Semantic Subtyping. Alain Frisch (ENS Paris) Giuseppe Castagna (ENS Paris) Véronique Benzaken (LRI U Paris Sud)

Transcription

1 Semantic Subtyping Alain Frisch (ENS Paris) Giuseppe Castagna (ENS Paris) Véronique Benzaken (LRI U Paris Sud) Semantic Subtyping - Groupe de travail BD LRI p.1/28

2 CDuce A functional language adapted to XML applications: Types are pervasives in Duce: Soundness of transformations Informative error messages Type-driven semantics, pattern matching Type-driven compilation and optimization This talk: theoretical foundation of Duce type system. Most of the technical difficulties in the subtyping relation, which is kept simple by using a semantic approach. Semantic Subtyping - Groupe de travail BD LRI p.2/28

3 Semantic and syntactic subtyping How to define a subtyping relation? Syntactic approach: a formal system (axioms,rules) Semantic approach: start with a denotational model of the language interpret types as subsets of the model define subtyping as inclusion of denotations derive a subtyping algorithm Semantic Subtyping - Groupe de travail BD LRI p.3/28

4 Advantages of semantic subtyping subtyping is complete w.r.t the intuitive interpretation when does not hold, it is possible to exhibit an element of the model in the interpretation of and not of ( error message) modularization of proofs: use the semantic interpretation, not the rules of the subtyping algorithm properties for free : transitivity of,... Whenever possible, avoid ad hoc rules, formulas and algorithms: derive them from computations. Semantic Subtyping - Groupe de travail BD LRI p.4/28

5 XDuce (H. Hosoya, B. Pierce, J. Vouillon) XDuce: a typed programming language for XML applications value = XML document (tree) type = regular tree language subtyping = inclusion of languages A powerful pattern-matching operation Type-driven semantics Recursive patterns to extract information in the middle of a document Exact propagation of the type from the matched expression to binding variables Semantic Subtyping - Groupe de travail BD LRI p.5/28

6 Semantic subtyping in XDuce Semantic subtyping: easy because no first class function, so typing of values does not depend on subtyping! Start from the typing judgment Interpretation Subtyping Semantic Subtyping - Groupe de travail BD LRI p.6/28

7 XDuce + higher-order Duce Integrate XDuce features to a general higher order functional language Extend XDuce semantic approach? But: typing of values now depends on subtyping! Classical semantic approach? Define an untyped denotational model But: semantics depends on typing! Semantic Subtyping - Groupe de travail BD LRI p.7/28

8 Circularities Type system PM on types Subtyping Typing values PM semantics Semantic Subtyping - Groupe de travail BD LRI p.8/28

9 Circularities Type system PM on types Subtyping Typing values PM semantics Model Semantic Subtyping - Groupe de travail BD LRI p.8/28

10 Circularities Type system PM on types Subtyping Typing values PM semantics Model Semantic Subtyping - Groupe de travail BD LRI p.8/28

11 Circularities Type system PM on types Subtyping Typing values PM semantics Model Semantic Subtyping - Groupe de travail BD LRI p.8/28

12 Types Semantic Subtyping - Groupe de travail BD LRI p.9/28

13 Type algebra Types: Constructors: basic, product, arrow types. Semantic Subtyping - Groupe de travail BD LRI p.10/28

14 Type algebra Types: Constructors: basic, product, arrow types. Arbitrary finite boolean combinations. Semantic Subtyping - Groupe de travail BD LRI p.10/28

15 Type algebra Types: Constructors: basic, product, arrow types. Arbitrary finite boolean combinations. Guarded recursive types. Semantic Subtyping - Groupe de travail BD LRI p.10/28

16 Subtyping and models by: We want to define : is subset of a structure For each type, prod fun with: basic prod must satisfy natural conditions: _ The interpretation function prod Semantic Subtyping - Groupe de travail BD LRI p.11/28

