The Science of Deep Specification (Appel et al., 2017) is an ambitious experiment in specification, rigorous testing, and formal verification “from internet RFCs down to transistors” of real-world systems such as web servers. The principal challenges lie in integrating disparate specification styles, legacy specifications, and testing and verification tools to build and reason about complex, multi-layered systems.
We report here on a first step toward realizing this vision: an in-depth case study demonstrating how to specify, test, and verify a simple networked server with the same fundamental interaction model as more sophisticated ones—it communicates with multiple clients via ordered, reliable TCP connections. Our server is implemented in C and verified, using the Verified Software Toolchain (Appel, 2014), against a formal “implementation model” written in Coq [(2018)]; this is further verified (in Coq) against a linear “one client at a time” specification of allowed behaviors. The main property we prove is that any trace that can be observed by a collection of concurrent clients interacting with the server over the network can be rearranged into a trace allowed by the linear specification. We also show how property-based random testing using Coq’s QuickChick plugin (Lampropoulos and Pierce, 2018) can be deployed in this setting, both for detecting disagreements between the implementation and specification and for validating the specification itself against legacy servers. We compile the server code with the CompCert verified compiler (Leroy, 2009) and run it on CertiKOS (Gu et al., 2016), a verified operating system with support for TCP socket operations.
Our verified server provides a simple “swap” interface that allows clients to send a new bytestring to the server and receive the currently stored one in exchange. It is simpler in many respects than a full-blown web server; in particular, it follows a much simpler protocol (no authentication, encryption, header parsing, etc.), which means that it can be implemented with much less code.
Moreover, the degree of vertical integration falls short of our ultimate ambitions for the DeepSpec project, since we stop at the CertiKOS interface (which we axiomatize) instead of going all the way down to transistors. On the other hand, the C implementation of our server is realistic enough that it offers a challenging test of how to integrate disparate Coq-based methodologies and tools for verifying and testing systems software. In particular, it uses a single-process, event-driven architecture (Pai et al., 1999), hides latency by buffering interleaved TCP communications from multiple clients, and is built on the POSIX socket API.
We describe our experiences integrating Coq, CompCert, VST, CertiKOS, and QuickChick to build a verified swap server. This is the first VST verification of a program that interacts with the external environment. It is also, to the best of our knowledge, the first verification of functional correctness of a networked server implemented in C. Our technical contributions are as follows:
First, we identify interaction trees (ITrees)—a Coq adaptation of structures known variously as “freer” (Kiselyov and Ishii, 2015), “general” (McBride, 2015), or “program” (Letan et al., 2018) monads—as a suitable unifying structure for expressing and relating specifications at different levels of abstraction (Section 3).
Second, we adapt standard notions of linearizability and observational refinement from the literature on concurrent data structures to give a simple specification methodology for networked servers that is suitable both for rigorous property-based testing and for formal verification. We call this variant network refinement (Section 4).
Third, we demonstrate practical techniques for both verifying (Section 5) and testing (Section 6) network refinement between a low-level implementation model and a simple linear specification. We also demonstrate testing against the compiled C implementation across a network interface.
Lastly, the ITrees embedded into both VST’s separation logic and CertiKOS’s socket model allow us to make progress on connecting the two developments. Though we leave completing the formal proofs as future work, we identify the challenges and describe preliminary results in Section 7.
Figure 1 shows the high-level architecture of the entire case study. This section surveys the major components, starting with the high-level, user-facing specification (the linear specification shown at the top of the figure) and working down to OS-level details.
Specifying the Swap Server
Informally, the intended behavior of the swap server is straightforward: any number of clients can connect and send “swap requests,” each containing a fixed-size message. The server acts as a one-element concurrent buffer: it retains the most recent message that it has received and, upon getting a swap request, updates its state with the new message and replies to the sender with the old one. The left-hand side of Figure 2 shows a simple example of correct behavior of a swap server.
Figure 3 shows the linear specification of the server’s behavior. It says that the server can either accept a connection with a new client (obs_connect) or else receive a message from a client over some established connection (obs_msg_to_server c), send back the current stored message (obs_msg_from_server c last_msg), and then start over with the last received message as the current state. The set of possible behaviors is represented as an interaction tree (of type itree specE unit). We will discuss the types used here in more details in Section 3.
Our main correctness theorem should relate the actual behavior of our server (the CompCert semantics of the C code) to this linear description of its desired behavior. Informally:
Theorem 1 ().
Any sequence of interactions with the swap server that can be observed by clients over the network could have been produced by the linear specification.
This theorem constrains the implementation to act as a swap server: it prevents the server from sending a message before it receives one, or while it has only received a partial message; it prevents the server from sending an arbitrary value in response to a request, or replying multiple times with the same value that has only been received once; it also prevents the server from sending a response to a client from which it has not received a request. However, the “over the network” clause is a significant caveat: the server communicates with clients via TCP, and even a correct implementation might thus exhibit a number of undesirable behaviors from the clients’ point of view. The network might drop all packets after a certain point, causing the server to appear to have stopped running, so the theorem allows the server to stop running at any point. Similarly, the network might delay messages and might reorder messages on different connections, so the theorem allows the server to respond to an earlier request after responding to a later request. However, as long as the server performs any network operations, those operations must be consistent with the protocol for a swap server. The right-hand side of Figure 2 shows another run of the system illustrating these possibilities; it should also be accepted by the top-level theorem.
Figure 4 shows the formal specification linking the linear specification (linear_spec), which describes interactions with one client at a time, to the C program (C_prog). It is split in two parts articulated around an implementation model (impl_model). It is another interaction tree that describes the network-level behavior of the C program more closely than the linear specification. Like the C program, the implementation model interleaves requests from multiple clients and accounts for the effects of the network. A refinement between the C program and the implementation model is formalized by the VST property ext_behavior. Then the implementation model is connected to the specification by a different network refinement layer (network_refines).
The linear specification is short and easy to understand, but an implementation that strictly followed it would be obliged to serve clients sequentially, which is not what real servers (including ours) want to do. Moreover, as shown on the right-hand side of Figure 2, the network may delay and reorder messages, so that, for example, the first two bytes of a message from client 1 might be received after the first byte of a message from client 2. The server should be able to account for this by buffering messages until they are complete. The second part of our server specification loosens the linear specification to account for the effects of communicating over a network; this also permits realistic implementations that serve multiple clients concurrently.
Network refinement states that every possible behavior of the implementation model is allowed by the linear specification, while accounting for message reordering and buffering that might be introduced by the network and/or server. Section 4 explains this process in more detail.
Our C implementation is a simple but reasonably performant server in a classical single-process, event-driven style (Pai et al., 1999). The implementation maintains a list of connection structures, each representing a state machine for one connection. Specifically, a connection structure contains (1) a state, which may be RECVING, SENDING, or DELETED; (2) a buffer for storing bytes that have been received on the connection; and (3) a buffer for storing bytes to send on the connection.
The main body of the server is a non-terminating loop (Figure 5); in each iteration, it uses the select system call 111For simplicity, we choose select over epoll, a more efficient version found in Linux. to check for pending connections to accept and for existing connections ready for receiving/sending bytes from/to, and processes them. A new connection is handled by initializing a new connection structure and adding it into the list, and an existing connection is processed by updating the read/write buffers and advancing the connection’s state appropriately. This buffering strategy lets the server interleave processing of multiple connections without having to wait for one client to send or receive a complete message.
Our C code is compiled by CompCert and should run on any operating system with POSIX sockets. We have tested it on CertiKOS, OSX, and Linux; our long-term aim is deeper integration with CertiKOS’s own formal verification.
Verifying the C code
To prove that the C implementation refines the implementation model (that is, that every possible network behavior of the C program is allowed by the implementation model), we use VST, a tool for proving correctness of C programs using separation logic. The VST predicate ext_behavior C_prog impl_model in Figure 4 relates the operational semantics of the C program C_prog to the interaction tree description given by impl_model. Section 3 describes the implementation model in more detail.
