principal type schemes for modular programs

Principal Type SchemesPrincipal Type Schemesfor Modular Programsfor Modular Programs

Derek Dreyer and Matthias Blume

Toyota Technological Institute at Chicago

ESOP 2007

Braga, Portugal

Principal Type Schemes for Principal Type Schemes for FunctionalFunctional Programs 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 forPrincipal Type Schemes forModularModular Programs? 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 WorkContributions 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)Example (a)


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

Example (a) is Well-TypedExample (a) is Well-TypedAccording to the DefinitionAccording to the Definition

What Went Wrong?What Went Wrong?

MacQueen’s GambitMacQueen’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 valueand is not uniquely typed, e.g. val f = id id.

MacQueen’s GambitMacQueen’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 valueand 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. 1st-class modules, modular type classes)

Our SolutionOur Solution


• Need a way of generalizing at the functor binding


• Idea: Generalized Functor Signatures (GFS)

– Allow functors to take implicit type arguments in addition to their explicit module arguments


• 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


…

Example (a) Typechecks!Example (a) Typechecks!

…

Still Typechecks!Still Typechecks!

…

ProblemProblem Solved!Solved!#1#1

Example (b)Example (b)

Example (b) is Well-TypedExample (b) is Well-TypedAccording to the DefinitionAccording to the Definition

Example (b) Using a GFSExample (b) Using a GFS

Example (b) Rejected!Example (b) Rejected!

Not in scope!


• 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!Example (b) Accepted!

No problem!


• Idea:



• How does that work and is it sound?


• Idea:



• 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?Isn’t It Complicated?


• No


• No

• Typing judgment for terms essentially same as Definition’s:


• No

• Traditional Definition-style typing judgment (a la Russo):


• No

• Our new declarative typing judgment:


• No

• Our new declarative typing judgment:

• Moreover, type inference becomes much simpler

Example (c)Example (c)

Example (c) is Not Well-TypedExample (c) is Not Well-TypedAccording to the DefinitionAccording to the Definition

Not in scope!

Distinguishing (b) and (c)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,In Our System,Example (c) is Well-TypedExample (c) is Well-Typed

No problem!

What Else Is In the PaperWhat 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”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!Obrigado!

principal type schemes for modular programs

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