Writeup
This commit is contained in:
Georges Dupéron
parent
63e1a77ef1
commit
e3074a37e4
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@ -30,76 +30,71 @@
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'none))
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|#
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(struct (A) Some ([f : A]) #:transparent)
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(struct (A) Some0 Some () #:transparent)
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(struct (A) Some1 Some () #:transparent)
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(struct (A) Some2 Some () #:transparent)
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(struct (A) Some3 Some () #:transparent)
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(struct (A) None ([f : A]) #:transparent)
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;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
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;; Actual good implementation:
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;(struct (A) Some ([f : A]) #:transparent)
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;(struct (A) Some0 Some () #:transparent)
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;(struct (A) Some1 Some () #:transparent)
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;(struct (A) Some2 Some () #:transparent)
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;(struct (A) Some3 Some () #:transparent)
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;(struct (A) None ([f : A]) #:transparent)
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;; A is:
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#;(U (Pairof Any (None (Listof (Some Any))))
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(Some Any))
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#;(Pairof Any (U (Some Any) (None (Listof Any))))
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(struct |0| ())
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(struct |1| ())
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(struct |2| ())
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(struct |3| ())
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(: f (builder-τ 4 2))
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(: f (∀ (A 0/K 0/X 1/K 1/X)
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(→ (∩ 0/K (U |0| |1| |2| |3|))
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0/X
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(∩ 1/K (U |0| |1| |2| |3|))
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1/X
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(List
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(∩
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(U
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(Pairof |0| (None (List (∩ 0/K (U |1| |2| |3|)) (∩ 1/K (U |1| |2| |3|)))))
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(∩ (Pairof 0/K (Some 0/X)) (Pairof |0| Any))
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(∩ (Pairof 1/K (Some 1/X)) (Pairof |0| Any)))
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A)
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(∩
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(U
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(Pairof |1| (None (List (∩ 0/K (U |0| |2| |3|)) (∩ 1/K (U |0| |2| |3|)))))
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(∩ (Pairof 0/K (Some 0/X)) (Pairof |1| Any))
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(∩ (Pairof 1/K (Some 1/X)) (Pairof |1| Any)))
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A)
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(∩
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(U
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(Pairof |2| (None (List (∩ 0/K (U |0| |1| |3|)) (∩ 1/K (U |0| |1| |3|)))))
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(∩ (Pairof 0/K (Some 0/X)) (Pairof |2| Any))
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(∩ (Pairof 1/K (Some 1/X)) (Pairof |2| Any)))
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A)
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(∩
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(U
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(Pairof |3| (None (List (∩ 0/K (U |0| |1| |2|)) (∩ 1/K (U |0| |1| |2|)))))
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(∩ (Pairof 0/K (Some 0/X)) (Pairof |3| Any))
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(∩ (Pairof 1/K (Some 1/X)) (Pairof |3| Any)))
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A)))))
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(define (f kx x ky y)
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(define (f oracle kx x ky y)
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(list (cond
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[((make-predicate |0|) kx) (cons kx (Some x))]
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[((make-predicate |0|) ky) (cons ky (Some y))]
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[else (cons |0| (None (list kx ky)))])
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[((make-predicate '|0|) kx)
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(ann ((inst oracle (∩ (Pairof 0/K (Some 0/X)) (Pairof '|0| Any))) (cons kx (Some x)))
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(∩ (Pairof 0/K (Some 0/X)) (Pairof '|0| Any) A))]
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[((make-predicate '|0|) ky)
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(ann ((inst oracle (∩ (Pairof 1/K (Some 1/X)) (Pairof '|0| Any))) (cons ky (Some y)))
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(∩ (Pairof 1/K (Some 1/X)) (Pairof '|0| Any) A))]
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[else
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((inst oracle (Pairof '|0| (None (List (∩ 0/K (U '|1| '|2| '|3|)) (∩ 1/K (U '|1| '|2| '|3|))))))
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(cons '|0| (None (list kx ky))))])
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(cond
