Unions of Reducibility Families for -Calculus with Orthogonal - - PowerPoint PPT Presentation

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Unions of Reducibility Families for -Calculus with Orthogonal - - PowerPoint PPT Presentation

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Introduction Reducibility Stability by Union Orthogonal Constructor Rewriting Conclusion Unions of Reducibility Families for -Calculus with Orthogonal Rewriting Colin Riba INRIA Sophia Antipolis Mditerrane Everest 16 March 2008

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Introduction Reducibility Stability by Union Orthogonal Constructor Rewriting Conclusion

Unions of Reducibility Families

for λ-Calculus with Orthogonal Rewriting Colin Riba

INRIA Sophia Antipolis – Méditerranée Everest

16 March 2008 Swansea

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Introduction Reducibility Stability by Union Orthogonal Constructor Rewriting Conclusion

General Motivations

Termination of extensions of typed λ-calculus.

◮ Proofs assistants (strong normalization). ◮ Functional programming.

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Introduction Reducibility Stability by Union Orthogonal Constructor Rewriting Conclusion

λ-Calculus with Rewriting

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Introduction Reducibility Stability by Union Orthogonal Constructor Rewriting Conclusion

λ-Calculus with Rewriting

◮ Terms

t, u ∈ Λ(Σ) ::= x | λx.t | t u | f(t1, . . . , tn) , where f ∈ Σn.

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Introduction Reducibility Stability by Union Orthogonal Constructor Rewriting Conclusion

λ-Calculus with Rewriting

◮ Terms

t, u ∈ Λ(Σ) ::= x | λx.t | t u | f(t1, . . . , tn) , where f ∈ Σn.

◮ A type system (eg. simple types).

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Introduction Reducibility Stability by Union Orthogonal Constructor Rewriting Conclusion

λ-Calculus with Rewriting

◮ Terms

t, u ∈ Λ(Σ) ::= x | λx.t | t u | f(t1, . . . , tn) , where f ∈ Σn.

◮ A type system (eg. simple types). ◮ Rewrite Rules of the form

f(l1, . . . , ln) →R r , where Γ ⊢ l1 : T1 . . . Γ ⊢ ln : Tn Γ ⊢ f(l1, . . . , ln) : T and Γ ⊢ r : T

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Introduction Reducibility Stability by Union Orthogonal Constructor Rewriting Conclusion

Strong Normalization

◮ Strong Normalization (SN)

No infinite sequence t1 → . . . → tn → . . .

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Introduction Reducibility Stability by Union Orthogonal Constructor Rewriting Conclusion

Strong Normalization

◮ Strong Normalization (SN)

No infinite sequence t1 → . . . → tn → . . .

◮ Tools to prove that

if ⊢ t : T then t ∈ SN

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Introduction Reducibility Stability by Union Orthogonal Constructor Rewriting Conclusion

Type Interpretation

Let →R be a rewrite relation on Λ(Σ).

◮ Interpretation of Types

T ∈ T → T ⊆ SN

◮ Adequacy

⊢ t : T = ⇒ t ∈ T

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Introduction Reducibility Stability by Union Orthogonal Constructor Rewriting Conclusion

Type Interpretation

Let →R be a rewrite relation on Λ(Σ).

◮ Reducibility Family

Red ⊆ P(Λ(Σ)) such that ∀X ∈ Red. X ⊆ X ⊆ SN

◮ Interpretation of Types

T ∈ T → T ∈ Red

◮ Adequacy

⊢ t : T = ⇒ t ∈ T

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Introduction Reducibility Stability by Union Orthogonal Constructor Rewriting Conclusion

Type Interpretation

Let →R be a rewrite relation on Λ(Σ).

◮ Reducibility Family

Red ⊆ P(Λ(Σ)) such that ∀X ∈ Red. X ⊆ X ⊆ SN

◮ Interpretation of Types

T ∈ T → T ∈ Red

◮ Adequacy

⊢ t : T = ⇒ t ∈ T

◮ Red is a complete lattice

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Introduction Reducibility Stability by Union Orthogonal Constructor Rewriting Conclusion

Different reducibility families:

◮ Tait’s Saturated Sets [Tai75] ◮ Girard’s Reducibility Candidates [Gir72] ◮ Biorthogonals [Gir87]

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Introduction Reducibility Stability by Union Orthogonal Constructor Rewriting Conclusion

Different reducibility families:

◮ Tait’s Saturated Sets [Tai75] ◮ Girard’s Reducibility Candidates [Gir72] ◮ Biorthogonals [Gir87]

Compare them wrt Stability by Union: ∅ = R ⊆ Red = ⇒

  • R ∈ Red

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Introduction Reducibility Stability by Union Orthogonal Constructor Rewriting Conclusion

Different reducibility families:

◮ Tait’s Saturated Sets [Tai75] ◮ Girard’s Reducibility Candidates [Gir72] ◮ Biorthogonals [Gir87]

Compare them wrt Stability by Union: ∅ = R ⊆ Red = ⇒

  • R ∈ Red

Property used eg. in [BR06, Abe06, Tat07].

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Introduction Reducibility Stability by Union Orthogonal Constructor Rewriting Conclusion

Outline

Introduction Reducibility Stability by Union Application to Orthogonal Constructor Rewriting Conclusion

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Introduction Reducibility Stability by Union Orthogonal Constructor Rewriting Conclusion

Outline

Introduction Reducibility Stability by Union Application to Orthogonal Constructor Rewriting Conclusion

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Introduction Reducibility Stability by Union Orthogonal Constructor Rewriting Conclusion

Reducibility

Let →R be a rewrite relation on Λ(Σ).

