Abstract Stone Duality (ASD) is a direct axiomatisation of general topology, in contrast to the traditional and all other contemporary approaches, which rely on a prior notion of discrete set, type or object of a topos.

ASD reconciles mathematical and computational viewpoints, providing an inherently computable calculus that does not sacrifice key properties of real analysis such as compactness of the closed interval. Previous theories of recursive analysis failed to do this because they were based on points; ASD succeeds because, like locale theory and formal topology, it is founded on the algebra of open subspaces.

ASD is presented as a lambda calculus, of which we provide a self-contained summary, as the foundational background has been investigated in earlier work.

The core of the paper constructs the real line using two-sided Dedekind cuts. We show that the closed interval is compact and overt, where these concepts are defined using quantifiers. Further topics, such as the Intermediate Value Theorem, are presented in a separate paper that builds on this one.

The interval domain plays an important foundational role. However, we see intervals as generalised Dedekind cuts, which underly the construction of the real line, not as sets or pairs of real numbers.

We make a thorough study of arithmetic, in which our operations are more complicated than Moore's, because we work constructively, and we also consider back-to-front (Kaucher) intervals.

Finally, we compare ASD with other systems of constructive and computable topology and analysis.

In this paper, we describe a new representation for deterministic rational-valued P systems that allows us to form a bridge between membrane computing and linear algebra. On the one hand, we prove that an efficient computation for these P systems can be described using linear algebra techniques. In particular, we show that the computation for getting a configuration in such P systems can be carried out by multiplying appropriate matrices. On the other hand, we also show that membrane computing techniques can be used to get the nth power of a given matrix.

We show that there is a computably enumerable function f (that is, computably approximable from below) that dominates almost all functions, and f ⊕ W is incomplete for all incomplete computably enumerable sets W. Our main methodology is the LR equivalence relation on reals: A ≡LRB if and only if the notions of A-randomness and B-randomness coincide. We also show that there are c.e. sets that cannot be split into two c.e. sets of the same LR degree. Moreover, a c.e. set is low for random if and only if it computes no c.e. set with this property.

In this paper we study quasi-metric spaces using domain theory. Our main objective in this paper is to study the maximal point space problem for quasi-metric spaces. Here we prove that quasi-metric spaces that satisfy certain completeness properties, such as Yoneda and Smyth completeness, can be modelled by continuous dcpo's. To achieve this goal, we first study the partially ordered set of formal balls (BX, ⊑) of a quasi-metric space (X, d). Following Edalat and Heckmann, we prove that the order properties of (BX, ⊑) are tightly connected to topological properties of (X, d). In particular, we prove that (BX, ⊑) is a continuous dcpo if (X, d) is algebraic Yoneda complete. Furthermore, we show that this construction gives a model for Smyth-complete quasi-metric spaces. Then, for a given quasi-metric space (X, d), we introduce the partially ordered set of abstract formal balls (BX, ⊑, ≺). We prove that if the conjugate space (X, d−1) of a quasi-metric space (X, d) is right K-complete, then the ideal completion of (BX, ⊑, ≺) is a model for (X, d). This construction provides a model for any Yoneda-complete quasi-metric space (X, d), as well as the Sorgenfrey line, Kofner plane and Michael line.

We introduce an extension of the pure lambda calculus by endowing the set of terms with the structure of a vector space, or, more generally, of a module, over a fixed set of scalars. Moreover, terms are subject to identities similar to the usual pointwise definition of linear combinations of functions with values in a vector space. We then study a natural extension of beta reduction in this setting: we prove it is confluent, then discuss consistency and conservativity over the ordinary lambda calculus. We also provide normalisation results for a simple type system.

We consider the structure of the intestinal epithelial tissue and of cell–cell junctions as the biological model inspiring a new class of P systems. First we define the concept of cell polarity, a formal property derived from epithelial cells, which present morphologically and functionally distinct regions of the plasma membrane. Then we show two preliminary results for this new model of computation: on the theoretical side, we show that P systems with cell polarity are computationally (Turing) complete; on the modelling side, we show that the transepithelial movement of glucose from the intestinal lumen into the blood can be described by such a formal system. Finally, we define tissue P systems with cell polarity, where each cell has fixed connections to the neighbouring cells and to the environment, according to both the cell polarity and specific cell–cell junctions.

We study computably enumerable (c.e.) prefix codes that are capable of coding all positive integers in an optimal way up to a fixed constant: these codes will be called universal. We prove various characterisations of these codes, including the following one: a c.e. prefix code is universal if and only if it contains the domain of a universal self-delimiting Turing machine. Finally, we study various properties of these codes from the points of view of computability, maximality and density.

We investigate, in a process algebraic setting, a new notion of correctness for service compositions, which we call strong service compliance: composed services are strong compliant if their composition is both deadlock and livelock free (this is the traditional notion of compliance), and whenever a message can be sent to invoke a service, it is guranteed to be ready to serve the invocation. We also define a new notion of refinement, called strong subcontract pre-order, suitable for strong compliance: given a composition of strong compliant services, we can replace any service with any other service in subcontract relation while preserving the overall strong compliance. Finally, we present a characterisation of the strong subcontract pre-order by resorting to the theory of a (should) testing pre-order.

