153 research outputs found
Internal Calculi for Separation Logics
We present a general approach to axiomatise separation logics with heaplet semantics with no external features such as nominals/labels. To start with, we design the first (internal) Hilbert-style axiomatisation for the quantifier-free separation logic SL(?, -*). We instantiate the method by introducing a new separation logic with essential features: it is equipped with the separating conjunction, the predicate ls, and a natural guarded form of first-order quantification. We apply our approach for its axiomatisation. As a by-product of our method, we also establish the exact expressive power of this new logic and we show PSpace-completeness of its satisfiability problem
Proof-theoretic Semantics for Intuitionistic Multiplicative Linear Logic
This work is the first exploration of proof-theoretic semantics for a substructural logic. It focuses on the base-extension semantics (B-eS) for intuitionistic multiplicative linear logic (IMLL). The starting point is a review of Sandqvist’s B-eS for intuitionistic propositional logic (IPL), for which we propose an alternative treatment of conjunction that takes the form of the generalized elimination rule for the connective. The resulting semantics is shown to be sound and complete. This motivates our main contribution, a B-eS for IMLL
, in which the definitions of the logical constants all take the form of their elimination rule and for which soundness and completeness are established
Bunched logics: a uniform approach
Bunched logics have found themselves to be key tools in modern computer science, in particular through the industrial-level program verification formalism Separation Logic. Despite this—and in contrast to adjacent families of logics like modal and substructural logic—there is a lack of uniform methodology in their study, leaving many evident variants uninvestigated and many open problems unresolved. In this thesis we investigate the family of bunched logics—including previously unexplored intuitionistic variants—through two uniform frameworks. The first is a system of duality theorems that relate the algebraic and Kripke-style interpretations of the logics; the second, a modular framework of tableaux calculi that are sound and complete for both the core logics themselves, as well as many classes of bunched logic model important for applications in program verification and systems modelling. In doing so we are able to resolve a number of open problems in the literature, including soundness and completeness theorems for intuitionistic variants of bunched logics, classes of Separation Logic models and layered graph models; decidability of layered graph logics; a characterisation theorem for the classes of bunched logic model definable by bunched logic formulae; and the failure of Craig interpolation for principal bunched logics. We also extend our duality theorems to the categorical structures suitable for interpreting predicate versions of the logics, in particular hyperdoctrinal structures used frequently in Separation Logic
Proof-theoretic Semantics and Tactical Proof
The use of logical systems for problem-solving may be as diverse as in
proving theorems in mathematics or in figuring out how to meet up with a
friend. In either case, the problem solving activity is captured by the search
for an \emph{argument}, broadly conceived as a certificate for a solution to
the problem. Crucially, for such a certificate to be a solution, it has be
\emph{valid}, and what makes it valid is that they are well-constructed
according to a notion of inference for the underlying logical system. We
provide a general framework uniformly describing the use of logic as a
mathematics of reasoning in the above sense. We use proof-theoretic validity in
the Dummett-Prawitz tradition to define validity of arguments, and use the
theory of tactical proof to relate arguments, inference, and search.Comment: submitte
A Complete Axiomatisation for Quantifier-Free Separation Logic
We present the first complete axiomatisation for quantifier-free separation
logic. The logic is equipped with the standard concrete heaplet semantics and
the proof system has no external feature such as nominals/labels. It is not
possible to rely completely on proof systems for Boolean BI as the concrete
semantics needs to be taken into account. Therefore, we present the first
internal Hilbert-style axiomatisation for quantifier-free separation logic. The
calculus is divided in three parts: the axiomatisation of core formulae where
Boolean combinations of core formulae capture the expressivity of the whole
logic, axioms and inference rules to simulate a bottom-up elimination of
separating connectives, and finally structural axioms and inference rules from
propositional calculus and Boolean BI with the magic wand
Defining Logical Systems via Algebraic Constraints on Proofs
We comprehensively present a program of decomposition of proof systems for
non-classical logics into proof systems for other logics, especially classical
logic, using an algebra of constraints. That is, one recovers a proof system
for a target logic by enriching a proof system for another, typically simpler,
logic with an algebra of constraints that act as correctness conditions on the
latter to capture the former; for example, one may use Boolean algebra to give
constraints in a sequent calculus for classical propositional logic to produce
a sequent calculus for intuitionistic propositional logic. The idea behind such
forms of reduction is to obtain a tool for uniform and modular treatment of
proof theory and provide a bridge between semantics logics and their proof
theory. The article discusses the theoretical background of the project and
provides several illustrations of its work in the field of intuitionistic and
modal logics. The results include the following: a uniform treatment of modular
and cut-free proof systems for a large class of propositional logics; a general
criterion for a novel approach to soundness and completeness of a logic with
respect to a model-theoretic semantics; and a case study deriving a
model-theoretic semantics from a proof-theoretic specification of a logic.Comment: submitte
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