Blame and coercion: together again for the first time (PLDI 2015) - PowerPoint PPT Presentation

Blame and coercion: together again for the first time (PLDI 2015) Jeremy Siek (Indiana) Peter Thiemann (Freiburg) Philip Wadler (Edinburgh) WG 2.8, Keffalonia, 25–29 May 2015

Part I Conclusion

Three calcluli • λ B Blame calculus Findler and Felleisen (2002) Wadler and Findler (2009) • λ C Coercion calculus Henglein (1994) • λ S Space-efficient coercion calculus Hermann, Tomb, Flanagan (2007) Siek and Wadler (2010) Garcia (2013)

Full abstraction Strong correctness property: Full abstraction ctx ctx if and only if | M | BC = C | N | BC • M = B N ctx ctx if and only if | M | CS = T | N | CS • M = C N Equivalences in λ B and λ C easily proved in λ S Key lemma Fundamental property of casts < : + < : < : − < : n Four subtyping relations: Translation between λ B and λ C explains < : + and < : −

Part II The Blame Calculus ( λ B )

Types and ground types Base types ι ι | A → B | ⋆ ::= Types A, B, C ι | ⋆ → ⋆ G, H ::= Ground types ι + ( ⋆ → ⋆ ) = ⋆

Compatibility Γ ⊢ M : A A ∼ B p Γ ⊢ ( M : A = ⇒ B ) : B A ′ ∼ A B ∼ B ′ A → B ∼ A ′ → B ′ ι ∼ ι A ∼ ⋆ ⋆ ∼ A Lemma 1. If A � = ⋆ then there is a unique G such that A ∼ G . Lemma 2. ∼ is reflexive and symmetric but not transitive.

Reductions p E [ V : ι ⇒ ι ] − → E [ V ] = ⇒ A ′ → B ′ ) W ] − p E [( V : A → B → = p p E [( V ( W : A ′ ⇒ A )) : B ⇒ B ′ ] = = p E [ V : ⋆ = ⇒ ⋆ ] − → E [ V ] p p p E [ V : A ⇒ ⋆ ] − → E [ V : A ⇒ G ⇒ ⋆ ] = = = if A � = ⋆, A � = G, A ∼ G p p p E [ V : ⋆ ⇒ A ] − → E [ V : ⋆ ⇒ G ⇒ A ] = = = if A � = ⋆, A � = G, A ∼ G p q E [ V : G ⇒ ⋆ ⇒ G ] − → E [ V ] = = p q E [ V : G ⇒ ⋆ ⇒ H ] − → blame q if G � = H = =

Part III The Coercion Calculus ( λ C )

Coercions and typing ⇒ A id A : A = ? p G : ⋆ = G ! : G = ⇒ ⋆ ⇒ G c : A ′ = ⇒ A d : B = ⇒ B ′ ⇒ A ′ → B ′ c → d : A → B = ⇒ B ⇒ C c : A = d : B = c ; d : A = ⇒ C A � = ⋆ A ∼ G G � = H ⊥ GpH : A = ⇒ B

Failure A � = ⋆ A ∼ G G � = H ⊥ GpH : A = ⇒ B ⊥ GpH : A = ⇒ B corresponds to p • • • M : A = ⇒ G = ⇒ ⋆ = ⇒ H = ⇒ B Lemma 3. (Failure) If A � = ⋆ and A ∼ G and G � = H then p 1 p 2 p 3 p 4 M : A = ⇒ G = ⇒ ⋆ = ⇒ H = ⇒ B − → blame p 3 .

Reductions E [ V � id A � ] − → E [ V ] E [( V � c → d � ) W ] − → E [( V ( W � c � )) � d � ] E [ V � G ! �� ? p G � ] − → E [ V ] E [ V � G ! �� ? p H � ] − → blame p if G � = H E [ V � c ; d � ] − → E [ V � c �� d � ] E [ V � ⊥ GpH � ] − → blame p

Part IV Space-efficient Blame Calculus ( λ S )

Coercions in normal form id ⋆ | ? p G ; i | i ::= Space-efficient coercions s, t g ; G ! | g | ⊥ GpH i ::= Intermediate coercions id ι | s → t g, h ::= Ground coercions Lemma 4. • If i : A = ⇒ B then A � = ⋆ . • If g : A = ⇒ B then A � = ⋆ and B � = ⋆ , and there is a unique G such that A ∼ G and B ∼ G .

Space-efficient composition id ι � id ι = id ι ( s → t ) � ( s ′ → t ′ ) = ( s ′ � s ) → ( t � t ′ ) id ⋆ � t = t ( g ; G ! ) � id ⋆ = g ; G ! ( ? p G ; i ) � t = ? p G ; ( i � t ) g � ( h ; H ! ) = ( g � h ) ; H ! ( g ; G ! ) � ( ? p G ; i ) = g � i ( g ; G ! ) � ( ? p H ; i ) = ⊥ GpH if G � = H ⊥ GpH � s = ⊥ GpH g � ⊥ GpH = ⊥ GpH

Reductions F [ U � id ι � ] − → F [ U ] E [( U � s → t � ) W ] − → E [( U ( W � s � )) � t � ] F [ U � id ⋆ � ] − → F [ U ] F [ M � s �� t � ] − → F [ M � s � t � ] F [ U � ⊥ GpH � ] − → blame p

