Typed recursion in the rewriting calculus Benjamin Wack joint work - - PowerPoint PPT Presentation

Typed recursion in the rewriting calculus

Benjamin Wack

joint work with C. Kirchner, L. Liquori, H. Cirstea

ρWorkshop, March 10th 2004, Nancy

The type system Typing fixpoints Encoding typed objects and TRS Logical failure

Typed recursion in the rewriting calculus Introducing types

The Type System I X : σ ∈ Γ Γ ⊢ X : σ (V ar)

Typed recursion in the rewriting calculus Introducing types

The Type System I X : σ ∈ Γ Γ ⊢ X : σ (V ar) Γ ⊢ T1 : σ τ Γ ⊢ T2 : σ Γ ⊢ T1 T2 : τ (Appl)

Typed recursion in the rewriting calculus Introducing types

The Type System I X : σ ∈ Γ Γ ⊢ X : σ (V ar) Γ ⊢ T1 : σ τ Γ ⊢ T2 : σ Γ ⊢ T1 T2 : τ (Appl) Γ, ∆ ⊢ T1 : σ Γ, ∆ ⊢ T2 : τ Γ ⊢ T1 →∆ T2 : σ τ (Abs)

Typed recursion in the rewriting calculus Introducing types

The Type System I X : σ ∈ Γ Γ ⊢ X : σ (V ar) Γ ⊢ T1 : σ τ Γ ⊢ T2 : σ Γ ⊢ T1 T2 : τ (Appl) Γ, ∆ ⊢ T1 : σ Γ, ∆ ⊢ T2 : τ Γ ⊢ T1 →∆ T2 : σ τ (Abs) Γ, ∆ ⊢ T1 : σ Γ ⊢ T2 : σ Γ, ∆ ⊢ T3 : τ Γ ⊢ [T1≪∆T2]T3 : τ (Match)

Typed recursion in the rewriting calculus Introducing types

The Type System II Γ ⊢ T1 : σ Γ ⊢ T2 : σ Γ ⊢ T1; T2 : σ (Struct)

Typed recursion in the rewriting calculus Introducing types

The Type System II Γ ⊢ T1 : σ Γ ⊢ T2 : σ Γ ⊢ T1; T2 : σ (Struct) Γ ⊢ T : σ α ∈ FV (Γ) Γ ⊢ T : ∀α.σ (Abs−∀) Γ ⊢ T : σ α ∈ FV (Γ) Γ ⊢ α → T : ∀α.σ (Abs−∀)

Typed recursion in the rewriting calculus Introducing types

The Type System II Γ ⊢ T1 : σ Γ ⊢ T2 : σ Γ ⊢ T1; T2 : σ (Struct) Γ ⊢ T : σ α ∈ FV (Γ) Γ ⊢ T : ∀α.σ (Abs−∀) Γ ⊢ T : σ α ∈ FV (Γ) Γ ⊢ α → T : ∀α.σ (Abs−∀) Γ ⊢ T : ∀α.σ Γ ⊢ T : σ{τ/α} (Appl−∀) Γ ⊢ T : ∀α.σ Γ ⊢ T τ : σ{τ/α} (Appl−∀)

Typed recursion in the rewriting calculus Introducing types

Typing properties

Well-typed matching: If Sol(P≺ ≺T ) = θ, then ∀X ∈ P, Γ ⊢ X : σ ⇒ Γ ⊢ Xθ : σ. Subject Reduction: If Γ ⊢ T1 : σ and T1 → →

ρ σ δ T2, then Γ ⊢ T2 : σ.

Uniqueness: If Γ ⊢ T : ϕ and Γ ⊢ T : ψ, then ϕ =α ψ. Decidability: (typechecking) Γ ⊢ T : ϕ ? (type inference) Γ ⊢ T : ?

• are decidable.

Typed recursion in the rewriting calculus Introducing types

The type system Typing fixpoints Encoding typed objects and TRS Logical failure

Typed recursion in the rewriting calculus A typed divergent term

Normalization failure f : (αα)α and Γ = X : αα, ω

△

= f •X → X•(f •X)

ω•(f •ω) ≡ (f •X → X•(f •X))•(f •ω) →ρ [f •X ≪ f •ω].(X•(f •X)) →σ ω•(f •ω) →ρ . . .

Typed recursion in the rewriting calculus A typed divergent term

Normalization failure (cont’d) f : (αα)α and Γ = X : αα, ω

△

= f •X → X•(f •X)

Γ ⊢ f : (αα)α Γ ⊢ X : αα (b) Γ ⊢ f •X : α Γ ⊢ X : αα (b) Γ ⊢ f •X : α Γ ⊢ X•(f •X) : α (a) ⊢ ω ≡ f •X → X•(f •X) : αα (a) ⊢ ω : αα ⊢ f : (αα)α (a) ⊢ ω : αα ⊢ f •ω : α ⊢ ω•(f •ω) : α

Typed recursion in the rewriting calculus A typed divergent term

Inductive types with “positive” occurrences

• In CaML, try to type:

type t = F of (t -> t);; let omega x = match x with (F y) -> y (F y);;

Typed recursion in the rewriting calculus In other typed formalisms with patterns

Inductive types with “positive” occurrences

• In CaML, try to type:

type t = F of (t -> t);; let omega x = match x with (F y) -> y (F y);;

• In CIC, the constructor F : (x1 : A1) . . . (xn : An).R is accepted only if R is

positive in each Ai:

• 1. R is positive in T if R does not occur in T;
• 2. R is positive in (R

t) if R does not occur in t;

• 3. R is positive in (x : A)C if R does not occur in A and R is positive in C.