VST’s model of program execution includes both conventional program state (memory, local variables, etc.) and external state, an abstract representation of the state of the environment in which the program is running. We connect the C program semantics to the implementation model by adding a predicate ITree(t) to VST’s separation logic, asserting that the environment expects the C program’s network behavior to match the interaction tree t. Section 5 describes this process.
Assumptions and modeling gaps
We have a complete proof (using VST) that the C implementation compiled with CompCert network-refines the linear specification—that is, a complete proof of the claim in Figure 4. This proof is grounded in axiomatic specifications of the OS-level system calls, and library functions like memset and the fdset macros. We rely on the soundness of the Coq proof assistant, plus the standard axioms of functional and propositional extensionality and proof irrelevance (Coq development team, 2017).
For this case study, our verification bottoms out at the interface between the application program and the operating system; we rely on the correctness of the OS’s socket library and of the OS itself. Since we are running on CertiKOS, the OS has actually been proved correct, but its correctness proofs and ours are not formally connected. That is, our specification of its socket API is axiomatized, but the axioms are partially validated by connection to the corresponding CertiKOS specifications (specifically, a VST specification of recv has been partly connected to the CertiKOS-level one; the other socket primitives remain to be connected). There are several remaining challenges with connecting VST to CertiKOS, ranging from the semantic—one critical technicality is connecting VST’s step-indexed view of memory with the flat memory model used by CertiKOS—to the technical—they use different versions of Coq. See Section 7 for a fuller description of what we have done to bridge these two formalizations. Also, because CertiKOS currently does not provide a verified TCP implementation, the best it can do is mediate between the VST axioms and some, possibly lower-level, axiomatization of the untrusted TCP stack. Filling these gaps is left to future work.
Testing network refinement
For our long-term goal of building verified systems software like web servers, rigorous testing will be crucial, for two reasons. First, even small web servers are fairly complex programs, and they take significant effort to verify; streamlining this effort by catching as many bugs as possible before spending much time on verification makes good economic sense, especially if the code can be automatically tested against the very same specification that will later be used in the verification effort. Second, programs like web servers must often fit into an existing ecosystem—a verified web server that interpreted the HTTP RFCs (e.g., Belshe et al. (2015)) differently from Apache and Nginx would not be used. Testing can be used to validate the formal specification against existing implementations.
For the present case study, we use QuickChick (Lampropoulos and Pierce, 2018), a Coq plugin for property-based testing based on the popular QuickCheck tool (Claessen and Hughes, 2000). We test both the compiled C code (by sending it messages over a network interface) and the implementation model (by exploring its behaviors within Coq) against the linear specification.
Supporting property-based testing requires executable specifications of the properties involved. Happily, interaction trees, which play a crucial role throughout our development, also work well with Coq-style program extraction, and hence with testing. Testing must also be performed “modulo network refinement” in the same way as verification. Section 6 describes this in more detail.
3. Interaction Trees
Components that interact with their environments appear at many levels in our development (see Figure 1). We use interaction trees (ITrees) as a general-purpose structure for specifying such components. ITrees are a Coq adaptation of similar concepts known variously as “freer,” “general,” or “program” monads (Kiselyov and Ishii, 2015; McBride, 2015; Letan et al., 2018). We defer a deeper comparison until Section 8.
Figure 6 defines the type itree E R. The definition is coinductive, so that it can represent potentially infinite sequences of interactions, as well as divergent behaviors. The parameter E is a type of external interactions—it defines the interface by which a computation interacts with its environment. R is the result of the computation: if the computation halts, it returns a value of type R.
There are three ways to construct an ITree. The Ret r constructor corresponds to the trivial computation that halts and yields the value r. The Tau t constructor corresponds to a silent step of computation, which does something internal that does not produce any visible effect and then continues as t. Representing silent steps explicitly with Tau allows us, for example, to represent diverging computation without violating Coq’s guardedness condition (Chlipala, 2017):
The final, and most interesting, way to build an ITree is with the Vis X e k constructor. Here, e : E X is a “visible” external effect (including any outputs provided by the computation to its environment) and X is the type of data that the environment provides in response to the event. The constructor also specifies a continuation, k, which produces the rest of the computation given the response from the environment. Vis creates branches in the interaction tree because k can behave differently for distinct values of type X.
Here is a small example that defines a type IO of output or input interactions, each of which works with natural numbers. It is then straightforward to define an ITree computation that loops forever, echoing each input received to the output:
Working with ITrees
Several properties of ITrees make them appealing as a structure for representing interactive computations. First, they are generic in the sense that, by varying the E parameter, they can be instantiated to work with different external interfaces. Moreover, such interfaces can be built compositionally: for example, we can combine a computation with external effects in E1 with a different computation with effects in E2, yielding a computation with effects in E1 +E2, the disjoint union of E1 and E2; there is a natural inclusion of ITrees with interface E1 into ITrees with interface E1 + E2. This approach is reminiscent of algebraic effects (Plotkin and Power, 2003). Our development exploits this flexibility to easily combine generic functionality, such as a nondeterministic choice effect (which provides the or operator used by the linear specification of Figure 3) with domain-specific interactions such as the network send and receive events. As with algebraic effects, we can write a handler or interpreter for some or all of the external interactions in an interface, for example to narrow the effects E1 + E2 down to just those in E1. Typically, such a handler will process the events of E2 and “internalize” them by replacing them with Tau steps.
Second, the type itree E is a monad (Moggi, 1989; Wadler, 1992), which makes it convenient to structure effectful computations using the conventions and notations of functional programming. We wrap the Ret constructor as a ret (return) function and use the sequencing notation x <- e ;; k for the monad’s bind. With a bit of wrapping and a loop combinator forever, we can rewrite the echo example with less syntactic clutter:
Third, the ITree definition works well with Coq’s extraction mechanism, allowing us to represent computations as ITrees and run them for testing purposes. Here again, the ability to provide a separate interpretation of events is useful, since its meaning can be defined outside of Coq. In the echo example, Output events could be linked to a console output or to an OS’s network-send system call. ITrees thus provide executable specifications.
One could, of course, simply consider such an extracted implementation to be the final artifact (as in, for example, Verdi (Wilcox et al., 2015)). However, we are interested in a verified C implementation for two main reasons. First, extracting Coq to OCaml generally involves a certain amount of hackery—substituting native OCaml data structures for less efficient Coq ones, interfacing with low-level operations such as I/O system calls, etc.—and this process is entirely unverified. Moreover, the extracted code relies on OCaml’s runtime and foreign-function interfaces, both of which would have to be formalized to obtain the same strong guarantees that we hope to achieve by connecting via C to CertiKOS. 222Compiling directly to native code using CertiCoq (Anand et al., 2017) would alleviate at least some of these concerns. Second, there is a potential performance gain from programming directly in a low-level imperative language that may, in the long run, be important for our eventual goal of verifying a high-performance web-server.
|r <- ei ;; k||r <- or e1 e2 ;; k||i|
|k x||r <- choose l ;; k||x l|
|r <- ret e ;; k||k e|
|b <- (a <- e ;; f a) ;; g b a <- e ;; b <- f a ;; g b|
Equivalence and Refinement
Intuitively, ITrees that encode the same computation should be considered equivalent. In particular, we want to equate ITrees that agree on their terminal behavior (they return the same value) and on Vis events; they may differ by inserting or removing any finite number of Tau constructors. This “equivalence up to Tau” is a form of weak bisimulation. We write t u when t and u are equivalent up to Tau. The monad laws for ITrees also hold modulo this notion of equivalence. (Some of the laws used in our development are shown in Figure 7.)
