Intersections and Unions of Session Types
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1 Intersections and Unions of Session Types Coşku Acay Frank Pfenning Carnegie Mellon University School of Computer Science ITRS 2016 C. Acay & F. Pfenning (CMU) Intersections and Unions of Session Types ITRS / 35
2 Overview 1 Background Message-passing Concurrency Session Types Subtyping Configurations and Reduction 2 Intersections and Unions Intersection Types Union Types Reinterpreting Choice 3 Algorithmic System 4 Metatheory C. Acay & F. Pfenning (CMU) Intersections and Unions of Session Types ITRS / 35
3 Plan 1 Background Message-passing Concurrency Session Types Subtyping Configurations and Reduction 2 Intersections and Unions Intersection Types Union Types Reinterpreting Choice 3 Algorithmic System 4 Metatheory C. Acay & F. Pfenning (CMU) Intersections and Unions of Session Types ITRS / 35
4 Setting Processes represented as nodes Channels go between processes and represented as edges Each channel is provided by a specific process (e.g. P provides c, Q provides d etc.): one-to-one correspondence between channels and processes g T c d e P Q R f S C. Acay & F. Pfenning (CMU) Intersections and Unions of Session Types ITRS / 35
5 Communication Processes compute internally Exchange messages along channels c d e P Q R
6 Communication Processes compute internally Exchange messages along channels c P d Q e R 3
7 Communication Processes compute internally Exchange messages along channels c d e P Q R 3 "aaa" end
8 Communication Processes compute internally Exchange messages along channels c d e P Q R 3 "aaa" end Note that communication is synchronous. C. Acay & F. Pfenning (CMU) Intersections and Unions of Session Types ITRS / 35
9 Higher-order Messages Processes can also send channels they own e R c d e P Q R
10 Higher-order Messages Processes can also send channels they own e R c d e P Q R e
11 Higher-order Messages Processes can also send channels they own e R c d e P Q R e C. Acay & F. Pfenning (CMU) Intersections and Unions of Session Types ITRS / 35
12 Session Types Don t want to send int if expecting string Don t try to receive if other process is not sending C. Acay & F. Pfenning (CMU) Intersections and Unions of Session Types ITRS / 35
13 Session Types Don t want to send int if expecting string Don t try to receive if other process is not sending Solution: Assign types to each channel C. Acay & F. Pfenning (CMU) Intersections and Unions of Session Types ITRS / 35
14 Session Types Don t want to send int if expecting string Don t try to receive if other process is not sending Solution: Assign types to each channel (from provider s perspective). C. Acay & F. Pfenning (CMU) Intersections and Unions of Session Types ITRS / 35
15 Session Types Don t want to send int if expecting string Don t try to receive if other process is not sending Solution: Assign types to each channel (from provider s perspective). c : A d : d int : string d string : B B B e : 1 P Q R
16 Session Types Don t want to send int if expecting string Don t try to receive if other process is not sending Solution: Assign types to each channel (from provider s perspective). c : A d : d int : string d string : B B B e : 1 P Q R 3
17 Session Types Don t want to send int if expecting string Don t try to receive if other process is not sending Solution: Assign types to each channel (from provider s perspective). c : A d : d int : string d string : B B B e : 1 P Q R 3 "aaa"
18 Session Types Don t want to send int if expecting string Don t try to receive if other process is not sending Solution: Assign types to each channel (from provider s perspective). c : A d : d int : string d string : B B B e : 1 P Q R 3 end "aaa"
19 Session Types Don t want to send int if expecting string Don t try to receive if other process is not sending Solution: Assign types to each channel (from provider s perspective). c : A d : d int : string d string : B B B e : 1 P Q R 3 end "aaa" C. Acay & F. Pfenning (CMU) Intersections and Unions of Session Types ITRS / 35
20 Why linear? Sessions are resources: communicating along a channel consumes the old type Contraction would violate type safety Weakening would work, but we keep things simple C. Acay & F. Pfenning (CMU) Intersections and Unions of Session Types ITRS / 35