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Introduction Reducibility Stability by Union Orthogonal Constructor Rewriting Conclusion

Reducibility

Let →R be a rewrite relation on Λ(Σ).

◮ Reducibility family

Red ⊆ P(Λ(Σ)) such that ∀X ∈ Red. X ⊆ X ⊆ SN

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Introduction Reducibility Stability by Union Orthogonal Constructor Rewriting Conclusion

Reducibility

Let →R be a rewrite relation on Λ(Σ).

◮ Reducibility family

Red ⊆ P(Λ(Σ)) such that ∀X ∈ Red. X ⊆ X ⊆ SN

◮ Interpretation of types

T ∈ T → T ∈ Red

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Introduction Reducibility Stability by Union Orthogonal Constructor Rewriting Conclusion

Reducibility

Let →R be a rewrite relation on Λ(Σ).

◮ Reducibility family

Red ⊆ P(Λ(Σ)) such that ∀X ∈ Red. X ⊆ X ⊆ SN

◮ Interpretation of types

T ∈ T → T ∈ Red

◮ Sufficient conditions on Red to get an adequate interpretation:

⊢ t : T = ⇒ t ∈ T

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Introduction Reducibility Stability by Union Orthogonal Constructor Rewriting Conclusion

Saturated Sets (Pure λ-Calculus)

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Introduction Reducibility Stability by Union Orthogonal Constructor Rewriting Conclusion

Saturated Sets (Pure λ-Calculus)

◮ Elimination Contexts

E[ ] ∈ E⇒ ::= [ ] | E[ ] t

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Introduction Reducibility Stability by Union Orthogonal Constructor Rewriting Conclusion

Saturated Sets (Pure λ-Calculus)

◮ Elimination Contexts

E[ ] ∈ E⇒ ::= [ ] | E[ ] t

◮ (Non) Interaction Properties

if E[x] →β v then the reduction is in E[ ], if E[(λx.t)u] →β v then the reduction is either in E[ ]

  • r in (λx.t)u.

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Introduction Reducibility Stability by Union Orthogonal Constructor Rewriting Conclusion

Saturated Sets (Pure λ-Calculus)

◮ Elimination Contexts

E[ ] ∈ E⇒ ::= [ ] | E[ ] t

◮ (Non) Interaction Properties

if E[x] →β v then the reduction is in E[ ], if E[(λx.t)u] →β v then the reduction is either in E[ ]

  • r in (λx.t)u.

◮ Weak Standardization

A β-reduct of (λx.t)u is either t[u/x] or (λx.t′)u′ with (t, u) →β (t′, u′).

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Introduction Reducibility Stability by Union Orthogonal Constructor Rewriting Conclusion

Saturated Sets (Pure λ-Calculus)

◮ Elimination Contexts

E[ ] ∈ E⇒ ::= [ ] | E[ ] t

◮ (Non) Interaction Properties

if E[x] →β v then the reduction is in E[ ], if E[(λx.t)u] →β v then the reduction is either in E[ ]

  • r in (λx.t)u.

◮ Weak Standardization

A β-reduct of (λx.t)u is either t[u/x] or (λx.t′)u′ with (t, u) →β (t′, u′).

◮ Consequence:

If E[t[u/x]] ∈ SN β and u ∈ SN β then E[(λx.t)u] ∈ SN β.

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Introduction Reducibility Stability by Union Orthogonal Constructor Rewriting Conclusion

Saturated Sets (Pure λ-Calculus)

◮ Elimination Contexts

E[ ] ∈ E⇒ ::= [ ] | E[ ] t

◮ (Non) Interaction Properties

if E[x] →β v then the reduction is in E[ ], if E[(λx.t)u] →β v then the reduction is either in E[ ]

  • r in (λx.t)u.

◮ S ⊆ SN β is a Saturated Set (S ∈ S

A T) iff (S A T1) if E[ ] ∈ SN β and x ∈ X then E[x] ∈ S, (S A T2β) if E[t[u/x]] ∈ S and u ∈ SN β then E[(λx.t)u] ∈ S.

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Introduction Reducibility Stability by Union Orthogonal Constructor Rewriting Conclusion

Rewriting

◮ Let R be a rewrite system on Λ(Σ) whose rules are of the form

f(l1, . . . , ln) →R r

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Introduction Reducibility Stability by Union Orthogonal Constructor Rewriting Conclusion

Rewriting

◮ Let R be a rewrite system on Λ(Σ) whose rules are of the form

f(l1, . . . , ln) →R r

◮ Saturated Sets

In addition to (S A T1) and (S A T2β) we need stability by reduction (if t ∈ S and t →βR u then u ∈ S),

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Introduction Reducibility Stability by Union Orthogonal Constructor Rewriting Conclusion

Rewriting

◮ Let R be a rewrite system on Λ(Σ) whose rules are of the form

f(l1, . . . , ln) →R r

◮ Saturated Sets

In addition to (S A T1) and (S A T2β) we need stability by reduction (if t ∈ S and t →βR u then u ∈ S), and that for all E[f(t1, . . . , tn)], E[f(t1, . . . , tn)]

βR

  • βR
  • =

⇒ E[f(t1, . . . , tn)] ∈ S u1 ∈ S . . . un ∈ S

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Introduction Reducibility Stability by Union Orthogonal Constructor Rewriting Conclusion

Girard’s Reducibility Candidates

◮ Consider a set of contexts E[ ] ∈ E and a rewrite relation →R.

