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Merge; make prose nicer
author Adam Chlipala <adamc@hcoop.net>
date Wed, 03 Sep 2008 13:45:59 -0400
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21 21
22 (** * Whence This Book? *) 22 (** * Whence This Book? *)
23 23
24 (** 24 (**
25 25
26 We would all like to have programs check that our programs are correct. Due in no small part to some bold but unfulfilled promises in the history of computer science, today most people who write software, practitioners and academics alike, assume that the costs of formal program verification outweigh the associated benefits. The purpose of this book is to convince you that the technology of program verification is mature enough today that it makes sense to use it in a support role in many kinds of research projects in computer science. Beyond the convincing, I also want to provide a handbook on practical engineering of certified programs with the Coq proof assistant. 26 We would all like to have programs check that our programs are correct. Due in no small part to some bold but unfulfilled promises in the history of computer science, today most people who write software, practitioners and academics alike, assume that the costs of formal program verification outweigh the benefits. The purpose of this book is to convince you that the technology of program verification is mature enough today that it makes sense to use it in a support role in many kinds of research projects in computer science. Beyond the convincing, I also want to provide a handbook on practical engineering of certified programs with the Coq proof assistant.
27 27
28 There are a good number of (though definitely not "many") tools that are in wide use today for building machine-checked mathematical proofs and machine-certified programs. This is my attempt at an exhaustive list of interactive "proof assistants" satisfying a few criteria. First, the authors of each tool must intend for it to be put to use for software-related applications. Second, there must have been enough engineering effort put into the tool that someone not doing research on the tool itself would feel his time was well spent using it. A third criterion is more of an empirical validation of the second: the tool must have a significant user community outside of its own development team. 28 There are a good number of (though definitely not "many") tools that are in wide use today for building machine-checked mathematical proofs and machine-certified programs. This is my attempt at an exhaustive list of interactive "proof assistants" satisfying a few criteria. First, the authors of each tool must intend for it to be put to use for software-related applications. Second, there must have been enough engineering effort put into the tool that someone not doing research on the tool itself would feel his time was well spent using it. A third criterion is more of an empirical validation of the second: the tool must have a significant user community outside of its own development team.
29 29
30 % 30 %
31 \begin{tabular}{rl} 31 \begin{tabular}{rl}
61 (** ** Based on a Higher-Order Functional Programming Language *) 61 (** ** Based on a Higher-Order Functional Programming Language *)
62 62
63 (** 63 (**
64 There is no reason to give up the familiar comforts of functional programming when you start writing certified programs. All of the tools I listed are based on functional programming languages, which means you can use them without their proof-related aspects to write and run regular programs. 64 There is no reason to give up the familiar comforts of functional programming when you start writing certified programs. All of the tools I listed are based on functional programming languages, which means you can use them without their proof-related aspects to write and run regular programs.
65 65
66 ACL2 is notable in this field for having only a %\textit{%#<i>#first-order#</i>#%}% language at its foundation. That is, you cannot work with functions over functions and all those other treats that functional programmers love. By giving up this facility, ACL2 can make broader assumptions about how well its proof automation will work, but we can generally recover the same advantages in other proof assistants when we happen to be programming in first-order fragments. 66 ACL2 is notable in this field for having only a %\textit{%#<i>#first-order#</i>#%}% language at its foundation. That is, you cannot work with functions over functions and all those other treats of functional programming. By giving up this facility, ACL2 can make broader assumptions about how well its proof automation will work, but we can generally recover the same advantages in other proof assistants when we happen to be programming in first-order fragments.
