Welcome
This is a new entry point in a series of electronic textbooks on
various aspects of
Software Foundations -- the mathematical
underpinnings of reliable software. Topics in the series include
basic concepts of logic, computer-assisted theorem proving, the
Coq proof assistant, functional programming, operational
semantics, logics for reasoning about programs, and static type
systems. The exposition is intended for a broad range of readers,
from advanced undergraduates to PhD students and researchers. No
specific background in logic or programming languages is assumed,
though a degree of mathematical maturity will be helpful.
The principal novelty of the series is that it is one hundred
percent formalized and machine-checked: each text is literally a
script for Coq. The books are 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.
The files in each book are organized into a sequence of core
chapters, covering about one semester's worth of material and
organized into a coherent linear narrative, plus a number of
"offshoot" chapters covering additional topics. All the core
chapters are suitable for both upper-level undergraduate and
graduate students.
This book,
Discrete Mathematics in Coq, lays groundwork for the
others, introducing the reader to the basic ideas of functional
programming, constructive logic, and the Coq proof assistant. It
also covers some of the material that's classically taught in
discrete math courses.
Overview
Building reliable software is really hard. The scale and
complexity of modern systems, the number of people involved, and
the range of demands placed on them make it extremely difficult to
build software that is even more-or-less correct, much less 100%
correct. At the same time, the increasing degree to which
information processing is woven into every aspect of society
greatly amplifies the cost of bugs and insecurities.
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
Software Foundations
series is focused on this last set of techniques.
The text is constructed around three 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.
Some suggestions for further reading can be found in the
Postscript chapter. Bibliographic information for all
cited works can be found in the file
Bib.
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 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.
By the way, in case you're wondering about the name, here's what
the official Coq web site at INRIA (the French national research
lab where Coq has mostly been developed) says about it: "Some
French computer scientists have a tradition of naming their
software as animal species: Caml, Elan, Foc or Phox are examples of
this tacit convention. In French, 'coq' means rooster, and it
sounds like the initials of the Calculus of Constructions (CoC) on
which it is based." The rooster is also the national symbol of
France, and C-o-q are the first three letters of the name of
Thierry Coquand, one of Coq's early developers.
Functional Programming
The term
functional programming refers both to a collection of
programming idioms that can be used in almost any programming
language and to a family of programming languages designed to
emphasize these idioms, including Haskell, OCaml, Standard ML,
F#, Scala, Scheme, Racket, Common Lisp, Clojure, Erlang, and Coq.
Functional programming has been developed over many decades --
indeed, its roots go back to Church's lambda-calculus, which was
invented in the 1930s, well before the first computers (at least
the first electronic ones)! But since the early '90s it has
enjoyed a surge of interest among industrial engineers and
language designers, playing a key role in high-value systems at
companies like Jane St. Capital, Microsoft, Facebook, and
Ericsson.
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.
Another reason for the current excitement about functional
programming is related to the first: functional programs are often
much easier to parallelize than their imperative counterparts. If
running a computation has no effect other than producing a result,
then it does not matter
where it is run. Similarly, if a data
structure is never modified destructively, then it can be copied
freely, across cores or across the network. Indeed, the
"Map-Reduce" idiom, which lies at the heart of massively
distributed query processors like Hadoop and is used by Google to
index the entire web is a classic example of functional
programming.
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.
Practicalities
Chapter Dependencies
A diagram of the dependencies between chapters and some suggested
paths through the material can be found in the file
deps.html.
System Requirements
Coq runs on Windows, Linux, and OS X. You will need:
- A current installation of Coq, available from the Coq home
page. These files have been tested with Coq 8.9.1.
- An IDE for interacting with Coq. Currently, there are three
choices:
- Proof General is an Emacs-based IDE. It tends to be
preferred by users who are already comfortable with
Emacs. It requires a separate installation (google
"Proof General").
Adventurous users of Coq within Emacs may also want to
check out extensions such as company-coq and
control-lock
- CoqIDE is a simpler stand-alone IDE. It is distributed
with Coq, so it should be available once you have Coq
installed. It can also be compiled from scratch, but on
some platforms this may involve installing additional
packages for GUI libraries and such.
- VS Code is a powerful and popular IDE for many
languages. There are plugins for working with Coq.
In this run of the course, we'll be using Emacs and
ProofGeneral. We'll spend the first day of class making sure
you're set up.
Exercises
Each chapter includes numerous exercises. Each is marked with a
"star rating," which can be interpreted as follows:
- One star: easy exercises that underscore points in the text
and that, for most readers, should take only a minute or two.
Get in the habit of working these as you reach them.
- Two stars: straightforward exercises (five or ten minutes).
- Three stars: exercises requiring a bit of thought (ten
minutes to half an hour).
- Four and five stars: more difficult exercises (half an hour
and up).
Also, some exercises are marked "optional." Doing just the
non-optional exercises should provide good coverage of the core
material. Optional exercises provide a bit of extra practice with
key concepts and introduce secondary themes that may be of
interest to some readers.
Please do not post solutions to the
exercises in a public place. Software Foundations is
widely used both for self-study and for university courses.
Having solutions easily available makes it much less useful for
courses, which typically have graded homework assignments. We
especially request that readers not post solutions to the
exercises anyplace where they can be found by search engines.
Downloading the Coq Files
Please download the Coq files from the main website.
(If you are using the book as part of a class, your professor may
give you access to a locally modified version of the files, which
you should use instead of the release version.)
Lecture Videos
Videos will be posted on Box.