ITrees that contain nondeterministic effects or that receive inputs from the environment denote a set of possible traces—(finite prefixes of) execution sequences that record each visible event together with the environment’s response. The definitions of trace and the predicate is_trace, which asserts that a trace belongs to an ITree, are shown in Figure 6. Subset inclusion of behaviors gives rise to a natural notion of ITree refinement, written t u, which says that the traces of t are a subset of those allowed by u. We use this refinement relation to allow an implementation to exhibit fewer behaviors than those permitted by its specification. Note that t u implies t u.
ITrees as specifications: the linear specification
Interaction trees provide a convenient yet rigorous way of formalizing specifications. We have already seen them in the linear specification of the swap server in Figure 3. The itree specE type there is an instance of itree whose visible events include nondeterministic choice as well as observations of swap request and response messages, which are events that include message content and connection ID information. The specification itself looks like a standard functional program that uses an effect monad to capture network interactions.
ITrees as specifications: the implementation model
We use the same itree datatype, this time instantiated with an event type implE which contains nondeterministic choice and a networking interface (e.g., accept, send, recv), to define the implementation model, which is a lower-level (but still purely functional) specification of the swap server that more closely resembles the C code. Figure 8 shows the body of the main loop from the implementation model.
In contrast to the linear specification, the implementation model maintains a list of connection structures instead of bare connection identifiers. Each structure records the state for some connection. The state indicates whether the server should be SENDING or RECVING on the connection (or whether the connection is closed). The state also records the contents of send and receive buffers. In each iteration of the loop, the server either accepts a new connection or services a connection that is in the SENDING or RECVING state. Servicing a connection in the SENDING state means sending some prefix of the bytes in the send buffer; servicing a connection in the RECVING state means receiving some bytes on the connection.
Note that the control flow of this model differs from both the linear specification and the C implementation. The linear specification bundles together request–response pairs and totally abstracts away from the details of buffering and interleaving communications among multiple clients. The relationship between the implementation model and the linear specification is given by network refinement, as we explain in the next section. For the C implementation, a single iteration of the main server loop in Figure 5 corresponds to multiple iterations of the select loop body of the model. Nevertheless, we can prove that the C behavior is a refinement of the implementation model, as we describe in Section 5.
4. Network Refinement
We show a “network refinement” relation between the implementation model and the linear specification. At a high level, this property is a form of observational refinement (He et al., 1986): the behaviors of the implementation that can be observed from across the network are included in those of the specification. Intuitively, this property is also an analog, in the network setting, of linearizability for concurrent data structures; we compare them in detail in Section 8.
We model a simple subset of the TCP socket interface, where connections carry bytestreams (the bytes sent on an individual connection are ordered); they are bidirectional (both ends can send bytes) and reliable (what is received is a prefix of what was sent). This network model is represented by a nondeterministic state machine where each connection carries a pair of buffers of “in flight” bytes, with labeled transitions for a client to open a connection, a server to accept it, and either party to send and receive bytes (Figures 9 and 10).
For example, there is a transition from network state ns to state ns’, labeled FromServer c b, if the connection c was previously accepted by the server (its status in ns is ACCEPTED) and the state ns’ is obtained from ns by adding byte b to the outgoing bytes on connection c.
We define a relation network_reordered_ ns ts tc : Prop between server- and client-side traces of network events ts and tc, which holds if they can be produced by an execution of the network starting from state ns. For the initial state with all connections closed, we define network_reordered ts tc = network_reordered_ initial_ns ts tc. The trace tc is a “disordering” of ts—i.e., tc is one possible trace a client may observe if the server generated the trace ts. Conversely, ts is a “reordering” of tc.
Network behavior of ITrees
As mentioned in Section 3, ITrees such as the implementation model (of type itree implE) and the linear specification (itree specE) define sets of event traces. From across the network, those events can appear disordered to the client, so the network behavior of an ITree is the set of possible disorderings of its traces (defined using network_reorder). Finally, the ITree impl_model network refines the linear_spec when the former’s network behavior is included in the latter’s; see Figure 11.
Proving network refinement
In order to prove that our implementation model network refines the linear specification, we establish logical proof rules for a generalization of network_refines, named nrefines_ (Figure 12). The nrefines\_relation is step-indexed (z : nat) to handle the server’s nonterminating loop; it relates a subtree of the implementation model impl to a record s of the current state of the network (get\_ns s: network\_state) and a subtree of the specification ITree (get\_spec s : itree specE unit).
Two example proof rules are shown in Figure 13. When the server performs a network operation, for example when it receives a byte on a connection c, we use a lemma such as nrefines_recv_byte_: we must prove that the connection c is open, and we then prove the nrefines_ relation on the continuation k b, with an updated network state in s’.
At any point in the proof, we can also generate a part of the reordered trace from the linear specification ITree get_spec s, using the nrefines_network_transition_ lemma. We actually use this rule at exactly two “linearization points” in the implementation model: right after the server accepts a new connection, and after it receives a complete message from a client and swaps it with the last stored message.
Using these rules, we prove the proposition forall z, nrefines_z s0 impl_model, where s0 is defined so that get_spec s0 =linear_spec and get_ns s0 is the initial network state, where all connections are closed; we can show this implies the second clause of the correctness theorem (Figure 4).
Embedding ITrees in VST
VST is a framework for proving separation logic specifications of C programs, based on the C semantics of the CompCert compiler. Its separation logic comes with a proof automation system, Floyd, that supplies tactics for symbolically executing a program while maintaining its pre- and postcondition (Cao et al., 2018). To support reasoning about external behavior in general—and the swap server’s invocations of OS/network primitives in particular—we extend VST’s logic with two abstract predicates (Penninckx et al., 2015); these are separation logic predicates that behave like resources but do not have a footprint in concrete memory. Instead they connect to VST’s model of external state, which in this case represents the allowed network behavior of the program. To make this possible, we made a small modification to the internals of VST to enable it to refer to the external state in assertions.
The first abstract predicate, ITree(t), injects an interaction tree t into a VST assertion (an mpred):
ITree t asserts that the observation traces of t (i.e., the traces that may be produced by a program satisfying the assertion ITree t) are included in the traces that are permitted by the external environment (here, the OS). The has_ext predicate asserts that the external state (here representing the network behavior the OS expects from the program) is exactly t’. The notation !!p lifts an ordinary Coq predicate p to a VST separation logic predicate, and && and EX are logical conjunction and existential quantification at the level of separation logic assertions.
While a detailed description of VST’s support for external state is beyond the scope of the present paper and will be reported elsewhere, we give some key properties of this embedding. Internal code execution does not depend on or alter external state, so every program step that is not a call to the socket API leaves the ITree predicate unchanged. The monad and equivalence laws from the abstract theory of interaction trees are reflected as (provable) entailments between ITree predicates (recall the refinement relation of Figure 7):
t u ITree u ITree t
This rule is contravariant because we can conform to the ITree u by producing some subset of its allowed behavior.
External calls to network and OS functions are equipped with specifications that reflect the evolution of interaction trees, in resource-consuming fashion: actions are “peeled off” from the ITree as execution proceeds, so that the interaction tree in the postcondition of an external function specification is a subtree of the tree in the precondition. The ITree found in the outermost precondition of a program is thus a sound approximation of all the program’s external interactions.
Hoare-logic specifications of system calls
This use of the ITree predicate can be seen in the VST axiom for the recv system call in Figure 14. The precondition of this rule requires that the ITree (r <- recvclient_conn (...);; k r), which starts with a recv event, be among the allowed behaviors of t, so a legal implementation of this specification is allowed to perform a recv call next. The postcondition either leaves the interaction tree t untouched, in the case that the call to recv failed, or says that the implementation may continue as k msg, in the case that the call to recv successfully returned a message msg.
