From C to Interaction trees: Specifying, Verifying, and Testing a networked server (CPP 2019 talk transcript)

Posted on January 14, 2019

This is a transcript of my talk at CPP 2019 (conference on Certified Programs and Proofs, co-hosted with POPL), presenting our efforts towards verifying a networked server.


This work is part of the Expedition in the Science of Deep Specification (DeepSpec). Led by four American universities (MIT, Yale, Princeton, Penn), DeepSpec aims to develop techniques to verify whole systems. DeepSpec encompasses many verification projects, and the work I’m going to present today is the first step towards connecting them together to verify a system from RFCs to transistors.

We will verify an application written in C using the Verified Software Toolchain (VST for short). It is a program logic for C, based on Hoare logic. One important reason for using VST is that the properties we prove are preserved by compilation with the CompCert compiler. This verified application will run on a verified OS, CertiKOS. And all this software will run on hardware implemented and verified with the Kami library.

Ultimately, we would like one theorem formalizing the correctness of the complete system. The main challenge of the DeepSpec project is to connect these components developed by different institutes to obtain such a unifying theorem.

Moreover, we would also like to make our specifications testable, as a way to relate specifications to the real world. For that, we will use QuickChick, a library for property-based testing: that allows to formulate specifications as automatically testable properties, producing concrete counterexamples when they are not satisfied.

Especially in the context of verification, testing provides two benefits:

  1. It can reveal bugs before proving, to gain confidence that the theorems being proved are indeed correct;

  2. and it can make specifications applicable to other implementations in the wild, ensuring that our interpretation of informal documents such as RFCs matches existing implementations. Implementations may also intentionally deviate from standard specifications, often for reasons related to performance or security.

A verified web server

As a final demo of the Deep Specification project, we aim to verify an HTTP server. By implementing such a widely used protocol we hope to show the applicability of verification at an industrial scale.

But today, rather than a project as complex as an HTTP server, we will instead focus on a simplified application. With this experiment we want to illustrate at a small scale the challenges in integrating different components to build a fully verified system.

And indeed, it still has the essential features allowing us to make the following contributions:

Case study: Swap server

Let’s see a simple example of what our server does. A swap server exchanges messages with one or many clients, and can store one message.

It is like an “echo server”, which repeats what it hears, but its responses are shifted to the next exchange. The key motivation for the swap server’s behavior is to make the simplest example of a stateful server.

Networked swap server

However, our server is implemented against a low-level socket interface. This exposes a wide range of interactions that the previous specification does not describe directly. The network, the hardware, and the OS can buffer messages internally in various ways. For example, this leads to messages being reordered, or delayed indefinitely.

These are behaviors that matter not only when we would run the server in production, but also during testing.

Network refinement

To account for the network, we adapt notions of observational refinement from the literature on concurrent and distributed systems.

The rough idea is to take a high-level specification of the server from earlier, and through the semantics of the network, we can define the observations that clients can make on the other side of the network.

In parallel, we also define the observable behavior of an implementation. Then, the correctness property relating the implementation and the specification is the inclusion of observable behaviors over the network, which we call network refinement.

Proof architecture

Here is a picture summarizing our system. Starting from the bottom, CertiKOS provides the environment to execute our server.

In particular, it provides a networking interface using sockets, against which our server is implemented. The aim is to show that the implementation respects the specification as illustrated in the previous slides.

First, we introduce an intermediate specification, the implementation model. The toplevel specification is now called the linear specification to distinguish those two forms of specifications. The implementation model is a low-level description of the server’s interactions with the socket interface, describing for example how the server buffers communications with many clients at the same time (whereas that kind of details doesn’t appear in the linear specification).

Using VST, we prove a refinement between the C program and the implementation model. Then, we prove that the implementation model network-refines the specification.

Of course, the fact that the C program refines the implementation model must rely on some semantics of the socket interface, which we wrote down first using VST.

Now, to relate the socket implementation to the VST specification, we also started formalizing the interface in CertiKOS. The implementation is however still not verified (thus we use a dashed arrow there in the diagram). The bridge between VST and CertiKOS is work-in-progress (as shown by the dotted line), and our paper gives a summary of our approach there.

As I said before, the long term goal of DeepSpec is to connect many components together into a whole verified system. This comes with various challenges that are already apparent in our simple experiment.

  1. To describe the system at different abstraction levels.

  2. To translate between varying specification styles already in use by the existing tools.

  3. To test the specification, both to reduce the engineering effort that will be required at a larger scale, and to be able to check whether our specification matches other existing implementations in the wild.

