Elaborating evaluation-order polymorphism
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1 Elaborating evaluation-order polymorphism Joshua Dunfield University of British Columbia ICFP
2 (prologue) ICFP in Canada for the first time since
3 (prologue) ICFP in Canada for the first time since but this is the only paper from Canada that got in 2
4 Evaluation order Begin at the beginning. the Algol-60 committee meets in Paris 3
5 Evaluation order Begin at the beginning. the Algol-60 committee meets in Paris invents call-by-name; decides to support it and call-by-value 3
6 Evaluation order Begin at the beginning. the Algol-60 committee meets in Paris invents call-by-name; decides to support it and call-by-value everyone goes home 3
7 Evaluation order Begin at the beginning. the Algol-60 committee meets in Paris invents call-by-name; decides to support it and call-by-value everyone goes home Peter Naur makes cbn the default (one of a few matters of detail ) 3
8 Evaluation order Naur had absorbed the Holy Ghost after the Paris meeting... there was nothing one could do... it was to be swallowed for the sake of loyalty. (F.L. Bauer) 4
9 And ever since, there has been moping Standard ML Haskell 5
10 Actually, I ll take your front yard too Cbn languages borrowing cbv Haskell strictness annotations Cbv languages borrowing cbn SML/NJ lazy keyword Scala => 6
11 WARNING: fake social science Programmers choose a language based, in part, on which evaluation order they usually need 7
12 WARNING: fake social science Programmers choose a language based, in part, on which evaluation order they usually need and then tend to solve problems suited to that language s evaluation order 7
13 WARNING: fake social science Programmers choose a language based, in part, on which evaluation order they usually need and then tend to solve problems suited to that language s evaluation order confirming their Objective Opinion about which evaluation order is better. 7
14 WARNING: fake social science Programmers choose a language based, in part, on which evaluation order they usually need and then tend to solve problems suited to that language s evaluation order confirming their Objective Opinion about which evaluation order is better. Is the question which evaluation order? that different from... 7
15 WARNING: fake social science Programmers choose a language based, in part, on which evaluation order they usually need and then tend to solve problems suited to that language s evaluation order confirming their Objective Opinion about which evaluation order is better. Is the question which evaluation order? that different from... Should a language have floating-point numbers... or integers?!? 7
16 A different question Can we design languages that are unbiased about evaluation order? 8
17 A different question Can we design languages that are unbiased about evaluation order? Types keep out bad programs 8
18 A different question Can we design languages that are unbiased about evaluation order? Types keep out bad programs Types are also about more programs! 8
19 A different question Can we design languages that are unbiased about evaluation order? Types keep out bad programs Types are also about more programs! Duplicate the types: two forms:, V N two forms: V, N evaluation-order polymorphism Da. τ: write once, instantiate a = V and/or a = N 8
20 Road map Impartial type system V V + V µ V N N + N µ N D e τ Source language (e) encode types Economical type system + µ V N D e S erase types er(e) : S Target language (M) Cbv type system + µ U (thunk) elaborate M : A D: for all over evaluation order 9
