A Kripke Logical Relation for Linear Functions:

The Story of a Free Theorem in the Presence of Non-termination

Phillip Mates & Amal AhmedNortheastern University

2

Programmers shouldn't have to think about compilers

3

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

7

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

41

Proving

know

42

Proving

know

43

Proving

showknow

44

Proving

know show

45

Proving

know:

showknow

46

Proving

know:

show:

showknow

47

Proving

know:

show:

showknow

48

Proving

49

Proving

know

50

Proving

know

know

51

Proving

show

know

know

52

show

know

Proving

know

behaviour?

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show

know

know

Proving

54

show

know

Proving

know

55

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

57

?

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

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