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Introduction Reducibility Stability by Union Orthogonal Constructor Rewriting Conclusion

Girard’s Reducibility Candidates

◮ Consider a set of contexts E[ ] ∈ E and a rewrite relation →R. ◮ A term t is Neutral if it interacts with no contexts E[ ] ∈ E:

if E[t] →R v then the reduction is either in E[ ] or in t.

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Introduction Reducibility Stability by Union Orthogonal Constructor Rewriting Conclusion

Girard’s Reducibility Candidates

◮ Consider a set of contexts E[ ] ∈ E and a rewrite relation →R. ◮ A term t is Neutral if it interacts with no contexts E[ ] ∈ E:

if E[t] →R v then the reduction is either in E[ ] or in t.

◮ C ⊆ SN is a Reducibility Candidate (C ∈ C

R) iff C is stable by reduction (if t ∈ C and t →R u then u ∈ C) and

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Introduction Reducibility Stability by Union Orthogonal Constructor Rewriting Conclusion

Girard’s Reducibility Candidates

◮ Consider a set of contexts E[ ] ∈ E and a rewrite relation →R. ◮ A term t is Neutral if it interacts with no contexts E[ ] ∈ E:

if E[t] →R v then the reduction is either in E[ ] or in t.

◮ C ⊆ SN is a Reducibility Candidate (C ∈ C

R) iff C is stable by reduction (if t ∈ C and t →R u then u ∈ C) and C has the neutral term property: for all neutral term t, t

R

  • R
  • =

⇒ t ∈ C u1 ∈ C . . . un ∈ C

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Introduction Reducibility Stability by Union Orthogonal Constructor Rewriting Conclusion

Girard’s Reducibility Candidates

◮ Consider a set of contexts E[ ] ∈ E and a rewrite relation →R. ◮ A term t is Neutral if it interacts with no contexts E[ ] ∈ E:

if E[t] →R v then the reduction is either in E[ ] or in t.

◮ C ⊆ SN is a Reducibility Candidate (C ∈ C

R) iff C is stable by reduction (if t ∈ C and t →R u then u ∈ C) and C has the neutral term property: for all neutral term t, t

R

  • R
  • =

⇒ t ∈ C u1 ∈ C . . . un ∈ C E[u1] ∈ SN . . . E[un] ∈ SN = ⇒ E[t] ∈ SN

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Introduction Reducibility Stability by Union Orthogonal Constructor Rewriting Conclusion

Biorthogonals [Gir87, Par97, DK00, Pit00, MV05]

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Introduction Reducibility Stability by Union Orthogonal Constructor Rewriting Conclusion

Biorthogonals [Gir87, Par97, DK00, Pit00, MV05]

◮ Choose a pole

t ⊥ ⊥ E[ ] ⇐ ⇒def E[t] ∈ SN

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Introduction Reducibility Stability by Union Orthogonal Constructor Rewriting Conclusion

Biorthogonals [Gir87, Par97, DK00, Pit00, MV05]

◮ Choose a pole

t ⊥ ⊥ E[ ] ⇐ ⇒def E[t] ∈ SN

◮ Given A ⊆ Λ(Σ) and P ⊆ E, let

A⊥

⊥

=def {E[ ] ∈ E | ∀t ∈ A. t ⊥ ⊥ E[ ]} P⊥

⊥

=def {t ∈ Λ(Σ) | ∀E[ ] ∈ P. t ⊥ ⊥ E[ ]}

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Introduction Reducibility Stability by Union Orthogonal Constructor Rewriting Conclusion

Biorthogonals [Gir87, Par97, DK00, Pit00, MV05]

◮ Choose a pole

t ⊥ ⊥ E[ ] ⇐ ⇒def E[t] ∈ SN

◮ Given A ⊆ Λ(Σ) and P ⊆ E, let

A⊥

⊥

=def {E[ ] ∈ E | ∀t ∈ A. t ⊥ ⊥ E[ ]} P⊥

⊥

=def {t ∈ Λ(Σ) | ∀E[ ] ∈ P. t ⊥ ⊥ E[ ]} A ⊆ Λ(Σ)

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Introduction Reducibility Stability by Union Orthogonal Constructor Rewriting Conclusion

Biorthogonals [Gir87, Par97, DK00, Pit00, MV05]

◮ Choose a pole

t ⊥ ⊥ E[ ] ⇐ ⇒def E[t] ∈ SN

◮ Given A ⊆ Λ(Σ) and P ⊆ E, let

A⊥

⊥

=def {E[ ] ∈ E | ∀t ∈ A. t ⊥ ⊥ E[ ]} P⊥

⊥

=def {t ∈ Λ(Σ) | ∀E[ ] ∈ P. t ⊥ ⊥ E[ ]} A ⊆ Λ(Σ) ↓⊥

⊥

A⊥

⊥ ⊆ E

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Introduction Reducibility Stability by Union Orthogonal Constructor Rewriting Conclusion

Biorthogonals [Gir87, Par97, DK00, Pit00, MV05]

◮ Choose a pole

t ⊥ ⊥ E[ ] ⇐ ⇒def E[t] ∈ SN

◮ Given A ⊆ Λ(Σ) and P ⊆ E, let

A⊥

⊥

=def {E[ ] ∈ E | ∀t ∈ A. t ⊥ ⊥ E[ ]} P⊥

⊥

=def {t ∈ Λ(Σ) | ∀E[ ] ∈ P. t ⊥ ⊥ E[ ]} A ⊆ Λ(Σ) ↓⊥

⊥

A⊥

⊥ ⊆ E

↓⊥

⊥

A⊥

⊥ ⊥ ⊥ ⊆ Λ(Σ)