Typed recursion in the rewriting calculus In other typed formalisms with patterns

The type system Typing fixpoints Encoding typed objects and TRS Logical failure

Typed recursion in the rewriting calculus Encodings

Detecting matching failures: the symbol stk

• 1. The relation P ⊑ A detects (some) definitive matching failures:

f ⊑ g f(An) ⊑ B if (B ≡ f(Bn) ) ∧ ∃i, Ai ⊑ Bi P ⊑ A if A ≡ ([Q ≪∆ A1].A2 ∧ Q ⊑ A1 ∨ P ⊑ A2)

Typed recursion in the rewriting calculus Encodings

Detecting matching failures: the symbol stk

• 1. The relation P ⊑ A detects (some) definitive matching failures:

f ⊑ g f(An) ⊑ B if (B ≡ f(Bn) ) ∧ ∃i, Ai ⊑ Bi P ⊑ A if A ≡ ([Q ≪∆ A1].A2 ∧ Q ⊑ A1 ∨ P ⊑ A2)

• 2. The relation →stk treats matching failures uniformly:

[P ≪∆ A].B →stk stk if P ⊑ A stk; A →stk A A; stk →stk A stk•A →stk stk

Typed recursion in the rewriting calculus Encodings

Rho and objects

• Object = record with an explicit account of self, i.e.

[mi = ς(Xi)ti]i∈I

△

=

(mi(Xi) → ti)i∈I

Typed recursion in the rewriting calculus Encodings

Rho and objects

• Object = record with an explicit account of self, i.e.

[mi = ς(Xi)ti]i∈I

△

=

(mi(Xi) → ti)i∈I

• Self-application = the application of an object to the object itself, i.e.

t1.t2

△

=

t1•t2(t1)

Typed recursion in the rewriting calculus Encodings

Rho and objects

• Object = record with an explicit account of self, i.e.

[mi = ς(Xi)ti]i∈I

△

=

(mi(Xi) → ti)i∈I

• Self-application = the application of an object to the object itself, i.e.

t1.t2

△

=

t1•t2(t1)

• Ex: t

△

= a(S) → b. Then: t.a

△

= t•a(t)

ρ

→ [a(S) ≪ a(t)]b

σ

→ b

Typed recursion in the rewriting calculus Encodings

Typed objects

• An object has type:

S : lab → φ ⊢ meth : (lab → φ) → lab ⊢ meth(S) : lab (Appl) ⊢ Tmeth : φ ⊢ meth(S) → Tmeth : lab → φ (Abst)

Typed recursion in the rewriting calculus Encodings

Typed objects

• An object has type:

S : lab → φ ⊢ meth : (lab → φ) → lab ⊢ meth(S) : lab (Appl) ⊢ Tmeth : φ ⊢ meth(S) → Tmeth : lab → φ (Abst)

• obj.meth

△

= obj•meth(obj) can be typed as follows:

⊢ obj : lab → φ ⊢ meth : (lab → φ) → lab ⊢ obj : lab → φ ⊢ meth(obj) : lab ⊢ obj•meth(obj) : φ

Typed recursion in the rewriting calculus Encodings

Encoding rewriting in the ρ-calculus

• 1. The following operator selects the first applicable rule of a set:

first(A1, A2, . . . , An)

△

= X → ((stk → An•X; I)• (. . . •(stk → A2•X; I)•(A1X))) first(A1, A2, . . . , An)•B → →

ρ σ δ

Aj+1•B if ∀i ≤ j, Ai•B → →

ρ σ δ stk

and Aj+1•B → →

ρ σ δ stk Typed recursion in the rewriting calculus Encodings

Encoding rewriting in the ρ-calculus

• 1. The following operator selects the first applicable rule of a set:

first(A1, A2, . . . , An)

△

= X → ((stk → An•X; I)• (. . . •(stk → A2•X; I)•(A1X))) first(A1, A2, . . . , An)•B → →

ρ σ δ

Aj+1•B if ∀i ≤ j, Ai•B → →

ρ σ δ stk

and Aj+1•B → →

ρ σ δ stk

• 2. The Term Rewrite System R = {ti → si} with signature {aj} is encoded by:

R (rec•S) → first      t1 → S•(rec•S)•s1, · · · , a1•X → S•(Rec•S)•(a1•S(rec•S)•X), · · · ,      , (Rec•S) → first   t1 → S•(rec•S)•s1, · · · , I  