21 Road map Impartial type system V V + V µ V N N + N µ N D e τ Source language (e) encode types Economical type system + µ V N D e S erase types er(e) : S Target language (M) Cbv type system + µ U (thunk) elaborate M : A D: for all over evaluation order V S and N S: suspension type 9
22 Road map Impartial type system V V + V µ V N N + N µ N D e τ Source language (e) encode types Economical type system + µ V N D e S erase types er(e) : S Target language (M) Cbv type system + µ U (thunk) elaborate M : A D: for all over evaluation order V S and N S: suspension type U A: thunk (unit A) 9
23 Source expression syntax Program variables x Source expressions e ::= () x λx. e e e 2 Λα. e e[τ] (e:τ) (e 1, e 2 ) proj k e inj k e case(e, x 1.e 1, x 2.e 2 ) Expression forms work for both V and N 10
24 Source expression syntax Program variables x Source expressions e ::= () x λx. e e e 2 Λα. e e[τ] (e:τ) (e 1, e 2 ) proj k e inj k e case(e, x 1.e 1, x 2.e 2 ) Expression forms work for both V and N Expression forms work for both V and N 10
25 Source expression syntax Program variables x Source expressions e ::= () x λx. e e e 2 Λα. e e[τ] (e:τ) (e 1, e 2 ) proj k e inj k e case(e, x 1.e 1, x 2.e 2 ) Expression forms work for both V and N Expression forms work for both V and N Expression forms work for both + V and + N 10
26 Impartial types Evaluation order vars. a Evaluation orders ɛ ::= V N a Type variables α Source types τ ::= 1 α α. τ Da. τ τ ɛ 1 τ 2 τ 1 ɛ τ 2 τ 1 + ɛ τ 2 µ ɛ α. τ 11
27 Impartial types: lists+streams Traditional encoding of datatypes into +, µ: datatype List α = Nil Cons of (α List α) 12
28 Impartial types: lists+streams Traditional encoding of datatypes into +, µ: datatype List α = Nil of 1 Cons of (α List α) 12
29 Impartial types: lists+streams Traditional encoding of datatypes into +, µ: datatype List α = Nil of 1 Cons of (α List α) datatype List α = Nil of 1 + Cons of (α List α) 12
30 Impartial types: lists+streams Traditional encoding of datatypes into +, µ: datatype List α = Nil of 1 Cons of (α List α) datatype List α = 1 + (α List α) 12
31 Impartial types: lists+streams Traditional encoding of datatypes into +, µ: datatype List α = Nil of 1 Cons of (α List α) datatype List α = 1 + (α List α) µβ. 1 + (α β) 12
32 Impartial types: lists+streams Traditional encoding of datatypes into +, µ: datatype List α = Nil of 1 Cons of (α List α) datatype List α = 1 + (α List α) µβ. 1 + (α β) Generalize over evaluation orders: type List a α = µ a β. ( 1 + a (α a β) ) 12
33 Impartial types: lists+streams type List a α = µ a β. ( 1 + a (α a β) ) List V α = µ V β. ( 1 + V (α V β) ) list of α List N α = µ N β. ( 1 + N (α N β) ) (terminable) stream of α 13
34 Impartial types: lists+streams type List a α = µ a β. ( 1 + a (α a β) ) List V α = µ V β. ( 1 + V (α V β) ) list of α List N α = µ N β. ( 1 + N (α N β) ) (terminable) stream of α Write map once, for both lists and streams: map : Da. α. (α V β) V (List a α) V (List a β) = Λα. fix map. λf. λxs. case(xs, x 1.inj 1 (), x 2.inj 2 (proj 1 x 2 ), (proj 2 x 2 ))) 13
35 Impartial typing Type safety holds for the simply-typed λ-calculus, regardless of evaluation order: Γ, x : A M : B Γ λx.m : A ɛ B -INTRO Γ M : A ɛ B Γ MN : B Γ N : A -ELIM 14
36 Impartial typing Type safety holds for the simply-typed λ-calculus, regardless of evaluation order: Γ, x : A M : B Γ λx.m : A ɛ B -INTRO Γ M : A ɛ B Γ MN : B Γ N : A -ELIM 14
