Type Soundness for Path Polymorphism

Transcription

1 Type Soundness for Path Polymorphism Andrés Ezequiel Viso 1,2 joint work with Eduardo Bonelli 1,3 and Mauricio Ayala-Rincón 4 1 CONICET, Argentina 2 Departamento de Computación, FCEyN, UBA, Argentina 3 Departamento de Ciencia y Tecnología, UNQ, Argentina 4 Departamentos de Matemática e Ciência da Computação, UnB, Brasil Logic and Foundations of Programming Languages Day UBA, Buenos Aires, Argentina May 21, 2018 Andrés E. Viso Type Soundness for Path Polymorphism 1 / 17

2 Index 1 Motivations PPC Typing Error 2 Type System Compatibility Safety 3 Conclusions 4 Future Work Andrés E. Viso Type Soundness for Path Polymorphism 2 / 17

3 Pure Pattern Calculus Every data structure is either an atom or a compound built by application [Jay09] Devise a type system for PPC [JK09] Pattern Matching Example (Pattern Matching) map {map} f {f } ( nil {} nil cons (vl x) xs {x,xs} cons (vl (f x)) (map f xs) ) Andrés E. Viso Type Soundness for Path Polymorphism 3 / 17

4 Pure Pattern Calculus Every data structure is either an atom or a compound built by application [Jay09] Devise a type system for PPC [JK09] Pattern Matching Path Polymorphism Example (Generic update query) upd {upd} f {f } ( vl y {y} vl (f y) x y {x,y} (upd f x) (upd f y) w {w} w ) Andrés E. Viso Type Soundness for Path Polymorphism 3 / 17

5 Pure Pattern Calculus Every data structure is either an atom or a compound built by application [Jay09] Devise a type system for PPC [JK09] Pattern Matching Path Polymorphism Pattern Polymorphism Example (Generic update query) upd {upd} z {z} f {f } ( z y {y} z (f y) x y {x,y} (upd z f x) (upd z f y) w {w} w ) Andrés E. Viso Type Soundness for Path Polymorphism 3 / 17

6 Pure Pattern Calculus Every data structure is either an atom or a compound built by application [Jay09] Devise a type system for PPC [JK09] Pattern Matching Path Polymorphism Pattern Polymorphism Matching failure is handled explicitly Example (Generic update query) upd {upd} z {z} f {f } ( z y {y} z (f y) x y {x,y} (upd z f x) (upd z f y) w {w} w ) Andrés E. Viso Type Soundness for Path Polymorphism 3 / 17

7 Calculus of Applicative Patterns We focus on a static restriction of PPC we call CAP Static Patterns Built-in case constructor (handles matching failure) Path Polymorphism Example (Generic update query) upd {upd} f {f } ( vl y {y} vl (f y) x y {x,y} (upd f x) (upd f y) w {w} w ) Evaluation from left to right {u/p i } = fail for all i < j {u/p j } = σ j j 1..n (β) (p i θi s i ) i 1..n u σ j s j Andrés E. Viso Type Soundness for Path Polymorphism 4 / 17

8 Typing Error Patterns are partial descriptions of data structures [Jay09] Matching failure Typing error (nil {} true) (cons x nil) Not all applications (p θ s) u make sense p u c d c d t c q θ t p u p 1 p 2 d p 1 p 2 d t p 1 p 2 q θ t Andrés E. Viso Type Soundness for Path Polymorphism 5 / 17

9 Type System The type of the compound does not determine the type of the components [Jay09] Example t = c x {x:bool} if x then 1 else 0 c y {y:nat} y + 1 Singleton types (c) Type application (@) [Pet11] Union types ( ) Recursive types (µ) Bool true false Nat µx.zero X t : Bool Nat) Nat Andrés E. Viso Type Soundness for Path Polymorphism 6 / 17

10 Compatibility We take advantage of the evaluation order... Given (p θ s q θ t) If p q 1 then q θ t is a dead branch 1 p q iff σ.σp = q Andrés E. Viso Type Soundness for Path Polymorphism 7 / 17

11 Compatibility We take advantage of the evaluation order... Given (p θ s q θ t) If p q 1 then q θ t is a dead branch If not p π q π a. y restriction required b. c d no overlapping (q p) c. q 1 q 2 no overlapping d. y restriction required p 1 p e. 2 d no overlapping 1 p q iff σ.σp = q Andrés E. Viso Type Soundness for Path Polymorphism 7 / 17