Most of the remaining constraints relate the program variables and the variables in the interaction tree to the corresponding state in memory. For example, the predicate data_at_ alloc_len buf_ptr says that buf_ptr points to a buffer of length alloc_len. The constraint lookup_socket st fd =ConnectedSocket client_conn says that the socket with identifier fd is in the connected state according to the API and is associated with the connection identifier client_conn appearing in the interaction tree.
This socket information is tracked by a second abstract predicate, SOCKAPI(st), which asserts that the external socket API memory can be abstracted as st, mapping file descriptors to socket states closed, opened, bound, listening, or connected. Bound and listening states are associated with an endpoint identifier in the network model, and connected states are associated with a connection identifier in the network model. The reason for modularly separating socket states from interaction trees is that the latter describe truly external behavior while the former concern the (private) contract between the server program and the OS. Specifically, the functions for creating sockets, binding them to addresses, and closing sockets (after shutdown) are not visible at the other end of the network and are hence specified to only operate over SOCKAPI abstract predicates. In general, system calls like recv that affect the network state carry specifications of the form
where is a boolean predicate distinguishing ITree-advancing (successful) invocations from failed invocations (which leave the ITree unmodified), by inspection of the implicitly quantified return value r.
Verifying the C implementation
Having defined the abstract predicates we need to describe the network behavior of the server, we can now prove that the C implementation refines the implementation model using VST’s separation logic. The goal is to prove that the implementation model impl_model is an envelope around the possible network behaviors of the C program, i.e., every execution of the C program performs only the socket operations described in impl_model; this is expressed by the predicate ext_behavior C_prog impl_model. This proof then composes with the network refinement proof between impl_model and the linear specification to give us the main theorem in Figure 4.
We prove ext_behavior C_prog impl_model by specifying and proving a Hoare triple for each function in the C implementation. We begin with axiomatized Hoare triples for the library functions, in particular those from the POSIX socket API; these triples modify the SOCKAPI state and possibly consume operations from the ITree, as described above.
We then specify Hoare triples for functions in the program, including embedded interaction trees where appropriate. Verification proceeds as in standard Hoare logic, including formulating an appropriate invariant for each loop. The most interesting invariant is for the main loop, shown in Figure 5; among other things, the invariant states that head points to a linked list l of connection structures, last_msg_store points to a buffer storing a message m, and the interaction tree under ITree is an infinite loop of select_loop_body (Figure 8)) started on (l, m); the server address and buffer size are constants.
Note that it is not immediate that the C loop body refines select_loop_body. The former iterates over all ready connections in process_connections, while the latter works on only one connection per iteration. However, each iteration in process_connections is itself an iteration of select_loop_body, so the inner invariant carries the same interaction tree. Conceptually, one iteration of the main loop in C corresponds to multiple iterations of the model.
Our overall approach to verifying software includes testing for errors in code and specifications before we invest too much effort in verification. For the swap server, we used QuickChick (Lampropoulos and Pierce, 2018), a property-based testing tool in Coq, to test both whether the C implementation satisfies the linear specification, and whether the implementation model refines the linear specification. These tests help establish confidence in all three artifacts.
Our testbed consists of a simple hand-written client, the server to be tested, and the linear specification that the server should satisfy. The client opens multiple TCP connections to simulate multiple clients communicating with the server over the network.
The testing process is straightforward: First, the client generates a random sequence of messages along randomly chosen TCP connections. The client then collects a trace of its interactions with the server—the messages that it sent and the responses that it received in return on each connection. Finally, the checker attempts to “explain” this trace by enumerating all the possible reorderings of this trace and checking whether any of them is, in fact, a trace of the linear specification. If such a trace is found, this test case passes, and another trace is generated. If none of the reorderings satisfies the specification, the tester reports that it has found a counterexample. Before actually displaying the counterexample, the tester attempts to shrink it using a greedy search process modeled on the one used in Haskell’s QuickCheck tool, successively throwing away bits of the counterexample and rechecking to see whether the remainder still fails.
We can also test that the implementation model refines the linear specification. The setup here is similar to the one for the C program, but simpler because we can execute both the client and server within a single Coq program rather than extracting a client from Coq and running it with the server and a network.
Testing the tester
Although we did not find any bugs, we assessed the effectiveness of testing using QuickChick’s mutation testing mode (DeMillo et al., 1978) to inject 12 different “plausible bugs” (of the sort commonly found in C: pointer errors, bad initialization, off-by-one errors, etc.) into the code and check that each could be detected during testing. The bugs are added to the C program as comments marking a section of “good code” and a “mutant” that can be substituted for it. QuickChick performs this substitution for each of the mutants in turn, generates random tests as usual, and reports how many tests it took to find a counterexample for each of the mutants.
We analyzed the running time and number of tests needed to capture the bugs, by repeating QuickChick for 29 times on each mutant. For five of the 12 mutants, the wrong behavior was caught by the very first test in each run. Six of the mutants passed the first test in some runs, but always failed by the second test. The most interesting mutant was changing the return value of the recv call. 3/4 of the runs caught the bug within four rounds, but others took up to nine rounds. This mutant sometimes causes the server not to respond, which is trivially correct because our specification does not deal with liveness. As a result, the tester discarded up to three thousand test cases where the server did not respond, and ran for up to five minutes before failing. The other mutants could fail within 0.4 second with 95% confidence.
It is hard to draw definite conclusions about the effectiveness of testing from a case study of this size, but the fact that we are able to detect a dozen different bugs, most quite quickly, is an encouraging sign that this approach to testing will provide significant value as the codebase and its specification become more complex. Reports in the literature of property-based random testing of similar kinds of systems (e.g., Dropbox (Hughes et al., 2016)) are also encouraging.
7. Connecting to CertiKOS
A key pillar of the proof of correctness of the C implementation is the specification of the socket operations such as send and recv. We took these specifications as axioms when proving the implementation model, but because we are running the server on top of CertiKOS, which has its own formal specification, we should be able to go one step better: we would like to prove that the socket operations as specified by CertiKOS satisfy the axioms used in the VST proof. This part of the case study is still in progress; we report here on what we’ve achieved so far and identify the challenges that remain.
The Socket API in CertiKOS
CertiKOS provides its own axiomatized specifications for the POSIX socket API. Unlike VST specifications, which are expressed as Hoare triples, CertiKOS specifications are written as state transition functions on the OS abstract state. This state is a record with a field for each piece of real or ghost state that the OS maintains. This includes, for example, buffers for received network messages, or socket statuses. To provide a common language with VST for expressing allowable network communications, we have modified CertiKOS’ state to also include an ITree for each user process.
A function like recv presents a challenge in that it depends on nondeterministic behavior by the network, but the specification must be a deterministic function. The standard solution used in CertiKOS is to parametrize the specification by an “environment context” (Gu et al., 2018), which acts as a deterministic oracle that takes a log of events and returns the next step taken by the environment. Because the only restriction on the environment context is that it is “valid” (e.g., for networks this could mean that receive events always have a corresponding earlier send event), properties proved about the specifications hold regardless of the particular choice of oracle. Equipped with such a network oracle, the specification of recv is fairly straightforward (Figure 15).
Bridging VST and CertiKOS memories
The other major gap between VST and CertiKOS is their treatment of memory. Both VST and CertiKOS build on CompCert’s memory model to describe the state of memory, but the changes they make to it are unrelated and incompatible. VST builds a step-indexed model on top of CompCert memories (Appel, 2014), to allow for “predicates in the heap”-based features, including recursive predicates and lock invariants. Hoare triples are interpreted as assertions on these step-indexed memories. On the other hand, the CompCert model corresponds to virtual memory, and treats independent memory allocations as belonging to separate, nonoverlapping “blocks”, while CertiKOS uses a “flat” memory model in which there is only one block to more accurately represent the kernel’s view of physical memory. To bridge this gap, we need to translate VST pre- and postconditions into assertions on ordinary, step-index-free CompCert memories (and vice versa), and transform predicates on multiple-block CompCert memories into predicates on CertiKOS’s flat memories (and vice versa).
