We prove strong completeness of a range of substructural logics with respect to a natural poset-b... more 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.
Despite the ubiquity of the concept of stewardship in environmental economics and accounting ther... more Despite the ubiquity of the concept of stewardship in environmental economics and accounting there is no field a-specific, generally accepted definition. We define the information steward as the agent/institution who enhances the system’s resilience and sustainability, by maintaining and extending the life of its nominal operational capacity. Unlike individual agents who are not able to individual mitigate systemic losses, the steward, whose function is the viability of the system as a whole, val-ues such damages that degrade the system at a higher rate by longer lower discount rate. In the presence of active attackers, individual agents ’ defensive expenditure is always lower that the expenditure undertaken under instructions from the informa-tion steward. The resulting configuration of defensive expenditure, although higher than the level that individual agents would have chosen based on their own valuation of their expected losses, ensures that the overall welfare of agents is at...
Stone-type duality theorems, which relate algebraic and relational/topological models, are import... more 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-intuiti...
Algorithmic proof-search is an essential enabling technology throughout informatics. Proof-search... more Algorithmic proof-search is an essential enabling technology throughout informatics. Proof-search is the proof-theoretic realization of the formulation of logic not as a theory of deduction but rather as a theory of reduction. Whilst deductive logics typically have a well-developed semantics of proofs, reductive logics are typically well-understood only operationally. Each deductive system can, typically, be read as a corresponding reductive system. We discuss some of the problems which must be addressed in order to provide a semantics of proof-searches of comparable value to the corresponding semantics of proofs. Just as the semantics of proofs is intimately related to the model theory of the underlying logic, so too should be the semantics of proof-searches. We discuss how to solve the problem of providing a semantics for proof-searches which adequately models both operational and logical aspects of the reductive system.
The concept of stewardship in environmental economics is an established tool for environmental an... more The concept of stewardship in environmental economics is an established tool for environmental and natural resource management and the mitigation of risk from climate change. Similar concepts are well-established in accounting and management. Despite the ubiquity of the concept of stewardship, there is no generally accepted definition. We define the information steward as the agent/institution who enhances the system’s resilience and sustainability, by maintaining and extending the life of its nominal operational capacity. Unlike individual agents who are not able to value systemic losses, the steward, whose function is the viability of the system as a whole, values such damages that degrade the system at a higher rate by adopting a lower discount rate. In the presence of deliberate attacks that degrade the information used/kept in the system, individual agents’ defensive expenditure is always lower that the expenditure undertaken under instructions from the information steward. The...
We present a substructural epistemic logic, based on Boolean BI, in which the epistemic modalitie... more 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 with epistemic agency. We give a labelled tableaux calculus and establish soundness and completeness with respect to the resource semantics. We illustrate the use of the logic by discussing an example of side-channels in access control using resource tokens. A Substructural Epistemic Resource Logic (Extended Version) Didier Galmiche Pierre Kimmel David Pym
Should public policy-makers set minimum levels of behaviour for individuals and corporations rega... more Should public policy-makers set minimum levels of behaviour for individuals and corporations regarding information security policies and investments? We consider a model in which a finite number of targets are at risk of attack, attacks are costly, and have a finite probability of success. One important innovation is an explicit model of the decisions of potential attackers on whether to mount attacks. The model shows how the behaviour of attackers and the nature of the technological environment can create a role for a policy-maker to coordinate optimal minimum levels of protective expenditure for firms.
Concurrent Kleene Algebra is an elegant tool for equational reasoning about concurrent programs. ... more Concurrent Kleene Algebra is an elegant tool for equational reasoning about concurrent programs. An important feature of concurrent programs that is missing from CKA is the ability to restrict legal interleavings. To remedy this we extend the standard model of CKA, namely pomsets, with a new feature, called boxes, which can specify that part of the system is protected from outside interference. We study the algebraic properties of this new model. Another drawback of CKA is that the language used for expressing properties of programs is the same as that which is used to express programs themselves. This is often too restrictive for practical purposes. We provide a logic, 'pomset logic', that is an assertion language for specifying such properties, and which is interpreted on pomsets with boxes. In contrast with other approaches, this logic is not state-based, but rather characterizes the runtime behaviour of a program. We develop the basic metatheory for the relationship betw...
We present a substructural epistemic logic, based on Boolean BI, in which the epistemic modalitie... more 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 with epistemic agency. We give a labelled tableaux calculus and establish soundness and completeness with respect to the resource semantics. We illustrate the use of the logic by discussing an example of side-channels in access control using resource tokens. A Substructural Epistemic Resource Logic (Extended Version) (Updating RN 16/08) Didier Galmiche Pierre Kimmel David Pym
The logic of bunched implications BI provides a logical analysis of a basic notion of resource ri... more The logic of bunched implications BI provides a logical analysis of a basic notion of resource rich enough to provide a ``pointer logic'' semantics for programs which manipulate mutable data structures. We develop a theory of semantic tableaux for BI, so providing an elegant basis for efficient theorem proving tools for BI. It is based on the use of an algebra of labels to solve the resource-distribution problem, the labels being the elements of resource models. For BI with inconsistency, the challenge consists in dealing with BI's Grothendieck topological models within such a proof-search method, based on labels. We prove soundness and completeness theorems for a resource tableaux method TBI with respect to this semantics and provide a way to build countermodels from so-called dependency graphs. As consequences, we have two strong new results for BI: the decidability of propositional BI and the finite model property with respect to Grothendieck topological semantics. In...
We introduce a substructural modal logic for reasoning about (optimal) resource allocation in mod... more We introduce a substructural modal logic for reasoning about (optimal) resource allocation in models of distributed systems. The underlying logic is a variant of the modal logic of bunched implications, and based on the same resource semantics, which is itself closely related to concurrent separation logic. By considering notions of cost, strategy, and utility, we are able to formulate characterizations of Pareto optimality, best responses, and Nash equilibrium within resource semantics. UCL DEPARTMENT OF COMPUTER SCIENCE
Abramsky: Two perspectives on logic and structure 3 1. Intrinsic: logic embodies structure e.g. p... more Abramsky: Two perspectives on logic and structure 3 1. Intrinsic: logic embodies structure e.g. propositions-as-types/proofs-as-programs. Bunched Logic Duality Separation Logic Further Work & Conclusions Intrinsic vs Descriptive Resource Reasoning Abramsky: Two perspectives on logic and structure 3 1. Intrinsic: logic embodies structure e.g. propositions-as-types/proofs-as-programs. 2. Descriptive: logic is used to talk about structure e.g. Kripke semantics of modal logic. Bunched Logic Duality Separation Logic Further Work & Conclusions Intrinsic vs Descriptive Resource Reasoning Abramsky: Two perspectives on logic and structure 3 1. Intrinsic: logic embodies structure e.g. propositions-as-types/proofs-as-programs. 2. Descriptive: logic is used to talk about structure e.g. Kripke semantics of modal logic. Linear Logic: control of weakening and contraction via exponentials ⇒ formulae are consumable resources (intrinsic).
We present a substructural epistemic logic, based on Boolean BI, in which the epistemic modalitie... more 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 with epistemic agency. We give a labelled tableaux calculus and establish soundness and completeness with respect to the resource semantics. We illustrate the use of the logic by discussing an example of side-channels in access control using resource tokens.
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Papers by David J Pym