# SLFPrefaceIntroduction to the course

Warning! Beta release!

# 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.
Note that this is not a course on how to specify and verify data structures and algorithms using Separation Logic---that will be the matter of yet-to-written chapters.
You are assumed to understand the material in Software Foundations Volume 1 (Logic Foundations), and the two chapters on Hoare Logic (Hoare and Hoare2) from Software Foundations Volume 2 (PL Foundations).
A good fraction of the contents from this course is described in the ICFP'20 paper: "Separation Logic for Sequential Programs", by Arthur Charguéraud. Its long version 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.

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 of the 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.
Central to Separation Logic is the frame rule, which is key to the modularity of the verification proofs. Its statement is 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 union H', producing the postcondition Q extended with that same resource H' unmodified.
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; such proofs are 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...), for verifying various kind of programs, ranging from low-level operating system kernels to high-level data structures and algorithms.
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 verification proof to be carried out through relatively concise proof scripts.
The purpose of this course is to explain how to set up such a construction of Separation Logic for sequential programs, embedded in Coq. To that end, we consider in this course:
• A minimalistic imperative programming language: 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.
• The simplest possible variant of Separation Logic.
For this core language and this core Separation Logic, we present in full the construction of a Separation Logic framework, all the way to presenting concise proof scripts for verifying programs manipulating linked lists.

## Course Summary

Before diving into the Coq files, the reader might be interested in material summarizing the contents of the course:
• SeqSepLogic.pdf is a LaTeX-formatted paper that gives a summary of most of the definitions involved in the course, yet not covering the chapters SLFBasic and SLFWPgen, which involve a weakest precondition generator.
• SLFSummary.pdf contains LaTeX-formatted slides presenting the most important definitions.
• SLFSummary.v contains Coq material, aimed to give a 1-hour tour of the key ingredients involved in the course.
• SLFMinimal.v contains a minimal proof of soundness of Separation Logic for sequential programs. It is aimed to argue for the simplicity of the soundness proof of Separation Logic when set up with the definitions considered in this course.

# Organization of the chapters

## Chapters

The "Foundations of Separation Logic" course includes the following chapters.
• SLFPreface: the present introduction.
• SLFBasic: introduction to Separation Logic through practical examples of specifications and verification proofs, on basic programs; this chapter is helpful for motivating Separation Logic.
• SLFHprop: definition of the core operators of Separation Logic, of Hoare triples, and of Separation Logic triples.
• SLFHimpl: definition of the entailment relation, statement and proof of its fundamental properties, and description of the simplification tactic for entailment.
• SLFRules: statement and proofs of the reasoning rules of Separation Logic, and examples of complete verification proofs.
• SLFWPsem: definition of the semantic notion of weakest precondition, and statement of reasoning rules in weakest-precondition style.
• SLFWPgen: presentation of a function that effectively computes weakest preconditions, and description of the construction of a set up that leads to concise verification proofs.
• SLFWand: introduction of the magic wand, of the ramified frame rule, and recursive computation of weakest precondition inside functions.
• SLFAffine: description of a generalized Separation Logic that supports the ability to freely discard certain types of heap predicates.
• SLFStruct: specification of array and record operations, and realization of these operations using pointer arithmetic.
• SLFRich: treatment of additional language constructs, including loops, assertions, and n-ary functions.

## Special chapters

• SLFSummary:This file contains the material for a one-hour talk that introduces, at a high level, the most important ideas from the course. This material is accompanied by LaTeX-generated slides to be found in the file SLFSummary.pdf.
• SLFDirect: This file provides the minimal set of definitions and lemmas required to build a practical program verification tool, without detour. This file is mostly self-contained; it depends only on the representation of variables, of finite maps, and on the implementation of the entailment simplification tactic (i.e., auxiliary files Var.v, Fmap.v, and SepSimpl.v). Note that the file SLFDirect contains a minimal amount of comments; explanations are given in the main course chapters.
• SLFExtra: This file recaps the definition and lemmas that are presented in the main course chapters but that are not included in the file SLFDirect. Again, this file serves as a reference, and does not contain further explanations.

