Welcome

This electronic book is Volume 6 of the Software Foundations series, which presents the mathematical underpinnings of reliable software. *COMPANION COURSE NOTES*: A large part of this course is also written up in traditional LaTeX-style presentation. It's available from http://www.chargueraud.org/teach/verif/slf_notes.pdf. This book will teach you about the foundations of Separation Logic, a practical approach to the modular verification of imperative programs. In particular, it 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). 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.

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 that 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 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 (at least 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 developed 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, e.g., results from number theory or graph theory;
• proof scripts may be easily maintained to reflect changes to the source code; and
• the fact that Separation Logic itself 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 proof assistants 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 in 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 notations, 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. It avoids, in particular, the need to distinguish between stack variables and heap- allocated variables. Advanced chapters later in the course explain how to add support for loops, records, arrays, and n-ary functions.

Multiple Reading Depths

All chapters except the first one are organized 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 detail.
• 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. These sections are intended for (1) readers who plan to continue studying Separation Logic beyond the present course, and (2) teachers using the course.

Chapters Overview

The first two chapters, Basic and Repr, give a primer on how to prove imperative programs in Separation Logic, i.e. they focus on the end user's perspective. The eight following chapters focus on the implementor's perspective, explaining how Separation Logic is defined and how a practical verification tool can be constructed. The last three chapters cover language extensions, from the perspective of both the user and the implementor. 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 reasonably aim to cover one teaching unit per week.

• (1) Basic: introduction to the core features of Separation Logic,
• (2) 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.
• (2) Himpl: definition of the entailment relation, statement and proofs of its fundamental properties, and description of the simplification tactic for entailment.
• (3) Triples: definition of Separation Logic triples in terms of the semantics of the programming language.
• (3) Rules: statement and proofs of the reasoning rules of Separation Logic, and example proofs of programs using these rules.
• (4) Wand: introduction of the magic wand operator and other Separation Logic operators, and to the ramified frame rule.
• (4) WPsem: definition of the semantic notion of weakest precondition, and statement of rules in weakest-precondition style.
• (5) WPgen: presentation of a function that effectively computes the weakest precondition of a term, independently of its specification.
• (5) WPsound: soundness proof for the weakest precondition generator (mostly optional).
• (6) Affine: description of a generalization of Separation Logic with affine heap predicates, which are useful, in particular, for handling garbage-collected programming languages.
• (6) Arrays: specification of both ML-style arrays with headers and C-style arrays with pointer arithmetic.
• (6) Records: representation predicate for records, allowing us to isolate arbitrary subsets of a record's fields.

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 earlier chapters.
• 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, up to the verification of the incr function.
• 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 in chapter Hprop. Each TLC tactic is introduced when it is first used. Most of these tactics are also presented in the chapter UseTactics of Software Foundations Volume 2 (Programming Language Foundations).

Practicalities

System Requirements

Install instructions for Coq and IDEs may be found on this page: https://www.chargueraud.org/teach/verif/install/install.html The files you are reading have been tested with Coq version 8.17.1 but may also work with other versions.

Feedback Welcome

If you intend to use this course either in class of for self-study, the author would love to hear from you. Just knowing in which contexts the course has been used and how much of the text students were able to cover is very valuable information. You can send feedback to slf --at-- chargueraud.org. 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.

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 fairly speculative. You feedback is very much welcome. Disclaimer: (for instructors) 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.

Recommended Citation Format

If you want to refer to this volume in your own writing, please do so as follows:
    @book {Chargueraud:SF₆,
    author = {Arthur Charguéraud},
    editor = {Benjamin C. Pierce},
    title = "Separation Logic Foundations",
    series = "Software Foundations",
    volume = "6",
    year = "2024",
    publisher = "Electronic textbook",
    note = {Version 2.2, \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.