The document presents a coeffect type system that generalizes effect systems. Coeffect systems track context-dependent computations using comonads rather than monads. The coeffect calculus models both per-variable and whole-context dependencies using indexed comonads. It can capture properties like variable liveness, implicit parameters, and distributed computing contexts.

1. Coeffects:A calculus of context-dependent computation
Tomas Petricek, Dominic Orchard and Alan Mycroft
University of Cambridge
{name.surname}@cl.cam.ac.uk|@tomaspetricek| @dorchard

2. Γ⊢푒:휏

3. Properties of computations
Effect systems
Effects on environment
Modeled using monads
Coeffect systems
Requirements on environment
Modeled using comonads
Γ⊢푒:휏&흈
Γ@풓⊢푒:휏

4. View from the extreme left
Variable related
Liveness
Bounded linear logic
Data-flow programming
Provenance
Environment related
Implicit parameters
Type classes
Distributed computing
Platform versioning

5. Variable related

6. Coeffect calculus for bounded reuse
Coeffect annotations
track number of variable uses
vectorof variable annotations + function annotation
richer structure than effect systems
푦:int@1⊢휆푥→푦+푥+푥∶푖푛푡→ퟐ 푖푛푡

7. Coeffect calculus for bounded reuse
structural rules = contraction + weakening + exchange 푥:휏@1⊢푥:휏 (푣푎푟) Γ,푥:휏1,푦:휏1@푅×푟,푠⊢푒∶휏2Γ,푧:휏1@푅×푟+푠⊢푒푥←푧,푦←푧:휏2(푐표푛푡푟) Γ,푥:휏1@푅×푠⊢푒:휏2Γ@푅⊢휆푥.푒∶휏1→푠 휏2(푎푏푠)

8. Coeffect calculus with historical values
Γ,푥:휏1,푦:휏1@푅×푟,푠⊢푒∶휏2Γ,푧:휏1@푅×max(푟,푠)⊢푒푥←푧,푦←푧:휏2(푐표푛푡푟)
푦:int@1⊢휆푥→푥+prev푥+푦∶푖푛푡→1 푖푛푡

9. Environment related

10. Coeffect calculus for implicit parameters
Tracking implicit parameters
Non-determinism can be fixed by annotation
@{?zone}⊢휆()→?time+?zone∶푢푛푖푡 {?zone} 푡푖푚푒
time} @{?time,?zone}⊢휆()→?time+?zone∶푢푛푖푡→∅ 푡푖푚푒

11. Coeffect calculus for implicit parameters
Tracking implicit parameters
Annotation does not mirror context structure
Not the same as reader monadΓ,푥:휏1@푟∪푠⊢푒:휏2Γ@푟⊢휆푥.푒∶휏1→푠 휏2

12. What is in the paper?

13. Unified system
Coeffect scalar structure (⊗,⊕,use,ign,≤)
Generalized semiringstructure
Individual variable or function annotations
Shape-indexed coeffect(⨲,,⊥)
Context splittingand merging
Per-variable or whole-context
⨲

14. Semantics using indexed comonads
Monadic semanticsComonadic semantics
It is not as simple
휆-calculus is asymmetric (many to one)
per-variable orwhole-context? Γ@푹⊢푒:휏:푪푹Γ→휏Γ⊢푒:휏&흈:Γ→푴흈휏

15. Why coeffects matter?
generalize interesting systems
liveness, data-flow, implicit parameters, type classes
comonadsare cool
model both per-variable and whole-context
indexed
{tomas.petricek, dominic.orchard, alan.mycroft}@cl.cam.ac.uk @tomaspetricek|@dorchard| #coeffects

16. EOF

17. Coeffect type system 푥:휏@use⊢푥:휏 (푣푎푟) Γ1@푅⊢푒1∶휏1→푡 휏2Γ2@푆⊢푒2∶휏1Γ1,Γ2@푅⨲푡⊗푆⊢푒1푒2:휏2(푎푝푝) Γ,푥:휏1@푅푠⊢푒:휏2Γ@푅⊢휆푥.푒∶휏1→푠 휏2(푎푏푠)
⨲ Γ,푥:휏1,푦:휏1@푅푟,푠⊢푒∶휏2Γ,푧:휏1@푅⨲푟⊕푠⊢푒푥←푧,푦←푧:휏2(푐표푛푡푟)
⨲

18. Effectsystems vs. coeffect systems
More powerful lambda abstraction
Richer structure
Sequential ⊗vs. point-wise ⊕composition
Vectors over variable context Γ,푥:휏1@푟∪푠⊢푒:휏2Γ@푟⊢휆푥.푒∶휏1→푠 휏2Γ,푥:휏1⊢푒:휏2&휎 Γ⊢휆푥.푒∶휏1→휎 휏2&∅

19. EOF

20. Why coeffects matter?
generalize interesting systems
liveness, data-flow, implicit parameters, type classes
comonadsare cool
model both per-variable and whole-context
indexed
{tomas.petricek, dominic.orchard, alan.mycroft}@cl.cam.ac.uk @tomaspetricek|@dorchard| #coeffects