PrefaceIntroduction to the Course

Welcome

This electronic book is Volume 6 of the Software Foundations series, which presents the mathematical underpinnings of reliable software.
In this volume, you will learn about the foundations of Separation Logic, a practical approach for the modular verification of imperative programs. In particular, this volume presents the building blocks for constructing a program verification tool. It does not, however, focus on reasoning about data structures and algorithms using Separation Logic. This aspect is covered to some extent by Volume 5 of Software Foundations, which presents Verifiable C, a program logic and proof system for C. For OCaml programs, this aspect will be covered in a yet-to-be-written volume presenting CFML, a tool that builds upon all the techniques presented in this volume.
You are only assumed to understand the material in Software Foundations Volume 1 (Logical Foundations), and the two chapters on Hoare Logic (Hoare and Hoare2) from Software Foundations Volume 2 (PL Foundations). The reading of Volume 5 is not a prerequisite. The exposition here is intended for a broad range of readers, from advanced undergraduates to PhD students and researchers.
A large fraction of the contents of this course is also written up in traditional LaTeX-style presentation, in the ICFP'20 article: Separation Logic for Sequential Programs, by Arthur Charguéraud. The long version of this paper is available from: http://www.chargueraud.org/research/2020/seq_seplogic/seq_seplogic.pdf.
This paper includes, in particular, a 5-page historical survey of contributions to mechanized presentations of Separation Logic, featuring 100+ citations. For a broader survey of Separation Logic, we recommend Peter O'Hearn's 2019 survey, which is available from: https://dl.acm.org/doi/10.1145/3211968-- including the supplemental material linked near the bottom of that page.

Separation Logic

Separation Logic is a program logic: it enables one to establish that a program satisfies its specification. Specifications are expressed using triples of the form {H} t {Q}. Whereas in Hoare logic the precondition H and the postcondition Q describe the whole memory state, in Separation Logic, H and Q describe only a fragment of the memory state. This fragment includes the resources necessary to the execution of t.
A key ingredient of Separation Logic is the frame rule, which enables modular proofs. It is stated as follows.
{ H } t { Q }  

{ H \* H' } t { Q \* H' }
The above rule asserts that if a term t executes correctly with the resources H and produces Q, then the term t admits the same behavior in a larger memory state, described by the union of H with a disjoint component H', producing the postcondition Q extended with that same resource H' unmodified. The star symbol \* denotes the separating conjunction operator of Separation Logic.
Separation Logic can be exploited in three kind of tools.
  • Automated proofs: the user provides only the code, and the tool locates sources of potential bugs. A good automated tool provides feedback that, most of time, is relevant.
  • Semi-automated proofs: the user provides not just the code, but also specifications and invariants. The tool then leverages automated solvers (e.g., SMT solvers) to discharge proof obligations.
  • Interactive proofs: the user provides not just the code and its specifications, but also a detailed proof script justifying the correctness of the code. These proofs may be worked on interactively using a proof assistant such as Coq.
The present course focuses on the third approach, that is, the integration of Separation Logic in an interactive proof assistant. This approach has been successfully put to practice throughout the world, using various proof assistants (Coq, Isabelle/HOL, HOL), targeting different languages (Assembly, C, SML, OCaml, Rust...), and for verifying various kind of programs, ranging from low-level operating system kernels to high-level data structures and algorithms.

Separation Logic in a proof assistant

The benefits of exploiting Separation Logic in a proof assistant include at least four major points:
  • higher-order logic provides virtually unlimited expressiveness that enables formulating arbitrarily complex specifications and invariants;
  • a proof assistant provides a unified framework to prove both the implementation details of the code and the underlying mathematical results form, e.g., results from theory or graph theory;
  • proof scripts may be easily maintained to reflect on a change to the source code;
  • the fact that Separation Logic is formalized in the proof assistant provides high confidence in the correctness of the tool.
Pretty much all the tools that leverage Separation Logic in a proof assistant are constructed following the same schema:
  • A formalization of the syntax and semantics of the source language. This is called a deep embedding of the programming language.
  • A definition of Separation Logic predicates as predicates from higher-order logic. This is called a shallow embedding of the program logic.
  • A definition of Separation Logic triples as a predicate, the statements of the reasoning rules as lemmas, and the proof of these reasoning rules with respect to the semantics.
  • An infrastructure that consists of lemmas, tactics and notation, allowing for the verification of concrete programs to be carried out through relatively concise proof scripts.
  • Applications of this infrastructure to the verification of concrete programs.
The purpose of this course is to explain how to set up such a construction. To that end, we consider in this course the simplest possible variant of Separation Logic, and apply it to a minimalistic imperative programming language. The language essentially consists of a lambda-calculus with references. This language admits a simple semantics and avoids in particular the need to distinguish between stack variables and heap- allocated variables. Advanced chapters from the course explains how to add support for loops, records, arrays, and n-ary functions.

Several Possible Depths of Reading

The material is organized in such a way that it can be easily adapted to the amount of time that the reader is ready to invest in the course.
The course contains 13 chapters, not counting the Preface and Postface. These chapters are organized in 3 major parts, as pictured in the dependency diagram.
  • The short curriculum includes only the 5 first chapters (ranging from chapter Basic to chapter Rules).
  • The medium curriculum includes 3 additional chapters (ranging from chapter WPsem to chapter Wand).
  • The full curriculum includes 5 more chapters (ranging from chapter Partial to chapter Rich).
In addition, each chapter is decomposed in three parts.
  • The First Pass section presents the most important ideas only. The course in designed in such a way that it is possible to read only the First Pass section of every chapter. The reader may be interested in going through all these sections to get the big picture, before revisiting each chapter in more details.
  • The More Details section presents additional material explaining in more depth the meaning and the consequences of the key results. This section also contains descriptions of the most important proofs. By default, readers would eventually read all this material.
  • The Optional Material section typically contains the remaining proofs, as well as discussions of alternative definitions. The Optional Material sections are all independent from each other. They would typically be of interest to readers who want to understand every detail, readers who are seeking for a deep understanding of a particular notion, and readers who are looking for answers to specific question.

