Started writing on phantom types as witnesses for invariants
This commit is contained in:
Georges Dupéron
parent
48bc1ed2f9
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@ -1,14 +1,14 @@
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3
info.rkt
3
info.rkt
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@ -11,7 +11,8 @@
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"srfi-lite-lib"
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"delay-pure"
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"backport-template-pr1514"
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"typed-map"))
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"typed-map"
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"scribble-lib"))
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(define build-deps '("scribble-lib"
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"racket-doc"
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"remember"
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@ -1,15 +1,19 @@
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#lang aful/unhygienic hyper-literate type-expander/lang
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@title[#;#:style #;(with-html5 manual-doc-style)
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#:tag "invariants-phantom"
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#:tag-prefix "phc-graph/invariants-phantom"]{Tracking checked contracts
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@require[scribble-math
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scribble-enhanced/doc
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"notations.rkt"]
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@title[#:style (with-html5 manual-doc-style)
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#:tag "inv-phantom"
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#:tag-prefix "phc-graph/inv-phantom"]{Tracking checked contracts
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via phantom types}
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@(chunks-toc-prefix
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'("(lib phc-graph/scribblings/phc-graph-implementation.scrbl)"
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"phc-graph/invariants-phantom"))
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"phc-graph/inv-phantom"))
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@section{Overview}
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@section{Introduction}
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The cautious compiler writer will no doubt want to check that the graph used
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to represent the program verifies some structural properties. For example, the
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@ -18,6 +22,8 @@ property is that the @racket[in-method] field of an instruction points back to
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the method containing it. We will use this second property as a running
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example in this section.
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@section{Implementation overview}
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It is possible to express with Typed/Racket that a @racket[Method] should
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contain a list of @racket[Instruction]s, and that @racket[Instruction]s should
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point to a @racket[Method]@note{We are not concerned here about the ability to
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@ -86,13 +92,80 @@ to wrap it in a struct type which is only accessible by the graph library:
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(code:comment "provide only the type, not the constructor or accessor")
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(provide Flag-Wrapper)]
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@; size → use a dummy function which errors when called
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@; (accepts an opaque token to prevent calling)
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We would like to be able to indicate that a graph node has validated several
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invariants. For that, we need a way to represent the type of a "set" of
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invariant witnesses. We also want some subtyping relationship between the
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sets: a set @${s₁} with more invariant witnesses should be a subtype of a
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subset @${s₂ ⊆ s₁}. We can order the invariant witnesses and use @racket[Rec]
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to build the type of a list of invariant witnesses, where some may be missing:
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@; Subtyping: multiple contracts with case→
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@chunk[<invariant-set-as-List+Rec>
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(define-type At-Least-InvB+InvD
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(Rec R₁ (U (Pairof Any R₁)
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(Pairof 'InvB (Rec R₂ (U (Pairof Any R₂)
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(Pairof 'InvD (Listof Any))))))))]
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@; Subtyping: when C1 ⇒ C2,
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@; make the marker for C1 a subtype of the marker for C2
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@chunk[<invariant-set-as-List+Rec-use>
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(ann '(InvA InvB InvC InvD InvE) At-Least-InvB+InvD)
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(ann '(InvB InvD) At-Least-InvB+InvD)
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(code:comment "Rejected, because it lacks 'InvD")
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(code:comment "(ann '(InvB InvC InvE) At-Least-InvB+InvD)")
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(code:comment "The elements must be in the right order,")
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(code:comment "this would be rejected by the typechecker:")
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(code:comment "(ann '(InvD InvB) At-Least-InvB+InvD)")]
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Another solution is to group the witnesses in an untagged union with
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@racket[U], and place it in a contravariant position:
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@chunk[<invariant-set-as-contravariant-U>
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(define-type At-Least-InvB+InvD
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(→ (U 'InvB 'InvD) Void))]
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This solution also has the advantage that the size of the run-time witness is
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constant, and does not depend on the number of checked contracts (unlike the
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representation using a list). In practice the function should never be called.
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It can however simply be implemented in a way which pleases the type checked
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as a function accepting anything and returning void.
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In addition to testifying that a graph node was checked against multiple,
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separate contracts, there might be some contracts which check stronger
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properties than others. A way to encode this relationship in the type system
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is to have subtyping relationships between the contract witnesses, so that
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@; TODO: get rid of the mathit
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@${\mathit{P}₁(x) ⇒ \mathit{P}₂(x) ⇒ \mathit{Inv}₁ @texsubtype \mathit{Inv}₂}:
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@chunk[<invariant-contract-subtyping>
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(struct InvWeak ())
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(struct InvStrong InvWeak ())]
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If the witnesses must appear in a contravariant position (when using
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@racket[U] to group them), the relationship must be reversed:
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@chunk[<invariant-contract-subtyping>
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(struct InvStrongContra ())
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(struct InvWeakContra InvStrongContra ())]
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Alternatively, it is possible to use a second contravariant position to
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reverse the subtyping relationship again:
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@chunk[<invariant-contract-subtyping>
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(struct InvWeak ())
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(struct InvStrong InvWeak ())
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(define InvWeakContra (→ InvWeak Void))
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(define InvStrongContra (→ InvStrong Void))]
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Finally, we note that the invariants should always be represented using a
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particular struct type, instead of using a symbol, so that name clashes are
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not a problem.
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@section{Types for some graph contracts}
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@chunk[<structural>
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(≡ [a : Nd] (get a f1 f2))
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(∈ [a : Nd] (get a f1 f2))
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(∉ [a : Nd] (get a f1 f2))
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(∉ [a : Nd] (get a (* f1 f2 f3 f4) (* f5 f6)))]
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@chunk[<*>
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(void)]
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7
notations.rkt
Normal file
7
notations.rkt
Normal file
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@ -0,0 +1,7 @@
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#lang racket
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(require scribble/base)
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(provide texsubtype)
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(define texsubtype "<:")
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@ -22,15 +22,13 @@
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[struct struct-field …] …)))]))
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(gs bt-fields
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257
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16
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(a b c)
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[sab a b]
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[sbc b c]
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[sabc a b c])
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;(define-type btac (bt-fields a c))
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#|
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(define-type btac (bt-fields a c))
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(check-equal?:
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(~> (ann (with-c (sab→tree 1 2) 'nine)
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@ -78,5 +76,4 @@
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(call-with-values
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#λ(tree→sbc (without-a (with-c (sab→tree 1 2) 3)))
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list)
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'(2 3))
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|#
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'(2 3))
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