Compact Closed Freyd Category and -calculus Ken Sakayori & - PowerPoint PPT Presentation

Compact Closed Freyd Category and π -calculus Ken Sakayori & Takeshi Tsukada (The University of Tokyo) 23 December 2019

This talk is about Categorical type theory correspondenceb/w ◼ Closed Freyd categories [Power & Thielecke, 99] whose premonoidal category is compact closed ◼ a variant of 𝜌 -calculus that is not limited to race/deadlock-free processes ◼ Different from Curry-Howard correspondence for session-typed calculi [Caires et al, 16], [Wadler, 12]

Closed Freyd / Compact closed category Closed Freyd category [Power & Thielecke, 99] ◼ Model of the computational 𝜇 -calculus [Moggi, 89] ◼ Models computation sensitive to evaluation order using the premonoidal structure ◼ 𝑔 ⊗ 𝑗𝑒 ; 𝑗𝑒 ⊗ 𝑕 ≠ 𝑗𝑒 ⊗ 𝑕 ; 𝑔 ⊗ 𝑗𝑒 Compact closed category ◼ Used to represent circuit diagrams, networks, etc... [Joyal and Street, 91] ◼ Has been used to model a simple process calculus [Abramsky et al. 96]

Closed Freyd / Compact closed category Closed Freyd category [Power & Thielecke, 99] ◼ Model of the computational 𝜇 -calculus [Moggi, 89] ◼ Models computation sensitive to evaluation order using the premonoidal structure ◼ 𝑔 ⊗ 𝑗𝑒 ; 𝑗𝑒 ⊗ 𝑕 ≠ 𝑗𝑒 ⊗ 𝑕 ; 𝑔 ⊗ 𝑗𝑒 Compact closed category ◼ Used to represent circuit diagrams, networks, etc... [Joyal and Street, 91] ◼ Has been used to model a simple process calculus [Abramsky et al. 96] How can we use these categories to model 𝜌 -calculus?

Key Observation Consider 𝜌 -calculus as a process passing calculus (instead of name passing calculus) [Sangiorgi 93] ◼ Regard as a function located at 𝑏 (i.e. ) ◼ Regard as ത 𝑏 applied to ◼ Then the reduction can be seen as

Key Observation Consider 𝜌 -calculus as a process passing calculus (instead of name passing calculus) [Sangiorgi 93] ◼ Regard as a function located at 𝑏 (i.e. ) ◼ Regard as ത 𝑏 applied to ◼ Abstraction 𝜇𝑦. 𝑄 , application ത 𝑏⟨𝑦⟩ (functional part) can be modeled using the closed Freyd structure Address assignment 𝑏@ (−) ( ≃ connection) ◼ can be modeled using the compact closed structure 𝑏 𝜇𝑦. 𝑄

Outline ◼ Introduction ◼ Syntax of the Target Calculus ◼ Categorical Semantics ◼ Equational Theory of the Calculus ◼ Connection w/ logic ◼ Conclusion

Outline ◼ Introduction ◼ Syntax of the Target Calculus ◼ Categorical Semantics ◼ Equational Theory of the Calculus ◼ Connection w/ logic ◼ Conclusion

Target calculus ( 𝝆 𝑮 -calculus) A subcalculus of the asynchronous 𝜌 -calculus Processes 𝑄, 𝑅 ∷= 0 ∣ 𝑦 Ԧ 𝑧 ∣ 𝑄 𝑅 ∣ ! 𝑦 Ԧ 𝑧 . 𝑄 ∣ 𝜉 𝑦 𝑧 𝑄 Only replicated inputs are allowed ◼ Reflects the idea that input prefixing represents a function that can be used arbitrary number of times (𝜉𝑦𝑧) means “ connect 𝑦 and 𝑧 ” ◼ Creates input and output endpoints of a channel ◼ Communications occur b/w names connected by 𝜉 NB Calculi with similar characteristics have been studied (e.g. [Honda &Laurent 10], [Laird 05] )

Types for the 𝝆 𝑮 -calculus Types ◼ Names are either used as input or output channel ◼ Types have duality ( ) Typing rules where Γ ∷= ⋅ ∣ Γ, 𝑦: 𝑈 and is the type for processes

Compact closed Freyd category (CCFC) Cartesian Compact closed Identity-on-obj. strict symmetric monoidal with the right adjoint Remark The Kleisli exponential can be defined by

Examples of CCFC Category of posets and downward-closed relations In ,

Interpretation The interpretation is a morphism from to in (the category of computation/ compact closed category) where

Interpretation (by example) unit morph. Notation

Interpretation (by example) adjunction iso. counit morph.

