What is Coq ?
Coq is a formal proof management system. It provides a formal language to write mathematical definitions, executable algorithms and theorems together with an environment for semi-interactive development of machine-checked proofs. Typical applications include the certification of properties of programming languages (e.g. the CompCert compiler certification project, or the Bedrock verified low-level programming library), the formalization of mathematics (e.g. the full formalization of the Feit-Thompson theorem or homotopy type theory) and teaching.
Welcome
This electronic book is a course on Software Foundations, the mathematical underpinnings of reliable software. Topics include basic concepts of logic, computer-assisted theorem proving, the Coq proof assistant, functional programming, operational semantics, Hoare logic, and static type systems.
The principal novelty of the course is that it is one hundred percent formalized and machine-checked: the entire text is literally a script for Coq. It is intended to be read alongside (or inside) an interactive session with Coq. All the details in the text are fully formalized in Coq, and most of the exercises are designed to be worked using Coq.
Overview
Computer scientists and software engineers have responded to these challenges by developing a whole host of techniques for improving software reliability, ranging from recommendations about managing software projects teams (e.g., extreme programming) to design philosophies for libraries (e.g., model-view-controller, publish-subscribe, etc.) and programming languages (e.g., object-oriented programming, aspect-oriented programming, functional programming, ...) to mathematical techniques for specifying and reasoning about properties of software and tools for helping validate these properties. The present course is focused on this last set of techniques.
The text weaves together five conceptual threads:
(1) basic tools from logic for making and justifying precise claims about programs;
(2) the use of proof assistants to construct rigorous logical arguments;
(3) functional programming, both as a method of programming that simplifies reasoning about programs and as a bridge between programming and logic;
(4) formal techniques for reasoning about the properties of specific programs (e.g., the fact that a sorting function or a compiler obeys some formal specification); and
(5) the use of type systems for establishing well-behavedness guarantees for all programs in a given programming language (e.g., the fact that well-typed Java programs cannot be subverted at runtime).
Each of these is easily rich enough to fill a whole course in its own right, and tackling all of them together naturally means that much will be left unsaid. Nevertheless, we hope readers will find that these themes illuminate and amplify each other and that bringing them together creates a good foundation for digging into any of them more deeply.
Logic
Logic is the field of study whose subject matter is proofs — unassailable arguments for the truth of particular propositions. Volumes have been written about the central role of logic in computer science. Manna and Waldinger called it "the calculus of computer science," while Halpern et al.'s paper On the Unusual Effectiveness of Logic in Computer Science catalogs scores of ways in which logic offers critical tools and insights. Indeed, they observe that, "As a matter of fact, logic has turned out to be significiantly more effective in computer science than it has been in mathematics. This is quite remarkable, especially since much of the impetus for the development of logic during the past one hundred years came from mathematics."
In particular, the fundamental tools of inductive proof are ubiquitous in all of computer science. You have surely seen them before, perhaps in a course on discrete math or analysis of algorithms, but in this course we will examine them much more deeply than you have probably done so far.
Proof Assistants
The flow of ideas between logic and computer science has not been unidirectional: CS has also made important contributions to logic. One of these has been the development of software tools for helping construct proofs of logical propositions. These tools fall into two broad categories:
- Automated theorem provers provide "push-button" operation: you give them a proposition and they return either true or false (or, sometimes, don't know: ran out of time). Although their capabilities are still limited to specific domains, they have matured tremendously in recent years and are used now in a multitude of settings. Examples of such tools include SAT solvers, SMT solvers, and model checkers.
- Proof assistants are hybrid tools that automate the more routine aspects of building proofs while depending on human guidance for more difficult aspects. Widely used proof assistants include Isabelle, Agda, Twelf, ACL2, PVS, and Coq, among many others.
This course is based around Coq, a proof assistant that has been under development since 1983 and that in recent years has attracted a large community of users in both research and industry. Coq provides a rich environment for interactive development of machine-checked formal reasoning. The kernel of the Coq system is a simple proof-checker, which guarantees that only correct deduction steps are ever performed. On top of this kernel, the Coq environment provides high-level facilities for proof development, including a large library of common definitions and lemmas, powerful tactics for constructing complex proofs semi-automatically, and a special-purpose programming language for defining new proof-automation tactics for specific situations.
Coq has been a critical enabler for a huge variety of work across computer science and mathematics:
- As a platform for modeling programming languages, it has become a standard tool for researchers who need to describe and reason about complex language definitions. It has been used, for example, to check the security of the JavaCard platform, obtaining the highest level of common criteria certification, and for formal specifications of the x86 and LLVM instruction sets and programming languages such as C.