17 Model condition for arrows A condition on motivated by: Intuition: extensive view of functions (=binary relations) Derived from simple set-theoretic computations Makes the safety proof work The type system induces a model of values (Turns out to hold in the revelance logic B+; M. Dezani) Semantic Subtyping - Groupe de travail BD LRI p.12/28

18 Is there at least one model!? Yes, this one: basic It is universal (largest subtyping relation). For this model, we derive an algorithm to compute. Semantic Subtyping - Groupe de travail BD LRI p.13/28

19 The language Semantic Subtyping - Groupe de travail BD LRI p.14/28

20 The language: syntax variable abstraction application constant pair pattern matching... Semantic Subtyping - Groupe de travail BD LRI p.15/28

21 Type system Semantic Subtyping - Groupe de travail BD LRI p.16/28...

22 Type system with... Need it for subject reduction but can discard it for typechecking. Semantic Subtyping - Groupe de travail BD LRI p.16/28

23 Results Classical syntactical results, such as admissibility of conjunction rule and subsumption elimination. Subject reduction for a small step semantics. A new interpretation of types as sets of values: This is indeed a model. It induces the same subtyping relation as the bootstrap model. This holds because of overloaded functions! Semantic Subtyping - Groupe de travail BD LRI p.17/28

24 The circle is now complete ) ( Semantic Subtyping - Groupe de travail BD LRI p.18/28

25 Subtyping algorithm As any coinductive relation, the subtyping algorithm can be expressed in an abstract way through a notion of simulation. A simulation is a subset of all types closed under rules like: The largest simulation is exactly the set of empty types. This abstract presentation separates implementation issues (such as caching) and the theoretical study of the algorithm. Semantic Subtyping - Groupe de travail BD LRI p.19/28

26 Using set-theoretic semantics Simple set-theoretic facts yield a number of optimizations for the subtyping algorithm, avoiding exponential explosion in many cases. E.g.: whenever Semantic Subtyping - Groupe de travail BD LRI p.20/28

27 Patterns Semantic Subtyping - Groupe de travail BD LRI p.21/28

28 Patterns capture type constraint conjunction alternative pair constant recursive pattern Result of matching : (failure) or a substitution Var. Semantic Subtyping - Groupe de travail BD LRI p.22/28

29 Pattern-matching: static semantics The pattern accepts a type characterized by: If by: Thm: it exists + algorithm and Var, type of the result for characterized Thm: it exists + algorithm Semantic Subtyping - Groupe de travail BD LRI p.23/28

30 Type system: pattern matching ) ( Semantic Subtyping - Groupe de travail BD LRI p.24/28

31 Type system: pattern matching ) ( Exhaustivity checking : set of values matched by Semantic Subtyping - Groupe de travail BD LRI p.24/28

32 Type system: pattern matching ) ( Exhaustivity checking : set of values matched by Dispatching Semantic Subtyping - Groupe de travail BD LRI p.24/28

33 Type system: pattern matching ) ( Exhaustivity checking : set of values matched by Dispatching Matching : typing environment for variables bound in Semantic Subtyping - Groupe de travail BD LRI p.24/28

34 Type system: pattern matching ) ( Exhaustivity checking : set of values matched by Dispatching Matching : typing environment for variables bound in Result Discard useless branches Semantic Subtyping - Groupe de travail BD LRI p.24/28

35 Typing patterns: an example Lists à la Lisp (head,tail) + terminator. Consider the following pattern and types: : If Otherwise: Then: If : Semantic Subtyping - Groupe de travail BD LRI p.25/28

36 Pattern algorithms (1) Semantic Subtyping - Groupe de travail BD LRI p.26/28

37 Pattern algorithms (2) Var if Var Var if Var Var if Var Var if if if Semantic Subtyping - Groupe de travail BD LRI p.27/28

38 Compiling pattern matching Naive solution: backtracking. Less naive, but untractable: bottom-up tree automata. Adopted solution: hybrid top-down/bottom-up automata, which avoid backtracking, and take profit of static type information. Complex algorithm: set-theoretic semantics of types and patterns helps a lot here! Semantic Subtyping - Groupe de travail BD LRI p.28/28