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Roadmap to Atsushi Ohori’s “ A Simple Semantics for ML Polymorphism ”

Roadmap to Atsushi Ohori’s “ A Simple Semantics for ML Polymorphism ”. Bill Harrison. ML polymorphism. Can apply polymorphic term for any type a. Know what it denotes at any instance…. …from the semantics of simply-typed - calculus. Ohori’s idea: simple semantics.

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Roadmap to Atsushi Ohori’s “ A Simple Semantics for ML Polymorphism ”

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  1. Roadmap to Atsushi Ohori’s“A Simple Semantics for ML Polymorphism” Bill Harrison

  2. ML polymorphism Can apply polymorphic term for any type a Know what it denotes at any instance… …from the semantics of simply-typed -calculus

  3. Ohori’s idea: simple semantics Starting from the semantics of STLC: …extend it to ML-like languages from simply-typed -calculus semantics

  4. Simple vs ? • An alternative [DamasMilner82] is to model type quantifiers with type abstraction as in System F[Girard,Reynolds] • However, this has drawbacks as the denotational semantics of System F is, well, hairy[GirardLafontTaylor]

  5. Semantics of Core-ML Semantics of Ground Core-ML Semantics Simply-typed T Overview of Ohori’s approach Introduce two languages • Core-ML: polymorphic, implicitly-typed -calculus • T: simply & explicitly-typed -calculus with “textbook” denotational • semantics …and define Core-ML in terms of T

  6. Today • Core-ML: syntactic matters • T: its syntax & semantics • In terms of frame models • Relating Core-ML to T • “Roadmap’’ handout contains the technical minutiae of the paper • Laid out in a more logical manner • Ohori’s paper is rich with material I won’t discuss today: • Comparisons with [Milner78,DamasMilner82], full abstraction, extensions to recursive terms & datatypes

  7. Core-ML types Types & Type Schemes Type assignments Type assignment schemes Substitutions

  8. Core-ML terms Raw terms Typings any ground judgment derivable in: Core-ML terms are typing schemes

  9. Type system  Note the different “” T terms Preterms explicit typing

  10. Background: Frame Models

  11. Denotational semantics of T This is a completely conventional, “textbook” semantics

  12. Relating Core-ML to T Translation from Core-ML typings to T …with the following coherence property: * Consequence of Theorem 6, Section 3.2, of Ohori89

  13. Just two more auxiliary definitions (I promise) The set of admissible type assignments under  is E.g., The set of types associated with a term is: N.b.,

  14. Denotational Semantics of Core-ML Terms: Typings:

  15. Relevance to Haskell98 (?!) where

  16. Conclusions • Simple & intuitive: make a polymorphic term denote the set of its ground instance denotations • Avoids the pitfalls of “backwards” explanations: ML polymorphism is simpler than System F, so its semantics should be simpler

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