ACTRIS COQ DEVELOPMENT

This directory contains the Coq mechanization of the Actris framework, first presented in the paper Actris: Session Type Based Reasoning in Separation Logic at POPL'20.

It has been built and tested with the following dependencies

• Coq 8.12.0
• 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 individual types contain the following:

• 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.v: The instantiation of protocols with the Iris logic, definition of iProto_own 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.
  • iProto_le: The subprotocol relation for protocols (notation ⊑).
• theories/channel/channel.v: The encoding of bidirectional channels in terms of Iris's HeapLang language, with specifications defined in terms of the dependent separation protocols. The relevant definitions and proof rules are as follows:
  • iProto_mapsto: endpoint ownership (notation ↣).
  • new_chan_spec, send_spec and recv_spec: proof rule for new_chan, send, and recv.
  • select_spec and branch_spec: proof rule for the derived (binary) select and branch operations.

Notation

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

Dependent Separation Protocols:

	POPL20 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

This notation is additionally used for the LMCS submission.

Semantic Session Types:

	CPP21 submission	Coq mechanization
Send	!_{X_1 .. X_n} A . S	<!! X_1 .. X_n> TY A ; S
Recv	?_{X_1 .. X_n} A . S	<?? X_1 .. X_n> TY A ; S
End	end	END
Select	(+){ Ss }	lty_choice SEND Ss
Branch	&{ Ss }	lty_choice RECV Ss
Dual	An overlined type	No notation
N-append	S^n	lty_napp S n

Coq tactics

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. Recursive session types can be defined using the mechanism defined in theories/logrel/model.v.
• theories/logrel/operators.v: Type definitions of unary and binary operators.
• theories/logrel/contexts.v: Definition of the semantic type contexts, which is used in the semantic typing relation. This also contains the rules for updating the context, which is used for distributing affine resources across the various parts of the proofs 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/subtyping_rules.v: Subtyping rules for term types and session types.
• 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/napp.v: Definition of session types iteratively being appended to themselves a finite number of times, with support for swapping e.g. a send over an arbitrary number of receives.

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 resources shared between multiple threads. An encoding of a list type is found in theories/logrel/lib/mutex.v, along with axillary lemmas, and a weakest precondition for llength, that converts ownership of a list type into a list reference predicate, with the values of the list made explicit.

Paper-specific remarks

For remarks about the CPP21 submission, see papers/POPL20.md. papers/CPP21.md. papers/LMCS.md.