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Introduction Reducibility Stability by Union Orthogonal Constructor Rewriting Conclusion

Biorthogonals [Gir87, Par97, DK00, Pit00, MV05]

◮ Choose a pole

t ⊥ ⊥ E[ ] ⇐ ⇒def E[t] ∈ SN

◮ Given A ⊆ Λ(Σ) and P ⊆ E, let

A⊥

⊥

=def {E[ ] ∈ E | ∀t ∈ A. t ⊥ ⊥ E[ ]} P⊥

⊥

=def {t ∈ Λ(Σ) | ∀E[ ] ∈ P. t ⊥ ⊥ E[ ]}

◮ (_)⊥ ⊥ ⊥ ⊥ is a closure operator.

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Introduction Reducibility Stability by Union Orthogonal Constructor Rewriting Conclusion

Biorthogonals [Gir87, Par97, DK00, Pit00, MV05]

◮ Choose a pole

t ⊥ ⊥ E[ ] ⇐ ⇒def E[t] ∈ SN

◮ Given A ⊆ Λ(Σ) and P ⊆ E, let

A⊥

⊥

=def {E[ ] ∈ E | ∀t ∈ A. t ⊥ ⊥ E[ ]} P⊥

⊥

=def {t ∈ Λ(Σ) | ∀E[ ] ∈ P. t ⊥ ⊥ E[ ]}

◮ (_)⊥ ⊥ ⊥ ⊥ is a closure operator.

Lemma

∅ = A ⊆ SN = ⇒ A⊥

⊥ ⊥ ⊥ ∈ C

R

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Introduction Reducibility Stability by Union Orthogonal Constructor Rewriting Conclusion

Outline

Introduction Reducibility Stability by Union Application to Orthogonal Constructor Rewriting Conclusion

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Introduction Reducibility Stability by Union Orthogonal Constructor Rewriting Conclusion

◮ Given a typed rewrite system R, ◮ find a reducibility family Red

which leads to an adequate type interpretation and such that ∅ = R ⊆ Red = ⇒

  • R ∈ Red

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Introduction Reducibility Stability by Union Orthogonal Constructor Rewriting Conclusion

Union Types

T1, T2 ∈ T ::= . . . | T1 ⊔ T2

◮ We put

T1 ⊔ T2 =def Red(T1 ∪ T2) This validates (⊔ I) Γ ⊢ t : Ti Γ ⊢ t : T1 ⊔ T2

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Introduction Reducibility Stability by Union Orthogonal Constructor Rewriting Conclusion

Union Types

T1, T2 ∈ T ::= . . . | T1 ⊔ T2

◮ We put

T1 ⊔ T2 =def Red(T1 ∪ T2) This validates (⊔ I) Γ ⊢ t : Ti Γ ⊢ t : T1 ⊔ T2

◮ If Red stable by union, we have

T1 ⊔ T2 = T1 ∪ T2 This is sufficient to validate (⊔ E) Γ, x : T1 ⊢ c : C Γ ⊢ t : T1 ⊔ T2 Γ, x : T2 ⊢ c : C Γ ⊢ c[t/x] : C

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Introduction Reducibility Stability by Union Orthogonal Constructor Rewriting Conclusion

Unsafe Interaction [Rib07b]

t1 + t2 →R t1 t1 + t2 →R t2 t1 =def λx.xaδ t2 =def λy.δ

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Introduction Reducibility Stability by Union Orthogonal Constructor Rewriting Conclusion

Unsafe Interaction [Rib07b]

t1 + t2 →R t1 t1 + t2 →R t2 t1 =def λx.xaδ t2 =def λy.δ t1 : T1 t2 : T2 t1 + t2 : T1 ⊔ T2 x : T1 ⊢ xx : C x : T2 ⊢ xx : C Because t1t1 ∈ SN and t2t2 ∈ SN

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Introduction Reducibility Stability by Union Orthogonal Constructor Rewriting Conclusion

Unsafe Interaction [Rib07b]

t1 + t2 →R t1 t1 + t2 →R t2 t1 =def λx.xaδ t2 =def λy.δ (⊔ E) t1 : T1 t2 : T2 t1 + t2 : T1 ⊔ T2 x : T1 ⊢ xx : C x : T2 ⊢ xx : C (t1 + t2)(t1 + t2) : C While (t1 + t2)(t1 + t2) → t1t2 → (λy.δ) a δ → δ δ / ∈ SN

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Introduction Reducibility Stability by Union Orthogonal Constructor Rewriting Conclusion

Unsafe Interaction [Rib07b]

t1 + t2 →R t1 t1 + t2 →R t2 t1 =def λx.xaδ t2 =def λy.δ (⊔ E) t1 : T1 t2 : T2 t1 + t2 : T1 ⊔ T2 x : T1 ⊢ xx : C x : T2 ⊢ xx : C (t1 + t2)(t1 + t2) : C While (t1 + t2)(t1 + t2) → t1t2 → (λy.δ) a δ → δ δ / ∈ SN Similar example with a confluent rewrite system.

