Banach Spaces as Data Types

We introduce the operators "modified limit" and "accumulation" on a Banach space, and we use this to define what we mean by being internally computable over the space. We prove that any externally computable function from a computable metric space to a computable Banach space is internally computable. We motivate the need for internal concepts of computability by observing that the complexity of the set of finite sets of closed balls with a nonempty intersection is not uniformly hyperarithmetical, and thus that approximating an externally computable function is highly complex.Comment: 20 page

Verification of the busy-forbidden protocol (using an extension of the cones and foci framework)

The busy-forbidden protocol is a new readers-writer lock with no resource contention between readers, which allows it to outperform other locks. For its verification, specifications of its implementation and its less complex external behavior are provided by the original authors but are only proven equivalent for up to 7 threads. We provide a general proof using the cones and foci proof framework, which rephrases whether two specifications are branching bisimilar in terms of proof obligations on relations between the data objects occurring in the implementation and specification. We provide an extension of this framework consisting of three additional properties on data objects, When these three additional properties also hold, the two systems are divergence-preserving branching bisimilar, a stronger version of the aforementioned relation that also distinguishes livelock. We prove this extension to be sound and use it to give a general equivalence proof for the busy-forbidden protocol

On Bisimilarity for Quasi-discrete Closure Spaces

Closure spaces, a generalisation of topological spaces, have shown to be a convenient theoretical framework for spatial model checking. The closure operator of closure spaces and quasi-discrete closure spaces induces a notion of neighborhood akin to that of topological spaces that build on open sets. For closure models and quasi-discrete closure models, in this paper we present three notions of bisimilarity that are logically characterised by corresponding modal logics with spatial modalities: (i) CM-bisimilarity for closure models (CMs) is shown to generalise Topo-bisimilarity for topological models. CM-bisimilarity corresponds to equivalence with respect to the infinitary modal logic IML that includes the modality ${\cal N}$ for ``being near''. (ii) CMC-bisimilarity, with `CMC' standing for CM-bisimilarity with converse, refines CM-bisimilarity for quasi-discrete closure spaces, carriers of quasi-discrete closure models. Quasi-discrete closure models come equipped with two closure operators, Direct ${\cal C}$ and Converse ${\cal C}$, stemming from the binary relation underlying closure and its converse. CMC-bisimilarity, is captured by the infinitary modal logic IMLC including two modalities, Direct ${\cal N}$ and Converse ${\cal N}$, corresponding to the two closure operators. (iii) CoPa-bisimilarity on quasi-discrete closure models, which is weaker than CMC-bisimilarity, is based on the notion of compatible paths. The logical counterpart of CoPa-bisimilarity is the infinitary modal logic ICRL with modalities Direct $\zeta$ and Converse $\zeta$, whose semantics relies on forward and backward paths, respectively. It is shown that CoPa-bisimilarity for quasi-discrete closure models relates to divergence-blind stuttering equivalence for Kripke structures.Comment: 32 pages, 14 figure