21 Linear Propositions as Session Types 1 send end and terminate A B send channel of type A and continue as B τ B send value of type τ and continue as B {lab k : A k } k I send lab i and continue as A i for some i I A B receive channel of type A and continue as B τ B receive value of type τ and continue as B &{lab k : A k } k I receive lab i and continue as A i for some i I µt.a t (equi-)recursive type C. Acay & F. Pfenning (CMU) Intersections and Unions of Session Types ITRS / 35
22 Linear Propositions as Session Types 1 send end and terminate A B send channel of type A and continue as B τ B send value of type τ and continue as B {lab k : A k } k I send lab i and continue as A i for some i I A B receive channel of type A and continue as B τ B receive value of type τ and continue as B &{lab k : A k } k I receive lab i and continue as A i for some i I µt.a t (equi-)recursive type Example: Queue Interface type queue = &{ enq : A -o queue, deg : +{ none : 1, some : A * queue } } C. Acay & F. Pfenning (CMU) Intersections and Unions of Session Types ITRS / 35
23 Proof Terms as Concurrent Processes P, Q, R ::= x P x ; Q x cut (spawn) c d id (forward) close c wait c ; P 1 send c (y P y ) ; Q x recv c ; R x A B, A B c.lab ; P case c of {lab k Q k } k I &{lab k : A k } k I, {lab k : A k } k I C. Acay & F. Pfenning (CMU) Intersections and Unions of Session Types ITRS / 35
24 Example: An Implementation of Queues type queue = &{ enq : A -o queue, deg : +{ none : 1, some : A * queue } } empty : queue q <- empty = case q of enq -> x <- recv q; e <- empty ; q <- elem x e deq -> q. none ; close q elem : A -o queue -o queue q <- elem x r = case q of enq -> y <- recv q; r. enq ; send r y; q <- elem x r deq -> q. some ; send q x; q <- r C. Acay & F. Pfenning (CMU) Intersections and Unions of Session Types ITRS / 35
25 Process Typing Typing judgement has the form Ψ η P :: (c : A) meaning process P offers along channel c the session A under the context Ψ. η tracks recursive variables. C. Acay & F. Pfenning (CMU) Intersections and Unions of Session Types ITRS / 35
26 Process Typing Typing judgement has the form Ψ η P :: (c : A) meaning process P offers along channel c the session A under the context Ψ. η tracks recursive variables. Some examples: c : A d c :: (d : A) id Ψ P c :: (c : A) Ψ, c : A Q c :: (d : D) Ψ, Ψ c P c ; Q c :: (d : D) cut close c :: (c : 1) 1R Ψ P :: (d : A) Ψ, c : 1 wait c ; P :: (d : A) 1L Ψ P :: (d : A) Ψ Q :: (c : B) Ψ, Ψ send c (d P d ) ; Q :: (c : A B) R C. Acay & F. Pfenning (CMU) Intersections and Unions of Session Types ITRS / 35
27 Subtyping Width and depth subtyping for n-ary choices Width: &{lab k : A k } k I &{lab k : A k } k J whenever J I Depth: &{lab k : A k } k I &{lab k : A k} k I whenever A i A i C. Acay & F. Pfenning (CMU) Intersections and Unions of Session Types ITRS / 35
28 Subtyping Width and depth subtyping for n-ary choices Width: &{lab k : A k } k I &{lab k : A k } k J whenever J I Depth: &{lab k : A k } k I &{lab k : A k} k I whenever A i A i Defined coinductively because of recursive types C. Acay & F. Pfenning (CMU) Intersections and Unions of Session Types ITRS / 35
29 Subtyping Width and depth subtyping for n-ary choices Width: &{lab k : A k } k I &{lab k : A k } k J whenever J I Depth: &{lab k : A k } k I &{lab k : A k} k I whenever A i A i Defined coinductively because of recursive types Introduced to process typing using subsumption: Ψ η P :: (c : A ) A A Ψ η P :: (c : A) SubR Ψ, c : A η P :: (d : B) A A Ψ, c : A η P :: (d : B) SubL C. Acay & F. Pfenning (CMU) Intersections and Unions of Session Types ITRS / 35
30 Configurations A processes by itself is not very useful in concurrent setting Need to be able to talk about interactions C. Acay & F. Pfenning (CMU) Intersections and Unions of Session Types ITRS / 35
31 Configurations A processes by itself is not very useful in concurrent setting Need to be able to talk about interactions Use a process configuration, which is simply a set of labelled processes: Ω = proc c1 (P 1 ),..., proc cn (P n ). C. Acay & F. Pfenning (CMU) Intersections and Unions of Session Types ITRS / 35