67 *) 67 *)
68 68
69 69
70 (** ** Dependent Types *) 70 (** ** Dependent Types *)
71 71
91 (** ** Convenient Programmable Proof Automation *) 91 (** ** Convenient Programmable Proof Automation *)
92 92
93 (** 93 (**
94 A commitment to a kernel proof language opens up wide possibilities for user extension of proof automation systems, without allowing user mistakes to trick the overall system into accepting invalid proofs. Almost any interesting verification problem is undecidable, so it is important to help users build their own procedures for solving the restricted problems that they encounter in particular implementations. 94 A commitment to a kernel proof language opens up wide possibilities for user extension of proof automation systems, without allowing user mistakes to trick the overall system into accepting invalid proofs. Almost any interesting verification problem is undecidable, so it is important to help users build their own procedures for solving the restricted problems that they encounter in particular implementations.
95 95
96 Twelf features no proof automation marked as a bonafide part of the latest release; there is some code included for testing purposes. The Twelf style is based on writing out all proofs in full detail. Because Twelf is specialized to the domain of syntactic metatheory proofs about programming languages and logics, it is feasible to use it to write those kinds of proofs manually. Outside that domain, the lack of automation can be a serious obstacle to productivity. Most kinds of program verification fall outside Twelf's forte. 96 Twelf features no proof automation marked as a bonafide part of the latest release; there is some automation code included for testing purposes. The Twelf style is based on writing out all proofs in full detail. Because Twelf is specialized to the domain of syntactic metatheory proofs about programming languages and logics, it is feasible to use it to write those kinds of proofs manually. Outside that domain, the lack of automation can be a serious obstacle to productivity. Most kinds of program verification fall outside Twelf's forte.
97 97
98 Of the remaining tools, all can support user extension with new decision procedures by hacking directly in the tool's implementation language (such as OCaml for Coq). Since ACL2 and PVS do not satisfy the de Bruijn criterion, overall correctness is at the mercy of the authors of new procedures. 98 Of the remaining tools, all can support user extension with new decision procedures by hacking directly in the tool's implementation language (such as OCaml for Coq). Since ACL2 and PVS do not satisfy the de Bruijn criterion, overall correctness is at the mercy of the authors of new procedures.
99 99
100 Isabelle/HOL and Coq both support coding new proof manipulations in ML in ways that cannot lead to the acceptance of invalid proofs. Additionally, Coq includes a domain-specific language for coding decision procedures in normal Coq source code, with no need to break out into ML. This language is called Ltac, and I think of it as the unsung hero of the proof assistant world. Not only does Ltac prevent you from making fatal mistakes, it also includes a number of novel programming constructs which combine to make a "proof by decision procedure" style very pleasant. We will meet these features in the chapters to come. 100 Isabelle/HOL and Coq both support coding new proof manipulations in ML in ways that cannot lead to the acceptance of invalid proofs. Additionally, Coq includes a domain-specific language for coding decision procedures in normal Coq source code, with no need to break out into ML. This language is called Ltac, and I think of it as the unsung hero of the proof assistant world. Not only does Ltac prevent you from making fatal mistakes, it also includes a number of novel programming constructs which combine to make a "proof by decision procedure" style very pleasant. We will meet these features in the chapters to come.
101 *) 101 *)
104 104
105 (** 105 (**
106 A surprising wealth of benefits follow from choosing a proof language that integrates a rich notion of computation. Coq includes programs and proof terms in the same syntactic class. This makes it easy to write programs that compute proofs. With rich enough dependent types, such programs are %\textit{%#<i>#certified decision procedures#</i>#%}%. In such cases, these certified procedures can be put to good use %\textit{%#<i>#without ever running them#</i>#%}%! Their types guarantee that, if we did bother to run them, we would receive proper "ground" proofs. 106 A surprising wealth of benefits follow from choosing a proof language that integrates a rich notion of computation. Coq includes programs and proof terms in the same syntactic class. This makes it easy to write programs that compute proofs. With rich enough dependent types, such programs are %\textit{%#<i>#certified decision procedures#</i>#%}%. In such cases, these certified procedures can be put to good use %\textit{%#<i>#without ever running them#</i>#%}%! Their types guarantee that, if we did bother to run them, we would receive proper "ground" proofs.
