Computable Cyclic Functions

This dissertation concerns computable analysis where the idea of a representation of a set is of central importance. The key ideas introduced are those commenting on the computable relationship between two newly constructed representations, a representation of integrable cyclic functions, and the continuous cyclic function representation. Also, the computable relationship of an absolutely convergent Fourier series representation is considered. It is observed that the representation of integrable cyclic functions gives rise to a much larger set of computable functions than obtained by the continuous cyclic function representation and that integration remains a computable operation, but that basic evaluation of the function is not computable. Many other representations are acknowledged enhancing the picture of the partial order structure on the space of representations of cyclic functions. The paper can also be seen as a foundation for the study of Fourier analysis in a computable universe and concludes with an investigation into the computability of the Fourier transform

A Provenance Tracking Model for Data Updates

For data-centric systems, provenance tracking is particularly important when the system is open and decentralised, such as the Web of Linked Data. In this paper, a concise but expressive calculus which models data updates is presented. The calculus is used to provide an operational semantics for a system where data and updates interact concurrently. The operational semantics of the calculus also tracks the provenance of data with respect to updates. This provides a new formal semantics extending provenance diagrams which takes into account the execution of processes in a concurrent setting. Moreover, a sound and complete model for the calculus based on ideals of series-parallel DAGs is provided. The notion of provenance introduced can be used as a subjective indicator of the quality of data in concurrent interacting systems.Comment: In Proceedings FOCLASA 2012, arXiv:1208.432

A Typed Model for Linked Data

The term Linked Data is used to describe ubiquitous and emerging semi-structured data formats on the Web. URIs in Linked Data allow diverse data sources to link to each other, forming a Web of Data. A calculus which models concurrent queries and updates over Linked Data is presented. The calculus exhibits operations essential for declaring rich atomic actions. The operations recover emergent structure in the loosely structured Web of Data. The calculus is executable due to its operational semantics. A light type system ensures that URIs with a distinguished role are used consistently. The main theorem verifies that the light type system and operational semantics work at the same level of granularity, so are compatible. Examples show that a range of existing and emerging standards are captured. Data formats include RDF, named graphs and feeds. The primitives of the calculus model SPARQL Query and the Atom Publishing Protocol. The subtype system is based on RDFS, which improves interoperability. Examples focuss on the SPARQL Update proposal for which a fine grained operational semantics is developed. Further potential high level languages are outlined for exploiting Linked Data

Local Type Checking for Linked Data Consumers

The Web of Linked Data is the cumulation of over a decade of work by the Web standards community in their effort to make data more Web-like. We provide an introduction to the Web of Linked Data from the perspective of a Web developer that would like to build an application using Linked Data. We identify a weakness in the development stack as being a lack of domain specific scripting languages for designing background processes that consume Linked Data. To address this weakness, we design a scripting language with a simple but appropriate type system. In our proposed architecture some data is consumed from sources outside of the control of the system and some data is held locally. Stronger type assumptions can be made about the local data than external data, hence our type system mixes static and dynamic typing. Throughout, we relate our work to the W3C recommendations that drive Linked Data, so our syntax is accessible to Web developers.Comment: In Proceedings WWV 2013, arXiv:1308.026

De Morgan Dual Nominal Quantifiers Modelling Private Names in Non-Commutative Logic

This paper explores the proof theory necessary for recommending an expressive but decidable first-order system, named MAV1, featuring a de Morgan dual pair of nominal quantifiers. These nominal quantifiers called `new' and `wen' are distinct from the self-dual Gabbay-Pitts and Miller-Tiu nominal quantifiers. The novelty of these nominal quantifiers is they are polarised in the sense that `new' distributes over positive operators while `wen' distributes over negative operators. This greater control of bookkeeping enables private names to be modelled in processes embedded as formulae in MAV1. The technical challenge is to establish a cut elimination result, from which essential properties including the transitivity of implication follow. Since the system is defined using the calculus of structures, a generalisation of the sequent calculus, novel techniques are employed. The proof relies on an intricately designed multiset-based measure of the size of a proof, which is used to guide a normalisation technique called splitting. The presence of equivariance, which swaps successive quantifiers, induces complex inter-dependencies between nominal quantifiers, additive conjunction and multiplicative operators in the proof of splitting. Every rule is justified by an example demonstrating why the rule is necessary for soundly embedding processes and ensuring that cut elimination holds.Comment: Submitted for review 18/2/2016; accepted CONCUR 2016; extended version submitted to journal 27/11/201

The Sub-Additives: A Proof Theory for Probabilistic Choice extending Linear Logic

Probabilistic choice, where each branch of a choice is weighted according to a probability distribution, is an established approach for modelling processes, quantifying uncertainty in the environment and other sources of randomness. This paper uncovers new insight showing probabilistic choice has a purely logical interpretation as an operator in an extension of linear logic. By forbidding projection and injection, we reveal additive operators between the standard with and plus operators of linear logic. We call these operators the sub-additives. The attention of the reader is drawn to two sub-additive operators: the first being sound with respect to probabilistic choice; while the second arises due to the fact that probabilistic choice cannot be self-dual, hence has a de Morgan dual counterpart. The proof theoretic justification for the sub-additives is a cut elimination result, employing a technique called decomposition. The justification from the perspective of modelling probabilistic concurrent processes is that implication is sound with respect to established notions of probabilistic refinement, and is fully compositional

Diamonds for Security: A Non-Interleaving Operational Semantics for the Applied Pi-Calculus

We introduce a non-interleaving structural operational semantics for the applied ?-calculus and prove that it satisfies the properties expected of a labelled asynchronous transition system (LATS). LATS have well-studied relations with other standard non-interleaving models, such as Mazurkiewicz traces or event structures, and are a natural extension of labelled transition systems where the independence of transitions is made explicit. We build on a considerable body of literature on located semantics for process algebras and adopt a static view on locations to identify the parallel processes that perform a transition. By lifting, in this way, work on CCS and ?-calculus to the applied ?-calculus, we lay down a principled foundation for reusing verification techniques such as partial-order reduction and non-interleaving equivalences in the field of security. The key technical device we develop is the notion of located aliases to refer unambiguously to a specific output originating from a specific process. This light mechanism ensures stability, avoiding disjunctive causality problems that parallel extrusion incurs in similar non-interleaving semantics for the ?-calculus

A Logical Account of Subtyping for Session Types

We study the notion of subtyping for session types in a logical setting, where session types are propositions of multiplicative/additive linear logic extended with least and greatest fixed points. The resulting subtyping relation admits a simple characterization that can be roughly spelled out as the following lapalissade: every session type is larger than the smallest session type and smaller than the largest session type. At the same time, we observe that this subtyping, unlike traditional ones, preserves termination in addition to the usual safety properties of sessions. We present a calculus of sessions that adopts this subtyping relation and we show that subtyping, while useful in practice, is superfluous in the theory: every use of subtyping can be "compiled away" via a coercion semantics.Comment: In Proceedings PLACES 2023, arXiv:2304.0543