PostscriptPostcript and bibliography

Looking back

You've now seen how to verify programs that use many of C's data structures (arrays, pointers, structs, integers) and control structures (functions, if-then-else, loops), and abstraction structures (data abstraction, module interfaces). The final exercise (hash tables) demonstrated all of these at once, in addition to a key concept: layering the proof using a functional model, so that proofs about the properties of the functional model (Hashfun) cleanly separate from the implementation proof (Verif_hash) showing that the C program refines the functional model.

Looking forward

If you want to verify some C programs on your own, you may take inspiration from some of these Verifiable C proofs:

Small examples

VST is distributed with a progs directory of many small C programs that demonstrate different features and methods of Verifiable C. If your VST installation is in the standard place, you can find this as a subdirectory of `coqc -where`/user-contrib/VST.

Modules

All the verifications in this volume are single-file C programs. Real software engineering in C is done with modules in .c files and interfaces in .h files. Since 2019, VST has a module system; the early version is described in [Beringer 2019] and the newer version is demonstrated in progs/VSUpile. The description in [Beringer 2019] mostly matches the proofs in progs/VSUpile, but the proofs handle data abstraction in a more advanced way than is described in the paper.

Input/output

To prove I/O using our Verifiable C logic, we treat the state of the outside world as a resource in the SEP clause, alongside (but separating from) in-memory conjuncts. Proved examples are in progs/io.c, progs/io_mem.c, and their proof files progs/verif_io.v, progs/verif_io_mem.v. Research papers describing these concepts include [Koh 2020] and [Mansky 2020].

Looking around

VST is not the only program verification system. How should you decide which system to use?

Static analyzers

There are many static analyzers -- too numerous to list here -- that attempt to check safety of your program: will it crash? Will it access an array out of bounds, or dereference an uninitialized pointer? Static analyzers can be useful in software engineering, but they have two major limitations:
  • They don't verify functional correctness -- that is, does your program produce the right answer, or interact with the right behavior?
  • They are incomplete. For example, proving that a[i] is an in-bounds array access requires proving that 0 i < N. Sometimes a static analyzer can deduce that from the program flow, but in general it is a functional correctness property that may require sophisticated invariants to prove.

Functional correctness verifiers -- functional languages

A good way to write proved-correct software is to program in a pure functional program logic, and use higher-order logic to prove correctness. For example:
  • Write pure functional programs in Coq, prove them correct in Coq, then extract them from Coq to OCaml or Haskell. See Volume 3 of Software Foundations: Verified Functional Algorithms.
  • In HOL systems (Higher-order Logic) such as Isabelle/HOL, you can do the same thing: write functional programs, prove them correct, extract, compile.
  • You can write Haskell programs, compile with ghc, and import into Coq for verification using hs-to-coq [Spector-Zabusky 2018].
  • ACL2 is an older system, that uses a first-order logic. That's less expressive, you don't get polymorphism or quantification, but there have been many successful applications of ACL2 in industry.

Functional correctness verifiers -- imperative languages

Functional programming has its limitations: it requires a garbage collector, usually written in an imperative language, and how did you prove that correct? In functional languages it is often harder to build (and reason about) low-latency code, or to access low-level primitives. For these and other reasons, some software is still written in low-level imperative languages such as C.
There are verifiers for high-level imperative languages (that require a garbage collector). For example, Dafny [Leino 2010] is a language and tool for Hoare-logic style verification. It's relatively simple to learn and elegant to use.
ML with mutable references and arrays is also a high-level imperative language. CFML is a system for reasoning about imperative ML programs using separation logic in Coq [Chargueraud 2010], soon to be described in another volume of Software Foundations. CakeML is a system for proving (and correctly compiling) ML programs in HOL [Kumar 2014].

Functional correctness verifiers -- C

For the canonical low-level imperative language -- C -- there are several systems:
Unlike VST, where (as you have seen) the proof script is separate from the program, in both Frama-C and VeriFast you prove the program by inserting assertions and invariants into the program. the tools complete the verification automatically -- or else, point out which assertions have failed, so you can adjust your assertions and invariants, and try again.
Each of these three systems has an assertion language, in which you express your function specifications, assertions, and invariants.
  • In VST, as you have seen, that language is a separation language (PROP/LOCAL/SEP) embedded in Coq, so that the PROP, LOCAL, and SEP clauses can all make use of the full expressive power of the Calculus of Inductive Constructions (CiC). You have seen a simple example of the expressive power of this approach, where we can use ordinary Coq proofs in Hashfun, and directly connect them to separation-logic proofs in Verif_hash.
  • Frama-C uses a weaker assertion language, expressed in C syntax. That's a much weaker logic to reason in, and it doesn't directly connect to a general-purpose logic (and proof assistant) like Coq. Also, since Frama-C is not a separation logic, it can be difficult to reason about data structures.
  • VeriFast uses a capable Dafny-like logic -- even more capable, since it is separation logic, not just Hoare logic. That means you can reason about data structures. There's no artificial limitation to a C-like syntax in the assertion language. Unfortunately, VeriFast's assertion language is not connected to a general-purpose logic (and proof assistant); that means you can do refinement proofs (this C program implements that functional model), but it's not so easy to reason about properties of your functional models.

Foundational soundness

Formal reasoning about source programs is a good thing -- but once you've proved your source program correct, how do you know that is compiled correctly? That is,
  • the compiler must not have bugs, and
  • the compiler must agree with your verifier on every detail of the semantics of the source language.
Of all the systems described above, only VST and CakeML can make this connection end-to-end, with machine-checked proofs.

Conclusion

This volume has given you a taste of formal verification for a low-level imperative language. Since C was not designed with verification in mind, it has many sharp corners and idiosyncrasies, which the verification tool cannot always hide from you. But C is a lingua franca in which you can express a wide variety of programming paradigms, and Verifiable C is expressive enough to allow to verify them. We wish you success in your future software verification efforts.


(* 2020-09-18 15:39 *)