Stone-Type Dualities for Separation Logics

Stone-type duality theorems, which relate algebraic and relational/topological models, are important tools in logic because -- in addition to elegant abstraction -- they strengthen soundness and completeness to a categorical equivalence, yielding a framework through which both algebraic and topological methods can be brought to bear on a logic. We give a systematic treatment of Stone-type duality for the structures that interpret bunched logics, starting with the weakest systems, recovering the familiar BI and Boolean BI (BBI), and extending to both classical and intuitionistic Separation Logic. We demonstrate the uniformity and modularity of this analysis by additionally capturing the bunched logics obtained by extending BI and BBI with modalities and multiplicative connectives corresponding to disjunction, negation and falsum. This includes the logic of separating modalities (LSM), De Morgan BI (DMBI), Classical BI (CBI), and the sub-classical family of logics extending Bi-intuitionistic (B)BI (Bi(B)BI). We additionally obtain as corollaries soundness and completeness theorems for the specific Kripke-style models of these logics as presented in the literature: for DMBI, the sub-classical logics extending BiBI and a new bunched logic, Concurrent Kleene BI (connecting our work to Concurrent Separation Logic), this is the first time soundness and completeness theorems have been proved. We thus obtain a comprehensive semantic account of the multiplicative variants of all standard propositional connectives in the bunched logic setting. This approach synthesises a variety of techniques from modal, substructural and categorical logic and contextualizes the "resource semantics" interpretation underpinning Separation Logic amongst them

Decidability and Complexity of Tree Share Formulas

Fractional share models are used to reason about how multiple actors share ownership of resources. We examine the decidability and complexity of reasoning over the "tree share" model of Dockins et al. using first-order logic, or fragments thereof. We pinpoint a connection between the basic operations on trees union, intersection, and complement and countable atomless Boolean algebras, allowing us to obtain decidability with the precise complexity of both first-order and existential theories over the tree share model with the aforementioned operations. We establish a connection between the multiplication operation on trees and the theory of word equations, allowing us to derive the decidability of its existential theory and the undecidability of its full first-order theory. We prove that the full first-order theory over the model with both the Boolean operations and the restricted multiplication operation (with constants on the right hand side) is decidable via an embedding to tree-automatic structures

Completeness for a First-order Abstract Separation Logic

Existing work on theorem proving for the assertion language of separation logic (SL) either focuses on abstract semantics which are not readily available in most applications of program verification, or on concrete models for which completeness is not possible. An important element in concrete SL is the points-to predicate which denotes a singleton heap. SL with the points-to predicate has been shown to be non-recursively enumerable. In this paper, we develop a first-order SL, called FOASL, with an abstracted version of the points-to predicate. We prove that FOASL is sound and complete with respect to an abstract semantics, of which the standard SL semantics is an instance. We also show that some reasoning principles involving the points-to predicate can be approximated as FOASL theories, thus allowing our logic to be used for reasoning about concrete program verification problems. We give some example theories that are sound with respect to different variants of separation logics from the literature, including those that are incompatible with Reynolds's semantics. In the experiment we demonstrate our FOASL based theorem prover which is able to handle a large fragment of separation logic with heap semantics as well as non-standard semantics.Comment: This is an extended version of the APLAS 2016 paper with the same titl