37 Impartial typing Type safety holds for the simply-typed λ-calculus, regardless of evaluation order: Γ, x : A M : B Γ λx.m : A ɛ B -INTRO Γ M : A ɛ B Γ MN : B Γ N : A -ELIM Evaluation order ɛ is implicit and depends on typing: typing rules can t be syntax-directed. How can we avoid choosing ɛ at random? 14
38 Impartial typing Type safety holds for the simply-typed λ-calculus, regardless of evaluation order: Γ, x : A M : B Γ λx.m : A ɛ B -INTRO Γ M : A ɛ B Γ MN : B Γ N : A -ELIM Evaluation order ɛ is implicit and depends on typing: typing rules can t be syntax-directed. How can we avoid choosing ɛ at random? Bidirectional typing 14
39 Bidirectional typing: why? Some past answers: to handle features beyond Damas Milner (Pierce & Turner 00; Dunfield & Pfenning 04; Dunfield & Krishnaswami 13;... ) for better (earlier) type error messages Here: to make typing more predictable. 15
40 Bidirectional typing in one slide Organize the flow of information from type annotations: Given γ, e, and a known type τ, check e: γ I e τ Given γ and e, synthesize a type for e: γ I e τ The type τ in the checking judgment e τ is a goal. 16
41 Impartial bidirectional typing Frank Pfenning s recipe: intro rules check, elim rules synthesize. γ, (x τ 1 ) I e τ 2 γ I (λx. e) τ 1 ɛ τ 2 I Intro γ I e 1 τ 1 ɛ τ 2 γ I e 2 τ 1 γ I (e e 2 ) τ 2 I Elim I Intro: The type τ 1 ɛ τ 2 must flow from an annotation. I Elim: The type τ ɛ 1 τ 2 must flow from an annotation, perhaps via γ. 17
42 Impartial bidirectional typing Frank Pfenning s recipe: intro rules check, elim rules synthesize. γ, (x τ 1 ) I e τ 2 γ I (λx. e) τ 1 ɛ τ 2 I Intro γ I e 1 τ 1 ɛ τ 2 γ I e 2 τ 1 γ I (e e 2 ) τ 2 I Elim I Intro: The type τ 1 ɛ τ 2 must flow from an annotation. I Elim: The type τ ɛ 1 τ 2 must flow from an annotation, perhaps via γ. 17
43 Impartial bidirectional typing Intro rules check, elim rules synthesize. Handle D like : γ, a evalorder I e τ γ I e Da. τ IDIntro γ I e Da. τ γ ɛ evalorder γ I e [ɛ/a]τ IDElim 18
44 Impartial bidirectional typing Intro rules check, elim rules synthesize. Handle D like : γ, a evalorder I e τ γ I e Da. τ IDIntro γ I e Da. τ γ ɛ evalorder γ I e [ɛ/a]τ IDElim Back to guessing? ɛ in IDElim Can force an eval. order by annotating e In practice, probably want the ability to set a default (lexically-scoped?) for IDElim. 18
45 Road map Impartial type system V V + V µ V N N + N µ N D e τ Source language (e) encode types Economical type system + µ V N D e S erase types er(e) : S Target language (M) Cbv type system + µ U (thunk) elaborate M : A D: for all over evaluation order V S and N S: suspension type U A: thunk (unit A) 19
46 Road map Impartial type system V V + V µ V N N + N µ N D e τ Source language (e) encode types Economical type system + µ V N D e S erase types er(e) : S Target language (M) Cbv type system + µ U (thunk) elaborate M : A D: for all over evaluation order V S and N S: suspension type U A: thunk (unit A) 19
47 Road map Impartial type system V V + V µ V N N + N µ N D e τ Source language (e) encode types Economical type system + µ V N D e S erase types er(e) : S Target language (M) Cbv type system + µ U (thunk) elaborate M : A D: for all over evaluation order V S and N S: suspension type U A: thunk (unit A) 19
48 Economical types = by-value + suspensions Factor out the differences between V, N connectives: Economical types S ::= 1 α α. S Da. S ɛ S S 1 S 2 S 1 S 2 S 1 + S 2 µα. S N S N suspend S 1 S V S V suspend S S 20
