Binary Session Types for Psi-Calculi (APLAS 2016)
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This document presents a generic type system for binary session types in the ψ-calculus setting. It defines a generic type system where types have addition and transition rules, and instances of the type system are defined by specifying types, terms, assertions, and conditions. The generic system includes type rules for processes and assertions. It proves a general fidelity theorem showing that known type systems can be represented as instances and fidelity holds for each instance. Further work includes improving the characterization of type safety for instances and handling other notions of duality.
![The role of assertions
Assertions can be used to represent e.g.
Active substitutions
(νx)(P | [x := M])
Name fusions
P | [a = b]
Assertions are composed using an operator called ⊗ and compared
using the equivalence relation .
Hans H¨uttel
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Binary Session Types for Psi-Calculi (APLAS 2016)
- 1. Binary session types for ψ-calculi APLAS 2016 Hanoi, Vietnam,November 2016 Hans H¨uttel Department of Computer Science Aalborg University Selma Lagerl¨ofs Vej 300 Denmark 23 November 2016 1 / 45
- 2. What are binary session types? A type discipline for communicating processes with channels due to Honda, Kubo and Vasconcelos. The type of a channel describes the values that can be transmitted along it. Different kinds of values can be transmitted on the same channel at different times. A channel has two endpoints. The type of a channel describes the protocol followed by the endpoints. Hans H¨uttel Binary session types for ψ-calculi 2
- 3. What are binary session types? (νc : S)(ac.c(x).cx = 3 P1 | a(y).y17.y(x).P(x) P2 ) The channel c has two endpoints, c+ and c−. In P1 c+ follows the protocol T: c+ :?Int.!Bool.end In P2 c− follows the dual protocol T c− :!Int.?Bool.end c has type S = (T, T). The endpoint types are dual; we say that c is balanced. Hans H¨uttel Binary session types for ψ-calculi 3
- 4. What are binary session types? Honda, Kubo and Vasconcelos added the notions of branching and selection to the π-calculus. a lk.Q selection | a {l1 : P1, . . . , lk : Pk, . . . ln : Pn} branching → Q | Pk The type of an endpoint used for selection: {l1, T1, . . . , ln : Tn} The type of an endpoint used for branching: &{l1, T1, . . . , ln : Tn} The type of a branching/selection name a is (T, T). Hans H¨uttel Binary session types for ψ-calculi 4
- 5. Adding types and type environments We define addition of types by (T1, T2) = T1 + T2. This corresponds to joining/separating endpoints. A type environment Γ is a function Γ : Names → Types. We define (Γ1 + Γ2)(x) = T1 + T2 if Γ1(x) = T1, Γ2(x) = T2 T1 if Γ1(x) = T1, Γ2(x) undefined T2 if Γ2(x) = T2, Γ1(x) undefined Hans H¨uttel Binary session types for ψ-calculi 5
- 6. Typing parallel composition by environment splitting The rule Γ1 P1 Γ2 P2 Γ1 + Γ2 P1 | P2 separates the end points of each channel. Hans H¨uttel Binary session types for ψ-calculi 6
- 7. A type rule for output Γ, c : T2 P Γ x : T1 Γ, c :!T1.T2 cx.P Hans H¨uttel Binary session types for ψ-calculi 7
- 8. Usual properties of binary session type systems Theorem (Fidelity) If Γ P, if all types in Γ and P are balanced and P τ −→ P then Γ P where all types in Γ and P are balanced and If c was used in the τ-step, then if Γ(c) = T, then Γ (c) = T where T is the “remains” of T Hans H¨uttel Binary session types for ψ-calculi 8
- 9. A plethora of binary session type systems Type systems for progress (Vieira and Vasconcelos) Type systems for refinement types (Baltazar, Mostrous and Vasconcelos) Type systems with subtyping (Gay and Hole) . . . Hans H¨uttel Binary session types for ψ-calculi 9
- 10. A generic type system The goal of this work is to present a generic type system such that Known binary session type systems can be represented as instances of the generic system A general fidelity theorem holds such that fidelity holds for every instance of the generic system So whenever we see a new type system in the future, all we need to do is to show that it is an instance of the generic system. Hans H¨uttel Binary session types for ψ-calculi 10
- 11. A generic process calculus framework: ψ-calculi Bengtson, Johansson, Parrow and Victor introduce ψ-calculi as a common generalization of many π-like process calculi. In a psi-calculus, any term can be used as a channel. Here is a ψ-calculus with numbers. 5 83 .84(x).84 x = 3 | 5(y).y + 1 17 .y(x).P(x) Hans H¨uttel Binary session types for ψ-calculi 11
