Preface
Welcome
This is the entry point to 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 and techniques for reasoning about programs,
static type systems, property-based random testing, and
verification of practical C code. 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,
Logical Foundations, lays groundwork for the others,
introducing the reader to the basic ideas of functional
programming, constructive logic, and the Coq proof assistant.
Overview
Building reliable software is really hard -- really hard. The
scale and complexity of modern systems, the number of people
involved, and the range of demands placed on them make it
challenging 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 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 tools.
This volume weaves together 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.
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 significantly 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 x_{86} 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 architecture.
- 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 it required 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 electronic
computers! 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
Street Capital, Microsoft, Facebook, Twitter, 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 relied on by another part of the
program. 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 and physically distribute 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.
Further Reading
This text is intended to be self contained, but readers looking
for a deeper treatment of particular topics will find some
suggestions for further reading in the
Postscript chapter.
Bibliographic information for all cited works can be found in the
file
Bib.
Practicalities
System Requirements
Coq runs on Windows, Linux, and macOS. The files in this book
have been tested with Coq 8.17.
You will need:
- A current installation of Coq, available from the Coq home page.
The "Coq Platform" usually offers the smoothest installation
experience.
If you use the VSCode + Docker option described below, you don't
need to install Coq separately.
- An IDE for interacting with Coq. There are several choices:
- The VSCoq extension for Visual Studio Code offers a simple
interface via a familiar IDE. This option is the
recommended default.
VSCoq can be used as an ordinary IDE or it can be combined
with Docker (see below) for a lightweight installation
experience.
- 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 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.
Users who like CoqIDE should consider running it with the
"asynchronous" and "error resilience" modes disabled:
coqide -async-proofs off \
-async-proofs-command-error-resilience off Foo.v & ]] *)
Using Coq with VSCode and Docker
The Visual Studio Code IDE can cooperate with the Docker
virtualization platform to compile Coq scripts without the need
for any separate Coq installation. To get things set up, follow
these steps:
- Install Docker from https://www.docker.com/get-started/ or
make sure your existing installation is up to date.
- Make sure Docker is running.
- Install VSCode from https://code.visualstudio.com and start it
running.
- Install VSCode's Remote Containers Extention from
https://marketplace.visualstudio.com/items?itemName=ms-vscode-remote.remote-containers
- Set up a directory for this SF volume by downloading the
provided .tgz file. Besides the .v file for each chapter,
this directory will contain a .devcontainer subdirectory with
instructions for VSCode about where to find an appropriate
Docker image and a _CoqProject file, whose presence triggers
the VSCoq extension.
- In VSCode, use File > Open Folder to open the new directory.
VSCode should ask you whether you want to run the project in the
associated Docker container. (If it does not ask you, you can
open the command palette by pressing F_{1} and run the command “Dev
Containers: Reopen in Container”.)
- Check that VSCoq is working by double-clicking the file
Basics.v from the list on the left (you should see a blinking
cursor in the window that opens; if not you can click in that
window to select it), and pressing alt+downarrow (on MacOS,
control+option+downarrow) a few times. You should see the
cursor move through the file and the region above the cursor get
highlighted.
- To see what other key bindings are available, press F_{1} and then
type Coq:, or visit the VSCoq web pages:
https://github.com/coq-community/vscoq/tree/vscoq1.
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).
Those using SF in a classroom setting should note that the autograder
assigns extra points to harder exercises:
1 star = 1 point
2 stars = 2 points
3 stars = 3 points
4 stars = 6 points
5 stars = 10 points
Some exercises are marked "advanced," and some are marked
"optional." Doing just the non-optional, non-advanced 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. Advanced exercises are for readers who want an extra
challenge and a deeper cut at the material.
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
A tar file containing the full sources for the "release version"
of this book (as a collection of Coq scripts and HTML files) is
available at
https://softwarefoundations.cis.upenn.edu.
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; you
should use that one instead of the public release version, so that
you get any local updates during the semester.
Chapter Dependencies
A diagram of the dependencies between chapters and some suggested
paths through the material can be found in the file
deps.html.
Recommended Citation Format
If you want to refer to this volume in your own writing, please
do so as follows:
@book {Pierce:SF_{1},
author = {Benjamin C. Pierce and
Arthur Azevedo de Amorim and
Chris Casinghino and
Marco Gaboardi and
Michael Greenberg and
Cătălin Hriţcu and
Vilhelm Sjöberg and
Brent Yorgey},
editor = {Benjamin C. Pierce},
title = "Logical Foundations",
series = "Software Foundations",
volume = "1",
year = "2023",
publisher = "Electronic textbook",
note = {Version 6.5, \URL{http://softwarefoundations.cis.upenn.edu}}
}
Resources
Sample Exams
A large compendium of exams from many offerings of
CIS5000 ("Software Foundations") at the University of Pennsylvania
can be found at
https://www.seas.upenn.edu/~cis5000/current/exams/index.html.
There has been some drift of notations over the years, but most of
the problems are still relevant to the current text.
Lecture Videos
Lectures for two intensive summer courses based on
Logical
Foundations (part of the DeepSpec summer school series) can be
found at
https://deepspec.org/event/dsss17and
https://deepspec.org/event/dsss18/. The video quality in the
2017 lectures is poor at the beginning but gets better in the
later lectures.
Note for Instructors and Contributors
If you plan to use these materials in your own teaching, or if you
are using software foundations for self study and are finding
things you'd like to help add or improve, your contributions are
welcome! You are warmly invited to join the private SF git repo.
In order to keep the legalities simple and to have a single point
of responsibility in case the need should ever arise to adjust the
license terms, sublicense, etc., we ask all contributors (i.e.,
everyone with access to the developers' repository) to assign
copyright in their contributions to the appropriate "author of
record," as follows:
- I hereby assign copyright in my past and future contributions
to the Software Foundations project to the Author of Record of
each volume or component, to be licensed under the same terms
as the rest of Software Foundations. I understand that, at
present, the Authors of Record are as follows: For Volumes 1
and 2, known until 2016 as "Software Foundations" and from
2016 as (respectively) "Logical Foundations" and "Programming
Foundations," and for Volume 4, "QuickChick: Property-Based
Testing in Coq," the Author of Record is Benjamin C. Pierce.
For Volume 3, "Verified Functional Algorithms," and volume 5,
"Verifiable C," the Author of Record is Andrew W. Appel. For
Volume 6, "Separation Logic Foundations," the author of record
is Arthur Chargueraud. For components outside of designated
volumes (e.g., typesetting and grading tools and other
software infrastructure), the Author of Record is Benjamin
Pierce.
To get started, please send an email to Benjamin Pierce,
describing yourself and how you plan to use the materials and
including (1) the above copyright transfer text and (2) your
github username.
We'll set you up with access to the git repository and developers'
mailing lists. In the repository you'll find the files
INSTRUCTORS and
CONTRIBUTING with further instructions.
Thanks
Development of the
Software Foundations series has been
supported, in part, by the National Science Foundation under the
NSF Expeditions grant 1521523,
The Science of Deep
Specification.