32 Configurations A processes by itself is not very useful in concurrent setting Need to be able to talk about interactions Use a process configuration, which is simply a set of labelled processes: Ω = proc c1 (P 1 ),..., proc cn (P n ). Typing judgement, = Ω :: Ψ, imposes a tree structure (ensures linearity) C. Acay & F. Pfenning (CMU) Intersections and Unions of Session Types ITRS / 35
33 Reduction Configurations reduce by interaction. Some examples: id : proc c (c d) {c = d} cut : proc c (x P x ; Q x ) { a.proc a (P a ) proc c (Q a )} one : proc c (close c) proc d (wait c ; P) {proc d (P)} C. Acay & F. Pfenning (CMU) Intersections and Unions of Session Types ITRS / 35
34 Plan 1 Background Message-passing Concurrency Session Types Subtyping Configurations and Reduction 2 Intersections and Unions Intersection Types Union Types Reinterpreting Choice 3 Algorithmic System 4 Metatheory C. Acay & F. Pfenning (CMU) Intersections and Unions of Session Types ITRS / 35
35 Intersections and Unions What if we want to track more properties of queues? C. Acay & F. Pfenning (CMU) Intersections and Unions of Session Types ITRS / 35
36 Intersections and Unions What if we want to track more properties of queues? Empty, non-empty, even length? C. Acay & F. Pfenning (CMU) Intersections and Unions of Session Types ITRS / 35
37 Intersections and Unions What if we want to track more properties of queues? Empty, non-empty, even length? These can be defined in the base system: type empty - queue = &{ enq : A -o queue, deg : +{ none : 1} } type nonempty - queue = &{ enq : A -o queue, deg : +{ some : A * queue } } C. Acay & F. Pfenning (CMU) Intersections and Unions of Session Types ITRS / 35
38 Intersections and Unions What if we want to track more properties of queues? Empty, non-empty, even length? These can be defined in the base system: type empty - queue = &{ enq : A -o queue, deg : +{ none : 1} } type nonempty - queue = &{ enq : A -o queue, deg : +{ some : A * queue } } However, there is no way to properly track them! C. Acay & F. Pfenning (CMU) Intersections and Unions of Session Types ITRS / 35
39 We cannot track multiple refinements. C. Acay & F. Pfenning (CMU) Intersections and Unions of Session Types ITRS / 35
40 We cannot track multiple refinements. Consider concat : queue -o queue -o queue that concatenates two queues. It has many types but no most general type: concat : empty - queue -o empty - queue -o empty - queue concat : queue -o nonempty - queue -o nonempty - queue concat : nonempty - queue -o queue -o nonempty - queue C. Acay & F. Pfenning (CMU) Intersections and Unions of Session Types ITRS / 35
41 Intersection Types Intersection of two types: A B c : A B if channel c offers both behaviors simultaneously C. Acay & F. Pfenning (CMU) Intersections and Unions of Session Types ITRS / 35
42 Intersection Types Intersection of two types: A B c : A B if channel c offers both behaviors simultaneously Ψ η P :: (c : A) Ψ η P :: (c : B) Ψ η P :: (c : A B) R Ψ, c : A η P :: (d : D) Ψ, c : A B η P :: (d : D) L 1 Ψ, c : B η P :: (d : D) Ψ, c : A B η P :: (d : D) L 2 C. Acay & F. Pfenning (CMU) Intersections and Unions of Session Types ITRS / 35
43 Intersections Solve the Previous Problem We can now specify multiple behavioral properties: concat : empty - queue -o empty - queue -o empty - queue and queue -o nonempty - queue -o nonempty - queue and nonempty - queue -o queue -o nonempty - queue C. Acay & F. Pfenning (CMU) Intersections and Unions of Session Types ITRS / 35
44 Union Types Union of two types: A B c : A B if channel c offers either behavior C. Acay & F. Pfenning (CMU) Intersections and Unions of Session Types ITRS / 35
45 Union Types Union of two types: A B c : A B if channel c offers either behavior Dual to intersections C. Acay & F. Pfenning (CMU) Intersections and Unions of Session Types ITRS / 35
46 Union Types Union of two types: A B c : A B if channel c offers either behavior Dual to intersections Ψ η P :: (c : A) Ψ η P :: (c : A B) R 1 Ψ η P :: (c : B) Ψ η P :: (c : A B) R 2 Ψ, c : A η P :: (d : D) Ψ, c : B η P :: (d : D) Ψ, c : A B η P :: (d : D) L C. Acay & F. Pfenning (CMU) Intersections and Unions of Session Types ITRS / 35
47 Reasons for Adding Unions Maintain the symmetry of the system Makes working with internal choice more convenient Interpretation of internal choice (we will explain later) C. Acay & F. Pfenning (CMU) Intersections and Unions of Session Types ITRS / 35