Performing this translation in general is an interesting research problem, but for this application, the specifications to be connected have a very particular form. The pre- and postconditions send and recv functions are each divided into two parts: a memory assertion on a single buffer, an array of bytes meant to hold the message, and an ITree assertion describing the external network behavior. This simplifies the task of connecting the VST and CertiKOS specs: we just need to relate the interaction tree to some component of the OS state, and translate an assertion on a single piece of memory into the flat memory model and back. (The other socket operations do not involve any changes to user memory, though they do modify kernel memory, which is abstracted to the C program via the SOCKAPI predicate.)
We have explored this approach by sketching the correspondence between the VST specification of recv and its CertiKOS specification. We translated the VST pre- and postcondition for recv into step-index-free predicates on CompCert memories and interaction trees by hand, and proved the correctness of the translation using the underlying logic of VST. We then wrote functions that transfer a single block of memory between the CompCert model and the flat model, and adapted the CertiKOS OS component representing the network state to use interaction trees, so that the two systems have a common language to describe network operations. The network component of the CertiKOS OS state is now a map that, for each user process, holds an interaction tree describing the network communication that that process is allowed to perform. Finally, we are in the process of proving that the CertiKOS specification for recv satisfies the step-index-free, flattened versions of the VST pre- and postcondition. This gives us a path to validating the axiomatized specifications of the socket API that we rely on for the correctness of the C implementation: they can be substantiated by connection to the (axiomatized) behavior of the socket operations in the underlying operating system.
8. Related Work
As mentioned in Section 3, our “interaction trees” are a Coq-compatible variation of ideas found elsewhere. Kiselyov and Ishii (2015) present a similar concept under the name “freer monad”. It is proposed as an improvement over a “free monad” type, which one might hope to define in Coq as follows:
Unfortunately, the recursive occurrence of free in the Vis constructor is not strictly positive, so this definition will be rejected by Coq. Thus in a total language, the choice for the Vis constructor to separate the effect E X from the continuation X -> itree E R is largely driven by necessity, whereas the work on freer monads proposes it as a matter of convenience and performance.
The McBride (2015) variant, which builds on earlier work by Hancock (2000), is called the “general monad.” It is defined inductively, and its effects interface replaces our single E : Type-> Type parameter with S : Type and a type family S ->Type to calculate the result type. It was introduced as a way to implement general recursive programs in a total language (Agda), by representing recursive calls as effects (i.e., Vis nodes). Our coinductively defined interaction trees also support a general (monadic) fixpoint combinator.
Letan et al. (2018) present the “program monad” to model components of complex computing systems. Like the general monad, it is defined inductively. Whereas our interpretation of ITrees is based on traces, they use a coinductively defined notion of “operational semantics” to provide the context in which to interpret programs, describing the state transitions and results associated with method calls/effects.
Our choice to use coinduction and the Tau constructor gives us a way to account for “silent” (internal) computation steps, and hence allows us to semantically distinguish terminating from silently-diverging computations (which is not easy with trace-based semantics, at least not without adding a “diverges” terminal component to some of the traces). Although liveness is explicitly not part of our correctness specification in this project (the spec is conditioned on there being visible output), it is conceivable to strengthen the specifications and account for Tau transitions as part of the C semantics, which might allow one to prove liveness properties (although VST does not currently support that). However, there are also costs to working with coinduction: our top-level programs are defined by CoFixpoint, and coinduction is generally not as easy to use in Coq as it could be (Chlipala, 2017; Hur et al., 2013).
Verifying effectful systems
A common approach to reasoning about effectful programs is to provide a model of the state of the outside world, with access mediated strictly through external functions. These functions may be given (possibly non-deterministic) semantics directly (Chlipala, 2015), or indirectly through an oracle (Gu et al., 2015; Férée et al., 2018). For example, in Férée et al. (2018), external functions are called through a Foreign Function Interface (FFI), and specification/verification is done with respect to an instantiated FFI oracle that records external calls and defines the state of the environment and the semantics of external functions. In their work, a TextIO library was verified with respect to a model of the file system. Similarly, our specifications in terms of Hoare triples assume a model of external socket API memory, i.e., the state under the SOCKAPI predicate, and describe how this state is transformed.
Stronger specifications of effectful programs can involve dynamics (“what has happened”) rather than statics (“what is the final state”). In such cases, a model of the external state is commonly extended with (or taken to be) a trace or history of past events, and specifications involve these traces. Leroy (2009); Malecha et al. (2011); Chajed et al. (2018); Hawblitzel et al. (2015), etc. use this approach.
Our specifications are based on interaction trees (which can be construed as sets of traces), with one major difference: interaction trees specify “what is allowed to happen”. Rather than reasoning about lists of events that have occurred in the past, our reasoning is based on the trees of events that are allowed to be produced in the future. One main advantage of using interaction trees is that it gives us a unifying structure for specification, testing, and verification, as detailed in Section 3. A similar underlying structure to interaction trees is used as specifications of distributed systems in an early version of F* (Swamy et al., 2011), but that work did not show how to use the structure for testing or how to do refinement. Gu et al. (2018) use environment contexts to specify past events as well as future events, but rather than starting with all possible traces and consuming them, valid traces are generated one event at a time by consulting an oracle. Although using this step-based approach instead of explicitly coinductive ITrees leads to different specification styles, it is possible to connect them as we discussed in Section 7.
Network refinement is closely related to linearizability (Herlihy and Wing, 1990), a correctness criterion for concurrent data structures. A data structure implementation is linearizable if, for every possible collection of client threads, the behavior of the data structure is indistinguishable from the behavior of a sequential implementation of the structure. Filipovic et al. (2009) related linearizability to contextual refinement. Network refinement is essentially this same idea of contextual refinement, but with network effects playing the role of relaxed memory. Our network model closely resembles TSO, and network refinement is similar to TSO-linearizability (Burckhardt et al., 2012).
Verifying networked servers
In one early attempt at server verification, Black (1998) verified security properties of the thttpd web server, based on axiomatized C semantics. That work did not establish the functional correctness of the web server, the axiomatic semantics was not testable, and it did not consider the effects of network reordering.
IronFleet (Hawblitzel et al., 2015) is a methodology for verifying distributed system implementations and it is similar to our approach in several ways: both verify the functional correctness of a networked system; both use a “one client at a time” style specification at the top-level; and both verify the correctness of a system implementation which interleaves its operations via linearizability. However, there are several major differences between IronFleet and our work: (1) We are concerned with testing, as it allows us to find implementation bugs early, and it also allows us to use the same specification for blackbox-testing of existing implementations. For these reasons, we choose the executable interaction trees to represent the specification. IronFleet focuses instead on reducing the burden of verification. It uses non-executable state machines, and it relies on tool support such as near-real-time IDE-integrated feedback for rapid verification. (2) Our work verifies C implementations. VST and CompCert ensure that the properties we have proved at the source-code level are preserved after the program has been compiled to assembly code. IronFleet verifies programs written in Dafny (Leino, 2010), and extracts them to C#. This means that both the extraction engine and the .NET compiler must be trusted. The authors of IronFleet also suggest an alternative strategy to reduce the trusted computing base, by first translating the programs to assembly code, and verifying the assembly code using an automatically translated specification (Hawblitzel et al., 2014). However, that still requires the specification translator to be trusted. (3) IronFleet is based on UDP, while our works is based on TCP. Nevertheless, we both need to consider packet reordering. The difference is that messages will not be reordered on each individual connection. (4) IronFleet uses TLA+ (Lamport, 2002) to prove liveness properties. The partial-correctness approach of separation logic makes it more difficult to reason about liveness.