- 12. Terms, assertions and conditions Processes use a syntax similar to that of π-calculus and also make use of T data terms M, N C conditions ϕ A assertions Ψ These are different for each instance. Hans H¨uttel Binary session types for ψ-calculi 12
- 13. The role of assertions Assertions can be used to represent e.g. Active substitutions (νx)(P | [x := M]) Name fusions P | [a = b] Assertions are composed using an operator called ⊗ and compared using the equivalence relation . Hans H¨uttel Binary session types for ψ-calculi 13
- 14. Processes in ψ-calculi Every process calculus that is a ψ-calculus has the same syntax. P ::= M(λx)N.P input with pattern (λx)N MN.P output P1 | P2 parallel composition (νx : T)P restriction ∗ P replication case ϕ1 : P1, . . . , ϕk : Pk conditional M l.P1 selection M {l1 : P1, . . . , lk : Pk} branching (|Ψ|) assertion Hans H¨uttel Binary session types for ψ-calculi 14
- 15. ψ-calculi: Semantics Bengtson et al. give a labelled semantics of ψ-calculi. Transitions are of the form Ψ P α −→ P Ψ is a global assertion. Think of it as the knowledge external to the process P. Hans H¨uttel Binary session types for ψ-calculi 15
- 16. Session channels in the generic system In a psi-calculus, any term can be used as a channel. 5 83 .84(x).84 x = 3 | 5(y).y + 1 17 .y + 3(x).P(x) How can we set up new session channels? We introduce names that are session constructors. Applying a session constructor to a term gives us a session channel. (5@c d, 83 .d@84(x).d@84 x = 3 | c@5(s, y).s@(y + 1) 17 .s@(y + 3)(x).P(x)) Hans H¨uttel Binary session types for ψ-calculi 16
- 17. Types in the generic type system Types have an addition operator defined. Types have transitions that describe the protocol steps followed by a channel: T1 !T2 −−→ T3 T4 ?T5 −−→ T6 T1 T2 −−−→ T3 T4 T5 −−−→ T6 Hans H¨uttel Binary session types for ψ-calculi 17
- 18. How to define an instance of the type system Define the types The set of types Define how to add types T1 + T2 Define transitions for types Define the missing type rules Type rules for terms: Γ, Ψ M : T Type rules for assertions: Γ, Ψ Ψ Type rules for conditions: Γ, Ψ ϕ Hans H¨uttel Binary session types for ψ-calculi 18
- 19. Typing processes Typings can depend on The types of names; these are recorded in Γ Global knowledge (e.g. identities on names); this is recorded in an assertion Ψ The type judgements for processes have the form Γ, Ψ P Hans H¨uttel Binary session types for ψ-calculi 19
- 20. An example rule from the generic system (Out) Γ1, Ψ1 min M : T1@c T1 !,T2 −−→ T3 Γ2, Ψ2 min N : T2 Γ3 + c : T3, Ψ3 P Γ1 + Γ2 + Γ3, Ψ1 Ψ2 Ψ3 MN.P Hans H¨uttel Binary session types for ψ-calculi 20
- 21. Fidelity, generalized Definition We write Γ, Ψ bal P if all types in Γ and in P are balanced. Definition Let α be an action. We let Γ ↑ α denote the type environment where the session type for the names used in α have progressed. Hans H¨uttel Binary session types for ψ-calculi 21
- 22. Fidelity, generalized Theorem Suppose we have Ψ0 P τ −→ P , that Γ, Ψ bal P and Ψ ≤ Ψ0. Then for some Ψ ≤ Ψ we have Γ ↑ τ, Ψ bal P . Hans H¨uttel Binary session types for ψ-calculi 22
- 23. Progress A type system due to Vieira and Vasconcelos that guarantees progress in the π-calculus. Type judgments are of the form Γ, P where is a well-founded order on names. The key insight is to represent these as assertions. Here, the representation of the type system makes use of psi-assertions to play the part of well-founded orders. Hans H¨uttel Binary session types for ψ-calculi 23
- 24. Other type systems that are instances of our generic system A type system for refinement types (Baltazar, Mostrous and Vasconcelos) A type system with subtyping (Gay and Hole) . . . Hans H¨uttel Binary session types for ψ-calculi 24
- 25. Conclusions A generic type system for binary session types in the psi-calculus setting. A general fidelity result Known type systems can be represented as instances of the generic system. Hans H¨uttel Binary session types for ψ-calculi 25
- 26. Further work There is no general theorem that tells us what it means to type-safe for instances. Can we find a way to improve on this? Our generic type system can capture both liveness and safety properties! Can we handle notions of duality other than the standard one? (Cf. Bernardi et al.) Hans H¨uttel Binary session types for ψ-calculi 26