Discretionary Access Control (DAC) systems provide powerful resource management mechanisms based on the selective distribution of capabilities to selected classes of principals. We study a type-based theory of DAC models for a process calculus that extends Cardelli, Ghelli and Gordon's pi-calculus with groups (Cardelli et al. 2005). In our theory, groups play the role of principals and form the unit of abstraction for our access control policies, and types allow the specification of fine-grained access control policies to govern the transmission of names, bound the (iterated) re-transmission of capabilities and predicate their use on the inability to pass them to third parties. The type system relies on subtyping to achieve a selective distribution of capabilities to the groups that control the communication channels. We show that the typing and subtyping relationships of the calculus are decidable. We also prove a type safety result, showing that in well-typed processes all names:

1. (i) flow according to the access control policies specified by their types; and

2. (ii) are received at the intended sites with the intended capabilities.

A restriction category is an abstract category of partial maps. A Boolean restriction category is a restriction category that supports classical (Boolean) reasoning. Such categories are models of loop-free dynamic logic that is deterministic in the sense that < α > Q ⊂ [α]Q. Classical restriction categories are restriction categories with a locally Boolean structure: it is shown that they are precisely full subcategories of Boolean restriction categories. In particular, a Boolean restriction category may be characterised as a classical restriction category with finite coproducts in which all restriction idempotents split.

Every restriction category admits a restriction embedding into a Boolean restriction category. Thus, every abstract category of partial maps admits a conservative extension that supports classical reasoning. An explicit construction of the classical completion of a restriction category is given.

Priority is a frequently used feature of many computational systems. In this paper we study the expressiveness of two process algebras enriched with different priority mechanisms. In particular, we consider a finite (that is, recursion-free) fragment of asynchronous CCS with global priority (FAP, for short) and Phillips' CPG (CCS with local priority), and contrast their expressive power with that of two non-prioritised calculi, namely the π-calculus and its broadcast-based version, called bπ. We prove, by means of leader-election-based separation results, that, under certain conditions, there exists no encoding of FAP in π-Calculus or CPG. Moreover, we single out another problem in distributed computing, which we call the last man standing problem (LMS for short), that better reveals the gap between the two prioritised calculi above and the two non-prioritised ones, by proving that there exists no parallel-preserving encoding of the prioritised calculi in the non-prioritised calculi retaining any sincere (complete but partially correct, that is, admitting divergence or premature termination) semantics.

In the representation approach (TTE) to computable analysis, the representations of an algebraic or topological structure for which the basic predicates and functions become computable are of particular interest. There are, however, many predicates (like equality of real numbers) and functions that are absolutely non-computable, that is, not computable for any representation. Many of these results can be deduced from a simple lemma. In this article we prove this lemma for multi-representations and apply it to a number of examples. As applications, we show that various predicates and functions on computable measure spaces are absolutely non-computable. Since all the arguments are topological, we prove that the predicates are not relatively open and the functions are not relatively continuous for any multi-representation.

We compare three categorical models of type theory with extensional constructs: setoids over extensional type theory; setoids over intensional type theory and a certain free exact category (the free ‘ΠW-pretopos’). By studying the amount of choice available in these categories, we are able show that they are distinct.

In this paper we investigate the expressive power of three alternative approaches to the definition of infinite behaviours in process calculi, namely, recursive definitions, replication and iteration. We prove several results discriminating between the calculi obtained from a core CCS by adding the three mechanisms mentioned above. These results are derived by considering the decidability of four basic properties: termination (that is, all computations are finite); convergence (that is, the existence of a finite computation); barb (that is, the ability to perform an action on a given channel) and weak bisimulation.

Our results, which are summarised in Table 1, show that the three calculi form a strict expressiveness hierarchy in that: all the properties mentioned are undecidable in CCS with recursion; only termination and barb are decidable in CCS with replication; all the properties are decidable in CCS with iteration.

As a corollary, we also obtain a strict expressiveness hierarchy with respect to weak bisimulation, since there exist weak bisimulation preserving encodings of iteration in replication and of replication in recursion, whereas there are no weak bisimulation preserving encodings in the other directions.

This paper investigates the complexity of Propositional Projection Temporal Logic with Star (PPTL*). To this end, Propositional Projection Temporal Logic (PPTL) is first extended to include projection star. Then, by reducing the emptiness problem of star-free expressions to the problem of the satisfiability of PPTL* formulas, the lower bound of the complexity for the satisfiability of PPTL* formulas is proved to be non-elementary. Then, to prove the decidability of PPTL*, the normal form, normal form graph (NFG) and labelled normal form graph (LNFG) for PPTL* are defined. Also, algorithms for transforming a formula to its normal form and LNFG are presented. Finally, a decision algorithm for checking the satisfiability of PPTL* formulas is formalised using LNFGs.