CSPEC (Chajed et al., 2018) is a framework for verifying concurrent software. CSPEC focuses on reducing the number of interleavings a verifier must consider. To do that, it provides a general verification framework built on mover types (Lipton, 1975). We may be able to use mover types to simplify the process of proving network refinement.
Verdi (Wilcox et al., 2015) is a framework for verified distributed systems that work under different fault and network models. Verified System Transformers transform a distributed system verified under one model to one that works in another. In particular, the Raft system transformer (Woos et al., 2016) transforms a given state machine (server) into a distributed system of servers that synchronize state using Raft messages, over a network that may drop, reorder, or duplicate messages. Any trace of Raft I/O messages produced by the distributed system can then be linearized to an I/O trace of the input state machine. Distributed systems and transformers are written in Coq and extracted to OCaml.
Ridge (2009) verified the functional correctness and linearizability of a networked, persistent message queue written in OCaml using the HOL4 theorem prover. In contrast to Verdi and Ridge’s work, our methodology focuses on testing and verifying C implementations, dealing with the full complexity of low-level programming including memory allocation and pointer aliasing.
There is more research on testing linearizability of concurrent or distributed systems than we can summarize here, including Vechev et al. (2009); Burckhardt et al. (2010); Scott et al. (2016); Shacham et al. (2011). Our work is distinguished by its focus on uniting testing and verification in the same framework. QuickCheck’s property-based testing methodology has been shown to be useful in formal verification (Lampropoulos and Pierce, 2018; Bulwahn, 2012). There are also many accounts of successfully applying property-based random testing to real-world systems. For example, Hughes and Bolinder (2011) used QuickCheck to test for race conditions in dets, a vital component of the Mnesia distributed database system; Arts et al. (2015) have applied the methodology to test the AUTOSAR Basic Software for Volvo Cars, and Hughes et al. (2016) have tested the linearizability of Dropbox, the distributed synchronization service.
9. Conclusions and Future Work
Starting from a C implementation and a “one client at a time” specification of swap server behavior, we have proved that every execution of the implementation correctly follows the specification. The proof breaks down into layers of refinements: from the C program to an implementation-level interaction tree, and from there, via network refinement to the linear interaction tree. We use VST to verify the C code, pure Coq to relate the trees, QuickChick to test our specifications and implementations, and CertiKOS to validate our specifications of network communication. The result is a proof of the correctness of the swap server from the linear specification down to the interface between the C program and the operating system.
Although this work represents significant progress toward the Deep Specification project’s goal of formally-verified systems software, much remains to be done. The verification of the swap server has tested the limits of VST, in terms of both scale and style of specifications. Previous VST verifications were self-contained libraries, but this swap server interacts with the OS through the socket API, requiring us to develop new features (the external assertions) that should be useful for verifying a variety of more realistic programs. The scale of this project forced us to debug and streamline VST’s existing automation.
A clear next step is to fully verify the socket API used by the server, by completing the proof that each VST socket axiom follows from the specification of the corresponding operation in CertiKOS. Doing so will require several more proofs along the lines of our verification of recv, bridging the gap between VST’s step-indexed memory and CertiKOS’s flat memory, as well as defining a suitable C-level abstraction of the kernel memory related to the socket operations. This will further extend the reach of our result, so that we rely only on the correctness of the operating system’s model of the socket API.
Many real-world web servers are multi-threaded, handling requests from different clients in separate threads. Some parts of our approach are already able to handle concurrency: the top-level specification ITree should be sequential regardless of the implementation, and VST and CertiKOS already support concurrent C programs (Mansky et al., 2017; Gu et al., 2018). Other parts will require adjustment: for instance, the implementation model may need to explicitly represent the concurrency allowed in the C program.
This work was funded by the National Science Foundation’s Expedition in Computing The Science of Deep Specification under the awards 1521602 (Appel), 1521539 (Weirich, Zdancewic, Pierce), and 1521523 (Shao), with additional support by the NSF projects Verified High Performance Data Structure Implementations, award 1005849 (Beringer, Mansky), and Random Testing for Language Design, award 1421243 (Pierce). We are grateful to all the members of the DeepSpec project for their collaboration and feedback, and we greatly appreciate the reviewers’ comments and suggestions.
- Anand et al. (2017) Abhishek Anand, Andrew Appel, Greg Morrisett, Zoe Paraskevopoulou, Randy Pollack, Olivier Savary Belanger, Matthieu Sozeau, and Matthew Weaver. 2017. CertiCoq: A verified compiler for Coq. In The Third International Workshop on Coq for Programming Languages (CoqPL).
- Appel (2014) Andrew W. Appel. 2014. Program Logics - for Certified Compilers. Cambridge University Press. http://www.cambridge.org/de/academic/subjects/computer-science/programming-languages-and-applied-logic/program-logics-certified-compilers?format=HB
- Appel et al. (2017) Andrew W. Appel, Lennart Beringer, Adam Chlipala, Benjamin C. Pierce, Zhong Shao, Stephanie Weirich, and Steve Zdancewic. 2017. Position paper: the science of deep specification. Philosophical Transactions of the Royal Society of London A: Mathematical, Physical and Engineering Sciences 375, 2104 (2017). https://doi.org/10.1098/rsta.2016.0331
- Arts et al. (2015) Thomas Arts, John Hughes, Ulf Norell, and Hans Svensson. 2015. Testing AUTOSAR software with QuickCheck. In Eighth IEEE International Conference on Software Testing, Verification and Validation, ICST 2015 Workshops, Graz, Austria, April 13-17, 2015. 1–4. https://doi.org/10.1109/ICSTW.2015.7107466
- Belshe et al. (2015) M. Belshe, R. Peon, and M. Thomson. 2015. Hypertext Transfer Protocol Version 2 (HTTP/2). RFC 7540. RFC Editor. http://www.rfc-editor.org/rfc/rfc7540.txt
- Bishop et al. (2005a) Steven Bishop, Matthew Fairbairn, Michael Norrish, Peter Sewell, Michael Smith, and Keith Wansbrough. 2005a. TCP, UDP, and Sockets: rigorous and experimentally-validated behavioural specification. Volume 1: Overview. Technical Report UCAM-CL-TR-624. Computer Laboratory, University of Cambridge. http://www.cl.cam.ac.uk/TechReports/UCAM-CL-TR-624.html 88pp.
- Bishop et al. (2005b) Steven Bishop, Matthew Fairbairn, Michael Norrish, Peter Sewell, Michael Smith, and Keith Wansbrough. 2005b. TCP, UDP, and Sockets: rigorous and experimentally-validated behavioural specification. Volume 2: The Specification. Technical Report UCAM-CL-TR-625. Computer Laboratory, University of Cambridge. http://www.cl.cam.ac.uk/TechReports/UCAM-CL-TR-625.html 386pp.
- Black (1998) Paul E. Black. 1998. Axiomatic Semantics Verification of a Secure Web Server. Ph.D. Dissertation. Provo, UT, USA. AAI9820483.
