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

Provability in BI's Sequent Calculus is Decidable

The logic of Bunched Implications (BI) combines both additive and multiplicative connectives, which include two primitive intuitionistic implications. As a consequence, contexts in the sequent presentation are not lists, nor multisets, but rather tree-like structures called bunches. This additional complexity notwithstanding, the logic has a well-behaved metatheory admitting all the familiar forms of semantics and proof systems. However, the presentation of an effective proof-search procedure has been elusive since the logic's debut. We show that one can reduce the proof-search space for any given sequent to a primitive recursive set, the argument generalizing Gentzen's decidability argument for classical propositional logic and combining key features of Dyckhoff's contraction-elimination argument for intuitionistic logic. An effective proof-search procedure, and hence decidability of provability, follows as a corollary.Comment: Submitted to CADE-2

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

Computational modelling of inertia friction welding

This study details the development and validation of a finite element methodology to robustly simulate the inertia friction welding (IFW) process. There are many difficulties involved in modelling IFW. These include the short and violent process to complete a weld, as well as the challenges in obtaining experimental data throughout the process to complement, validate and inform the modelling effort. The objectives here are to model the macroscale multiphysical process leading to an accurate prediction of key process output variables, ultimately leading to a reliable method for predicting the post weld microstructure

Syntheses and Properties of Salts of Chromophores with Ferrocenyl Electron Donor Groups and Quaternary Nitrogen Acceptors

A series of five new dipolar cations has been synthesized with ferrocenyl (Fc) electron donor groups connected to N-arylpyridinium, N-methylquinolinium, N-methylbenzothiazolium, or N-methylacridinium acceptors. Together with their known N-methylpyridinium analogue, these chromophores have been characterized as their PF_6^− salts by using various techniques including electronic absorption spectroscopy and cyclic voltammetry. Nine single-crystal X-ray structures have been determined, including two polymorphs of one salt obtained from a single crystallization experiment, and two of these are polar materials. A highly favorable degree of dipolar alignment for bulk NLO effects is observed in one case. Molecular quadratic nonlinear optical (NLO) responses have been determined by using femtosecond hyper-Rayleigh scattering (HRS) at 1300 nm and also via Stark (electroabsorption) spectroscopic studies on the intense π → π^* intraligand and d → π^* metal-to-ligand charge-transfer bands. A broad correlation between the electron acceptor strength and the HRS-derived first hyperpolarizabilities β and the static first hyperpolarizabilities β0 estimated from the Stark data is evident. This is the first time that meaningful (albeit indirectly determined) β_0 data have been reported for Fc compounds, allowing quantitative comparisons with the chromophore in the technologically important material (E)-4′-(dimethylamino)-N-methyl-4-stilbazolium (DAS) tosylate. The observed β_0 values are in several cases similar to that of [DAS]PF_6, and possibly even larger in one instance

COVID-19 and cardiac rehabilitation

The British Association for Cardiovascular Prevention and Rehabilitation (BACPR), the British Cardiovascular Society (BCS) and the British Heart Foundation (BHF) have issued a joint position statement ‘Retention of cardiac rehabilitation services during the COVID-19 pandemic’

Critical review of self-reported functional ankle instability measures.

a b s t r a c t Objective: Determine which ankle instability questionnaire predicts subject's ankle instability status based on a minimum accepted criteria for FAI (MC_FAI). Design: Cross-sectional study. Setting: Large Midwestern University. Participants: College aged subjects (n ¼ 1127 19.6 AE 2.1 years) from a university population were recruited for this study. Any volunteer, regardless of ankle injury history was included in the study. Main outcome measures: The independent variables were the score on three self-report ankle instability questionnaires: Ankle Instability Instrument, Cumberland Ankle Instability Tool, and Identification of Functional Ankle Instability. Subjects completed the questionnaires for their dominant limb during a single testing session. The dependent variable was created based on the previously established MC_FAI. This was established as at least one ankle sprain and at least one episode of giving way. Data were modeled using a chi-square and multinomial logistic regression. 95% confidence intervals were calculated for the resulting odds ratios. Results: A test of the full model with all three predictors against MC_FAI revealed that only the IdFAI (X 2 ¼ 457.09, p ¼ .001) had a significant relationship with the outcome variable. The IdFAI had an overall prediction rate of 87.8%. Conclusions: This analysis illustrates that IdFAI is a good overall option for predicting ankle stability status by self-reported questionnaire