We propose interaction trees as a unifying structure to represent effectful computations across our development while addressing those challenges.

Interaction trees (aka. free monads)1

Intuitively, a program interacting with the outside world can be described as a tree, where nodes correspond to effects performed by the program, with one branch for each possible result of that effect. In this example, the program reads a bit, and if it is zero it goes to the left, if it is one it goes to the right, and so on, and it may return some value as a final result.

We can define interaction trees with many different kinds of effects. The simple example on the slide uses effects to read and write bits. Effects can be easily defined for any particular application as a sum type, whose constructors correspond to the effects that can be performed, and the type index gives the result type of each effect.


CoInductive itree (E : Type -> Type) (R : Type) : Type :=
| Vis : forall Y, E Y -> (Y -> itree E R) -> itree E R
| Ret : R -> itree E R
| Tau : itree E R -> itree E R

Programs such as web servers can run for indefinitely long, so their interactions trees can be infinite. This is why we declare the type of itrees to be coinductive. It is parameterized by an effect type E and a result type R.

An itree can perform an effect, of type E Y, with a continuation expecting a response of type Y from the environment. We can see this as a node of type E Y whose arity is the cardinality of Y.

An itree can terminate, returning a result of type R.

An itree can take a silent step of internal computation, without any visible effect. That last constructor is actually necessary to satisfy Coq’s guardedness condition when constructing nontrivial infinite trees.

To come back to the initial point, itrees provide a general-purpose representation that can be used at different abstraction levels and that seems compatible with many existing verification frameworks. At the same time since it is a simple coinductive type, so it can be easily extracted for testing.

Swap server: linear specification

Here is the linear specification of a swap server, as an interaction tree defined in Coq. It is a recursively defined itree parameterized by a set of open connections and the contents of the store, first nondeterministically choosing a connection c, receiving a new_msg from it, and sending back the last_msg from the store, before looping again with the new_msg in the store. Since it is defined monadically, it looks almost like a reference implementation, except for the nondeterminism at the top.

The implementation model is also an itree, but more complex since it reflects the C program’s logic of buffering and interleaving exchanges on multiple connections.

Refinement in VST

The other half of the proof is to show that the behavior of the C program is contained in the behavior of the implementation model, using VST’s logic. VST is a Hoare-style logic for C programs: specifications are given as pre- and postconditions on the program’s memory. Perhaps surprisingly, the implementation model appears in the precondition, as part of an ITree predicate. The idea is that it describes the interactions that are initially allowed by the external environment. A valid program under this precondition is one that follows those allowed behaviors.

When the program performs an effect corresponding to a node in the allowed itree, that itree gets replaced by the branch associated with the result of the effect. Here you can see a more concrete example of a C program allowed to perform a recv(), a send(), followed by effects described by the itree t.

{ ITree(msg <- Recv c ;; (* Coq *)
        Send c msg ;;
        t) * ... }

  recv(c, buf, len); // C
  send(c, buf, len);

{ ITree(t) * ... (* Coq *) }

The top-level correctness theorem

We can now state the theorem formalizing the correctness of our server which we proved. It relates the C program to the high-level linear specification as follows. The refinement relation I just described provides a model of the C program’s use of the socket interface as an itree. This implementation model is existentially quantified: it is a detail of the proof, rather than the toplevel specification. Finally, we show that the implementation model network-refines the linear specification: seen from across the network, the observable behavior of the former is included in that of the latter.

Theorem correct_server :
  exists impl_model,
            refines C_prog impl_model /\
    network_refines impl_model linear_spec.

Next steps

All this is only the first step of our project. There is a lot more work needed to apply the lessons we learned to a full HTTP server.

In the swap server we only talked to the socket interface. We will soon need to tackle the question of composing more systems together, such as a filesystem or a cryptographic library. The flexibility of itrees seems particularly well-suited to this kind of composition.

The development of the swap server tested the limits of VST, motivating some new features to model external effects, and there is certainly more work to make the integration with CertiKOS smoother.

We currently have a quite simplistic model of the network. We are certainly interested in techniques that will allow us to scale up, towards more complicated servers that talk to more realistic networks.

Speaking of interaction trees, we are working on putting together a general purpose library of itrees. At the moment our focus is on how to use paco to make coinduction compose better.

Thanks for listening/reading!

  1. If you have a functor F and take its free monad Free F, its inhabitants are what we call interaction trees here. Furthermore, to actually get an free monad, you should make the itree type defined here inductive instead of coinductive.↩︎