- Bulwahn (2012) Lukas Bulwahn. 2012. The New Quickcheck for Isabelle - Random, Exhaustive and Symbolic Testing under One Roof. In Certified Programs and Proofs - Second International Conference, CPP 2012, Kyoto, Japan, December 13-15, 2012. Proceedings. 92–108. https://doi.org/10.1007/978-3-642-35308-6_10
- Burckhardt et al. (2010) Sebastian Burckhardt, Chris Dern, Madanlal Musuvathi, and Roy Tan. 2010. Line-up: a complete and automatic linearizability checker. In Proceedings of the 2010 ACM SIGPLAN Conference on Programming Language Design and Implementation, PLDI 2010, Toronto, Ontario, Canada, June 5-10, 2010. 330–340. https://doi.org/10.1145/1806596.1806634
- Burckhardt et al. (2012) Sebastian Burckhardt, Alexey Gotsman, Madanlal Musuvathi, and Hongseok Yang. 2012. Concurrent Library Correctness on the TSO Memory Model. In Programming Languages and Systems - 21st European Symposium on Programming, ESOP 2012, Held as Part of the European Joint Conferences on Theory and Practice of Software, ETAPS 2012, Tallinn, Estonia, March 24 - April 1, 2012. Proceedings. 87–107. https://doi.org/10.1007/978-3-642-28869-2_5
- Cao et al. (2018) Qinxiang Cao, Lennart Beringer, Samuel Gruetter, Josiah Dodds, and Andrew W. Appel. 2018. VST-Floyd: A Separation Logic Tool to Verify Correctness of C Programs. https://doi.org/10.1007/s10817-018-9457-5
- Chajed et al. (2018) Tej Chajed, Frans Kaashoek, Butler Lampson, and Nickolai Zeldovich. 2018. Verifying a concurrent mail server with CSPEC. In 13th USENIX Symposium on Operating Systems Design and Implementation (OSDI 18). USENIX Association, Carlsbad, CA. https://www.usenix.org/conference/osdi18/presentation/chajed
- Chlipala (2015) Adam Chlipala. 2015. From Network Interface to Multithreaded Web Applications: A Case Study in Modular Program Verification. In Proceedings of the 42nd Annual ACM SIGPLAN-SIGACT Symposium on Principles of Programming Languages, POPL 2015, Mumbai, India, January 15-17, 2015. 609–622. https://doi.org/10.1145/2676726.2677003
- Chlipala (2017) Adam Chlipala. 2017. Infinite Data and Proofs. In Certified Programming with Dependent Types. MIT Press. http://adam.chlipala.net/cpdt/html/Cpdt.Coinductive.html
- Claessen and Hughes (2000) Koen Claessen and John Hughes. 2000. QuickCheck: a lightweight tool for random testing of Haskell programs. In Proceedings of the Fifth ACM SIGPLAN International Conference on Functional Programming (ICFP ’00), Montreal, Canada, September 18-21, 2000. 268–279. https://doi.org/10.1145/351240.351266
- DeMillo et al. (1978) R. A. DeMillo, R. J. Lipton, and F. G. Sayward. 1978. Hints on Test Data Selection: Help for the Practicing Programmer. Computer 11, 4 (April 1978), 34–41. https://doi.org/10.1109/C-M.1978.218136
- Férée et al. (2018) Hugo Férée, Johannes Åman Pohjola, Ramana Kumar, Scott Owens, Magnus O Myreen, and Son Ho. 2018. Program Verification in the Presence of I/O: Semantics, verified library routines, and verified applications. In 10th Working Conference on Verified Software: Theories, Tools, and Experiments.
- Filipovic et al. (2009) Ivana Filipovic, Peter W. O’Hearn, Noam Rinetzky, and Hongseok Yang. 2009. Abstraction for Concurrent Objects. In Programming Languages and Systems, 18th European Symposium on Programming, ESOP 2009, Held as Part of the Joint European Conferences on Theory and Practice of Software, ETAPS 2009, York, UK, March 22-29, 2009. Proceedings. 252–266. https://doi.org/10.1007/978-3-642-00590-9_19
- Gu et al. (2015) Ronghui Gu, Jérémie Koenig, Tahina Ramananandro, Zhong Shao, Xiongnan (Newman) Wu, Shu-Chun Weng, Haozhong Zhang, and Yu Guo. 2015. Deep Specifications and Certified Abstraction Layers. In Proceedings of the 42nd Annual ACM SIGPLAN-SIGACT Symposium on Principles of Programming Languages (POPL ’15). ACM, New York, NY, USA, 595–608. https://doi.org/10.1145/2676726.2676975
- Gu et al. (2016) Ronghui Gu, Zhong Shao, Hao Chen, Xiongnan (Newman) Wu, Jieung Kim, Vilhelm Sjöberg, and David Costanzo. 2016. CertiKOS: An Extensible Architecture for Building Certified Concurrent OS Kernels. In 12th USENIX Symposium on Operating Systems Design and Implementation, OSDI 2016, Savannah, GA, USA, November 2-4, 2016. 653–669. https://www.usenix.org/conference/osdi16/technical-sessions/presentation/gu
- Gu et al. (2018) Ronghui Gu, Zhong Shao, Jieung Kim, Xiongnan (Newman) Wu, Jérémie Koenig, Vilhelm Sjöberg, Hao Chen, David Costanzo, and Tahina Ramananandro. 2018. Certified concurrent abstraction layers. In Proceedings of the 39th ACM SIGPLAN Conference on Programming Language Design and Implementation, PLDI 2018, Philadelphia, PA, USA, June 18-22, 2018. 646–661. https://doi.org/10.1145/3192366.3192381
- Hancock (2000) Peter Hancock. 2000. Ordinals and interactive programs. Ph.D. Dissertation. University of Edinburgh, UK. http://hdl.handle.net/1842/376
- Hawblitzel et al. (2015) Chris Hawblitzel, Jon Howell, Manos Kapritsos, Jacob R. Lorch, Bryan Parno, Michael L. Roberts, Srinath T. V. Setty, and Brian Zill. 2015. IronFleet: proving practical distributed systems correct. In Proceedings of the 25th Symposium on Operating Systems Principles, SOSP 2015, Monterey, CA, USA, October 4-7, 2015. 1–17. https://doi.org/10.1145/2815400.2815428
- Hawblitzel et al. (2014) Chris Hawblitzel, Jon Howell, Jacob R. Lorch, Arjun Narayan, Bryan Parno, Danfeng Zhang, and Brian Zill. 2014. Ironclad Apps: End-to-End Security via Automated Full-System Verification. In 11th USENIX Symposium on Operating Systems Design and Implementation, OSDI ’14, Broomfield, CO, USA, October 6-8, 2014. 165–181. https://www.usenix.org/conference/osdi14/technical-sessions/presentation/hawblitzel
- He et al. (1986) Jifeng He, C. A. R. Hoare, and Jeff W. Sanders. 1986. Data Refinement Refined. In ESOP 86, European Symposium on Programming, Saarbrücken, Federal Republic of Germany, March 17-19, 1986, Proceedings. 187–196. https://doi.org/10.1007/3-540-16442-1_14
- Herlihy and Wing (1990) Maurice Herlihy and Jeannette M. Wing. 1990. Linearizability: A Correctness Condition for Concurrent Objects. ACM Trans. Program. Lang. Syst. 12, 3 (1990), 463–492. https://doi.org/10.1145/78969.78972
- Hughes et al. (2016) John Hughes, Benjamin C. Pierce, Thomas Arts, and Ulf Norell. 2016. Mysteries of DropBox: Property-Based Testing of a Distributed Synchronization Service. In 2016 IEEE International Conference on Software Testing, Verification and Validation, ICST 2016, Chicago, IL, USA, April 11-15, 2016. 135–145. https://doi.org/10.1109/ICST.2016.37
- Hughes and Bolinder (2011) John M. Hughes and Hans Bolinder. 2011. Testing a database for race conditions with QuickCheck. In Proceedings of the 10th ACM SIGPLAN workshop on Erlang, Tokyo, Japan, September 23, 2011. 72–77. https://doi.org/10.1145/2034654.2034667
- Hur et al. (2013) Chung-Kil Hur, Georg Neis, Derek Dreyer, and Viktor Vafeiadis. 2013. The Power of Parameterization in Coinductive Proof. In Proceedings of the 40th Annual ACM SIGPLAN-SIGACT Symposium on Principles of Programming Languages (POPL ’13). ACM, New York, NY, USA, 193–206. https://doi.org/10.1145/2429069.2429093
- Kiselyov and Ishii (2015) Oleg Kiselyov and Hiromi Ishii. 2015. Freer monads, more extensible effects. In Proceedings of the 8th ACM SIGPLAN Symposium on Haskell, Haskell 2015, Vancouver, BC, Canada, September 3-4, 2015. 94–105. https://doi.org/10.1145/2804302.2804319
- Lamport (2002) Leslie Lamport. 2002. Specifying Systems: The TLA+ Language and Tools for Hardware and Software Engineers. Addison-Wesley.
