Intuitionistic Layered Graph Logic

Abstract

Abstract. 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.

Modular Tableaux Calculi for Separation Theories

“…Technical Approach. The present work generalizes methods pioneered on tableaux systems for a range of logics including and related to BI and BBI [20,22,28,34] to specify modular tableaux calculi for the breadth of separation theories in the literature, proved sound and complete uniformly and parametrically in choice of separation theory. While previous systems implicitly implement a systematic method for constructing tableaux proof theory for bunched logics, subtle but significant changes must be made to additionally capture separation theories.…”

“…BI and BBI are archetypal examples of bunched logics; systems given by combining the standard additives of classical or intutionistic propositional logic with the multiplicatives of a substructural logic. This idea has been developed to give logics for reasoning about concurrency [23] and the layering structure of complex systems [17,18,22], Hennessey-Milner-style process logics for reasoning about security and systems modelling [1,19] and modal and epistemic systems for reasoning about reachability/knowledge subject to the availability of resources [20,26].…”

A Substructural Epistemic Resource Logic

“…Although we have shown how ERL is sufficiently expressive to describe a security problem and check some of its behavioural properties, the modelling approach described so far quite limited to capturing specific situations in a more-or-less ad hoc manner. One approach to analysing the relationship between policy and system architecture is to reason in terms of layers, as developed in [4,5,10], using logics that are similar to, but weaker than, BI. In this set-up, a policy architecture is layered over a system architecture.…”

Bunched Hypersequent Calculi for Distributive Substructural Logics

“…The logic BI [30,33] of bunched implication extends the language of DFL e with ⊥, and a connective → for intuitionistic implication. The interest in BI and its extensions is due to the fact that these logics allow us to reason about resource composition and systems modelling and provide a basis for an assertion language of separation logic [21], and even more recently, for reasoning about systems architecture layers [13]. An analytic calculus for BI is obtained by the addition of the following rules to sDFL e [33].…”

“…We then consider the case of extensions of the logic of bunched implication BI. Aside from its theoretical interest, the recent applications [13] of BI-related logics illustrates the importance of having available general methods for the construction of analytic calculi. Extensions of BI by a certain class of axioms including restricted weakening or contraction are presented.…”