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[((make-predicate |1|) kx) (cons kx (Some x))]
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[((make-predicate |1|) ky) (cons ky (Some y))]
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[else (cons |1| (None (list kx ky)))])
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[((make-predicate '|1|) kx)
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(ann ((inst oracle (∩ (Pairof 0/K (Some 0/X)) (Pairof '|1| Any))) (cons kx (Some x)))
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(∩ (Pairof 0/K (Some 0/X)) (Pairof '|1| Any) A))]
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[((make-predicate '|1|) ky)
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(ann ((inst oracle (∩ (Pairof 1/K (Some 1/X)) (Pairof '|1| Any))) (cons ky (Some y)))
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(∩ (Pairof 1/K (Some 1/X)) (Pairof '|1| Any) A))]
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[else
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((inst oracle (Pairof '|1| (None (List (∩ 0/K (U '|0| '|2| '|3|)) (∩ 1/K (U '|0| '|2| '|3|))))))
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(cons '|1| (None (list kx ky))))])
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(cond
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[((make-predicate |2|) kx) (cons kx (Some x))]
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[((make-predicate |2|) ky) (cons ky (Some y))]
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[else (cons |2| (None (list kx ky)))])
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[((make-predicate '|2|) kx)
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(ann ((inst oracle (∩ (Pairof 0/K (Some 0/X)) (Pairof '|2| Any))) (cons kx (Some x)))
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(∩ (Pairof 0/K (Some 0/X)) (Pairof '|2| Any) A))]
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[((make-predicate '|2|) ky)
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(ann ((inst oracle (∩ (Pairof 1/K (Some 1/X)) (Pairof '|2| Any))) (cons ky (Some y)))
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(∩ (Pairof 1/K (Some 1/X)) (Pairof '|2| Any) A))]
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[else
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((inst oracle (Pairof '|2| (None (List (∩ 0/K (U '|0| '|1| '|3|)) (∩ 1/K (U '|0| '|1| '|3|))))))
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(cons '|2| (None (list kx ky))))])
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(cond
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[((make-predicate |3|) kx) (cons kx (Some x))]
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[((make-predicate |3|) ky) (cons ky (Some y))]
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[else (cons |3| (None (list kx ky)))]))
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((λ () (error "not yet"))))
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[((make-predicate '|3|) kx)
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(ann ((inst oracle (∩ (Pairof 0/K (Some 0/X)) (Pairof '|3| Any))) (cons kx (Some x)))
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(∩ (Pairof 0/K (Some 0/X)) (Pairof '|3| Any) A))]
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[((make-predicate '|3|) ky)
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(ann ((inst oracle (∩ (Pairof 1/K (Some 1/X)) (Pairof '|3| Any))) (cons ky (Some y)))
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(∩ (Pairof 1/K (Some 1/X)) (Pairof '|3| Any) A))]
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[else
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((inst oracle (Pairof '|3| (None (List (∩ 0/K (U '|0| '|1| '|2|)) (∩ 1/K (U '|0| '|1| '|2|))))))
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(cons '|3| (None (list kx ky))))])))
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@ -1,101 +1,230 @@
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#lang dotlambda/unhygienic type-expander/lang
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#|hyper-literate #:♦ #:no-auto-require (dotlambda/unhygienic
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#lang hyper-literate #:꩜ #:no-auto-require (dotlambda/unhygienic
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. type-expander/lang)
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♦chunk[<*>|#
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(provide builder-τ
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None
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Some
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Some?
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Some-f
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propagate-τ)
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꩜title[#:tag "generalized-flex-ctor"]{Generalised constructor functions for
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flexible structures}
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(require racket/require
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(for-syntax (subtract-in racket/base subtemplate/override)
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syntax/stx
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racket/list
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racket/function
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subtemplate/override)
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(for-meta 2 racket/base))
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꩜(require hyper-literate/diff1)
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꩜(init)
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(struct (T) Some ([f : T]))
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(struct (T) None ([f : T]))
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꩜chunk[<*>
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(provide builder-τ
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None
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Some
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Some?
