Post on 04-Nov-2021
A Kripke Logical Relation for Linear Functions:
The Story of a Free Theorem in the Presence of Non-termination
Phillip Mates & Amal AhmedNortheastern University
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Programmers shouldn't have to think about compilers
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Fully abstract compiler
Source Target
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Fully abstract compiler
Source Target
preserves equivalence
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Fully abstract compiler
Source Target
reflects equivalence
preserves equivalence
&
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Verifying a CPS translation fully abstract
Source
Target
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Outline
Finding a fully abstract CPS translation w/ • Standard CPS
• Polymorphic CPS ( only in a pure setting)
• Linear + polymorphic CPS
Proving full-abstraction
Towards a semantic model for linearly-treated functions
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Standard CPS isn't fully abstract
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Standard CPS isn't fully abstract
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Standard CPS isn't fully abstract
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Standard CPS isn't fully abstract
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Standard CPS isn't fully abstract
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Standard CPS isn't fully abstract
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Standard CPS isn't fully abstract
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Standard CPS isn't fully abstract
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How can we modify the standardCPS translation to be fully abstract?
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Standard CPS
Ahmed & Blume's polymorphic CPS
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What happens when we add non-termination?
Standard CPS
Ahmed & Blume's polymorphic CPS
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Linear + polymorphic CPS
Standard CPS
Ahmed & Blume's polymorphic CPS
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Outline
Finding a fully abstract CPS translation w/
Proving full-abstraction• Polymorphic CPS proof
• Scaling to a non-terminating setting using linearity
Towards a semantic model for linearly-treated functions
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Proving polymorphic CPS fully abstract
Ahmed + Blume's approach relies ona type isomorphism:
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Proving polymorphic CPS fully abstract
The type isomorphism relies ona parametric condition:
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Parametric condition fails in the presence of non-termination
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Parametric condition fails in the presence of non-termination
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Parametric condition fails in the presence of non-termination
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Parametric condition fails in the presence of non-termination
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A step back
Ahmed & Blume's polymorphic CPS
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A step back
Ahmed & Blume's polymorphic CPS
requires
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A step back
Ahmed & Blume's polymorphic CPS
requires
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A step back
Ahmed & Blume's polymorphic CPS
requires
Linear + polymorphic CPS
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A step back
Ahmed & Blume's polymorphic CPS
requires
Linear + polymorphic CPS
requires
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A step back
Ahmed & Blume's polymorphic CPS
requires
Linear + polymorphic CPS
requires linear
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A “linear” parametric conditionin presence of non-termination
linear
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How do you prove thisnew parametric condition,
which uses continuations linearly?
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Free theorem: use a logical relation
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Free theorem: use a logical relation
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Free theorem: use a logical relation
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Free theorem: use a logical relation
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Free theorem: use a logical relation
?
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Proving
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Proving
know
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Proving
know
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Proving
showknow
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Proving
know show
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Proving
know:
showknow
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Proving
know:
show:
showknow
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Proving
know:
show:
showknow
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Proving
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Proving
know
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Proving
know
know
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Proving
show
know
know
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show
know
Proving
know
behaviour?
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show
know
know
Proving
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show
know
Proving
know
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show
know
Proving
=know
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Outline
Finding a fully abstract CPS translation w/
Proving full-abstraction
Towards a semantic model for linearly-treated functions
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?
Function semantics: case 1
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for all possible inputs
Function semantics: case 1
unrestrictedusage
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Function semantics: case 2
for all possible inputs &treat inputs linearly
unrestrictedusage
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for particular inputs
Function semantics: case 3
linearly used
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Function semantics: case 4
for particular inputs &treat input linearly
linearly used
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Relating linearly-treated functions
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Relating linearly-treated functions
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Relating linearly-treated functions
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Relating linearly-treated functions
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Relating linearly-treated functions
where
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Linearity by default
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Kripke logical relation for linearity
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Kripke logical relation for linearity
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Kripke logical relation for linearity
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Kripke logical relation for linearity
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Kripke logical relation for linearity
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Kripke logical relation for linearity
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Kripke logical relation for linearity
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Kripke logical relation for linearity
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Related Work
Fully abstract CPS for PCF [Laird]
• Uses game semantics proof
Logical relation for linear free theorems [Zhao et. al.]
• Open logical relation to ensure preservation of linear resources
TT logical relation for Lily [Bierman et. al.]
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Conclusion
Fully abstract CPS in a language with rec. types using operational proof techniques
requires
which requires
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Conclusion
Fully abstract CPS in a language with rec. types using operational proof techniques
requires
which requires
Kripke logical relation that can distinguishlinearly-treated: unrestricted:
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Why polymorphism:ST boundary semantics