List of Chapters

The first two chapters, namely chapters Basic and Repr give a primer on how to prove imperative programs in Separation Logic, thus focusing on the end user's perspective. All the other chapters focus on the implementor's perspective, explaining how Separation Logic is defined and how a practical verification tool can be constructed.
The list of chapters appears below. The numbering corresponds to teaching units: if the chapters were taught as part of a University course, one could presumably aim to cover one teaching unit per week.
  • (1) Basic: introduction to the core features of Separation Logic, illustrated using short programs manipulating references.
  • (1) Repr: introduction to representation predicates in Separation Logic, in particular for describing mutable lists and trees.
  • (2) Hprop: definition of the core operators of Separation Logic, of Hoare triples, and of Separation Logic triples.
  • (2) Himpl: definition of the entailment relation, statement and proofs of its fundamental properties, and description of the simplification tactic for entailment.
  • (3) Rules: statement and proofs of the reasoning rules of Separation Logic, and example proofs of programs using these rules.
  • (4) WPsem: definition of the semantic notion of weakest precondition, and statement of rules in weakest-precondition style.
  • (4) WPgen: presentation of a function that effectively computes the weakest precondition of a term, independently of its specification.
  • (5) Wand: introduction of the magic wand operator and of the ramified frame rule, and extension of the weakest-precondition generator for handling local function definitions.
  • (5) Affine: description of a generalization of Separation Logic with affine heap predicates, which are useful, in particular, for handling garbage-collected programming languages.
  • (6) Struct: specification of array and record operations, and encoding of these operations using pointer arithmetic.
  • (6) Rich: description of the treatment of additional language constructs, including loops, assertions, and n-ary functions.
  • (7) Nondet: definition of triples for non-deterministic programming languages.
  • (7) Partial: definition of triples for partial correctness only, i.e., for not requiring termination proofs.

Other Distributed Files

The chapters listed above depend on a number of auxiliary files, which the reader does not need to go through, but might be interested in looking at, either by curiosity, or for checking out a specific implementation detail.
  • LibSepReference: a long file that defines the program verification tool that is used in the first two chapters, and whose implementation is discussed throughout the other chapters. Each chapter from the course imports this module, as opposed to importing the chapters that precedes it.
  • LibSepVar: a formalization of program variables, together with a bunch of notations for parsing variables.
  • LibSepFmap: a formalization of finite maps, which are used for representing the memory state.
  • LibSepSimpl: a functor that implements a powerful tactic for automatically simplifying entailments in Separation Logic.
  • LibSepMinimal: a minimalistic formalization of a soundness proof for Separation Logic, corresponding to the definitions and proofs presented in the ICFP'20 paper mentioned earlier.
  • All other Lib* files are imports from the TLC library, which is described next.
The TLC library is a collection of general purpose theory and tactics developed over the years by Arthur Charguéraud. The TLC library is exploited in this course to streamline the presentation. TLC provides, in particular, extensions for classical logic and tactics that are particularly well-suited for meta-theory. Prior knowledge of TLC is not required, and all exercises can be completed without using TLC tactics.
The classical logic aspects of TLC are presented at the moment they appear in the course. The TLC tactics are also briefly described upon their first occurrence. Moreover, most of these tactics are presented in the chapter UseTactics of Software Foundations Volume 2 (Programming Language Foundations).

Practicalities

Feedback Welcome

If you intend to use this course either in class of for self-study, the author, Arthur Charguéraud, would love to hear from you. Just knowing in which context the course has been used and how much of the course students were able to cover is very valuable information.
If you plan on providing any non-small amount of feedback, do not hesitate to ask the author to be added as contributor to the github repository. In particular, please do not hesitate to improve the formulation of the English sentences throughout this volume, as the author is not a native speaker.

Exercises

Each chapter includes numerous exercises. The star rating scheme is described in the Preface of Software Foundations Volume 1 (Logical Foundations).
Disclaimer: the difficulty ratings currently in place are highly speculative! You feedback is very much welcome.
Disclaimer: the auto-grading system has not been tested for this volume. If you are interested in using auto-grading for this volume, please contact the author.

System Requirements

The Preface of Software Foundations Volume 1 (Logical Foundations) describes how to install Coq. The files you are reading have been tested with Coq 8.12.

Note for CoqIDE Users

CoqIDE typically works better with its asynchronous proof mode disabled. To load all the course files in CoqIDE, use the following command line.
  coqide -async-proofs off -async-proofs-command-error-resilience off -Q . Basic.v &

Recommended Citation Format

If you want to refer to this volume in your own writing, please do so as follows:
   @book {Charguéraud:SF6,
   author = {Arthur Charguéraud},
   title = "Separation Logic Foundations",
   series = "Software Foundations",
   volume = "6",
   year = "2021",
   publisher = "Electronic textbook",
   note = {Version 1.0, \URL{http://softwarefoundations.cis.upenn.edu} },
   }

Thanks

The development of the technical infrastructure for 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.
(* 2021-01-27 11:54 *)