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Some-f
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propagate-τ)
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(define-type-expander BinaryTree
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(syntax-parser
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[(_ leafⱼ …)
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;; TODO: implement BinaryTree.
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#'(List leafⱼ …)]))
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(require racket/require
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(for-syntax (subtract-in racket/base subtemplate/override)
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syntax/stx
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racket/list
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racket/function
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subtemplate/override)
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(for-meta 2 racket/base))
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(define-syntax (def-SomeNone* stx)
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(syntax-case stx ()
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[(_ Some n)
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(with-syntax ([(Someᵢ …) (map (λ (i) (format-id #'Some "Some~a" i))
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(range n))])
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#`(begin
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(provide Someᵢ …)
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(struct (T) Someᵢ Some ()) …))]))
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(struct (T) Some ([f : T]))
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(struct (T) None ([f : T]))
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(def-SomeNone* Some 4)
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(define-type-expander BinaryTree
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(syntax-parser
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[(_ leafⱼ …)
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;; TODO: implement BinaryTree.
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#'(List leafⱼ …)]))
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(define-type-expander builder-τ
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(syntax-parser
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[(_ n m)
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#:with (Nᵢ …) (range n)
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#:with (Mⱼ …) (range m)
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#:with (Someᵢ …) (map (λ (n) (format-id #'HERE? "Some~a" n)) (Nᵢ …))
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#:with ((Someᵢⱼ …) …) (map λ.(map (const %) (Mⱼ …)) (#'Someᵢ …))
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;#:with ((Noneᵢⱼ …) …) (map (const #'(Noneᵢ …)) (Nᵢ …))
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#:with ((Kᵢⱼ …) …) (map (const #'(Kⱼ …)) (Nᵢ …))
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#:with ((Xᵢⱼ …) …) (map (const #'(Xⱼ …)) (Nᵢ …))
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#:with ((Nᵢⱼ …) …) (map (λ (ni) (map (const ni) (Xⱼ …))) (Nᵢ …))
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#:with ((Nⱼᵢ …) …) (map (const #'(Nᵢ …)) (Mⱼ …))
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(define Ns (Nᵢ …))
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(define Ms (Mⱼ …))
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(define Somes (Someᵢ …))
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;(define/with-syntax exceptⱼ (remove Mⱼ Ns)) …
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; (define/with-syntax ((exceptᵢⱼ …) …)
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; (map (const (exceptⱼ …)) (Nᵢ …)))
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(define/with-syntax (exceptᵢ …) ((remove Nᵢ Ns) …))
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(define/with-syntax (((exceptᵢⱼ …) …) …)
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((map (const (remove Someᵢ Somes)) Ms) …))
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(define-syntax (def-SomeNone* stx)
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(syntax-case stx ()
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[(_ Some n)
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(with-syntax ([(Someᵢ …) (map (λ (i) (format-id #'Some "Some~a" i))
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(range n))])
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#`(begin
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(provide Someᵢ …)
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(struct (T) Someᵢ Some ()) …))]))
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(define/with-syntax result
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#'(∀ (A (?@ Kⱼ Xⱼ) …)
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(→ (?@ (∩ Kⱼ (Some Any)) (∩ Kⱼ Xⱼ)) …
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(BinaryTree
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(∩ (U (Pairof Nᵢ
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;; If Kⱼ is Nᵢ, then {∩ Kᵢⱼ {U . exceptᵢⱼ}} will
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;; Nothing. We multiply the Nothing together by
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;; building a List out of them (a single occurrence
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;; of Nothing should collapse the list), so that the
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;; result should be Nothing only if there is no Kⱼ
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;; equal to Nᵢ. To force TR to propagate Nothing as
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;; much as possible, we intersect it with
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;; (None Any), which should be a no-op, but makes
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;; sure the simplifier which runs with ∩ kicks in.
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;; Therefore, the (None whatever) should appear only
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;; if there is indeed no key provided for that leaf.
