42 research outputs found
Intuitionistic layered graph logic
Models of complex systems are widely used in the physical and social sciences, and the concept of layering, typically building upon graph-theoretic structure, is a common feature. We describe an intuitionistic substructural logic that gives an account of layering. As in bunched systems, the logic includes the usual intuitionistic connectives, together with a non-commutative, non-associative conjunction (used to capture layering) and its associated implications. We give soundness and completeness theorems for labelled tableaux and Hilbert-type systems with respect to a Kripke semantics on graphs. To demonstrate the utility of the logic, we show how to represent a range of systems and security examples, illuminating the relationship between services/policies and the infrastructures/architectures to which they are applied
Structural Interactions and Absorption of Structural Rules in BI Sequent Calculus
Development of a contraction-free BI sequent calculus, be it in the sense of
G3i or G4i, has not been successful in literature. We address the open problem
by presenting such a sequent system. In fact our calculus involves no
structural rules
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
Coalgebraic completeness-via-canonicity for distributive substructural logics
We prove strong completeness of a range of substructural logics with respect
to a natural poset-based relational semantics using a coalgebraic version of
completeness-via-canonicity. By formalizing the problem in the language of
coalgebraic logics, we develop a modular theory which covers a wide variety of
different logics under a single framework, and lends itself to further
extensions. Moreover, we believe that the coalgebraic framework provides a
systematic and principled way to study the relationship between resource models
on the semantics side, and substructural logics on the syntactic side.Comment: 36 page
Nominal String Diagrams
We introduce nominal string diagrams as string diagrams internal in the category of nominal sets. This requires us to take nominal sets as a monoidal category, not with the cartesian product, but with the separated product. To this end, we develop the beginnings of a theory of monoidal categories internal in a symmetric monoidal category. As an instance, we obtain a notion of a nominal PROP as a PROP internal in nominal sets. A 2-dimensional calculus of simultaneous substitutions is an application
A Stone-type Duality Theorem for Separation Logic Via its Underlying Bunched 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 theorems for the structures that interpret bunched logics, starting with the weakest systems, recovering the familiar Boolean BI, and concluding with Separation Logic. Our results encompass all the known existing algebraic approaches to Separation Logic and prove them sound with respect to the standard store-heap semantics. We additionally recover soundness and completeness theorems of the specific truth-functional models of these logics as presented in the literature. This approach synthesises a variety of techniques from modal, substructural and categorical logic and contextualises the ‘resource semantics’ interpretation underpinning Separation Logic amongst them. As a consequence, theory from those fields — as well as algebraic and topological methods — can be applied to both Separation Logic and the systems of bunched logics it is built upon. Conversely, the notion of indexed resource frame (generalizing the standard model of Separation Logic) and its associated completeness proof can easily be adapted to other non-classical predicate logics
Focused Proof-search in the Logic of Bunched Implications
The logic of Bunched Implications (BI) freely combines additive and
multiplicative connectives, including implications; however, despite its
well-studied proof theory, proof-search in BI has always been a difficult
problem. The focusing principle is a restriction of the proof-search space that
can capture various goal-directed proof-search procedures. In this paper, we
show that focused proof-search is complete for BI by first reformulating the
traditional bunched sequent calculus using the simpler data-structure of nested
sequents, following with a polarised and focused variant that we show is sound
and complete via a cut-elimination argument. This establishes an operational
semantics for focused proof-search in the logic of Bunched Implications.Comment: 18 pages conten
On Malfunction, Mechanisms and Malware Classification
Malware has been around since the 1980s and is a large and expensive security concern today, constantly growing over the past years. As our social, professional and financial lives become more digitalised, they present larger and more profitable targets for malware. The problem of classifying and preventing malware is therefore urgent, and it is complicated by the existence of several specific approaches. In this paper, we use an existing malware taxonomy to formulate a general, language independent functional description of malware as transformers between states of the host system and described by a trust relation with its components. This description is then further generalised in terms of mechanisms, thereby contributing to a general understanding of malware. The aim is to use the latter in order to present an improved classification method for malware