README.md

ACTRIS COQ DEVELOPMENT

This directory contains the artifact for the paper "Actris: Session Type Based Reasoning in Separation Logic".

It has been built and tested with the following dependencies

• Coq 8.11.1
• The version of Iris in the opam file

In order to build, install the above dependencies and then run make -j [num CPU cores] to compile Actris.

Theory of Actris

The theory of Actris (semantics of channels, the model, and the proof rules) can be found in the directory theories/channel. The files correspond to the following parts of the paper:

• theories/channel/channel.v: The definitional semantics of bidirectional channels in terms of Iris's HeapLang language.
• theories/channel/proto_model.v: The construction of the model of Dependent Separation Protocols as the solution of a recursive domain equation.
• theories/channel/proto_channel.v: The instantiation of protocols with the Iris logic, definition of the connective ↣ for channel endpoint ownership, and lemmas corresponding to the Actris proof rules. The relevant definitions and proof rules are as follows:
  • iProto Σ: The type of protocols.
  • iProto_message: The constructor for sends and receives.
  • iProto_end: The constructor for terminated protocols.
  • mapsto_proto: endpoint ownership ↣.
  • new_chan_proto_spec: proof rule for new_chan.
  • send_proto_spec and send_proto_spec_packed: proof rules for send, the first version is more convenient to use in Coq, but otherwise the same as the latter, which is the rule in the paper.
  • recv_proto_spec and recv_proto_spec_packed: proof rules for recv, the first version is more convenient to use in Coq, but otherwise the same as the latter, which is the rule in the paper.
  • select_spec: proof rule for select.
  • branch_spec: proof rule for branch.

Notation

The following table gives a mapping between the notation in the paper and the Coq mechanization:

	Paper	Coq mechanization
Send	! x_1 .. x_n <v>{ P }. prot	<!> x_1 .. x_n, MSG v {{ P }}; prot
Recv	? x_1 .. x_n <v>{ P }. prot	<?> x_1 .. x_n, MSG v {{ P }}; prot
End	end	END
Select	prot_1 {Q_1}⊕{Q_2} prot_2	prot_1 <{Q_1}+{Q_2}> prot_2
Branch	prot_1 {Q_1}&{Q_2} prot_2	prot_1 <{Q_1}&{Q_2}> prot_2
Append	prot_1 · prot_2	prot_1 <++> prot_2
Dual	An overlined protocol	No notation

Weakest preconditions and Coq tactics

The presentation of Actris logic in the paper makes use of Hoare triples. In Coq, we make use of weakest preconditions because these are more convenient for interactive theorem proving using the the proof mode tactics. To state concise program specifications, we use the notion of Texan Triples from Iris, which provides a convenient "Hoare triple"-like syntax around weakest preconditions:

{{{ P }}} e {{{ x .. y, RET v; Q }}} :=
  □ ∀ Φ, P -∗ ▷ (∀ x .. y, Q -∗ Φ v) -∗ WP e {{ Φ }}

In order to prove programs using Actris, one can make use of a combination of Iris's symbolic execution tactics for HeapLang programs and Actris's symbolic execution tactics for message passing. The Actris tactics are as follows:

• wp_send (t1 .. tn) with "selpat": symbolically execute send c v by looking up ownership of a send protocol H : c ↣ <!> y1 .. yn, MSG v; {{ P }}; prot in the proof mode context. The tactic instantiates the variables y1 .. yn using the terms t1 .. tn and uses selpat to prove P. If fewer terms t are given than variables y, they will be instantiated using existential variables (evars). The tactic will put H : c ↣ prot back into the context.
• wp_recv (x1 .. xn) as "ipat": symbolically execute recv c by looking up H : c ↣ <?> y1 .. yn, MSG v; {{ P }}; prot in the proof mode context. The variables y1 .. yn are introduced as x1 .. xn, and the predicate P is introduced using the introduction pattern ipat. The tactic will put H : c ↣ prot back into the context.
• wp_select with "selpat": symbolically execute select c b by looking up H : c ↣ prot1 {Q1}<+>{Q2} prot2 in the proof mode context. The selection pattern selpat is used to resolve either Q1 or Q2, based on the chosen branch b. The tactic will put H : c ↣ prot1 or H : c ↣ prot2 back into the context based on the chosen branch b.
• wp_branch as ipat1 | ipat2: symbolically execute branch c e1 e2 by looking up H : c ↣ prot1 {Q1}<&>{Q2} prot2 in the proof mode context. The result of the tactic involves two subgoals, in which Q1 and Q2 are introduced using the introduction patterns ipat1 and ipat2, respectively. The tactic will put H : c ↣ prot1 and H : c ↣ prot2 back into the contexts of the two respectively goals.

The above tactics implicitly perform normalization of the protocol prot in the hypothesis H : c ↣ prot. For example, wp_send also works if there is a channel with the protocol iProto_dual ((<?> y1 .. yn, MSG v; {{ P }}; END) <++> prot). Concretely, the normalization performs the following actions:

• It re-associates appends (<++>), and removes left-identities (END) of it.
• It moves appends (<++>) into sends (<!>), receives (<?>), selections (<+>) and branches (<&>).
• It distributes duals (iProto_dual) over append (<++>).
• It unfolds prot1 into prot2 if there is an instance of the type class ProtoUnfold prot1 prot2. When defining a recursive protocol, it is useful to define a ProtoUnfold instance to obtain automatic unfolding of the recursive protocol. For example, see sort_protocol_br_unfold in theories/examples/sort_br_del.v.