107 107
108 The critical ingredient for this technique, many of whose instances are referred to as %\textit{%#<i>#proof by reflection#</i>#%}%, is a way of inducing non-trivial computation inside of logical propositions during proof checking. Further, most of these instances require dependent types to make it possible to state the appropriate theorems. Of the proof assistants I listed, only Coq really provides this support. 108 The critical ingredient for this technique, many of whose instances are referred to as %\textit{%#<i>#proof by reflection#</i>#%}%, is a way of inducing non-trivial computation inside of logical propositions during proof checking. Further, most of these instances require dependent types to make it possible to state the appropriate theorems. Of the proof assistants I listed, only Coq really provides this support.
109 *)
110
111
112 (** * Why Not a Different Dependently-Typed Language? *)
113
114 (**
115 The logic and programming language behind Coq belongs to a type-theory ecosystem with a good number of other thriving members. %Agda\footnote{\url{http://appserv.cs.chalmers.se/users/ulfn/wiki/agda.php}}%#<a href="http://appserv.cs.chalmers.se/users/ulfn/wiki/agda.php">Agda</a># and %Epigram\footnote{\url{http://www.e-pig.org/}}%#<a href="http://www.e-pig.org/">Epigram</a># are the most developed tools among the alternatives to Coq, and there are others that are earlier in their lifecycles. All of the languages in this family feel sort of like different historical offshoots of Latin. The hardest conceptual epiphanies are, for the most part, portable among all the languages. Given this, why choose Coq for certified programming?
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117 I think the answer is simple. None of the competition has well-developed systems for tactic-based theorem proving. Agda and Epigram are designed and marketed more as programming languages than proof assistants. Dependent types are great, because they often help you prove deep theorems without doing anything that feels like proving. Nonetheless, almost any interesting certified programming project will benefit from some activity that deserves to be called proving, and many interesting projects absolutely require semi-automated proving, if the sanity of the programmer is to be safeguarded. Informally, proving is unavoidable when any correctness proof for a program has a structure that does not mirror the structure of the program itself. An example is a compiler correctness proof, which probably proceeds by induction on program execution traces, which have no simple relationship with the structure of the compiler or the structure of the programs it compiles. In building such proofs, a mature system for scripted proof automation is invaluable.
118
119 On the other hand, Agda, Epigram, and similar tools have less implementation baggage associated with them, and so they tend to be the default first homes of innovations in practical type theory. Some significant kinds of dependently-typed programs are much easier to write in Agda and Epigram than in Coq. The former tools may very well be superior choices for projects that do not involve any "proving." Anecdotally, I have gotten the impression that manual proving is orders of magnitudes more costly then manual coping with Coq's lack of programming bells and whistles. In this book, I will devote significant time to patterns for programming with dependent types in Coq as it is today, and I will also try to mention related innovations in Agda and Epigram. We can hope that the type theory community is tending towards convergence on the right set of features for practical programming with dependent types, and that we will eventually have a single tool embodying those features.
109 *) 120 *)
110 121
111 122
112 (** * Engineering with a Proof Assistant *) 123 (** * Engineering with a Proof Assistant *)
113 124
125 (** 136 (**
126 I try to keep the required background knowledge to a minimum in this book. I will assume familiarity with the material from usual discrete math and logic courses taken by all undergraduate computer science majors, and I will assume that readers have significant experience programming in one of the ML dialects, in Haskell, or in some other, closely-related language. Experience with only dynamically-typed functional languages might lead to befuddlement in some places, but a reader who has come to grok Scheme deeply will probably be fine. 137 I try to keep the required background knowledge to a minimum in this book. I will assume familiarity with the material from usual discrete math and logic courses taken by all undergraduate computer science majors, and I will assume that readers have significant experience programming in one of the ML dialects, in Haskell, or in some other, closely-related language. Experience with only dynamically-typed functional languages might lead to befuddlement in some places, but a reader who has come to grok Scheme deeply will probably be fine.