- As an environment for developing formally certified software and hardware, Coq has been used, for example, to build CompCert, a fully-verified optimizing compiler for C, and CertiKos, a fully verified hypervisor, for proving the correctness of subtle algorithms involving floating point numbers, and as the basis for CertiCrypt, an environment for reasoning about the security of cryptographic algorithms. It is also being used to build verified implementations of the open-source RISC-V processor.
- As a realistic environment for functional programming with dependent types, it has inspired numerous innovations. For example, the Ynot system embeds "relational Hoare reasoning" (an extension of the Hoare Logic we will see later in this course) in Coq.
- As a proof assistant for higher-order logic, it has been used to validate a number of important results in mathematics. For example, its ability to include complex computations inside proofs made it possible to develop the first formally verified proof of the 4-color theorem. This proof had previously been controversial among mathematicians because part of it included checking a large number of configurations using a program. In the Coq formalization, everything is checked, including the correctness of the computational part. More recently, an even more massive effort led to a Coq formalization of the Feit-Thompson Theorem — the first major step in the classification of finite simple groups.
Functional Programming
The most basic tenet of functional programming is that, as much as possible, computation should be pure, in the sense that the only effect of execution should be to produce a result: it should be free from side effects such as I/O, assignments to mutable variables, redirecting pointers, etc. For example, whereas an imperative sorting function might take a list of numbers and rearrange its pointers to put the list in order, a pure sorting function would take the original list and return a new list containing the same numbers in sorted order.
A significant benefit of this style of programming is that it makes programs easier to understand and reason about. If every operation on a data structure yields a new data structure, leaving the old one intact, then there is no need to worry about how that structure is being shared and whether a change by one part of the program might break an invariant that another part of the program relies on. These considerations are particularly critical in concurrent systems, where every piece of mutable state that is shared between threads is a potential source of pernicious bugs. Indeed, a large part of the recent interest in functional programming in industry is due to its simpler behavior in the presence of concurrency.
For purposes of this course, functional programming has yet another significant attraction: it serves as a bridge between logic and computer science. Indeed, Coq itself can be viewed as a combination of a small but extremely expressive functional programming language plus a set of tools for stating and proving logical assertions. Moreover, when we come to look more closely, we find that these two sides of Coq are actually aspects of the very same underlying machinery — i.e., proofs are programs.
Program Verification
Approximately the first third of Software Foundations is devoted to developing the conceptual framework of logic and functional programming and gaining enough fluency with Coq to use it for modeling and reasoning about nontrivial artifacts. In the middle third, we turn our attention to two broad topics of critical importance in building reliable software (and hardware): techniques for proving specific properties of particular programs and for proving general properties of whole programming languages.
For both of these, the first thing we need is a way of representing programs as mathematical objects, so we can talk about them precisely, plus ways of describing their behavior in terms of mathematical functions or relations. Our main tools for these tasks are abstract syntax and operational semantics, a method of specifying programming languages by writing abstract interpreters. At the beginning, we work with operational semantics in the so-called "big-step" style, which leads to simple and readable definitions when it is applicable. Later on, we switch to a lower-level "small-step" style, which helps make some useful distinctions (e.g., between different sorts of nonterminating program behaviors) and which is applicable to a broader range of language features, including concurrency.
The first programming language we consider in detail is Imp, a tiny toy language capturing the core features of conventional imperative programming: variables, assignment, conditionals, and loops.
We study two different ways of reasoning about the properties of Imp programs. First, we consider what it means to say that two Imp programs are equivalent in the intuitive sense that they exhibit the same behavior when started in any initial memory state. This notion of equivalence then becomes a criterion for judging the correctness of metaprograms — programs that manipulate other programs, such as compilers and optimizers. We build a simple optimizer for Imp and prove that it is correct.
Second, we develop a methodology for proving that a given Imp program satisfies some formal specifications of its behavior. We introduce the notion of Hoare triples — Imp programs annotated with pre- and post-conditions describing what they expect to be true about the memory in which they are started and what they promise to make true about the memory in which they terminate — and the reasoning principles of Hoare Logic, a domain-specific logic specialized for convenient compositional reasoning about imperative programs, with concepts like "loop invariant" built in.
This part of the course is intended to give readers a taste of the key ideas and mathematical tools used in a wide variety of real-world software and hardware verification tasks.
Type Systems
Our final major topic, covering approximately the last third of the course, is type systems, which are powerful tools for establishing properties of all programs in a given language.
Type systems are the best established and most popular example of a highly successful class of formal verification techniques known as lightweight formal methods. These are reasoning techniques of modest power — modest enough that automatic checkers can be built into compilers, linkers, or program analyzers and thus be applied even by programmers unfamiliar with the underlying theories. Other examples of lightweight formal methods include hardware and software model checkers, contract checkers, and run-time monitoring techniques.
This also completes a full circle with the beginning of the book: the language whose properties we study in this part, the simply typed lambda-calculus, is essentially a simplified model of the core of Coq itself!