A Simple Semantics for Polymorphic Recursion*

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A Simple Semantics for Polymorphic Recursion

• William L. Harrison, Ph.D
• Department of Computer Science
• University of Missouri, Columbia

This research supported in part by subcontract
GPACS0016, System Information Assurance II,
through OGI/Oregon Health Sciences University.
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What is polymorphic recursion?
Compact list-like structure Okasaki98
size runs in O(log n), list length in O(n)
Polymorphic recursion useful in generic
programming Hinze00,HinzeJeuring02
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polymorphic recursion aka non-uniform
recursion
size Seq a?Int
size Seq (a,a)?Int
length a?Int
length a?Int
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Roadmap

• Ohori 89 conservatively extends semantics of STLC
  to Core-ML
• Harrison/Kieburtz 05 extend to model logic of
  Haskell demand
• Present work models recursion in Haskell
• Remaining aspects seem to fit well in this
  framework
• In particular, type classes
• Goal H98 semantics akin to Definition of
  Standard ML

Goal Haskell 98 Semantics
Simple Model for Polymorphic Recursion APLAS05
Demand in Haskell 98 JFP05
Simple Model for ML Polymorphism Ohori89
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Background Frame Models
is a frame model iff
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Simple Model for ML polymorphism Ex (?x.x)
?a.a?a
Denotes at any instance
Simple Model conservatively extends model of
simply-typed ?-calculus
An alternative HarperMitchell93 is to model
type quantifiers with type abstraction as in
System F/polymorphic ?-calculus Girard,Reynolds
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Language for Polymorphic Recursion

• Language combines features of Kfoury93
  Ohori89
• special pfix binder for polymorphic recursive
  definitions
• stratified type language simple, open, and
  universal

Abstract Syntax for PR
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Type Syntax
Stratified Types
? is an instantiation
if, and only if
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Typing Rules for PR
PR rules Kfoury93
Hindley-Milner
Type system is syntax-oriented Kfoury93 I.e.,
derivations are unique modulo application
order of GEN rule.
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Typing Rules for PR
Leaf in the type derivation of size
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Pointed CPO Frames

• is a pcpo frame if each D?
  is a pointed cpo w.r.t. , , and

Ex.
For ? Int?Int,
Where continuity and fix are defined
conventionally
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Where do we solve equations such as
?
size Seq a?Int
size Seq (a,a)?Int
lives in DSeq(?)?Int
lives in DSeq(?? ?)?Int
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Defining pcpo frame P in terms of a given pcpo
frame D
Frame Objects are type-indexed sets of values
Other Structure (lub p.o. defined pointwise)
By Theorem 1 in the paper, P is a pcpo frame
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Frame Semantics of PR
VAR
PFIX
where
Semantics defined by induction on type derivations
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Application Haskell Type Classes
The Eq class
defines () at infinitely many types
Without Poly-Rec, any Haskell program uses
finitely many, statically determinable instances
of () Jones94 not true with Poly-Rec
With the Simple Model, dictionaries are just
denotations
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Historical Note
Polymorphic Recursion entered Haskell through a
backdoor when certain expert programmers
noticed that the following variety of kludge
worked.
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Related Work

• Mycroft 84 introduced PR gave a model based on
  ideals MacQueen, et al. 84
• Cousot 97 formulated a hierarchy of type systems
  (including Mycrofts) in terms of lattice of
  abstract interpretations of untyped lambda
  calculus
• Henglein 93 showed PR type inference is
  undecidable
• Kfoury, et al. 93, gave several type systems for
  PR ours is called ML/1
• Models of Type classes WadlerBlott89, Thatte94,
  StuckeySulzmann04 and Higher-order polymorphism
  Jones93

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Conclusions

• Textbook semantics
• Substance of denotations change (type awareness)
• Form of semantic definitions mostly familiar
• Conservative extension of Simple Model of ML
  Polymorphism
• Straightforward, albeit non-trivial
• Present work models recursion in Haskell
• Remaining aspects seem to fit well type classes
  higher-order polymorphism

Goal Haskell 98 Semantics
Simple Model for Polymorphic Recursion APLAS05
Demand in Haskell 98 JFP05
Simple Model for ML Polymorphism Ohori89