12 Compatibility We take advantage of the evaluation order... Given (p θ s q θ t) If p q 1 then q θ t is a dead branch If not p π q π a. y restriction required b. c d no overlapping (q p) c. q 1 q 2 no overlapping d. y restriction required p 1 p e. 2 d no overlapping...and state this in the compatibility predicate, ovl(p : A, q : B). 1 p q iff σ.σp = q Andrés E. Viso Type Soundness for Path Polymorphism 7 / 17

13 Compatibility Example (Case a.) f {f :Bool A} ( vl z {z:bool} vl (f z) x y {x:vl,y:nat} x y ) p π q π a. y restriction required b. c d no overlapping (q p) c. q 1 q 2 no overlapping d. y restriction required p 1 p e. 2 d no overlapping Andrés E. Viso Type Soundness for Path Polymorphism 8 / 17

14 Compatibility Example (Case d.) f {f :vl Bool A} g {g:vl@nat A} ( x y {x:vl,y:bool} f x y z {z:vl@nat} g z ) p π q π a. y restriction required b. c d no overlapping (q p) c. q 1 q 2 no overlapping d. y restriction required p 1 p e. 2 d no overlapping Andrés E. Viso Type Soundness for Path Polymorphism 8 / 17

15 Compatibility Definition (Compatibility) p : A and q : B are compatible iff ovl(p : A, q : B) = B µ A Typing scheme (θ i p p i : A i ) i 1..n [p i : A i ] i 1..n compatible (Γ, θ i s i : B) i 1..n Γ (p i θi s i ) i 1..n : i 1..n A i B Andrés E. Viso Type Soundness for Path Polymorphism 9 / 17

16 Compatibility Lemma (Compatibility Lemma) Suppose Γ u : B, θ p p : A, θ p q : B and {u/p } is successful. Then, ovl(p : A, q : B) holds. Given a well-typed application (p θ s q θ t) u the lemma states that a branch should be more general than those to its left, or a dead branch otherwise (B µ A by compatibility). Andrés E. Viso Type Soundness for Path Polymorphism 10 / 17

17 Safety Reduction preserves typing Proposition (Subject Reduction) If Γ s : A and s s, then Γ s : A. And it does not get stuck Proposition (Progress) If s : A and s is not a value, then s s.t. s s. Andrés E. Viso Type Soundness for Path Polymorphism 11 / 17

18 Conclusions We proposed a type system for a calculus that supports path polymorphism The two fundamental properties were proved to hold: subject reduction and progress We developed a notion of pattern compatibility that turns out to be crucial for the achieved results Invertibility of subtyping of recursive types in the presence of associative, commutative and idempotent unions was addressed too Andrés E. Viso Type Soundness for Path Polymorphism 12 / 17

19 Other work Type-checking 2 Syntax-directed formalization of the type system Implementation of type-checking algorithms Disambiguation of typing rules Equivalence and subtype checking Compatibility check Efficiency of typing Checking equivalence and subtyping modulo ACI operators [JPZ02][DPR05] 2 Juan Edi s MSc Thesis at UBA Andrés E. Viso Type Soundness for Path Polymorphism 13 / 17

20 Work in progress Strong Normalization Recursive types are not SN Y C : (A A) A All known restrictions work with weak equivalence Our invertibility result holds for strong equivalence SN with strong equivalence has been an open problem for over 15 years [Cop98] Options Analyze SN with strong equivalence for recursive types in a simplified framework Revisit our results considering weak type equivalence Andrés E. Viso Type Soundness for Path Polymorphism 14 / 17

21 Future Work Polymorphic type extensions Intersection types Parametric polymorphism à la System F <: Curry Howard correspondence for the proposed system Comparison with Dagand s work in Ornaments Comparison with Datatype-Generic Programming based on Category Theory Extend this analysis to the dynamic patterns case Andrés E. Viso Type Soundness for Path Polymorphism 15 / 17

22 References I Mario Coppo. Recursive types: the syntactic and semantic approaches. In Theories of Types and Proofs, pages Kyoto University, Roberto Di Cosmo, François Pottier, and Didier Rémy. Subtyping recursive types modulo associative commutative products. In Seventh International Conference on Typed Lambda Calculi and Applications (TLCA 05), Nara, Japan, April C. Barry Jay. Pattern Calculus: Computing with Functions and Structures. Springer, C. Barry Jay and Delia Kesner. First-class patterns. J. Funct. Program., 19(2): , Andrés E. Viso Type Soundness for Path Polymorphism 16 / 17

