Principal Type Schemes for Modular Programs Derek Dreyer and Matthias Blume Toyota Technological Institute at Chicago ESOP 2007 Braga, Portugal

Principal Type Schemes for Functional Programs Damas and Milner’s classic POPL’82 paper about implicit ML-style “let-polymorphism” Declarative semantics:  ` e :  –Clean, but non-deterministic: e may have many types  Algorithm W:  ` e ) (  ;  ) –Computes the principal, “most general” type of e

Principal Type Schemes for Modular Programs? Definition of Standard ML joins Damas-Milner’s declarative rules with the rules of the ML module system Implementations of SML employ various generalizations of Algorithm W to work in the presence of modules Is Damas-Milner being generalized properly? –Does SML have principal types?

Contributions of This Work A set of example programs on which no SML typechecker accurately matches the Definition –Illustrate why the Definition is difficult to implement A novel declarative system for ML-style polymorphism in the presence of modules that is easy to implement –Principal types theorem proved –Backward-compatible with SML –Elegant application of previous ideas/techniques

Example (a)

Value restriction: f ’s type cannot be polymorphically generalized because id id is not a syntactic value.

Example (a)

Example (a) is Well-Typed According to the Definition

What Went Wrong?

MacQueen’s Gambit SML/NJ’s policy (according to Dave MacQueen): –Core and module languages should not mix –Reject module-level bindings where r.h.s. is not a value and is not uniquely typed, e.g. val f = id id.

MacQueen’s Gambit SML/NJ’s policy (according to Dave MacQueen): –Core and module languages should not mix –Reject module-level bindings where r.h.s. is not a value and is not uniquely typed, e.g. val f = id id. Disadvantages: –Rejects perfectly good, noncontrived examples, too. E.g. val L = ref nil. –May not scale to languages where module and core are intertwined (e.g. 1 st -class modules, modular type classes)

Our Solution

Need a way of generalizing  at the functor binding

Our Solution Idea: Generalized Functor Signatures (GFS) –Allow functors to take implicit type arguments in addition to their explicit module arguments

Our Solution Idea: Generalized Functor Signatures (GFS) –Allow functors to take implicit type arguments in addition to their explicit module arguments

Our Solution Idea: Generalized Functor Signatures (GFS) –Allow functors to take implicit type arguments in addition to their explicit module arguments Implicit functors were also useful for modular type classes

Our Solution

…

…

Example (a) Typechecks! …

Still Typechecks! …

Problem Solved! #1

Example (b)

Example (b) is Well-Typed According to the Definition

Example (b) Using a GFS

Example (b) Rejected! Not in scope!

Our Solution Idea: –Expand the definition of “in scope” –Allow inferred types to mention abstract types that are not defined until later in the program

Example (b) Accepted! No problem!

Our Solution Idea: –Expand the definition of “in scope” –Allow inferred types to mention abstract types that are not defined until later in the program How does that work and is it sound?

Our Solution Idea: –Expand the definition of “in scope” –Allow inferred types to mention abstract types that are not defined until later in the program How does that work and is it sound? –Using Dreyer’s RTG type system (ICFP 05), which was designed as a foundation for recursive modules –Soundness proved via progress/preservation

Isn’t It Complicated?

No

Isn’t It Complicated? No Typing judgment for terms essentially same as Definition’s:

Isn’t It Complicated? No Traditional Definition-style typing judgment (a la Russo):

Isn’t It Complicated? No Our new declarative typing judgment:

Isn’t It Complicated? No Our new declarative typing judgment: Moreover, type inference becomes much simpler

Example (c)

Example (c) is Not Well-Typed According to the Definition Not in scope!

Distinguishing (b) and (c) Involves tracking dependencies between abstract types and unification variables –Only 1.5 out of 9 SML implementations get it right Russo’s thesis (2000) gives an inference algorithm based on Miller’s technique of unification under a mixed prefix –But does not prove that it works –Algorithm doesn’t accept Example (a)

In Our System, Example (c) is Well-Typed No problem!

What Else Is In the Paper Full formalization of declarative semantics and inference algorithm –Hybrid of Definition and Harper-Stone semantics –Type soundness proven by reduction to RTG (reduction in tech report) Principal types theorem stated (proof in tech report)

“Benchmarks” Reject All: SML/NJ, ML-Kit, TILT, SML.NET, Hamlet Mixed Bag: Poly/ML, Alice, Moscow ML (interactive mode) MLton: Success relies on whole-program compilation, defunctorization coupled with typechecking

“Benchmarks” Reject All: SML/NJ, ML-Kit, TILT, SML.NET, Hamlet Mixed Bag: Poly/ML, Alice, Moscow ML (interactive mode) MLton: Success relies on whole-program compilation, defunctorization coupled with typechecking

Obrigado!