Example (Reduction)

Example (Reduction)

Example (Reduction)

Example (Reduction)

Example (Reduction)

Properties of interpretation Prop. (Soundness wrt. reductions) implies If and Thm. (Soundness wrt. equational theory) If then for every 𝐾. next topic

Outline ◼ Introduction ◼ Syntax of the Target Calculus ◼ Categorical Semantics ◼ Equational Theory of the Calculus ◼ Connection w/ logic ◼ Conclusion

What is the corresponding equational logic? ◼ Is it a known behavioral equivalence? ◼ Barbed congruence, bisimilarity, testing equivalences... ◼ Are the rules appropriate from operational viewpoint? ◼ What corresponds to composition?

What is the corresponding equational logic? ◼ Is it a known behavioral equivalence? ◼ Barbed congruence, bisimilarity, testing equivalences... No ◼ Are the rules appropriate from operational viewpoint? Yes, except for a single rule ◼ What corresponds to composition? Parallel composition + hiding

Equational rules All but (E-Eta) are quite standard Rules (Rules for structural congruence are omitted) (E-Beta) (E-GC) (E-FOut) (E-Eta) where

Equational rules (E-Beta) ◼ Rule similar to the reduction relation ◼ If 𝐷 = , we have ◼ The side condition ensures that there is only one input that is waiting for the output ◼ 𝐷 is not limited to the contexts of the form [ ] | Q

Equational rules (E-Beta) (E-GC) ◼ Garbage collection law

Equational rules (E-Beta) (E-GC) (E-FOut) where ◼ Well-studied law that equates a free output with a bound output + forwarder ◼ cf. Translation from the 𝜌 -calculus to the internal 𝜌 -calculus [Boreale 98]

Equational rules (E-Beta) (E-GC) (E-FOut) (E-Eta) where

Problem of the 𝜽 -rule is not valid from the operational viewpoint Example should not be equal to because these can be distinguished by

Remarks on The argument that shows ◼ is widely applicable to i/o-typed asynchronous 𝜌 -calculi, not specific to 𝜌 𝐺 ◼ uses the existence of race ◼ e.g. ◼ cf. session-typed calculus corresponding to linear logic [Caires et al. 16] is race-free

Operational Properties Prop. If without using (E-Eta) then and are weak barbed congruent “contextual equivalence” for the 𝜌 -calculus Prop. If then and are may-testing equivalent ◼ May-testing is a rather coarse equivalence

Is the 𝜽 -rule necessary? Yes, in order to make the term model a category provided that ◼ morphisms = processes modulo some “well - behaved” equivalence ◼ composition = “parallel composition + hiding” Without this rule we obtain a semicategory (cf. 𝛾 -theory of the 𝜇 -calculus [Hayashi, 95] )

Is the 𝜽 -rule necessary? Yes, in order to make the term model a category Without this rule the we obtain a semicategory (cf. 𝛾 -theory of the 𝜇 -calculus [Hayashi, 95]) Reason: says that is ◼ a left identity ◼ However, is a right identity i.e, holds for most of the behavioral equivalences

Theory/model correspondence Thm. Processes modulo the equational theory forms a CCFC that classifies CCFCs that satisfy the following strictness condition: The canonical isomorphisms are identities The i/o-type only have type of the following form

Digression: Operational/Categorical reading of From a (traditional) operational viewpoint, is a process that works as a buffer Whereas the category theoretic observation suggests us to treat as a “wire” instead of a “buffer” cannot keep a message, it should transmit a ◼ message w/o making any “observational event” ◼ In this setting, a “buffer” may be represented as where 𝜐 is a special constant represents an “event”

Digression: Operational/Categorical reading of From a (traditional) operational viewpoint, is a process that works as a buffer Whereas the category theoretic observation suggests us to treat as a “wire” instead of a “buffer” cannot keep a message, it should transmit a ◼ message w/o making any “observational event” ◼ In this setting, a “buffer” may be represented as where 𝜐 is a special constant represents an “event” It might be possible to translate the conventional 𝜌 -calculus into 𝜌 𝐺 -calculus with additional constants

Outline ◼ Introduction ◼ Syntax of the Target Calculus ◼ Categorical Semantics ◼ Equational Theory of the Calculus ◼ Connection w/ logic ◼ Conclusion

Connection with Logic (CCFC) (L/NL-model) (MELL) ???

Connection with Logic Conjecture (informal) (CCFC) (L/NL-model) (MELL) MELL + + since we are using compact closed categories ◼ because 𝜌 𝐺 allows to duplicate names with ◼ input type (= ? modality) ◼ Similar rule has been considered in [Atkey et al. 16]