Overview
Building reliable software is hard. The scale and complexity of
modern systems, the number of people involved in building them,
and the range of demands placed on them render 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 continually 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 and organizing programming 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) the idea of
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 topics is easily rich enough to fill a whole course
in its own right, so tackling all of them together naturally means
that much will be left unsaid. Nevertheless, we hope readers will
find that the themes illuminate and amplify each other and that
bringing them together creates a foundation from which it will be
easy to dig into any of them more deeply. Some suggestions for
further reading can be found in the
Postscript chapter.
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 notion of inductive proofs is
ubiquitous in all of computer science. You have surely seen them
before, in contexts from discrete math to 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
in just one direction: 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,
false, or ran out of time. Although their capabilities
are limited to fairly specific sorts of reasoning, they have
matured tremendously in recent years and are used now in a
huge variety 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, mostly in France, 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 performed. On top of this
kernel, the Coq environment provides high-level facilities for
proof development, including powerful tactics for constructing
complex proofs semi-automatically, and a large library of common
definitions and lemmas.
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,
Coq has been used, for example, to build CompCert, a
fully-verified optimizing compiler for C, 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.
- As a realistic environment for functional programming with
dependent types, it has inspired numerous innovations. For
example, the Ynot project at Harvard embedded "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 says: "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, before there were even any 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 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: the computation
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.
One 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 programs, 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 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 with 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 the book 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. From this point on, we
increasingly turn our attention to two broad topics of critical
importance to the enterprise of 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, together with ways of describing their
behavior in terms of mathematical functions or relations. Our
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 somewhat simpler and more readable definitions when it is
applicable. Later on, we switch to a more detailed "small-step"
style, which helps make some useful distinctions between different
sorts of "nonterminating" program behaviors and 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 yield 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 particular Imp
programs satisfy formal specifications of their behavior. We
introduce the notion of
Hoare triples — Imp programs annotated
with pre- and post-conditions describing what should 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, a powerful set of 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 property monitoring techniques for detecting when some
component of a system is not behaving according to specification.
This topic brings us full circle: 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!