CARVALHO’S ITS M0 THE INFINITARY CALCULUS Λ001 TRUNCATION AND APPROXIMABILITY SEQUENCES AS INTERSECTION TYPES CONCLUSION
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).
INTRODUCTION GARDNER/DE CARVALHO’S ITS M0 THE INFINITARY CALCULUS Λ001 TRUNCATION AND APPROXIMABILITY SEQUENCES AS INTERSECTION TYPES CONCLUSION
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).
INTRODUCTION GARDNER/DE CARVALHO’S ITS M0 THE INFINITARY CALCULUS Λ001 TRUNCATION AND APPROXIMABILITY SEQUENCES AS INTERSECTION TYPES CONCLUSION
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).
INTRODUCTION GARDNER/DE CARVALHO’S ITS M0 THE INFINITARY CALCULUS Λ001 TRUNCATION AND APPROXIMABILITY SEQUENCES AS INTERSECTION TYPES CONCLUSION
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).
INTRODUCTION GARDNER/DE CARVALHO’S ITS M0 THE INFINITARY CALCULUS Λ001 TRUNCATION AND APPROXIMABILITY SEQUENCES AS INTERSECTION TYPES CONCLUSION
PLAN
INTRODUCTION
GARDNER/DE CARVALHO’S ITS M0
THE INFINITARY CALCULUS Λ001
TRUNCATION AND APPROXIMABILITY
SEQUENCES AS INTERSECTION TYPES
CONCLUSION
INTRODUCTION GARDNER/DE CARVALHO’S ITS M0 THE INFINITARY CALCULUS Λ001 TRUNCATION AND APPROXIMABILITY SEQUENCES AS INTERSECTION TYPES 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).
INTRODUCTION GARDNER/DE CARVALHO’S ITS M0 THE INFINITARY CALCULUS Λ001 TRUNCATION AND APPROXIMABILITY SEQUENCES AS INTERSECTION TYPES CONCLUSION
QUESTIONS
Thank you for your attention !
INTRODUCTION GARDNER/DE CARVALHO’S ITS M0 THE INFINITARY CALCULUS Λ001 TRUNCATION AND APPROXIMABILITY SEQUENCES AS INTERSECTION TYPES CONCLUSION
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.
INTRODUCTION GARDNER/DE CARVALHO’S ITS M0 THE INFINITARY CALCULUS Λ001 TRUNCATION AND APPROXIMABILITY SEQUENCES AS INTERSECTION TYPES CONCLUSION
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.
INTRODUCTION GARDNER/DE CARVALHO’S ITS M0 THE INFINITARY CALCULUS Λ001 TRUNCATION AND APPROXIMABILITY SEQUENCES AS INTERSECTION TYPES CONCLUSION
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).
INTRODUCTION GARDNER/DE CARVALHO’S ITS M0 THE INFINITARY CALCULUS Λ001 TRUNCATION AND APPROXIMABILITY SEQUENCES AS INTERSECTION TYPES CONCLUSION
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).
INTRODUCTION GARDNER/DE CARVALHO’S ITS M0 THE INFINITARY CALCULUS Λ001 TRUNCATION AND APPROXIMABILITY SEQUENCES AS INTERSECTION TYPES CONCLUSION
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).
INTRODUCTION GARDNER/DE CARVALHO’S ITS M0 THE INFINITARY CALCULUS Λ001 TRUNCATION AND APPROXIMABILITY SEQUENCES AS INTERSECTION TYPES CONCLUSION
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).
INTRODUCTION GARDNER/DE CARVALHO’S ITS M0 THE INFINITARY CALCULUS Λ001 TRUNCATION AND APPROXIMABILITY SEQUENCES AS INTERSECTION TYPES CONCLUSION
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).
INTRODUCTION GARDNER/DE CARVALHO’S ITS M0 THE INFINITARY CALCULUS Λ001 TRUNCATION AND APPROXIMABILITY SEQUENCES AS INTERSECTION TYPES CONCLUSION
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).
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