49 Economical types = by-value + suspensions Factor out the differences between V, N connectives: Economical types S ::= 1 α α. S Da. S ɛ S S 1 S 2 S 1 S 2 S 1 + S 2 µα. S N S N suspend S 1 S V S V suspend S S Encode impartial types as economical types: τ 1 1 = 1 ɛ τ 2 = ( ɛ τ 1 ) τ 2 τ 1 + ɛ τ 2 = ɛ ( τ 1 + τ 2 ) τ 1 ɛ τ 2 = ( ɛ τ 1 ) ( ɛ τ 2 ) Da. τ = Da. τ µ ɛ α. τ = µα. ɛ τ α. τ = α. τ α = α 20
50 Economical types: lists+streams type List a α = µβ. a ( 1 + (α β) ) List V α = µβ. V ( 1 + (α β) ) list of α List N α = µβ. N ( 1 + (α β) ) (terminable) stream of α Write map once, for both lists and streams: map : Da. α. (α β) (List a α) (List a β) = Λα. fix map. λf. λxs. case(xs, x 1.inj 1 (), x 2.inj 2 (proj 1 x 2 ), (proj 2 x 2 ))) (body of map unchanged from the impartial system) 21
51 Road map Impartial type system V V + V µ V N N + N µ N D e τ Source language (e) encode types Economical type system + µ V N D e S erase types er(e) : S Target language (M) Cbv type system + µ U (thunk) elaborate M : A D: for all over evaluation order V S and N S: suspension type U A: thunk (unit A) 22
52 Road map Impartial type system V V + V µ V N N + N µ N D e τ Source language (e) encode types Economical type system + µ V N D e S erase types er(e) : S Target language (M) Cbv type system + µ U (thunk) elaborate M : A D: for all over evaluation order V S and N S: suspension type U A: thunk (unit A) 22
53 Road map Impartial type system V V + V µ V N N + N µ N D e τ Source language (e) encode types Economical type system + µ V N D e S erase types er(e) : S Target language (M) Cbv type system + µ U (thunk) elaborate M : A D: for all over evaluation order V S and N S: suspension type U A: thunk (unit A) 22
54 Elaboration For the metatheory, elaboration not bidirectional. For every economical typing with e S or e S we can erase types from e and elaborate to M er(e) : S M 23
55 Elaboration: rules for Γ e : S M Γ e : V S M Γ e : N S thunk M elab Intro Γ e : V S M Γ e : S M elab Elim V Γ e : N S M Γ e : S force M elab Elim N inj 1 () : N (1 + ) thunk (inj 1 ()) Given y : N (1 + ), y : (1 + ) force y 24
56 Elaboration: rules for Γ e : S M Γ e : V S M Γ e : N S thunk M elab Intro Γ e : V S M Γ e : S M elab Elim V Γ e : N S M Γ e : S force M elab Elim N inj 1 () : N (1 + ) thunk (inj 1 ()) Given y : N (1 + ), y : (1 + ) force y 24
57 Elaboration: rules for D Γ e : [V/a]S M 1 Γ e : [N/a]S M 2 Γ e : (Da. S) (M 1, M 2 ) elabdintro Γ e : (Da. S) M Γ e : [V/a]S proj 1 M Γ e : [N/a]S proj 2 M elabdelim map : Da. (map V, map N ) Given xs : List N β, xs (proj 2 map) f xs 25
58 Elaboration: rules for D Γ e : [V/a]S M 1 Γ e : [N/a]S M 2 Γ e : (Da. S) (M 1, M 2 ) elabdintro Γ e : (Da. S) M Γ e : [V/a]S proj 1 M Γ e : [N/a]S proj 2 M elabdelim map : Da. (map V, map N ) Given xs : List N β, xs (proj 2 map) f xs 25
59 Elaboration: rules for D Γ e : [V/a]S M 1 Γ e : [N/a]S M 2 Γ e : (Da. S) (M 1, M 2 ) elabdintro Γ e : (Da. S) M Γ e : [V/a]S proj 1 M Γ e : [N/a]S proj 2 M elabdelim map : Da. (map V, map N ) Given xs : List N β, xs (proj 2 map) f xs Only instantiating Da with N will succeed. 25
60 Metatheory Impartial type system V V + V µ V N N + N µ N D e τ Source language (e) encode types Economical type system + µ V N D e S erase types er(e) : S Target language (M) Cbv type system + µ U (thunk) elaborate M : A 26
61 Metatheory: consistency Impartial type system V V + V µ V N N + N µ N D e τ Source language (e) encode types Economical type system + µ V N D e S erase types er(e) : S Target language (M) Cbv type system + µ U (thunk) elaborate standard cbv evaluation ( 0 steps) M : A If er(e) : S M and M W W : A 27