Typed recursion in the rewriting calculus Encodings

Example

• Addition over Peano integers:

plus

△

=

S → add(0, y) → y; S → add(suc(x), y) → suc

• (S•S)•add(x, y)
• Typed recursion in the rewriting calculus

Encodings

Example

• Addition over Peano integers:

plus

△

=

S → add(0, y) → y; S → add(suc(x), y) → suc

• (S•S)•add(x, y)
• (plus • plus) • add(N, M)

→ →ρ

σ δ

[0 ≪ N].M; [0 ≪ N − 1].(M +1) · · · [0 ≪ 0].(M +N); [suc•x ≪ 0]. . . . →stk M + N

Typed recursion in the rewriting calculus Encodings

Example

• Addition over Peano integers:

plus

△

=

S → add(0, y) → y; S → add(suc(x), y) → suc

• (S•S)•add(x, y)
• (plus • plus) • add(N, M)

→ →ρ

σ δ

[0 ≪ N].M; [0 ≪ N − 1].(M +1) · · · [0 ≪ 0].(M +N); [suc•x ≪ 0]. . . . →stk M + N

• Fill in the blanks with your favorite rewrite system...

func

△

=

S → ; S → (S•S) •

• Typed recursion in the rewriting calculus

Encodings

Example

• Addition over Peano integers:

plus

△

=

S → add(0, y) → y; S → add(suc(x), y) → suc

• (S•S)•add(x, y)
• (plus • plus) • add(N, M)

→ →ρ

σ δ

[0 ≪ N].M; [0 ≪ N − 1].(M +1) · · · [0 ≪ 0].(M +N); [suc•x ≪ 0]. . . . →stk M + N

• Fill in the blanks with your favorite rewrite system...

func

△

=

S → len([ ]) → 0 ; S → len(Cons(x, l)) → suc

• (S•S) • len(l)
• Typed recursion in the rewriting calculus

Encodings

Example

• Addition over Peano integers:

plus

△

=

S → add(0, y) → y; S → add(suc(x), y) → suc

• (S•S)•add(x, y)
• (plus • plus) • add(N, M)

→ →ρ

σ δ

[0 ≪ N].M; [0 ≪ N − 1].(M +1) · · · [0 ≪ 0].(M +N); [suc•x ≪ 0]. . . . →stk M + N

• Fill in the blanks with your favorite rewrite system... provided it is convergent

and ground reducible if you want completeness. func

△

=

S → len([ ]) → 0 ; S → len(Cons(x, l)) → suc

• (S•S) • len(l)
• Typed recursion in the rewriting calculus

Encodings

The type system Typing fixpoints Encoding typed objects and TRS Logical failure

Typed recursion in the rewriting calculus Logical considerations

Logical inconsistency

• As is, the Curry-Howard isomorphism is not valid:

Γ, ∆ ⊢ T1 : σ Γ, ∆ ⊢ T2 : τ Γ ⊢ T1 →∆ T2 : σ τ (Abs) Γ, ∆ ⊢ σ Γ, ∆ ⊢ τ Γ ⊢ σ τ (→ I)

Typed recursion in the rewriting calculus Logical considerations

Logical inconsistency

• As is, the Curry-Howard isomorphism is not valid:

Γ, ∆ ⊢ T1 : σ Γ, ∆ ⊢ T2 : τ Γ ⊢ T1 →∆ T2 : σ τ (Abs) Γ, ∆ ⊢ σ Γ, ∆ ⊢ τ Γ ⊢ σ τ (→ I)

• Thus, for instance,

⊢ (h⊥αα(X⊥) → X⊥) : α α ⊥ ⊢ (h⊥αα(X⊥) → X⊥)•(Y α → Y α) : ⊥

Typed recursion in the rewriting calculus Logical considerations

Logical inconsistency

• As is, the Curry-Howard isomorphism is not valid:

Γ, ∆ ⊢ T1 : σ Γ, ∆ ⊢ T2 : τ Γ ⊢ T1 →∆ T2 : σ τ (Abs) Γ, ∆ ⊢ σ Γ, ∆ ⊢ τ Γ ⊢ σ τ (→ I)

• Thus, for instance,

⊢ (h⊥αα(X⊥) → X⊥) : α α ⊥ ⊢ (h⊥αα(X⊥) → X⊥)•(Y α → Y α) : ⊥

• How to fix it ?

Γ, Xi : ϕi ⊢ B : ψ Γ ⊢ A → B : ( ϕi) ψ (Abs) , FV (A) = {Xϕi

i }

Typed recursion in the rewriting calculus Logical considerations

Dependent type discipline

Γ, ∆ ⊢ T1 : σ Γ, ∆ ⊢ T2 : τ Γ ⊢ T1 : ∆ → T2 : στ (Abs) Γ ⊢ T1 : στ Γ ⊢ T2 : σ Γ ⊢ T1 T2 : τ (Appl) Γ, ∆ ⊢ T2 : τ Γ ⊢ (T1 : ∆) → T2 : (T1 : ∆)τ (Abs) Γ ⊢ T1 : (T11 : ∆)τ Γ ⊢ T2 : σ Γ, ∆ ⊢ T11 : σ Γ ⊢ T1 T2 : [T11≪∆T2].τ (Appl)

Typed recursion in the rewriting calculus Logical considerations