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Introduction Reducibility Stability by Union Orthogonal Constructor Rewriting Conclusion

Unsafe Interaction [Rib07b]

t1 + t2 →R t1 t1 + t2 →R t2 (⊔ E) t1 : T1 t2 : T2 t1 + t2 : T1 ⊔ T2 x : T1 ⊢ c : C x : T2 ⊢ c : C c[(t1 + t2)/x] : C

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Introduction Reducibility Stability by Union Orthogonal Constructor Rewriting Conclusion

Unsafe Interaction [Rib07b]

t1 + t2 →R t1 t1 + t2 →R t2 (⊔ E) t1 : T1 t2 : T2 t1 + t2 : T1 ⊔ T2 x : T1 ⊢ c : C x : T2 ⊢ c : C c[(t1 + t2)/x] : C But        Safe and Safe        does not imply Safe

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Introduction Reducibility Stability by Union Orthogonal Constructor Rewriting Conclusion

Unsafe Interaction [Rib07b]

t1 + t2 →R t1 t1 + t2 →R t2 (⊔ E) t1 : T1 t2 : T2 t1 + t2 : T1 ⊔ T2 x : T1 ⊢ c : C x : T2 ⊢ c : C c[(t1 + t2)/x] : C But        Safe and Safe        does not imply Safe Prevents from having T1 ⊔ T2 = T1 ∪ T2.

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Introduction Reducibility Stability by Union Orthogonal Constructor Rewriting Conclusion

Sufficient Conditions for T1 ⊔ T2 = T1 ∪ T2

Let →R be a rewrite relation on Λ(Σ) and E be a set of elimination contexts.

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Introduction Reducibility Stability by Union Orthogonal Constructor Rewriting Conclusion

Sufficient Conditions for T1 ⊔ T2 = T1 ∪ T2

Let →R be a rewrite relation on Λ(Σ) and E be a set of elimination contexts.        Safe and Safe        implies Safe

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Introduction Reducibility Stability by Union Orthogonal Constructor Rewriting Conclusion

Sufficient Conditions for T1 ⊔ T2 = T1 ∪ T2

Let →R be a rewrite relation on Λ(Σ) and E be a set of elimination contexts.        Safe and Safe        implies Safe OK if "

  • "

i.e. if t neutral has a "principal reduct" u: t

  • (u1

. . . un . . . )

  • u

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Introduction Reducibility Stability by Union Orthogonal Constructor Rewriting Conclusion

Reducibility Candidates [Rib07a]

◮ Neutral Term Property

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Introduction Reducibility Stability by Union Orthogonal Constructor Rewriting Conclusion

Reducibility Candidates [Rib07a]

◮ Neutral Term Property ◮ Characterize the membership to a candidate

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Introduction Reducibility Stability by Union Orthogonal Constructor Rewriting Conclusion

Reducibility Candidates [Rib07a]

◮ Neutral Term Property ◮ Characterize the membership to a candidate ◮ A Value is an observable term, ie a term which interacts with

some contexts E[ ] ∈ E.

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Introduction Reducibility Stability by Union Orthogonal Constructor Rewriting Conclusion

Reducibility Candidates [Rib07a]

◮ Neutral Term Property ◮ Characterize the membership to a candidate ◮ A Value is an observable term, ie a term which interacts with

some contexts E[ ] ∈ E.

◮ Weak Observational Preorder

Let u C

R t iff every value of u is a value of t.

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Introduction Reducibility Stability by Union Orthogonal Constructor Rewriting Conclusion

Reducibility Candidates [Rib07a]

◮ Neutral Term Property ◮ Characterize the membership to a candidate ◮ A Value is an observable term, ie a term which interacts with

some contexts E[ ] ∈ E.

◮ Weak Observational Preorder

Let u C

R t iff every value of u is a value of t.

Lemma

If C is a reducibility candidate, then C is downward closed wrt. C

R

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Introduction Reducibility Stability by Union Orthogonal Constructor Rewriting Conclusion

Reducibility Candidates [Rib07a]

◮ Neutral Term Property ◮ Characterize the membership to a candidate ◮ A Value is an observable term, ie a term which interacts with

some contexts E[ ] ∈ E.

◮ Weak Observational Preorder

Let u C

R t iff every value of u is a value of t.

Lemma

If C is a reducibility candidate, then C is downward closed wrt. C

R ◮ But not every such C is a reducibility candidate.

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Introduction Reducibility Stability by Union Orthogonal Constructor Rewriting Conclusion

Stability by Union of Reducibility Candidates [Rib07a]

Theorem

The following are equivalent: (i) C R is stable by union, (ii) C R is the set of all non-empty subsets C of SN that are downward closed wrt. C

R.

(iii) for every t which is non-normal, strongly normalizing and neutral, there is a term u such that t →R u and t C

R u

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Introduction Reducibility Stability by Union Orthogonal Constructor Rewriting Conclusion

Stability by Union of Reducibility Candidates [Rib07a]

Theorem

The following are equivalent: (i) C R is stable by union, (ii) C R is the set of all non-empty subsets C of SN that are downward closed wrt. C

R.

(iii) for every t which is non-normal, strongly normalizing and neutral, there is a term u such that t →R u and t C

R u ◮ u is a strong principal reduct of t.

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Introduction Reducibility Stability by Union Orthogonal Constructor Rewriting Conclusion

Stability by Union of Reducibility Candidates [Rib07a]

Theorem

The following are equivalent: (i) C R is stable by union, (ii) C R is the set of all non-empty subsets C of SN that are downward closed wrt. C

R.

(iii) for every t which is non-normal, strongly normalizing and neutral, there is a term u such that t →R u and t C

R u ◮ u is a strong principal reduct of t. ◮ This holds for the λ-calculus with products and sums

(also [Tat07]).