Compare: Herman, Tomb, and Flanagan (2007) F [ M � c �� d � ] − → F [ M � c ; d � ] ( c ; d ) ; e = c ; ( d ; e ) id A ; c = c c ; id A = c ( c → d ) ; ( c ′ → d ′ ) = ( c ′ ; c ) → ( d ; d ′ ) G ! ; ? G = id G G ! ; ? H = ⊥ if G � = H ⊥ ; c = ⊥ c ; ⊥ = ⊥

Compare: Siek and Wadler (2010) ι l � ι m = ι l ( P → l Q ) � ( P ′ → m Q ′ ) = ( P ′ � P ) → l ( Q � Q ′ ) ⋆ � P = P P � ⋆ = P P G m � Q H p = ⊥ pG m if G � = H ⊥ pG m � Q = ⊥ pG m P G l � ⊥ pG m = ⊥ pG l P G l � ⊥ pH q = ⊥ qG l if G � = H

Compare: Siek and Wadler (2010) P G l means P ∼ G and the top-level blame label in P is l . If there is no top-level blame label in P , then l is ǫ . ι ǫ corresponds to id ι ι p corresponds to ? p ι ; id ι P → ǫ Q corresponds to P → Q P → p Q corresponds to ? p ( ⋆ → ⋆ ) ; ( P → Q ) ⋆ corresponds to id ⋆ ⊥ pG ǫ corresponds to ⊥ GpH ⊥ pG q corresponds to ? q G ; ⊥ GpH

Compare: Garcia (2013) N [ [ id ⋆ ] ] = id ⋆ N [ [ id ι ] ] = id ι [ ⊥ pG ] ] = ⊥ p N [ [ ⊥ pGq ] ] = ? q G ; ⊥ p N [ N [ [ G ! ] ] = G ! [ G ? p ] ] = ? p G N [ [ G ? p ! ] ] = ? p G ; G ! N [ N [ c 1 → ¨ ] = N [ ] → N [ [¨ c 2 ] [¨ c 1 ] [¨ c 2 ] ] N [ c 1 → ! ¨ ] = ( N [ ] → N [ ]) ; ( ⋆ → ⋆ ) ! [¨ c 2 ] [¨ c 1 ] [¨ c 2 ] c 1 ? p → ¨ ] = ? p ( ⋆ → ⋆ ) ; ( N [ N [ ] → N [ [¨ c 2 ] [¨ c 1 ] [¨ c 2 ] ]) c 1 ? p → ! ¨ ] = ? p ( ⋆ → ⋆ ) ; ( N [ N [ ] → N [ ]) ; ( ⋆ → ⋆ ) ! [¨ c 2 ] [¨ c 1 ] [¨ c 2 ]

Part V Full abstraction

Contextual equivalence Definition 5 (Contextual equivalence) . Two terms are contextually ctx = B N , if for any context C , either equivalent, written M 1. both converge to a value, C [ M ] − → ∗ B V and C [ N ] − → ∗ B W , for some values V and W . 2. both allocate blame to the same label, C [ M ] − → ∗ B blame p and C [ N ] − → ∗ B blame p , for some label p , or 3. both diverge, C [ M ] ↑ B and C [ N ] ↑ B . The same definition applies, mutatis mutandis, for λ C and λ S .

Full abstraction The best previous result (Siek and Wadler (2010)): Theorem 6 (Contextual equivalence without the context) . | M | BT ↑ T • M ↑ B if and only if Our result: Theorem 7 (Full abstraction) . ctx ctx | M | BC = C | N | BC • M = B N if and only if ctx ctx | M | CS = T | N | CS • M = C N if and only if

A key lemma Lemma 8 (Equivalences) . The following hold in λ C . ctx 1. M � id � = C M ctx 2. M � c ; d � = C M � c �� d � ctx ctx 3. M � c ; id � = C M � c � = C M � id ; c � 4. M � ( c → d ) ; ( c ′ → d ′ ) � = C M � ( c ′ ; c ) → ( d ; d ′ ) � ctx ctx 5. M � c → d � = C M � ( c → id ) ; ( id → d ) � ctx 6. M � c → d � = C M � ( id → c ) ; ( d → id ) � Proof. Trivial to prove using full abstraction from λ C to λ S . [Tricky to prove otherwise; probably requires a custom bisimulation.]

Fundamental property of casts Lemma 9. If A & B < : n C then ⇒ B | BS = | A p p p ⇒ C | BS � | C ⇒ B | BS | A = = = Proof. Easy induction on A , B , and C . Corollary 10 (Fundamental Property of Casts) . Let M be a term of λ B . If A & B < : n C then p p p ctx ⇒ B ⇒ C ⇒ B M : A = = B M : A = = Proof. Immediate from Lemma 4 and full abstraction for λ C and λ S . [Required a custom bisimulation and six lemmas in Siek and Wadler (2010)!]

Part VI Conclusion

Three calcluli • λ B Blame calculus Findler and Felleisen (2002) Wadler and Findler (2009) • λ C Coercion calculus Henglein (1994) • λ S Space-efficient coercion calculus Hermann, Tomb, Flanagan (2007) Siek and Wadler (2010) Garcia (2013)

Full abstraction Strong correctness property: Full abstraction ctx ctx if and only if | M | BC = C | N | BC • M = B N ctx ctx if and only if | M | CS = T | N | CS • M = C N Equivalences in λ B and λ C easily proved in λ S Key lemma Fundamental property of casts < : + Four subtyping relations ( < : < : − < : n ) Translation between λ B and λ C explains < : + and < : −