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 ρ Workshop, 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 ⊢ T 1 : σ � τ Γ ⊢ T 2 : σ ( Appl ) Γ ⊢ T 1 T 2 : τ Typed recursion in the rewriting calculus Introducing types

The Type System I X : σ ∈ Γ Γ ⊢ X : σ ( V ar ) Γ ⊢ T 1 : σ � τ Γ ⊢ T 2 : σ ( Appl ) Γ ⊢ T 1 T 2 : τ Γ , ∆ ⊢ T 1 : σ Γ , ∆ ⊢ T 2 : τ ( Abs ) Γ ⊢ T 1 → ∆ T 2 : σ � τ Typed recursion in the rewriting calculus Introducing types

The Type System I X : σ ∈ Γ Γ ⊢ X : σ ( V ar ) Γ ⊢ T 1 : σ � τ Γ ⊢ T 2 : σ ( Appl ) Γ ⊢ T 1 T 2 : τ Γ , ∆ ⊢ T 1 : σ Γ , ∆ ⊢ T 2 : τ ( Abs ) Γ ⊢ T 1 → ∆ T 2 : σ � τ Γ , ∆ ⊢ T 1 : σ Γ ⊢ T 2 : σ Γ , ∆ ⊢ T 3 : τ ( Match ) Γ ⊢ [ T 1 ≪ ∆ T 2 ] T 3 : τ Typed recursion in the rewriting calculus Introducing types

The Type System II Γ ⊢ T 1 : σ Γ ⊢ T 2 : σ ( Struct ) Γ ⊢ T 1 ; T 2 : σ Typed recursion in the rewriting calculus Introducing types

The Type System II Γ ⊢ T 1 : σ Γ ⊢ T 2 : σ ( Struct ) Γ ⊢ T 1 ; T 2 : σ Γ ⊢ T : σ α �∈ FV (Γ) Γ ⊢ T : σ α �∈ FV (Γ) ( Abs −∀ ) ( Abs −∀ ) Γ ⊢ T : ∀ α.σ Γ ⊢ α → T : ∀ α.σ Typed recursion in the rewriting calculus Introducing types

The Type System II Γ ⊢ T 1 : σ Γ ⊢ T 2 : σ ( Struct ) Γ ⊢ T 1 ; T 2 : σ Γ ⊢ T : σ α �∈ FV (Γ) Γ ⊢ T : σ α �∈ FV (Γ) ( Abs −∀ ) ( Abs −∀ ) Γ ⊢ T : ∀ α.σ Γ ⊢ α → T : ∀ α.σ Γ ⊢ T : ∀ α.σ Γ ⊢ T : ∀ α.σ Γ ⊢ T : σ { τ/α } ( Appl −∀ ) Γ ⊢ T τ : σ { τ/α } ( Appl −∀ ) Typed recursion in the rewriting calculus Introducing types

Typing properties Well-typed matching: If S ol ( P≺ ≺T ) = θ , then ∀ X ∈ P , Γ ⊢ X : σ ⇒ Γ ⊢ Xθ : σ . Subject Reduction: If Γ ⊢ T 1 : σ and T 1 �→ → δ T 2 , then Γ ⊢ T 2 : σ . ρ σ Uniqueness: If Γ ⊢ T : ϕ and Γ ⊢ T : ψ , then ϕ = α ψ . Decidability: � (typechecking) Γ ⊢ T : ϕ ? are decidable. (type inference) Γ ⊢ T : ? 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 : ( α � α ) � α and Γ = X : α � α , 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 : ( α � α ) � α and Γ = X : α � α , ω = f • X → X • ( f • X ) ( b ) Γ ⊢ f : ( α � α ) � α Γ ⊢ X : α � α Γ ⊢ X : α � α Γ ⊢ f • X : α ( b ) Γ ⊢ f • X : α Γ ⊢ X • ( f • X ) : α ( a ) ⊢ ω ≡ f • X → X • ( f • X ) : α � α ( a ) ( a ) ⊢ f : ( α � α ) � α ⊢ ω : α � α ⊢ ω : α � α ⊢ 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 : ( x 1 : A 1 ) . . . ( x n : A n ) .R is accepted only if R is positive in each A i : 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 ( A n ) ⊑ B if ( B ≡ f ( B n ) ) ∧ ∃ i, A i �⊑ B i P �⊑ A if A ≡ ([ Q ≪ ∆ A 1 ] .A 2 ∧ Q �⊑ A 1 ∨ P �⊑ A 2 ) 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 ( A n ) ⊑ B if ( B ≡ f ( B n ) ) ∧ ∃ i, A i �⊑ B i P �⊑ A if A ≡ ([ Q ≪ ∆ A 1 ] .A 2 ∧ Q �⊑ A 1 ∨ P �⊑ A 2 ) 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 . △ [ m i = ς ( X i ) t i ] i ∈ I ( m i ( X i ) → t i ) i ∈ I = Typed recursion in the rewriting calculus Encodings