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Biorthogonals

◮ Biorthogonals are reducibility candidates.

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Introduction Reducibility Stability by Union Orthogonal Constructor Rewriting Conclusion

Biorthogonals

◮ Biorthogonals are reducibility candidates. ◮ In general, biorthogonals are not stable by union.

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Introduction Reducibility Stability by Union Orthogonal Constructor Rewriting Conclusion

Biorthogonals

◮ Biorthogonals are reducibility candidates. ◮ In general, biorthogonals are not stable by union. ◮ Is the closure by union of biorthogonals a reducibility family ?

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Introduction Reducibility Stability by Union Orthogonal Constructor Rewriting Conclusion

Biorthogonals [Rib07b]

◮ Is the closure by union of biorthogonals a reducibility family ?

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Introduction Reducibility Stability by Union Orthogonal Constructor Rewriting Conclusion

Biorthogonals [Rib07b]

◮ Is the closure by union of biorthogonals a reducibility family ? ◮ Let u SN t iff

for all E[ ] ∈ E, if E[u] ∈ SN then E[t] ∈ SN

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Introduction Reducibility Stability by Union Orthogonal Constructor Rewriting Conclusion

Biorthogonals [Rib07b]

◮ Is the closure by union of biorthogonals a reducibility family ? ◮ Let u SN t iff

for all E[ ] ∈ E, if E[u] ∈ SN then E[t] ∈ SN

Theorem

The following are equivalent: (i) unions of biorthogonals are reducibility candidates, (ii) for every t which is non-normal, strongly normalizing and neutral, there is a term u such that t →R u and u SN t

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Introduction Reducibility Stability by Union Orthogonal Constructor Rewriting Conclusion

Biorthogonals [Rib07b]

◮ Is the closure by union of biorthogonals a reducibility family ? ◮ Let u SN t iff

for all E[ ] ∈ E, if E[u] ∈ SN then E[t] ∈ SN

Theorem

The following are equivalent: (i) unions of biorthogonals are reducibility candidates, (ii) for every t which is non-normal, strongly normalizing and neutral, there is a term u such that t →R u and u SN t u is a principal reduct of t.

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Introduction Reducibility Stability by Union Orthogonal Constructor Rewriting Conclusion

Comparison [Rib07b]

Lemma

∀t, u ∈ SN. t C

R u

= ⇒ u SN t

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Introduction Reducibility Stability by Union Orthogonal Constructor Rewriting Conclusion

Comparison [Rib07b]

Lemma

∀t, u ∈ SN. t C

R u

= ⇒ u SN t Every strong principal reduct is a principal reduct.

Lemma

C R stable by union = ⇒ O ⊆ C R

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Introduction Reducibility Stability by Union Orthogonal Constructor Rewriting Conclusion

Comparison [Rib07b]

Lemma

∀t, u ∈ SN. t C

R u

= ⇒ u SN t Every strong principal reduct is a principal reduct.

Lemma

C R stable by union = ⇒ O ⊆ C R The converse is false, consider p →R λx.c1 p →R λx.c2 ci →R d Indeed, p C

R λx.ci

but λx.ci SN p

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Outline

Introduction Reducibility Stability by Union Application to Orthogonal Constructor Rewriting Conclusion

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Introduction Reducibility Stability by Union Orthogonal Constructor Rewriting Conclusion

Constructor Rewriting

◮ Constructors are symbols c of type

Γ ⊢ t1 : T1 . . . Γ ⊢ tn : Tn Γ ⊢ c(t1, . . . , tn) : B

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Introduction Reducibility Stability by Union Orthogonal Constructor Rewriting Conclusion

Constructor Rewriting

◮ Constructors are symbols c of type

Γ ⊢ t1 : T1 . . . Γ ⊢ tn : Tn Γ ⊢ c(t1, . . . , tn) : B

◮ Constructor Patterns

p ::= x | c(p1, . . . , pn)

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Introduction Reducibility Stability by Union Orthogonal Constructor Rewriting Conclusion

Constructor Rewriting

◮ Constructors are symbols c of type

Γ ⊢ t1 : T1 . . . Γ ⊢ tn : Tn Γ ⊢ c(t1, . . . , tn) : B

◮ Constructor Patterns

p ::= x | c(p1, . . . , pn)

◮ Rewrite Rules

f(p1, . . . , pn) →R r where p1, . . . , pn are constructor patterns.

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Introduction Reducibility Stability by Union Orthogonal Constructor Rewriting Conclusion

Values

◮ Destructors

For each n-ary c with n ≥ 1 and each i ∈ {1, . . . , n},

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Introduction Reducibility Stability by Union Orthogonal Constructor Rewriting Conclusion

Values

◮ Destructors

For each n-ary c with n ≥ 1 and each i ∈ {1, . . . , n}, dc,i(c(x1, . . . , xn)) →D xi

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Introduction Reducibility Stability by Union Orthogonal Constructor Rewriting Conclusion

Values

◮ Destructors

For each n-ary c with n ≥ 1 and each i ∈ {1, . . . , n}, dc,i(c(x1, . . . , xn)) →D xi For each nullary c, dcc →D ℧

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Introduction Reducibility Stability by Union Orthogonal Constructor Rewriting Conclusion

Values

◮ Destructors

For each n-ary c with n ≥ 1 and each i ∈ {1, . . . , n}, dc,i(c(x1, . . . , xn)) →D xi For each nullary c, dcc →D ℧

◮ Elimination Contexts

E[ ] ∈ E⇒C ::= [ ] | E[ ] t | d(E[ ])

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Introduction Reducibility Stability by Union Orthogonal Constructor Rewriting Conclusion

Values

◮ Destructors

For each n-ary c with n ≥ 1 and each i ∈ {1, . . . , n}, dc,i(c(x1, . . . , xn)) →D xi For each nullary c, dcc →D ℧

◮ Elimination Contexts

E[ ] ∈ E⇒C ::= [ ] | E[ ] t | d(E[ ])

◮ Values

λx.t c(t1, . . . , tn)

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Introduction Reducibility Stability by Union Orthogonal Constructor Rewriting Conclusion

External Redexes (CCERSs) [KOO01]

Let R be a constructor rewrite system.

