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.

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
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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.
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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.
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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).
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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
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6. Typing parallel composition by environment splitting
The rule
Γ1 P1 Γ2 P2
Γ1 + Γ2 P1 | P2
separates the end points of each channel.
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7. A type rule for output
Γ, c : T2 P Γ x : T1
Γ, c :!T1.T2 cx.P
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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
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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)
. . .
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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.
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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)
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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.
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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 .
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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
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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.
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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))
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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
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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
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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
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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
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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.
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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 .
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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.
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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)
. . .
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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.
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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.)
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