48 Reasons for Adding Unions Maintain the symmetry of the system Makes working with internal choice more convenient Interpretation of internal choice (we will explain later) We can also write things like: type queue = empty - queue or nonempty - queue C. Acay & F. Pfenning (CMU) Intersections and Unions of Session Types ITRS / 35
49 Reinterpreting Choice Consider &{inl : A, inr : B} C. Acay & F. Pfenning (CMU) Intersections and Unions of Session Types ITRS / 35
50 Reinterpreting Choice Consider &{inl : A, inr : B} This type says: I will act as A if you send me inl and I will act as B if you send me inr. C. Acay & F. Pfenning (CMU) Intersections and Unions of Session Types ITRS / 35
51 Reinterpreting Choice Consider &{inl : A, inr : B} This type says: I will act as A if you send me inl and I will act as B if you send me inr. Interpreting and as gives &{inl : A, inr : B} &{inl : A} &{inr : B}. C. Acay & F. Pfenning (CMU) Intersections and Unions of Session Types ITRS / 35
52 Reinterpreting Choice - General Case Generalizing to n-ary choice and dualising gives: &{lab k : A k } k I k I &{lab k : A k } {lab k : A k } k I k I {lab k : A k } Easy to verify these definitions satisfy the typing rules. C. Acay & F. Pfenning (CMU) Intersections and Unions of Session Types ITRS / 35
53 Reinterpreting Choice - General Case Generalizing to n-ary choice and dualising gives: &{lab k : A k } k I k I &{lab k : A k } {lab k : A k } k I k I {lab k : A k } Easy to verify these definitions satisfy the typing rules. Suggests treating intersections and unions as implicit choice. C. Acay & F. Pfenning (CMU) Intersections and Unions of Session Types ITRS / 35
54 Plan 1 Background Message-passing Concurrency Session Types Subtyping Configurations and Reduction 2 Intersections and Unions Intersection Types Union Types Reinterpreting Choice 3 Algorithmic System 4 Metatheory C. Acay & F. Pfenning (CMU) Intersections and Unions of Session Types ITRS / 35
55 Algorithmic Subtyping Idea: make L {1,2} and R {1,2} invertible so we can apply eagerly. A {1,2} B A 1 A 2 B L {1,2} A B {1,2} A B 1 B 2 R {1,2} α, A 1, A 2 β α, A 1 A 2 β L α β, A 1, A 2 α β, A 1 A 2 R C. Acay & F. Pfenning (CMU) Intersections and Unions of Session Types ITRS / 35
56 Algorithmic Subtyping Idea: make L {1,2} and R {1,2} invertible so we can apply eagerly. A {1,2} B A 1 A 2 B L {1,2} A B {1,2} A B 1 B 2 R {1,2} α, A 1, A 2 β α, A 1 A 2 β L α β, A 1, A 2 α β, A 1 A 2 R Also admits distributivity: (A 1 B) (A 2 B) (A 1 A 2 ) B (A 1 A 2 ) B (A 1 B) (A 2 B) Turns out to be necessary for soundness of algorithmic typing. C. Acay & F. Pfenning (CMU) Intersections and Unions of Session Types ITRS / 35
57 Algorithmic Type Checking Make L {1,2} and R {1,2} invertible so we can apply eagerly. C. Acay & F. Pfenning (CMU) Intersections and Unions of Session Types ITRS / 35
58 Algorithmic Type Checking Make L {1,2} and R {1,2} invertible so we can apply eagerly. Delay subtyping to id Label cut with it s type C. Acay & F. Pfenning (CMU) Intersections and Unions of Session Types ITRS / 35
59 Algorithmic Type Checking Make L {1,2} and R {1,2} invertible so we can apply eagerly. Delay subtyping to id Label cut with it s type Ψ, c : A 1,2 η P :: (d : D) Ψ, c : A 1 A 2 η P :: (d : D) L1,2 Ψ, c : (α, A, B) η P :: (d : β) Ψ, c : (α, A B) η P :: (d : β) L Ψ η P :: (c : A 1,2 ) Ψ η P :: (c : A 1 A 2 ) R 1,2 Ψ η P :: (c : A, B, α) Ψ η P :: (c : A B, α) R C. Acay & F. Pfenning (CMU) Intersections and Unions of Session Types ITRS / 35
60 Algorithmic Type Checking Make L {1,2} and R {1,2} invertible so we can apply eagerly. Delay subtyping to id Label cut with it s type Ψ, c : A 1,2 η P :: (d : D) Ψ, c : A 1 A 2 η P :: (d : D) L1,2 Ψ, c : (α, A, B) η P :: (d : β) Ψ, c : (α, A B) η P :: (d : β) L Ψ η P :: (c : A 1,2 ) Ψ η P :: (c : A 1 A 2 ) R 1,2 Ψ η P :: (c : A, B, α) Ψ η P :: (c : A B, α) R α β c : α η d c :: (d : β) id Ψ η P c :: (c : A) Ψ, c : A η Q c :: (d : α) Ψ, Ψ η c : A P c ; Q c :: (d : α) cut C. Acay & F. Pfenning (CMU) Intersections and Unions of Session Types ITRS / 35
61 Results Theorem (Completeness of Algorithmic Subtyping) Algorithmic subtyping is complete with respect to declarative subtyping. Theorem (Equivalence of Algorithmic Typing) Algorithmic typing is sound and complete with respect to declarative typing. C. Acay & F. Pfenning (CMU) Intersections and Unions of Session Types ITRS / 35