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(None (List {∩ Kᵢⱼ {U (exceptᵢⱼ Any) …}} …)))
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(∩ Kᵢⱼ (Someᵢⱼ Xᵢⱼ))
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…)
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A)
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…))))
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(displayln (syntax->datum #'result))
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#'result]))
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(def-SomeNone* Some 4)
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(define-type propagate-τ
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(Pairof Any
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(U (None (Listof Natural))
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(Some Any))))
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<builder-τ>
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; ../../../.racket/snapshot/pkgs/typed-racket-lib/typed-racket/types/overlap.rkt
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;:40:0: mask-accessor: contract violation
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; expected: mask?
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; given: #f
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;(define-type τ-4-2 (builder-τ 4 2))
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<propagate-τ>]
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We first define the builder function's type. Since this type is rather
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complex, we define it using a macro. The type expander takes two arguments.
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The first argument ꩜racket[n] indicates the total number of fields which
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appear in the program (i.e. on the number of leaves of the generated tree
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type), and the second argument ꩜racket[m] indicates how many key/value pairs
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the function accepts.
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꩜chunk[<builder-τ>
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(define-type-expander builder-τ
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(syntax-parser
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[(_ n m)
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<builder-τ-with-1>
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<builder-τ-with-2>
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<builder-τ-with-3>
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#'|<builder-function-type''>|]))]
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We start by defining a few syntax pattern variables which will be used in the
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later definitions. The lists ꩜racket[Nᵢ] and ꩜racket[Mⱼ] range over the field
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and argument indices, respectively:
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꩜chunk[<builder-τ-with-1>
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#:with (Nᵢ …) (range n)
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#:with (Mⱼ …) (range m)]
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The builder function takes a number of keys and values, and builds a (promise
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for) a binary tree where the leaves corresponding to those keys contain given
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value, and other leaves contain ꩜racket[None]. We could write (a simplified
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form of) the builder function type as follows:
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꩜chunk[<builder-function-type>
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(∀ ({?@ Kⱼ Xⱼ} …)
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(→ (code:comment "; Keys and values:")
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{?@ (∩ Kⱼ (U 'NSymⱼᵢ …)) Xⱼ} …
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;; Result type:
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(BinaryTree |<Some or None>| …)))]
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We expect each key ꩜racket[Kⱼ] to be a symbol of the shape ꩜racket[|0|],
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꩜racket[|1|], ꩜racket[|2|] and so on:
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꩜chunk[<builder-τ-with-2>
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#:with (NSymᵢ …) ((string->symbol (format "~a" Nᵢ)) …)
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#:with ((NSymⱼᵢ …) …) (map (const (NSymᵢ …)) (Mⱼ …))]
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The type of each leaf of the binary tree should be ꩜racket[(Some Xⱼ)] if a
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corresponding ꩜racket[Kⱼ] matches the leaf name, and ꩜racket[None] otherwise.
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꩜chunk[|<Some or None>|
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(U |<(Some Xⱼ) if Kⱼ = NSymᵢ>|
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|<None if ∀ k ∈ Kⱼ, k ≠ NSymᵢ>|)]
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This type-level conditional is achieved via a trick involving intersection
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types. The ꩜racket[Kⱼ] type should be a singleton type containing exactly one
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of the ꩜racket['NSymᵢ …] symbols. For a given leaf with index ꩜racket[i], if
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the ꩜racket[Kⱼ] key is the type ꩜racket['NSymᵢ], then the intersection type
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꩜racket[(∩ Kⱼ 'NSymᵢ)] is ꩜racket['NSymᵢ]. Conversely, if the ꩜racket[Kⱼ] key
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is not ꩜racket['NSymᵢ], the intersection will be the bottom type
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꩜racket[Nothing]. No values inhabit the bottom type, and Typed Racket can
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determine that there is no pair whose first (or second) element has the type
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꩜racket[Nothing], since no concrete value could be used to construct such a
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pair.
