Sub-classical Boolean Bunched Logics and the Meaning of Par

We investigate intermediate logics between the bunched logics Boolean BI and Classical BI, obtained by combining classical propositional logic with various flavours of Hyland and De Paiva\u27s full intuitionistic linear logic. Thus, in addition to the usual multiplicative conjunction (with its adjoint implication and unit), our logics also feature a multiplicative disjunction (with its adjoint co-implication and unit). The multiplicatives behave "sub-classically", in that disjunction and conjunction are related by a weak distribution principle, rather than by De Morgan equivalence. We formulate a Kripke semantics, covering all our sub-classical bunched logics, in which the multiplicatives are naturally read in terms of resource operations. Our main theoretical result is that validity according to this semantics coincides with provability in a corresponding Hilbert-style proof system. Our logical investigation sheds considerable new light on how one can understand the multiplicative disjunction, better known as linear logic\u27s "par", in terms of resource operations. In particular, and in contrast to the earlier Classical BI, the models of our logics include the heap-like memory models of separation logic, in which disjunction can be interpreted as a property of intersection operations over heaps

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

A bunch of sessions:a propositions-as-sessions interpretation of bunched implications in channel-based concurrency

The emergence of propositions-as-sessions, a Curry-Howard correspondence between propositions of Linear Logic and session types for concurrent processes, has settled the logical foundations of message-passing concurrency. Central to this approach is the resource consumption paradigm heralded by Linear Logic. In this paper, we investigate a new point in the design space of session type systems for message-passing concurrent programs. We identify O’Hearn and Pym’s Logic of Bunched Implications (BI) as a fruitful basis for an interpretation of the logic as a concurrent programming language. This leads to a treatment of non-linear resources that is radically different from existing approaches based on Linear Logic. We introduce a new π-calculus with sessions, called πBI; its most salient feature is a construct called spawn, which expresses new forms of sharing that are induced by structural principles in BI. We illustrate the expressiveness of πBI and lay out its fundamental theory: type preservation, deadlock-freedom, and weak normalization results for well-typed processes; an operationally sound and complete typed encoding of an affine λ-calculus; and a non-interference result for access of resources

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

Resource semantics: logic as a modelling technology

The Logic of Bunched Implications (BI) was introduced by O'Hearn and Pym. The original presentation of BI emphasised its role as a system for formal logic (broadly in the tradition of relevant logic) that has some interesting properties, combining a clean proof theory, including a categorical interpretation, with a simple truth-functional semantics. BI quickly found significant applications in program verification and program analysis, chiefly through a specific theory of BI that is commonly known as 'Separation Logic'. We survey the state of work in bunched logics - which, by now, is a quite large family of systems, including modal and epistemic logics and logics for layered graphs - in such a way as to organize the ideas into a coherent (semantic) picture with a strong interpretation in terms of resources. One such picture can be seen as deriving from an interpretation of BI's semantics in terms of resources, and this view provides a basis for a systematic interpretation of the family of bunched logics, including modal, epistemic, layered graph, and process-theoretic variants, in terms of resources. We explain the basic ideas of resource semantics, including comparisons with Linear Logic and ideas from economics and physics. We include discussions of BI's λ-calculus, of Separation Logic, and of an approach to distributed systems modelling based on resource semantics

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

Human decision-making in computer security incident response

Background: Cybersecurity has risen to international importance. Almost every organization will fall victim to a successful cyberattack. Yet, guidance for computer security incident response analysts is inadequate. Research Questions: What heuristics should an incident analyst use to construct general knowledge and analyse attacks? Can we construct formal tools to enable automated decision support for the analyst with such heuristics and knowledge? Method: We take an interdisciplinary approach. To answer the first question, we use the research tradition of philosophy of science, specifically the study of mechanisms. To answer the question on formal tools, we use the research tradition of program verification and logic, specifically Separation Logic. Results: We identify several heuristics from biological sciences that cybersecurity researchers have re-invented to varying degrees. We consolidate the new mechanisms literature to yield heuristics related to the fact that knowledge is of clusters of multi-field mechanism schema on four dimensions. General knowledge structures such as the intrusion kill chain provide context and provide hypotheses for filling in details. The philosophical analysis answers this research question, and also provides constraints on building the logic. Finally, we succeed in defining an incident analysis logic resembling Separation Logic and translating the kill chain into it as a proof of concept. Conclusion: These results benefits incident analysis, enabling it to expand from a tradecraft or art to also integrate science. Future research might realize our logic into automated decision-support. Additionally, we have opened the field of cybersecuity to collaboration with philosophers of science and logicians