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Introduction Reducibility Stability by Union Orthogonal Constructor Rewriting Conclusion

External Redexes (CCERSs) [KOO01]

Let R be a constructor rewrite system.

◮ A subterm (position) u occurs in a Redex Argument of a term t

if either

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Introduction Reducibility Stability by Union Orthogonal Constructor Rewriting Conclusion

External Redexes (CCERSs) [KOO01]

Let R be a constructor rewrite system.

◮ A subterm (position) u occurs in a Redex Argument of a term t

if either t has a subterm (λx.t1)t2 and u occurs in some ti or

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Introduction Reducibility Stability by Union Orthogonal Constructor Rewriting Conclusion

External Redexes (CCERSs) [KOO01]

Let R be a constructor rewrite system.

◮ A subterm (position) u occurs in a Redex Argument of a term t

if either t has a subterm (λx.t1)t2 and u occurs in some ti or t has a subterm f(l1σ, . . . , lnσ) and u occurs in some liσ where f(l1, . . . , ln) →R r

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Introduction Reducibility Stability by Union Orthogonal Constructor Rewriting Conclusion

External Redexes (CCERSs) [KOO01]

Let R be a constructor rewrite system.

◮ A subterm (position) u occurs in a Redex Argument of a term t

if either t has a subterm (λx.t1)t2 and u occurs in some ti or t has a subterm f(l1σ, . . . , lnσ) and u occurs in some liσ where f(l1, . . . , ln) →R r

◮ A redex (position) u is External in a term t

if no residual of u occurs in a redex argument of a reduct of t.

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Introduction Reducibility Stability by Union Orthogonal Constructor Rewriting Conclusion

External Redexes (CCERSs) [KOO01]

Let R be a constructor rewrite system.

◮ A subterm (position) u occurs in a Redex Argument of a term t

if either t has a subterm (λx.t1)t2 and u occurs in some ti or t has a subterm f(l1σ, . . . , lnσ) and u occurs in some liσ where f(l1, . . . , ln) →R r

◮ A redex (position) u is External in a term t

if no residual of u occurs in a redex argument of a reduct of t.

◮ If t →βR u by contracting an external redex of t,

then u is an External Reduct of t.

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Introduction Reducibility Stability by Union Orthogonal Constructor Rewriting Conclusion

External Reducts are Strong Principal [Rib08]

Let R be an orthogonal constructor rewrite system.

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Introduction Reducibility Stability by Union Orthogonal Constructor Rewriting Conclusion

External Reducts are Strong Principal [Rib08]

Let R be an orthogonal constructor rewrite system.

Lemma

Let t be a neutral term and u be an external reduct of t. If t reduces to a value v then u reduces to v.

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Introduction Reducibility Stability by Union Orthogonal Constructor Rewriting Conclusion

External Reducts are Strong Principal [Rib08]

Let R be an orthogonal constructor rewrite system.

Lemma

Let t be a neutral term and u be an external reduct of t. If t reduces to a value v then u reduces to v.

◮ If t is neutral and u is an external reduct of t then t C R u.

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Introduction Reducibility Stability by Union Orthogonal Constructor Rewriting Conclusion

External Reducts are Strong Principal [Rib08]

Let R be an orthogonal constructor rewrite system.

Lemma

Let t be a neutral term and u be an external reduct of t. If t reduces to a value v then u reduces to v.

◮ If t is neutral and u is an external reduct of t then t C R u. ◮ Theorem [KOO01]

If R is orthogonal then every reducible term has an external redex.

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Introduction Reducibility Stability by Union Orthogonal Constructor Rewriting Conclusion

External Reducts are Strong Principal [Rib08]

Let R be an orthogonal constructor rewrite system.

Lemma

Let t be a neutral term and u be an external reduct of t. If t reduces to a value v then u reduces to v.

◮ If t is neutral and u is an external reduct of t then t C R u. ◮ Theorem [KOO01]

If R is orthogonal then every reducible term has an external redex.

Corollary

If R is an orthogonal constructor rewrite system then C R is stable by union.

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Outline

Introduction Reducibility Stability by Union Application to Orthogonal Constructor Rewriting Conclusion

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Introduction Reducibility Stability by Union Orthogonal Constructor Rewriting Conclusion

Conclusion

We have studied different reducibility families, and compared them wrt. stability by union.

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Introduction Reducibility Stability by Union Orthogonal Constructor Rewriting Conclusion

Conclusion

We have studied different reducibility families, and compared them wrt. stability by union.

◮ Some rewrite systems do not admit reducibility families stable by

union.

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Introduction Reducibility Stability by Union Orthogonal Constructor Rewriting Conclusion

Conclusion

We have studied different reducibility families, and compared them wrt. stability by union.

