Developing a Conceptual Framework for Cloud Security Assurance

Postprin

Base-extension semantics for modal logic

In proof-theoretic semantics, meaning is based on inference. It may seen as the mathematical expression of the inferentialist interpretation of logic. Much recent work has focused on base-extension semantics, in which the validity of formulas is given by an inductive definition generated by provability in a ‘base’ of atomic rules. Base-extension semantics for classical and intuitionistic propositional logic have been explored by several authors. In this paper, we develop base-extension semantics for the classical propositional modal systems K, KT, K4 and S4, with as the primary modal operator. We establish appropriate soundness and completeness theorems and establish the duality between and a natural presentation of ♦. We also show that our semantics is in its current form not complete with respect to euclidean modal logics. Our formulation makes essential use of relational structures on bases

Regulation of neuronal excitability through pumilio-dependent control of a sodium channel gene

Dynamic changes in synaptic connectivity and strength, which occur during both embryonic development and learning, have the tendency to destabilize neural circuits. To overcome this, neurons have developed a diversity of homeostatic mechanisms to maintain firing within physiologically defined limits. In this study, we show that activity-dependent control of mRNA for a specific voltage-gated Na+ channel [encoded by paralytic (para)] contributes to the regulation of membrane excitability in Drosophila motoneurons. Quantification of para mRNA, by real-time reverse-transcription PCR, shows that levels are significantly decreased in CNSs in which synaptic excitation is elevated, whereas, conversely, they are significantly increased when synaptic vesicle release is blocked. Quantification of mRNA encoding the translational repressor pumilio (pum) reveals a reciprocal regulation to that seen for para. Pumilio is sufficient to influence para mRNA. Thus, para mRNA is significantly elevated in a loss-of-function allele of pum (pumbemused), whereas expression of a full-length pum transgene is sufficient to reduce para mRNA. In the absence of pum, increased synaptic excitation fails to reduce para mRNA, showing that Pum is also necessary for activity-dependent regulation of para mRNA. Analysis of voltage-gated Na+ current (INa) mediated by para in two identified motoneurons (termed aCC and RP2) reveals that removal of pum is sufficient to increase one of two separable INa components (persistent INa), whereas overexpression of a pum transgene is sufficient to suppress both components (transient and persistent). We show, through use of anemone toxin (ATX II), that alteration in persistent INa is sufficient to regulate membrane excitability in these two motoneurons

On character amenability of Banach algebras

AbstractWe continue our work [E. Kaniuth, A.T. Lau, J. Pym, On φ-amenability of Banach algebras, Math. Proc. Cambridge Philos. Soc. 144 (2008) 85–96] in the study of amenability of a Banach algebra A defined with respect to a character φ of A. Various necessary and sufficient conditions of a global and a pointwise nature are found for a Banach algebra to possess a φ-mean of norm 1. We also completely determine the size of the set of φ-means for a separable weakly sequentially complete Banach algebra A with no φ-mean in A itself. A number of illustrative examples are discussed

Definite Formulae, Negation-as-Failure, and the Base-extension Semantics of Intuitionistic Propositional Logic

Proof-theoretic semantics (P-tS) is the paradigm of semantics in which meaning in logic is based on proof (as opposed to truth). A particular instance of P-tS for intuitionistic propositional logic (IPL) is its base-extension semantics (B-eS). This semantics is given by a relation called support, explaining the meaning of the logical constants, which is parameterized by systems of rules called bases that provide the semantics of atomic propositions. In this paper, we interpret bases as collections of definite formulae and use the operational view of the latter as provided by uniform proof-search -- the proof-theoretic foundation of logic programming (LP) -- to establish the completeness of IPL for the B-eS. This perspective allows negation, a subtle issue in P-tS, to be understood in terms of the negation-as-failure protocol in LP. Specifically, while the denial of a proposition is traditionally understood as the assertion of its negation, in B-eS we may understand the denial of a proposition as the failure to find a proof of it. In this way, assertion and denial are both prime concepts in P-tS.Comment: submitte

Proof-theoretic Semantics and Tactical Proof

The use of logical systems for problem-solving may be as diverse as in proving theorems in mathematics or in figuring out how to meet up with a friend. In either case, the problem solving activity is captured by the search for an \emph{argument}, broadly conceived as a certificate for a solution to the problem. Crucially, for such a certificate to be a solution, it has be \emph{valid}, and what makes it valid is that they are well-constructed according to a notion of inference for the underlying logical system. We provide a general framework uniformly describing the use of logic as a mathematics of reasoning in the above sense. We use proof-theoretic validity in the Dummett-Prawitz tradition to define validity of arguments, and use the theory of tactical proof to relate arguments, inference, and search.Comment: submitte

From Proof-theoretic Validity to Base-extension Semantics for Intuitionistic Propositional Logic

Proof-theoretic semantics (P-tS) is the approach to meaning in logic based on \emph{proof} (as opposed to truth). There are two major approaches to P-tS: proof-theoretic validity (P-tV) and base-extension semantics (B-eS). The former is a semantics of arguments, and the latter is a semantics of logical constants in a logic. This paper demonstrates that the B-eS for intuitionistic propositional logic (IPL) encapsulates the declarative content of a basic version of P-tV. Such relationships have been considered before yielding incompleteness results. This paper diverges from these approaches by accounting for the constructive, hypothetical setup of P-tV. It explicates how the B-eS for IPL works

Defining Logical Systems via Algebraic Constraints on Proofs

We comprehensively present a program of decomposition of proof systems for non-classical logics into proof systems for other logics, especially classical logic, using an algebra of constraints. That is, one recovers a proof system for a target logic by enriching a proof system for another, typically simpler, logic with an algebra of constraints that act as correctness conditions on the latter to capture the former; for example, one may use Boolean algebra to give constraints in a sequent calculus for classical propositional logic to produce a sequent calculus for intuitionistic propositional logic. The idea behind such forms of reduction is to obtain a tool for uniform and modular treatment of proof theory and provide a bridge between semantics logics and their proof theory. The article discusses the theoretical background of the project and provides several illustrations of its work in the field of intuitionistic and modal logics. The results include the following: a uniform treatment of modular and cut-free proof systems for a large class of propositional logics; a general criterion for a novel approach to soundness and completeness of a logic with respect to a model-theoretic semantics; and a case study deriving a model-theoretic semantics from a proof-theoretic specification of a logic.Comment: submitte

Proof-theoretic Semantics for Intuitionistic Multiplicative Linear Logic

This work is the first exploration of proof-theoretic semantics for a substructural logic. It focuses on the base-extension semantics (B-eS) for intuitionistic multiplicative linear logic (IMLL). The starting point is a review of Sandqvist's B-eS for intuitionistic propositional logic (IPL), for which we propose an alternative treatment of conjunction that takes the form of the generalized elimination rule for the connective. The resulting semantics is shown to be sound and complete. This motivates our main contribution, a B-eS for IMLL, in which the definitions of the logical constants all take the form of their elimination rule and for which soundness and completeness are established.Comment: 27 page