Understanding and evolving the Rust programming language

Rust is a young systems programming language that aims to fill the gap between high-level languages—which provide strong static guarantees like memory and thread safety—and low-level languages—which give the programmer fine-grained control over data layout and memory management. This dissertation presents two projects establishing the first formal foundations for Rust, enabling us to better understand and evolve this important language: RustBelt and Stacked Borrows. RustBelt is a formal model of Rust’s type system, together with a soundness proof establishing memory and thread safety. The model is designed to verify the safety of a number of intricate APIs from the Rust standard library, despite the fact that the implementations of these APIs use unsafe language features. Stacked Borrows is a proposed extension of the Rust specification, which enables the compiler to use the strong aliasing information in Rust’s types to better analyze and optimize the code it is compiling. The adequacy of this specification is evaluated not only formally, but also by running real Rust code in an instrumented version of Rust’s Miri interpreter that implements the Stacked Borrows semantics. RustBelt is built on top of Iris, a language-agnostic framework, implemented in the Coq proof assistant, for building higher-order concurrent separation logics. This dissertation begins by giving an introduction to Iris, and explaining how Iris enables the derivation of complex high-level reasoning principles from a few simple ingredients. In RustBelt, this technique is exploited crucially to introduce the lifetime logic, which provides a novel separation-logic account of borrowing, a key distinguishing feature of the Rust type system.Rust ist eine junge systemnahe Programmiersprache, die es sich zum Ziel gesetzt hat, die Lücke zu schließen zwischen Sprachen mit hohem Abstraktionsniveau, die vor Speicher- und Nebenläufigkeitsfehlern schützen, und Sprachen mit niedrigem Abstraktionsniveau, welche dem Programmierer detaillierte Kontrolle über die Repräsentation von Daten und die Verwaltung des Speichers ermöglichen. Diese Dissertation stellt zwei Projekte vor, welche die ersten formalen Grundlagen für Rust zum Zwecke des besseren Verständnisses und der weiteren Entwicklung dieser wichtigen Sprache legen: RustBelt und Stacked Borrows. RustBelt ist ein formales Modell des Typsystems von Rust einschließlich eines Korrektheitsbeweises, welcher die Sicherheit von Speicherzugriffen und Nebenläufigkeit zeigt. Das Modell ist darauf ausgerichtet, einige komplexe Komponenten der Standardbibliothek von Rust zu verifizieren, obwohl die Implementierung dieser Komponenten unsichere Sprachkonstrukte verwendet. Stacked Borrows ist eine Erweiterung der Spezifikation von Rust, die es dem Compiler ermöglicht, den Quelltext mit Hilfe der im Typsystem kodierten Alias-Informationen besser zu analysieren und zu optimieren. Die Tauglichkeit dieser Spezifikation wird nicht nur formal belegt, sondern auch an echten Programmen getestet, und zwar mit Hilfe einer um Stacked Borrows erweiterten Version des Interpreters Miri. RustBelt basiert auf Iris, welches die Konstruktion von Separationslogiken für beliebige Programmiersprachen im Beweisassistenten Coq ermöglicht. Diese Dissertation beginnt mit einer Einführung in Iris und erklärt, wie komplexe Beweismethoden mit Hilfe weniger einfacher Bausteine hergeleitet werden können. In RustBelt wird diese Technik für die Umsetzung der „Lebenszeitlogik“ verwendet, einer Erweiterung der Separationslogik mit dem Konzept von „Leihgaben“ (borrows), welche eine wichtige Rolle im Typsystem von Rust spielen.This research was supported in part by a European Research Council (ERC) Consolidator Grant for the project "RustBelt", funded under the European Union’s Horizon 2020 Framework Programme (grant agreement no. 683289)

When champions meet: Rethinking the Bohr--Einstein debate

Einstein's philosophy of physics (as clarified by Fine, Howard, and Held) was predicated on his Trennungsprinzip, a combination of separability and locality, without which he believed objectification, and thereby "physical thought" and "physical laws", to be impossible. Bohr's philosophy (as elucidated by Hooker, Scheibe, Folse, Howard, Held, and others), on the other hand, was grounded in a seemingly different doctrine about the possibility of objective knowledge, namely the necessity of classical concepts. In fact, it follows from Raggio's Theorem in algebraic quantum theory that - within an appropriate class of physical theories - suitable mathematical translations of the doctrines of Bohr and Einstein are equivalent. Thus - upon our specific formalization - quantum mechanics accommodates Einstein's Trennungsprinzip if and only if it is interpreted a la Bohr through classical physics. Unfortunately, the protagonists themselves failed to discuss their differences in this constructive way, since their debate was dominated by Einstein's ingenious but ultimately flawed attempts to establish the "incompleteness" of quantum mechanics. This aspect of their debate may still be understood and appreciated, however, as reflecting a much deeper and insurmountable disagreement between Bohr and Einstein on the knowability of Nature. Using the theological controversy on the knowability of God as a analogy, Einstein was a Spinozist, whereas Bohr could be said to be on the side of Maimonides. Thus Einstein's off-the-cuff characterization of Bohr as a 'Talmudic philosopher' was spot-on.Comment: 22 pages. Argument sharpened and references update

A Formal C Memory Model for Separation Logic

The core of a formal semantics of an imperative programming language is a memory model that describes the behavior of operations on the memory. Defining a memory model that matches the description of C in the C11 standard is challenging because C allows both high-level (by means of typed expressions) and low-level (by means of bit manipulation) memory accesses. The C11 standard has restricted the interaction between these two levels to make more effective compiler optimizations possible, on the expense of making the memory model complicated. We describe a formal memory model of the (non-concurrent part of the) C11 standard that incorporates these restrictions, and at the same time describes low-level memory operations. This formal memory model includes a rich permission model to make it usable in separation logic and supports reasoning about program transformations. The memory model and essential properties of it have been fully formalized using the Coq proof assistant

Classical BI: Its Semantics and Proof Theory

We present Classical BI (CBI), a new addition to the family of bunched logics which originates in O'Hearn and Pym's logic of bunched implications BI. CBI differs from existing bunched logics in that its multiplicative connectives behave classically rather than intuitionistically (including in particular a multiplicative version of classical negation). At the semantic level, CBI-formulas have the normal bunched logic reading as declarative statements about resources, but its resource models necessarily feature more structure than those for other bunched logics; principally, they satisfy the requirement that every resource has a unique dual. At the proof-theoretic level, a very natural formalism for CBI is provided by a display calculus \`a la Belnap, which can be seen as a generalisation of the bunched sequent calculus for BI. In this paper we formulate the aforementioned model theory and proof theory for CBI, and prove some fundamental results about the logic, most notably completeness of the proof theory with respect to the semantics.Comment: 42 pages, 8 figure