62 Plan 1 Background Message-passing Concurrency Session Types Subtyping Configurations and Reduction 2 Intersections and Unions Intersection Types Union Types Reinterpreting Choice 3 Algorithmic System 4 Metatheory C. Acay & F. Pfenning (CMU) Intersections and Unions of Session Types ITRS / 35
63 Type Safety We have proved progress and preservation for the system extended with intersections and unions. C. Acay & F. Pfenning (CMU) Intersections and Unions of Session Types ITRS / 35
64 Type Safety We have proved progress and preservation for the system extended with intersections and unions. Progress deadlock-freedom Type preservation session fidelity C. Acay & F. Pfenning (CMU) Intersections and Unions of Session Types ITRS / 35
65 Progress Theorem (Progress) If = Ω :: Ψ then either 1 Ω Ω for some Ω, or 2 Ω is poised. Proof. By induction on = Ω :: Ψ followed by a nested induction on the typing of the root process. When two processes are involved, we also need inversion on client s typing. A process is poised if it is waiting to communicate with its client. C. Acay & F. Pfenning (CMU) Intersections and Unions of Session Types ITRS / 35
66 Type Preservation Theorem (Preservation) If = Ω :: Ψ and Ω Ω then = Ω :: Ψ. Proof. By inversion on Ω Ω, followed by induction on the typing judgments of the involved processes. Each branch requires a hand-rolled induction hypothesis. C. Acay & F. Pfenning (CMU) Intersections and Unions of Session Types ITRS / 35
67 Conclusion and Highlights We introduced intersection and union types to a session-typed process calculus and demonstrated their usefulness. Unions work naturally. The elimination rule we give has been shown unsound in the presence of effects (even non-termination). C. Acay & F. Pfenning (CMU) Intersections and Unions of Session Types ITRS / 35
68 Conclusion and Highlights We introduced intersection and union types to a session-typed process calculus and demonstrated their usefulness. Unions work naturally. The elimination rule we give has been shown unsound in the presence of effects (even non-termination). More general than refinement system of Freeman and Pfenning C. Acay & F. Pfenning (CMU) Intersections and Unions of Session Types ITRS / 35
69 Conclusion and Highlights We introduced intersection and union types to a session-typed process calculus and demonstrated their usefulness. Unions work naturally. The elimination rule we give has been shown unsound in the presence of effects (even non-termination). More general than refinement system of Freeman and Pfenning Subtyping resembles Gentzen s multiple conclusion calculus C. Acay & F. Pfenning (CMU) Intersections and Unions of Session Types ITRS / 35
70 Conclusion and Highlights We introduced intersection and union types to a session-typed process calculus and demonstrated their usefulness. Unions work naturally. The elimination rule we give has been shown unsound in the presence of effects (even non-termination). More general than refinement system of Freeman and Pfenning Subtyping resembles Gentzen s multiple conclusion calculus Algorithmic typing mirrors subtyping C. Acay & F. Pfenning (CMU) Intersections and Unions of Session Types ITRS / 35
71 Future Work Simple: integrate a functional language, extend to shared channels and asynchronous communication C. Acay & F. Pfenning (CMU) Intersections and Unions of Session Types ITRS / 35
72 Future Work Simple: integrate a functional language, extend to shared channels and asynchronous communication More interesting: Add polymorphism and abstract types Polymorphism is non-trivial with equirecursive types C. Acay & F. Pfenning (CMU) Intersections and Unions of Session Types ITRS / 35
73 Future Work Simple: integrate a functional language, extend to shared channels and asynchronous communication More interesting: Add polymorphism and abstract types Polymorphism is non-trivial with equirecursive types Applications other than refinements? C. Acay & F. Pfenning (CMU) Intersections and Unions of Session Types ITRS / 35
74 The End C. Acay & F. Pfenning (CMU) Intersections and Unions of Session Types ITRS / 35
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