Rho and objects • Object = record with an explicit account of self, i . e . △ [ m i = ς ( X i ) t i ] i ∈ I ( m i ( X i ) → t i ) i ∈ I = • Self-application = the application of an object to the object itself, i . e . △ t 1 .t 2 t 1 • t 2 ( t 1 ) = Typed recursion in the rewriting calculus Encodings

Rho and objects • Object = record with an explicit account of self, i . e . △ [ m i = ς ( X i ) t i ] i ∈ I ( m i ( X i ) → t i ) i ∈ I = • Self-application = the application of an object to the object itself, i . e . △ t 1 .t 2 t 1 • t 2 ( t 1 ) = ρ σ △ △ • Ex: t = a ( S ) → b . Then: t.a = t • a ( t ) �→ [ a ( S ) ≪ a ( t )] b �→ b Typed recursion in the rewriting calculus Encodings

Typed objects • An object has type: S : lab → φ ⊢ meth : ( lab → φ ) → lab ( Appl ) ⊢ meth ( S ) : lab ⊢ T meth : φ ( Abst ) ⊢ meth ( S ) → T meth : lab → φ Typed recursion in the rewriting calculus Encodings

Typed objects • An object has type: S : lab → φ ⊢ meth : ( lab → φ ) → lab ( Appl ) ⊢ meth ( S ) : lab ⊢ T meth : φ ( Abst ) ⊢ meth ( S ) → T meth : lab → φ △ • obj.meth = obj • meth ( obj ) can be typed as follows: ⊢ meth : ( lab → φ ) → lab ⊢ obj : lab → φ ⊢ 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: △ X → (( stk → A n • X ; I ) • ( . . . • ( stk → A 2 • X ; I ) • ( A 1 X ))) first ( A 1 , A 2 , . . . , A n ) = first ( A 1 , A 2 , . . . , A n ) • B A j +1 • B if ∀ i ≤ j, A i • B �→ and A j +1 • B � �→ �→ → → δ stk → δ stk ρ σ δ ρ σ ρ σ 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: △ X → (( stk → A n • X ; I ) • ( . . . • ( stk → A 2 • X ; I ) • ( A 1 X ))) first ( A 1 , A 2 , . . . , A n ) = first ( A 1 , A 2 , . . . , A n ) • B A j +1 • B if ∀ i ≤ j, A i • B �→ and A j +1 • B � �→ �→ → → δ stk → δ stk ρ σ δ ρ σ ρ σ 2. The Term Rewrite System R = { t i �→ s i } with signature { a j } is encoded by:   t 1 → S • ( rec • S ) • s 1 , · · · ,   ( rec • S ) → first R �  ,   a 1 • X → S • ( Rec • S ) • ( a 1 • S ( rec • S ) • X ) ,    · · · ,  t 1 → S • ( rec • S ) • s 1 ,  ( Rec • S ) → first · · · ,   I Typed recursion in the rewriting calculus Encodings

Example • Addition over Peano integers: � S → add (0 , y ) → y ; � △ plus = � � S → add ( suc ( x ) , y ) → suc ( S • S ) • add ( x, y ) Typed recursion in the rewriting calculus Encodings

Example • Addition over Peano integers: � S → add (0 , y ) → y ; � △ plus = � � S → add ( suc ( x ) , y ) → suc ( S • S ) • add ( x, y ) ( plus • plus ) • add ( N, M ) �→ �→ ρ [0 ≪ N ] .M ; [0 ≪ N − 1] . ( M +1) · · · [0 ≪ 0] . ( M + N ); [ suc • x ≪ 0] . . . . σ δ �→ stk M + N Typed recursion in the rewriting calculus Encodings

Example • Addition over Peano integers: � S → add (0 , y ) → y ; � △ plus = � � S → add ( suc ( x ) , y ) → suc ( S • S ) • add ( x, y ) ( plus • plus ) • add ( N, M ) �→ �→ ρ [0 ≪ N ] .M ; [0 ≪ N − 1] . ( M +1) · · · [0 ≪ 0] . ( M + N ); [ suc • x ≪ 0] . . . . σ δ �→ stk M + N • Fill in the blanks with your favorite rewrite system... � S → � ; △ func = S → ( S • S ) • Typed recursion in the rewriting calculus Encodings