127 138
128 A good portion of the book is about how to formalize programming languages, compilers, and proofs about them. I depart significantly from today's most popular methodology for pencil-and-paper formalism among programming languages researchers. There is no need to be familiar with operational semantics, preservation and progress theorems, or any of the material found in courses on programming language semantics but not in basic discrete math and logic courses. I will use operational semantics very sparingly, and there will be no preservation or progress proofs. Instead, I will use a style that seems to work much better in Coq, which can be given the fancy-sounding name %\textit{%#<i>#foundational type-theoretic semantics#</i>#%}% or the more populist name %\textit{%#<i>#semantics by definitional compilers#</i>#%}%. 139 A good portion of the book is about how to formalize programming languages, compilers, and proofs about them. I depart significantly from today's most popular methodology for pencil-and-paper formalism among programming languages researchers. There is no need to be familiar with operational semantics, preservation and progress theorems, or any of the material found in courses on programming language semantics but not in basic discrete math and logic courses. I will use operational semantics very sparingly, and there will be no preservation or progress proofs. Instead, I will use a style that seems to work much better in Coq, which can be given the fancy-sounding name %\textit{%#<i>#foundational type-theoretic semantics#</i>#%}% or the more populist name %\textit{%#<i>#semantics by definitional compilers#</i>#%}%.
129 *) 140 *)
141
142
143 (** * Using This Book *)
144
145 (**
146 This book is generated automatically from Coq source files using the wonderful coqdoc program. The latest PDF version is available at:
147 %\begin{center}\url{http://adam.chlipala.net/cpdt/cpdt.pdf}\end{center}%#<blockquote><tt><a href="http://adam.chlipala.net/cpdt/cpdt.pdf">http://adam.chlipala.net/cpdt/cpdt.pdf</a></tt></blockquote>#
148 There is also an online HTML version available, with a hyperlink from each use of an identifier to that identifier's definition:
149 %\begin{center}\url{http://adam.chlipala.net/cpdt/html/}\end{center}%#<blockquote><tt><a href="http://adam.chlipala.net/cpdt/html/">http://adam.chlipala.net/cpdt/html/</a></tt></blockquote>#
150
151 The chapters of this book are named like "Module Foo," rather than having proper names, because literally the entire document is generated by coqdoc, which by default bases chapter structure on the module structure of the development being documented. This chapter is headed "Module Intro" because it comes from a module named [Intro], which comes from a fascinating source file %\texttt{%#<tt>#Intro.v#</tt>#%}% containing nothing but specially-formatted coqdoc comments.
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153 The source code to the book is also freely available at:
154 %\begin{center}\url{http://adam.chlipala.net/cpdt/cpdt.tgz}\end{center}%#<blockquote><tt><a href="http://adam.chlipala.net/cpdt/cpdt.tgz">http://adam.chlipala.net/cpdt/cpdt.tgz</a></tt></blockquote>#
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156 There, you can find all of the code appearing in this book, with prose interspersed in comments, in exactly the order that you find in this document. You can step through the code interactively with your chosen graphical Coq interface. The code also has special comments indicating which parts of the chapters make suitable starting points for interactive class sessions, where the class works together to construct the programs and proofs. The included Makefile has a target %\texttt{%#<tt>#templates#</tt>#%}% for building a fresh set of class template files automatically from the book source.
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158 I believe that a good graphical interface to Coq is crucial for using it productively. I use the %Proof General\footnote{\url{http://proofgeneral.inf.ed.ac.uk/}}%#<a href="http://proofgeneral.inf.ed.ac.uk/">Proof General</a># mode for Emacs, which supports a number of other proof assistants besides Coq. There is also the standalone CoqIDE program developed by the Coq team. I like being able to combine certified programming and proving with other kinds of work inside the same full-featured editor, and CoqIDE has had a good number of crashes and other annoying bugs in recent history, though I hear that it is improving. In the initial part of this book, I will reference Proof General procedures explicitly, in introducing how to use Coq, but most of the book will be interface-agnostic, so feel free to use CoqIDE if you prefer it.
159 *)