## Teaching units

If you plan to use the material for teaching students, the following teaching units would probably make sense:
• SLFBasic
• SLFProp and SLFHimpl
• SLFRules
• SLFWPsem and SLFWPgen
• A presentation of the main ideas from the chapters SLFAffine and SLFStruct and SLFRich, with students reading the advanced contents if they are interested.

# Organization of each chapter

Each chapter contains three parts:
• the "chapter in a rush" part, which presents the main take-away messages,
• the "detailed contents" part, which presents important technical results,
• the "optional contents" part, intended for those who seek a deeper understanding, including in particular discussion of alternative definitions.
The course is organized in such a way that:
• reading only the "in a rush" parts of every chapter should make sense,
• all the "optional contents" parts are essentially independent: up to a few exceptions, these parts may be read, partially read, or skipped, without consequences on the understanding of the other chapters.
For the first chapter, the "detailed contents" material consists of exercises that are interleaved with the "chapter in a rush" material. For the other chapters, the tree parts are consecutive (i.e., not interleaved).

## Exercises

Each chapter includes numerous exercises. The star rating scheme is described in the Preface of Volume 1.
Disclaimer: the difficulty ratings currently in place are highly speculative. They will get tuned in subsequent releases.

# Practicalities

## System Requirements

The Preface of Volume 1 describes the Coq installation you will need. This edition was built with

## Note for CoqIDE users

CoqIDE 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 . SLF SLF*.v &

## TLC: tactics and libraries

The proofs are carried out using tactics from the TLC library. These tactics are very useful. Most of the tactics used in this course are described in the chapter UseTactics from the "Programming Language Foundations" (PLF) course. They are also briefly introduced in the present course.
The first two chapters, SLFBasic and SLFList, are careful to use as few TLC tactics as possible, and to explain the ones that are used. In the other chapters, TLC tactics are used in proof scripts to improve conciseness, however familiarity with these tactics should not be necessary to follow through the proofs. All exercises can be carried out without using TLC tactics.
Note that a few proofs also rely occasionally on lemmas from the TLC library, for example extensionality properties, or results on lists. Such lemmas are described whenever they are relevant to a proof.

## Imports between files

To simplify the compilation process, copies of the source files from the TLC libraries are included in the present folder. There is no need to look at these files, which are named Lib*.v.
The chapters introduce Separation Logic definitions and lemmas layer by layer. Several chapters import definitions from the previous layer. Other chapters instead import definitions from the files SLFDirect.v and SLFExtra.v, which summarize all the definitions of the course. Which definitions are imported should be essentially transparent to the reader.
There is one notable exception: the definition of the core operators of Separation Logic are set Opaque in SLFDirect. Doing so benefits to abstraction: one may no longer "unfold" the core definitions. Instead, one must work exclusively using the high-level lemmas that characterize the useful properties of the definitions.

# Dissemination

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 http://arthur.chargueraud.org/teach/verif
(If you are using the book as part of a class, your teacher may give you access to a locally modified version of the files, which you should use instead of the release version.)

## 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},
title = "Separation Logic Foundations",
series = "Software Foundations",
volume = "6",
year = "2020",
publisher = "Electronic textbook",
note = {Version 1.0, \URL{http://softwarefoundations.cis.upenn.edu} },
}

## For instructors and contributors

If you intend to use this course for teaching, please notify Arthur Charguéraud.
If you plan to use these materials in your own course, you will undoubtedly find things you'd like to change, improve, or add. Your contributions are welcome! Please see the Preface to Logical Foundations for instructions.
In particular, please do not hesitate to improve the formulation of the English sentences throughout this volume.

# 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.

(* 2020-09-03 15:43:08 (UTC+02) *)