A proof-theoretic analysis of the classical propositional matrix method

The matrix method, due to Bibel and Andrews, is a proof procedure designed for automated theorem-proving. We show that underlying this method is a fully structured combinatorial model of conventional classical proof theory. © 2012 The Author, 2012. Published by Oxford University Press

A calculus and logic of bunched resources and processes

Mathematical modelling and simulation modelling are fundamental tools of engineering, science, and social sciences such as economics, and provide decision-support tools in management. Mathematical models are essentially deployed at all scales, all levels of complexity, and all levels of abstraction. Models are often required to be executable, as a simulation, on a computer. We present some contributions to the process-theoretic and logical foundations of discrete-event modelling with resources and processes. Building on previous work in resource semantics, process calculus, and modal logic, we describe a process calculus with an explicit representation of resources in which processes and resources co-evolve. The calculus is closely connected to a substructural modal logic that may be used as a specification language for properties of models. In contrast to earlier work, we formulate the resource semantics, and its relationship with process calculus, in such a way that we obtain soundness and completeness of bisimulation with respect to logical equivalence for the naturally full range of logical connectives and modalities. We give a range of examples of the use of the process combinators and logical structure to describe system structure and behaviour

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

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

A System of Interaction and Structure II: The Need for Deep Inference

This paper studies properties of the logic BV, which is an extension of multiplicative linear logic (MLL) with a self-dual non-commutative operator. BV is presented in the calculus of structures, a proof theoretic formalism that supports deep inference, in which inference rules can be applied anywhere inside logical expressions. The use of deep inference results in a simple logical system for MLL extended with the self-dual non-commutative operator, which has been to date not known to be expressible in sequent calculus. In this paper, deep inference is shown to be crucial for the logic BV, that is, any restriction on the ``depth'' of the inference rules of BV would result in a strictly less expressive logical system

Discrete Choice, Social Interaction, and Policy in Encryption Technology Adoption

We introduce a model for examining the factors that lead to the adoption of new encryption technologies. Building on the work of Brock and Durlauf, the model describes how agents make choices, in the presence of social interaction, between competing technologies given their relative cost, functionality, and usability. We apply the model to examples about the adoption of encryption in communication (email and messaging) and storage technologies (self-encrypting drives) and also consider our model’s predictions for the evolution of technology adoption over time

Practicing a Science of Security: A Philosophy of Science Perspective

Our goal is to refocus the question about cybersecurity research from 'is this process scientific' to 'why is this scientific process producing unsatisfactory results'. We focus on five common complaints that claim cybersecurity is not or cannot be scientific. Many of these complaints presume views associated with the philosophical school known as Logical Empiricism that more recent scholarship has largely modified or rejected. Modern philosophy of science, supported by mathematical modeling methods, provides constructive resources to mitigate all purported challenges to a science of security. Therefore, we argue the community currently practices a science of cybersecurity. A philosophy of science perspective suggests the following form of practice: structured observation to seek intelligible explanations of phenomena, evaluating explanations in many ways, with specialized fields (including engineering and forensics) constraining explanations within their own expertise, inter-translating where necessary. A natural question to pursue in future work is how collecting, evaluating, and analyzing evidence for such explanations is different in security than other sciences

Categorical Proof-theoretic Semantics

In proof-theoretic semantics, model-theoretic validity is replaced by proof-theoretic validity. Validity of formulae is defined inductively from a base giving the validity of atoms using inductive clauses derived from proof-theoretic rules. A key aim is to show completeness of the proof rules without any requirement for formal models. Establishing this for propositional intuitionistic logic raises some technical and conceptual issues. We relate Sandqvist’s (complete) base-extension semantics of intuitionistic propositional logic to categorical proof theory in presheaves, reconstructing categorically the soundness and completeness arguments, thereby demonstrating the naturality of Sandqvist’s constructions. This naturality includes Sandqvist’s treatment of disjunction that is based on its second-order or elimination-rule presentation. These constructions embody not just validity, but certain forms of objects of justifications. This analysis is taken a step further by showing that from the perspective of validity, Sandqvist’s semantics can also be viewed as the natural disjunction in a category of sheaves

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

You've left me no choices: Security economics to inform behaviour intervention support in organizations

Security policy-makers (influencers) in an organization set security policies that embody intended behaviours for employees (as decision-makers) to follow. Decision-makers then face choices, where this is not simply a binary decision of whether to comply or not, but also how to approach compliance and secure working alongside other workplace pressures, and limited resources for identifying optimal security-related choices. Conflict arises due to information asymmetries present in the relationship, where influencers and decision-makers both consider costs, gains, and losses in ways which are not necessarily aligned. With the need to promote ‘good enough’ decisions about security-related behaviours under such constraints, we hypothesize that actions to resolve this misalignment can benefit from constructs from both traditional economics and behavioural economics. Here we demonstrate how current approaches to security behaviour provisioning in organizations mirror rational-agent economics, even where behavioural economics is embodied in the promotion of individual security behaviours. We develop and present a framework to accommodate bounded security decision-making, within an ongoing programme of behaviours which must be provisioned for and supported. We also point to applications of the framework in negotiating sustainable security behaviours, such as policy concordance and just security cultures