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

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

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

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

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

A Substructural Epistemic Resource Logic: Theory and Modelling Applications

We present a substructural epistemic logic, based on Boolean BI, in which the epistemic modalities are parametrized on agents' local resources. The new modalities can be seen as generalizations of the usual epistemic modalities. The logic combines Boolean BI's resource semantics --- we introduce BI and its resource semantics at some length --- with epistemic agency. We illustrate the use of the logic in systems modelling by discussing some examples about access control, including semaphores, using resource tokens. We also give a labelled tableaux calculus and establish soundness and completeness with respect to the resource semantics

Intuitionistic Layered Graph Logic: Semantics and Proof Theory

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 called ILGL that gives an account of layering. The logic is a bunched system, combining 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 a labelled tableaux system with respect to a Kripke semantics on graphs. We then give an equivalent relational semantics, itself proven equivalent to an algebraic semantics via a representation theorem. We utilise this result in two ways. First, we prove decidability of the logic by showing the finite embeddability property holds for the algebraic semantics. Second, we prove a Stone-type duality theorem for the logic. By introducing the notions of ILGL hyperdoctrine and indexed layered frame we are able to extend this result to a predicate version of the logic and prove soundness and completeness theorems for an extension of the layered graph semantics . We indicate the utility of predicate ILGL with a resource-labelled bigraph model

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