Semantic Session Type System

The logical relation for type safety of a semantic session type system is contained in the directory theories/logrel. The logical relation is defined across the following files:

• theories/logrel/model.v: Definition of the notions of a semantic term type and a semantic session type in terms of unary Iris predicates (on values) and Actris protocols, respectively. Also provides the required Coq definitions for creating recursive term/session types.
• theories/logrel/term_types.v: Definitions of the following semantic term types: basic types (integers, booleans, unit), sums, products, copyable/affine functions, universally and existentially quantified types, unique/shared references, and session-typed channels.
• theories/logrel/session_types.v: Definitions of the following semantic session types: sending and receiving with session polymorphism, n-ary choice. Session type duality is also defined here. As mentioned, recursive session types can be defined using the mechanism defined in theories/logrel/model.v.
• theories/logrel/environments.v: Definition of semantic type environments, which are used in the semantic typing relation. This also contains the rules for splitting and copying of environments, which is used for distributing affine resources across the various parts of the program inside the typing rules.
• theories/logrel/term_typing_judgment.v: Definition of the semantic typing relation, as well as the proof of type soundness, showing that semantically well-typed programs do not get stuck.
• theories/logrel/subtyping.v: Definition of the semantic subtyping relation for both term and session types. This file also defines the notion of copyability of types in terms of subtyping.
• theories/logrel/term_typing_rules.v: Semantic typing lemmas (typing rules) for the semantic term types.
• theories/logrel/session_typing_rules.v: Semantic typing lemmas (typing rules) for the semantic session types.
• theories/logrel/subtyping_rules.v: Subtyping rules for term types and session types.

An extension to the basic type system is given in theories/logrel/lib/mutex.v, which defines mutexes as a type-safe abstraction. Mutexes are implemented using spin locks and allow one to gain exclusive ownership of resource shared between multiple threads.

The logical relation is used to show that two example programs are semantically well-typed:

• theories/logrel/examples/pair.v: This program performs two sequential receives and stores the results in a pair. It is shown to be semantically well-typed by applying the semantic typing rules.
• theories/logrel/examples/double.v: This program performs two ``racy'' parallel receives on the same channel from two different threads, using locks to allow the channel to be shared. This program cannot be shown to be well-typed using the semantic typing rules. Therefore, a manual proof of the well-typedness is given.
• theories/examples/subprotocols.v: Contains an example of a subprotocol assertion between two protocols that sends references.

Examples

The examples can be found in the direction theories/examples.

The following list gives a mapping between the examples in the paper and their mechanization in Coq:

1. Introduction: theories/examples/basics.v
2. Tour of Actris
   • 2.3 Basic: theories/examples/sort.v
   • 2.4 Higher-Order Functions: theories/examples/sort.v
   • 2.5 Branching: theories/examples/sort_br_del.v
   • 2.6 Recursion: theories/examples/sort_br_del.v
   • 2.7 Delegation: theories/examples/sort_br_del.v
   • 2.8 Dependent: theories/examples/sort_fg.v
3. Manifest sharing via locks
   • 3.1 Sample program: theories/examples/basics.v
   • 3.2 Distributed mapper: theories/examples/map.v
4. Case study: map reduce:
   • Utilities for shuffling/grouping: theories/utils/group.v
   • Implementation and verification: theories/examples/map_reduce.v

Differences between the formalization and the paper

There are a number of small differences between the paper presentation of Actris and the formalization in Coq, that are briefly discussed here.

• Notation

  See the section "Notation" above.

• Weakest preconditions versus Hoare triples

  See the section "Weakest preconditions and Coq tactics" above.

• Connectives for physical ownership of channels

  In the paper, physical ownership of a channel is formalized using a single connective (c1,c2) ↣ (vs1,vs2), while the mechanization has two connectives for the endpoints and one for connecting them, namely:

  • chan_own γ Left vs1 and chan_own γ Right vs1
  • is_chan N γ c1 c2

  Here, γ is a ghost name and N an invariant name. This setup is less intuitive but gives rise to a more practical Jacobs/Piessens-style spec of recv that does not need a closing view shift (to handle the case that the buffer is empty).

• Later modalities in primitive rules for channels

  The primitive rules for send and recv (send_spec and recv_spec in theories/channel/channel.v) contain three later (▷) modalities, which are omitted for brevity's sake in the paper. These later modalities expose that these operations perform at least three steps in the operational semantics, and are needed to deal with the three levels of indirection in the invariant for protocols:

  1. the ▶ in the model of protocols,
  2. the higher-order ghost state used for ownership of protocols, and
  3. the opening of the protocol invariant.

• Protocol subtyping

  The mechanization has introduced the notion of "protocol subtyping", which allows one to strengthen/weaken the predicates of sends/receives, respectively. This achieved using the relation iProto_le p p', and the additional rule c ↣ p -∗ iProto_le p p' -∗ c ↣ p'. To support "protocol subtyping", the definition of c ↣ p in the model is changed to be closed under iProto_le.