62 Metatheory: consistency Impartial type system V V + V µ V N N + N µ N D e τ Source language (e) encode types Economical type system + µ V N D e S cbv + cbn evaluation ( 0 steps) erase types er(e) : S e : S Target language (M) Cbv type system + µ U (thunk) elaborate standard cbv evaluation ( 0 steps) elaborate M : A W : A If er(e) : S M and M W then er(e) e and e : S W. 27
63 Metatheory: N-freeness Theorem. If an impartial typing judgment with τ is N-free, then the corresponding economical judgment with τ is N-free. Theorem. If an economical typing judgment with e is N-free then er(e) elaborates to M, and M is N-free. Theorem (Multi-step consistency).... Moreover, if M is N-free then we can derive e e without by-name reductions. A judgment with D is not N-free (subformula property). 28
64 History Church: normal-order (leftmost-outermost) reduction Bernays (1936), reviewing Church & Rosser (1936): require arguments in normal form (not head-normal form) Naur et al. (1960): Algol-60 with cbn/cbv on per-argument basis (resentful committee members: HOPL) Wadsworth (1971), Henderson & Morris (1976), Friedman & Wise (1976): lazy evaluation 29
65 Immediate ancestors Chen et al., ICFP 2011: elaboration from intersection types (we just didn t call them that), for incremental computation ICFP 2012: elaboration from intersection types; like D, but more general and less predictable: here we only intersect V- and N-instantiations 30
66 Immediate ancestors Chen et al., ICFP 2011: elaboration from intersection types (we just didn t call them that), for incremental computation ICFP 2012: elaboration from intersection types; like D, but more general and less predictable: here we only intersect V- and N-instantiations 30
67 What s next? an implementation? let? lexically-scoped default ɛ? call-by-need Could these techniques resolve other fundamental differences between PLs? 31
68 What s next? an implementation? let? lexically-scoped default ɛ? call-by-need Could these techniques resolve other fundamental differences between PLs? One big happy family (united against the OOP enemy?) 31
69 Conclusion: Evaluation-order peace may be on its way! Paper and proofs: arxiv.org/abs/ Special thanks to: the cats of Metro Vancouver and Cambridge, England Shameless advertisement: I m looking for a research/teaching job in Canada. 32
70 Conclusion: Evaluation-order peace may be on its way! Paper and proofs: arxiv.org/abs/ Special thanks to: the cats of Metro Vancouver and Cambridge, England Shameless advertisement: I m looking for a research/teaching job in Canada. 32
71 Conclusion: Evaluation-order peace may be on its way! Paper and proofs: arxiv.org/abs/ Special thanks to: the cats of Metro Vancouver and Cambridge, England Shameless advertisement: I m looking for a research/teaching job in Canada. 32
72 33
73 Target for elaboration Standard cbv, plus thunks (type U B) G T M : B G T thunk M : U B TUIntro G T M 1 : U B G T force M 1 : B TUElim thunk type U B could be 1 B distinguished only for the metatheory 34
74 Source operational semantics Define by-value and by-name reductions: (λx. e 1 v 2 RV [v 2 /x]e 1 (λx. e 1 e 2 RN [e 2 /x]e 1 Allow reductions in the appropriate eval. contexts: C V [e] RV C V [e ] C N [e] RN C N [e ] Example: (λx. [ ] is a C V but not a C N 35
75 Source operational semantics The relation is too big, because it s not connected to typing. (λx. [ ] is a C V but not a C N In fact, is much too big in the paper, which defines C N incorrectly. But the metatheory doesn t care. Guess I m even more cbv-biased than I thought! 36
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