23 References II Somesh Jha, Jens Palsberg, and Tian Zhao. Efficient type matching. In Mogens Nielsen and Uffe Engberg, editors, Foundations of Software Science and Computation Structures, 5th International Conference, FOSSACS 2002 (ETAPS 2002) Grenoble, France, April 8-12, 2002, Proceedings, volume 2303 of LNCS, pages Springer, Barbara Petit. Semantics of typed lambda-calculus with constructors. Logical Methods in Computer Science, 7(1), Andrés E. Viso Type Soundness for Path Polymorphism 17 / 17

24 CAP Patterns, terms, data structures and matchable forms p ::= x (matchable) c (constant) p p (compound) d ::= c (constant) d t (compound) t ::= x (variable) c (constant) t t (application) p θ t... p θ t (abstraction) m ::= d (data struct) p θ t... p θ t (abstraction) Andrés E. Viso Type Soundness for Path Polymorphism 18 / 17

25 CAP Matching operation {u/x } {u/x} {c/c } {} {u v/p q } {u/p } {v/q } if u v is a matchable form {u/p } fail if u is a matchable form {u/p } wait Reduction rule {u/p i } = fail for all i < j {u/p j } = σ j j 1..n (β) (p i θi s i ) i 1..n u σ j s j Andrés E. Viso Type Soundness for Path Polymorphism 19 / 17

26 Types Datatypes and types syntax D ::= α (datatype variable) c (atom) A (compound) D D (union) µα.d (recursion) A ::= X (type variable) D (datatype) A A (type abstraction) A A (union) µx.a (recursion) Andrés E. Viso Type Soundness for Path Polymorphism 20 / 17

27 Type equivalence 3 (e-idem) A A µ A (e-comm) A B µ B A A (B C) µ (A B) C (e-assoc) (e-fold) µv.a µ {µv.a/v } A A µ {A/V } B µv.b contractive (e-contr) A µ µv.b 3 Reflexivity, transitivity, symmetry and congruence rules omitted. Andrés E. Viso Type Soundness for Path Polymorphism 21 / 17

28 Subtyping rules 4 A subtyping context consists of assumptions of the form X Y Σ, V µ W V µ W (s-hyp) A µ B (s-eq) Σ A µ B Σ A µ C Σ A B µ C Σ B µ C (s-union-l) Σ A µ B Σ A µ B C (s-union-r1) Σ A µ C Σ A µ B C (s-union-r2) Σ, V µ W A µ B W / fv(a) V / fv(b) (s-rec) Σ µv.a µ µw.b 4 Reflexivity, transitivity and congruence rules omitted. Andrés E. Viso Type Soundness for Path Polymorphism 22 / 17

29 Compatibility a ɛ {a} (A 1 A 2 ) ɛ { }, (A 1 A 2 ) iπ A i π, i {1, 2} (A 1 A 2 ) π A 1 π A 2 π (µv.a ) π ({µv.a /V } A ) π Maximal positions maxpos(p) { π P π P.π = ππ π ɛ } Mismatching positions mmpos(p, q) {π π maxpos(pos(p) pos(q)) p π q π } Andrés E. Viso Type Soundness for Path Polymorphism 23 / 17

30 Compatibility p : A is compatible with q : B (p : A q : B) iff: p q = B µ A p q = ( π mmpos(p, q).a π B π ) = B A. Alternately, let ovl(p : A, q : B) p q ( π mmpos(p, q).a π B π ) Then, p i : A i p j : A j iff ovl(p : A, q : B) = B µ A A list of patterns [p i : A i ] i 1..n is compatible if i, j 1..n.i < j = p i : A i p j : A j Andrés E. Viso Type Soundness for Path Polymorphism 24 / 17

31 Typing schemes Patterns are typed separately with special judgment p θ(x) = A (p-match) θ p x : A (p-const) θ p c : c θ p p : D θ p q : A (p-comp) θ p p q : A Andrés E. Viso Type Soundness for Path Polymorphism 25 / 17

32 Typing schemes Rules for variables, data structures and subtyping Γ(x) = A (t-var) Γ x : A (t-const) Γ c : c Γ r : D Γ u : A (t-comp) Γ r u : A Γ s : A Γ s : A A µ A (t-subs) The subtyping relation handles union and recursive types Andrés E. Viso Type Soundness for Path Polymorphism 26 / 17

33 Typing schemes Special rules for abstraction and application (θ i p p i : A i ) i 1..n [p i : A i ] i 1..n compatible (Γ, θ i s i : B) i 1..n (t-abs) Γ (p i θi s i ) i 1..n : i 1..n A i B Γ r : i 1..n A i B Γ u : A k k 1..n (t-app) Γ r u : B Andrés E. Viso Type Soundness for Path Polymorphism 27 / 17