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꩜chunk[|<(Some Xⱼ) if Kⱼ = NSymᵢ>|
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(Pairof (∩ Kᵢⱼ 'NSymᵢⱼ)
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(Some Xᵢⱼ))
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…]
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where ꩜racket[Kᵢⱼ], ꩜racket[Xᵢⱼ] and ꩜racket[NSymᵢⱼ] are defined as follows:
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꩜chunk[<builder-τ-with-3>
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#:with ((Kᵢⱼ …) …) (map (const (Kⱼ …)) (Nᵢ …))
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#:with ((Xᵢⱼ …) …) (map (const (Xⱼ …)) (Nᵢ …))
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#:with ((NSymᵢⱼ …) …) (map λni.(map (const ni) (Xⱼ …)) (NSymᵢ …))]
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We use this fact to construct a pair above. Its first element is either
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꩜racket['NSymᵢ] when ꩜racket[Kⱼ] is ꩜racket['NSymᵢ], and ꩜racket[Nothing]
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otherwise. The second element of the pair contains our expected
|
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꩜racket[(Some Xⱼ)] type, but the whole pair is collapsed to ꩜racket[Nothing]
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when ꩜racket[Kⱼ] is not ꩜racket['NSymᵢ].
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We use a similar approach to conditionally produce the ꩜racket[None] element,
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but instead of intersecting ꩜racket[Kⱼ] with ꩜racket['NSymᵢ], we intersect it
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with the complement of ꩜racket['NSymᵢ]. Typed Racket lacks the possibility to
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negate a type, so we manually compute the complement of ꩜racket['NSymᵢ] in the
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set of possible keys (that is, ꩜racket['NSymᵢ …]).
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꩜chunk[<builder-τ-with-3>
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#:with NSyms (NSymᵢ …)
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#:with Ms (Mⱼ …)
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#:with (exceptᵢ …) ((remove NSymᵢ NSyms) …)
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#:with (((exceptᵢⱼ …) …) …) ((map (const exceptᵢ) Ms) …)]
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|
||||
If ꩜racket[Kⱼ] is ꩜racket[Nᵢ], then ꩜racket[{∩ Kⱼ {U . exceptᵢ}}] will be
|
||||
꩜racket[Nothing]. We take the Cartesian product of the intersections by
|
||||
building a ꩜racket[List] out of them. A single occurrence of ꩜racket[Nothing]
|
||||
will collapse the whole list to ꩜racket[Nothing], because the Cartesian
|
||||
product of the empty set and any other set will produce the empty set.
|
||||
|
||||
The resulting type should therefore be ꩜racket[Nothing] only if there is no
|
||||
꩜racket[Kⱼ] equal to ꩜racket[Nᵢ], and be the list of symbols
|
||||
꩜racket[(List . exceptᵢ)] otherwise.
|
||||
|
||||
꩜chunk[|<None if ∀ k ∈ Kⱼ, k ≠ NSymᵢ>|
|
||||
(Pairof 'NSymᵢ
|
||||
(None (List {∩ Kᵢⱼ {U 'exceptᵢⱼ …}} …)))]
|
||||
|
||||
This approach relies on the fact that occurrences of ꩜racket[Nothing] within
|
||||
structs and pairs containing collapse the entire struct or pair type to
|
||||
꩜racket[Nothing]. Unfortunately, current versions of Typed Racket perform this
|
||||
simplification step in some situations but not others:
|
||||
|
||||
꩜itemlist[
|
||||
꩜item{It simplifies polymorphic types when they are defined;}
|
||||