◮ Some rewrite systems do not admit reducibility families stable by

union.

◮ Sufficient conditions to have a reducibility family stable by union.

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Introduction Reducibility Stability by Union Orthogonal Constructor Rewriting Conclusion

Conclusion

We have studied different reducibility families, and compared them wrt. stability by union.

◮ Some rewrite systems do not admit reducibility families stable by

union.

◮ Sufficient conditions to have a reducibility family stable by union. ◮ Investigation of the structure of Girard’s Candidates.

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Introduction Reducibility Stability by Union Orthogonal Constructor Rewriting Conclusion

Conclusion

We have studied different reducibility families, and compared them wrt. stability by union.

◮ Some rewrite systems do not admit reducibility families stable by

union.

◮ Sufficient conditions to have a reducibility family stable by union. ◮ Investigation of the structure of Girard’s Candidates. ◮ For the combination of λ-calculus with orthogonal constructor

rewriting, Girard’s Candidates are stable by union.

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Introduction Reducibility Stability by Union Orthogonal Constructor Rewriting Conclusion

Conclusion

We have studied different reducibility families, and compared them wrt. stability by union.

◮ Some rewrite systems do not admit reducibility families stable by

union.

◮ Sufficient conditions to have a reducibility family stable by union. ◮ Investigation of the structure of Girard’s Candidates. ◮ For the combination of λ-calculus with orthogonal constructor

rewriting, Girard’s Candidates are stable by union.

◮ In [Rib07b], we studied a type system with (⊔ E) such that for

simple rewrite systems R, the following are equivalent:

(i) terms typable using (⊔ E) are Strongly Normalizing, (ii) the interpretation _ : T → P⋆(SN)⊥

⊥ ⊥ ⊥ is adequate.

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Conclusion

More generally, can be an interesting point of view to connect notions from denotational semantics, rewriting theory and computational interpretations of logics.

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Introduction Reducibility Stability by Union Orthogonal Constructor Rewriting Conclusion

Conclusion

More generally, can be an interesting point of view to connect notions from denotational semantics, rewriting theory and computational interpretations of logics. In particular:

◮ Logical account of stability by union: can λ-calculus for classical

logic admit stable by union type interpretations ?

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Introduction Reducibility Stability by Union Orthogonal Constructor Rewriting Conclusion

Conclusion

More generally, can be an interesting point of view to connect notions from denotational semantics, rewriting theory and computational interpretations of logics. In particular:

◮ Logical account of stability by union: can λ-calculus for classical

logic admit stable by union type interpretations ?

◮ Link with notions such as sequentiality and stability.

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Introduction Reducibility Stability by Union Orthogonal Constructor Rewriting Conclusion

Conclusion

More generally, can be an interesting point of view to connect notions from denotational semantics, rewriting theory and computational interpretations of logics. In particular:

◮ Logical account of stability by union: can λ-calculus for classical

logic admit stable by union type interpretations ?

◮ Link with notions such as sequentiality and stability. ◮ Connect typability (eq. with union/intersection types) to

properties such as standardization.

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Introduction Reducibility Stability by Union Orthogonal Constructor Rewriting Conclusion

Thank you for your attention !

http://www.loria.fr/~riba/

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Introduction Reducibility Stability by Union Orthogonal Constructor Rewriting Conclusion

  • A. ABEL – “Semi-Continuous Sized Types and Termination”,

Proceedings of CSL ’06, LNCS, vol. 4207, Springer, 2006,

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F . BLANQUI et C. RIBA – “Combining Typing and Size Constraints for Checking the Termination of Higher-Order Conditional Rewrite Systems”, Proceedings of LPAR’06, LNAI, vol. 4246, 2006.

  • V. DANOS et J.-L. KRIVINE – “Disjunctive Tautologies as

Synchronisation Schemes”, Proceedings of CSL ’00, LNCS, vol. 1862, 2000, p. 292–301. J.-Y. GIRARD – “Interprétation Fonctionnelle et Élimination des Coupures de l’Arithmétique d’Ordre Supérieur”, Thèse, Université Paris 7, 1972. — , “Linear Logic”, Theoretical Computer Science 50 (1987),

  • p. 1–102.

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  • Z. KHASIDASHVILI, M. OGAWA et V. VAN OOSTROM –

“Perpetuality ans Uniform Normalization in Orthogonal Rewrite Systems”, Information and Computation 164 (2001), no. 1,

  • p. 118–152.

P .-A. MELLIÈS et J. VOUILLON – “Recursive Polymorphic Types and Parametricity in an Operational Framework”, Proceedings of LiCS’05, IEEE Computer Society, 2005, p. 82–91.

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Classical Natural Deduction”, Journal of Symbolic Logic 62 (1997), no. 4, p. 1461–1479.

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Equivalence”, Mathematical Structures in Computer Science 10 (2000), p. 321–359.

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Proceedings of FoSSaCS’07, LNCS, vol. 4423, 2007.

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— , “Strong Normalization as Safe Interaction”, Proceedings of LiCS’07, IEEE Computer Society, 2007, p. 13–22. — , “Unions of Reducibility Candidates for Orthogonal Constructor Rewriting”, Available on HAL-INRIA http://hal.inria.fr, 2008.

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Species”, Logic Colloquium (R. PARIKH, éd.), LNCS, vol. 453, 1975, p. 240–251.

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Second-Order Existential Quantification”, Proceedings of TLCA’07, LNCS, vol. 4583, Springer, 2007, p. 366–380.

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