- Lampropoulos and Pierce (2018) Leonidas Lampropoulos and Benjamin C. Pierce. 2018. QuickChick: Property-Based Testing in Coq. Electronic textbook. https://softwarefoundations.cis.upenn.edu/qc-current/index.html
K. Rustan M. Leino.
Dafny: An Automatic Program Verifier for Functional
Logic for Programming, Artificial Intelligence, and Reasoning - 16th International Conference, LPAR-16, Dakar, Senegal, April 25-May 1, 2010, Revised Selected Papers. 348–370. https://doi.org/10.1007/978-3-642-17511-4_20
- Leroy (2009) Xavier Leroy. 2009. Formal verification of a realistic compiler. Commun. ACM 52, 7 (2009), 107–115. https://doi.org/10.1145/1538788.1538814
- Letan et al. (2018) Thomas Letan, Yann Régis-Gianas, Pierre Chifflier, and Guillaume Hiet. 2018. Modular Verification of Programs with Effects and Effect Handlers in Coq. In Formal Methods - 22nd International Symposium, FM 2018, Held as Part of the Federated Logic Conference, FloC 2018, Oxford, UK, July 15-17, 2018, Proceedings. 338–354. https://doi.org/10.1007/978-3-319-95582-7_20
- Lipton (1975) Richard J. Lipton. 1975. Reduction: A Method of Proving Properties of Parallel Programs. Commun. ACM 18, 12 (1975), 717–721. https://doi.org/10.1145/361227.361234
- Malecha et al. (2011) Gregory Malecha, Greg Morrisett, and Ryan Wisnesky. 2011. Trace-based Verification of Imperative Programs with I/O. J. Symb. Comput. 46, 2 (Feb. 2011), 95–118. https://doi.org/10.1016/j.jsc.2010.08.004
- Mansky et al. (2017) William Mansky, Andrew W. Appel, and Aleksey Nogin. 2017. A Verified Messaging System. PACMPL 1, OOPSLA, Article 87 (Oct. 2017), 28 pages. https://doi.org/10.1145/3133911
- Coq development team (2017) Coq development team. 2017. Coq and Axioms. https://github.com/coq/coq/wiki/CoqAndAxioms
- Coq development team (2018) Coq development team. 2018. The Coq proof assistant reference manual. LogiCal Project. http://coq.inria.fr Version 8.8.1.
- McBride (2015) Conor McBride. 2015. Turing-Completeness Totally Free. In Mathematics of Program Construction - 12th International Conference, MPC 2015, Königswinter, Germany, June 29 - July 1, 2015. Proceedings. 257–275. https://doi.org/10.1007/978-3-319-19797-5_13
- Moggi (1989) Eugenio Moggi. 1989. Computational lambda-calculus and monads. 14–23. Full version, titled Notions of Computation and Monads, in Information and Computation, 93(1), pp. 55–92, 1991.
- Pai et al. (1999) Vivek S. Pai, Peter Druschel, and Willy Zwaenepoel. 1999. Flash: An efficient and portable Web server. In Proceedings of the 1999 USENIX Annual Technical Conference, June 6-11, 1999, Monterey, California, USA. 199–212. http://www.usenix.org/events/usenix99/full_papers/pai/pai.pdf
- Penninckx et al. (2015) Willem Penninckx, Bart Jacobs, and Frank Piessens. 2015. Sound, Modular and Compositional Verification of the Input/Output Behavior of Programs. In Programming Languages and Systems - 24th European Symposium on Programming, ESOP 2015, Held as Part of the European Joint Conferences on Theory and Practice of Software, ETAPS 2015, London, UK, April 11-18, 2015. Proceedings. 158–182. https://doi.org/10.1007/978-3-662-46669-8_7
- Plotkin and Power (2003) Gordon D. Plotkin and John Power. 2003. Algebraic Operations and Generic Effects. Applied Categorical Structures 11, 1 (2003), 69–94. https://doi.org/10.1023/A:1023064908962
- Ridge (2009) Thomas Ridge. 2009. Verifying distributed systems: the operational approach. In Proceedings of the 36th ACM SIGPLAN-SIGACT Symposium on Principles of Programming Languages, POPL 2009, Savannah, GA, USA, January 21-23, 2009. 429–440. https://doi.org/10.1145/1480881.1480934
- Ridge et al. (2009) Thomas Ridge, Michael Norrish, and Peter Sewell. 2009. TCP, UDP, and Sockets: Volume 3: The Service-level Specification. Technical Report UCAM-CL-TR-742. University of Cambridge, Computer Laboratory. 305pp.
- Scott et al. (2016) Colin Scott, Aurojit Panda, Vjekoslav Brajkovic, George C. Necula, Arvind Krishnamurthy, and Scott Shenker. 2016. Minimizing Faulty Executions of Distributed Systems. In 13th USENIX Symposium on Networked Systems Design and Implementation, NSDI 2016, Santa Clara, CA, USA, March 16-18, 2016. 291–309. https://www.usenix.org/conference/nsdi16/technical-sessions/presentation/scott
- Shacham et al. (2011) Ohad Shacham, Nathan Grasso Bronson, Alex Aiken, Mooly Sagiv, Martin T. Vechev, and Eran Yahav. 2011. Testing atomicity of composed concurrent operations. In Proceedings of the 26th Annual ACM SIGPLAN Conference on Object-Oriented Programming, Systems, Languages, and Applications, OOPSLA 2011, part of SPLASH 2011, Portland, OR, USA, October 22 - 27, 2011. 51–64. https://doi.org/10.1145/2048066.2048073
- Swamy et al. (2011) Nikhil Swamy, Juan Chen, Cédric Fournet, Pierre-Yves Strub, Karthikeyan Bhargavan, and Jean Yang. 2011. Secure distributed programming with value-dependent types. In Proceeding of the 16th ACM SIGPLAN international conference on Functional Programming, ICFP 2011, Tokyo, Japan, September 19-21, 2011. 266–278. https://doi.org/10.1145/2034773.2034811
- Vechev et al. (2009) Martin T. Vechev, Eran Yahav, and Greta Yorsh. 2009. Experience with Model Checking Linearizability. In Model Checking Software, 16th International SPIN Workshop, Grenoble, France, June 26-28, 2009. Proceedings. 261–278. https://doi.org/10.1007/978-3-642-02652-2_21
- Wadler (1992) Philip Wadler. 1992. Monads for functional programming. In Program Design Calculi, Proceedings of the NATO Advanced Study Institute on Program Design Calculi, Marktoberdorf, Germany, July 28 - August 9, 1992. 233–264. https://doi.org/10.1007/978-3-662-02880-3_8
- Wilcox et al. (2015) James R. Wilcox, Doug Woos, Pavel Panchekha, Zachary Tatlock, Xi Wang, Michael D. Ernst, and Thomas E. Anderson. 2015. Verdi: a framework for implementing and formally verifying distributed systems. In Proceedings of the 36th ACM SIGPLAN Conference on Programming Language Design and Implementation, Portland, OR, USA, June 15-17, 2015. 357–368. https://doi.org/10.1145/2737924.2737958
- Woos et al. (2016) Doug Woos, James R. Wilcox, Steve Anton, Zachary Tatlock, Michael D. Ernst, and Thomas E. Anderson. 2016. Planning for change in a formal verification of the Raft consensus protocol. In Proceedings of the 5th ACM SIGPLAN Conference on Certified Programs and Proofs, Saint Petersburg, FL, USA, January 20-22, 2016. 154–165. https://doi.org/10.1145/2854065.2854081