꩜item{When a polymorphic type is instantiated, the parts which are directly
|
||||
affected by the intersection with a polymorphic type variable are subject to
|
||||
this simplification;}
|
||||
꩜item{However, occurrences of ꩜racket[Nothing] which occur as a result of
|
||||
instantiating the type variable do not propagate outside of the intersection
|
||||
itself. This means that given the following type:
|
||||
꩜racketblock[(define-type (Foo A) (U (Pairof (∩ Integer A) String) Boolean))]
|
||||
its instantiation ꩜racket[(Foo Symbol)] will produce the type
|
||||
꩜racket[(U (Pairof Nothing String) Boolean)], but this type will not be
|
||||
simplified to ꩜racket[(U Nothing Boolean)] or equivalently to
|
||||
꩜racket[Boolean].}]
|
||||
|
||||
To force Typed Racket to propagate ꩜racket[Nothing] outwards as much as
|
||||
we need, we intersect the whole form with a polymorphic type ꩜racket[A]:
|
||||
|
||||
꩜hlite[|<builder-function-type'>|
|
||||
{/(∀(+ _ / _ _)(→ _ _ ooo (bt + (∩ / _ + A) / _)))}
|
||||
(∀ (A {?@ Kⱼ Xⱼ} …)
|
||||
(→ (code:comment "; Keys and values:")
|
||||
{?@ (∩ Kⱼ (U 'NSymⱼᵢ …)) Xⱼ} …
|
||||
;; Result type:
|
||||
(BinaryTree (∩ |<Some or None>| A) …)))]
|
||||
|
||||
The type ꩜racket[propagate-τ] defined below is used to instantiate ꩜racket[A],
|
||||
and is carefully picked so that its intersection will in no way change the
|
||||
result type (i.e. the intersection with ꩜racket[A] will be an identity
|
||||
operation where it is used). In other words, ꩜racket[propagate-τ] has the same
|
||||
shape as the leaves of the binary tree. This intersection however forces the
|
||||
simplification step to be performed on the affected parts once the type is
|
||||
instantiated.
|
||||
|
||||
꩜chunk[<propagate-τ>
|
||||
(define-type propagate-τ
|
||||
(Pairof Any
|
||||
(U (None (Listof Natural))
|
||||
(Some Any))))]
|
||||
|
||||
꩜;Use chunkref instead of ꩜racket[|<Some or None>|] ?
|
||||
|
||||
The implementation of the builder function will need to convert values with
|
||||
the ꩜racket[|<Some or None>|] type to values of type
|
||||
꩜racket[(∩ |<Some or None>| A)]. Since the intersection could, in principle,
|
||||
be any subtype of ꩜racket[|<Some or None>|], we request that the caller
|
||||
supplies a proof that the conversion is possible. This proof takes the form of
|
||||
an ꩜racket[oracle] function, which transforms an element with the type
|
||||
꩜racket[propagate-τ] (which is a supertype of every possible
|
||||
꩜racket[|<Some or None>|] type) into an element with the type
|
||||
꩜racket[(∩ A B)], where ꩜racket[B] is the original type of the value to
|
||||
upgrade.
|
||||
|
||||
꩜chunk[<oracle-type>
|
||||
(∀ (B) (→ (∩ B
|
||||
(Pairof Any (U (Some Any) (None (Listof Any)))))
|
||||
(∩ A B)))]
|
||||
|
||||
The builder function type is updated as follows:
|
||||
|
||||
꩜hlite[|<builder-function-type''>|
|
||||
{/(∀(_ _ _)(→ + _ _ / _ _ ooo _))}
|
||||
(∀ (A {?@ Kⱼ Xⱼ} …)
|
||||
(→ (code:comment "; Oracle:")
|
||||
<oracle-type>
|
||||
(code:comment "; Keys and values:")
|
||||
{?@ (∩ Kⱼ (U 'NSymⱼᵢ …)) Xⱼ} …
|
||||
;; Result type:
|
||||
(BinaryTree (∩ |<Some or None>| A) …)))]
|
||||
|
||||
|
||||
;]
|
||||
|
||||
|
|
|
|||
|
|
@ -17,4 +17,5 @@ the @other-doc['(lib "phc-graph/scribblings/phc-graph.scrbl")] document.
|
|||
@include-section[(submod "../graph-info.hl.rkt" doc)]
|
||||
@include-section[(submod "../graph-type.hl.rkt" doc)]
|
||||
@include-section[(submod "../graph.hl.rkt" doc)]
|
||||
@include-section[(submod "../main-draft.hl.rkt" doc)]
|
||||
@include-section[(submod "../main-draft.hl.rkt" doc)]
|
||||
@include-section[(submod "../flexible-with-generalized-ctor.hl.rkt" doc)]
|
||||
|
|
|
|||
Loading…
Reference in New Issue
Block a user