Michael P. Fourman

The Logic of Topoi

Publisher Summary The chapter presents correspondence between topoi and theories that makes precise Lawveres claim that the notion of topos summarizes in objective categorical form the essence of higher-order logic. Elementary topoi turn out to correspond to the theories in a quite natural logic, formally an intuitionistic type theory. The chapter develops a part of topos theory according to Lawvere and Tierne. The development is reminiscent of algebraic logic. The Cartesian-closed structure gives certain finite types and encapsulates the fact that definable maps are closed under composition, λ-abstraction, and pairing. It has been concluded that important morphisms among topoi are geometric morphisms. Reyes has shown how a Grothendieck topos may be viewed as the extension of sets obtained by adding a generic model for a suitable (possibly infinitary) first-order theory. Geometric morphisms, then, arise naturally from a consideration of models (in Grothendieck topoi) of such theories. This approach, however, depends on some fixed base topos (in this case sets) and fails to explicate the notion of topos in abstracto.

Partial Functions in a Total Setting

We discuss a scheme for defining and reasoning about partial recursive functions within a classical two-valued logic in which all terms denote. We show how a total extension of the partial function introduced by a recursive declaration may be axiomatized within a classical logic, and illustrate by an example the kind of reasoning that our scheme supports. By presenting a naive set-theoretic semantics, we show that the system we propose is logically consistent. Our work is motivated largely by the pragmatic issues arising from mechanical theorem proving – we discuss some of the practical benefits and limitations of our scheme for mechanical verification of software and hardware systems.

Proof and synthesis

The authors argue that the next generation of computer-aided design (CAD) tools will represent and manipulate behavior. Behavioral tools are necessary to provide fast reliable design of complex systems. An approach to high-level synthesis is described which has grown out of formal verification using mathematical logic. It is shown that a general-purpose proof strategy combined with simple design rules can be used to represent and automate a simple design strategy. It is noted that the example shown here is clearly trivial. It is concluded that developing a reasonably comprehensive library of proof strategies and of representational abstractions is necessary to produce a useful tool.<<ETX>>

Notions of Choice Sequence

We use sheaf models to undertake a constructive analysis of the effects of admitting non-constructive choice sequences to mathematics. § PREAMBLE “A choice sequence is an infinite sequence of natural numbers whose terms are generated in succession; in the process of generating them, free choices may play a part. At one extreme, the selection of each term may be totally determined in advance by some effective rule: a sequence generated by such a rule is a lawlike sequence. At the other extreme, we have a sequence the selection of each term of which is totally unrestricted: these are the lawless sequences. In between are those choice sequences the selection of whose terms is partially restricted in advance, but not completely determined.” Dummett (Elements, p. 418)

The “world's simplest axiom of choice” fails

We use topos-theoretic methods to show that intuitionistic set theory with countable or dependent choice does not imply that every family, all of whose elements are doubletons and which has at most one element, has a choice function.

A Proposed Categorical Semantics for ML Modules

We present a simple categorical semantics for ML signatures, structures and functors. Our approach relies on realizablity semantics in the category of assemblies. Signatures and structures are modelled as objects in slices of the category of assemblies. Instantiation of signatures to structures and hence functor application is modelled by pullback.

Datatypes in L2

We describe the axiomatisation of a subset of Standard MLs datatypes in L2 (the LAMBDA Logic). The subset includes parameterisation and mutual recursion but has restrictions on the use of function type construction. We sketch a set-theoretic model for these datatypes. Finally, we briefly discuss the relationship between L2s datatypes and datatypes in HOL.

Modular design as algebraic composition

The aim of this research is to study how high-level mathematical abstractions could be used for reasoning about hardware design. These abstractions will be used as the foundation for a CAD tool for VLSI. Of course, different algebraic abstractions are appropriate on different levels of the design hierarchy. Uncovering formalising and integrating these will be difficult but is necessary for the development of future CAD tools. We have developed a general algebra, applicable at all levels of the hierarchy, for composing small modules into large designs. Having a common composition algebra throughout allows transformations between levels (both refining transformations downwards and verification and abstraction transformations upwards) to be algebraic morphisms. In this paper, we present a sketch of the composition algebra, and closely follow a design at the behavioural level. It is hoped to give an indication both of the use of composition and of the kind of verification that can be done at the highest levels.

Measuring the Broadband Access Divide

Decreases in a Gini index for broadband uptake have been interpreted as evidence of a narrowing digital divide. Nevertheless, a significant divide persists. How should we measure the divide? We propose two related indices, introduced in the context of health inequality by Wagstaff et al. (1991, 2005), as measures for the depth and breadth of the digital divide. We show how these quantify the contribution of the digital divide to social inequalities and cycles of deprivation. Depth measures the barriers to digital inclusion presented by existing deprivation. Breadth measures the degree to which the digital divide tends to reinforce existing inequalities. We report briefly on two applications, one local, one global, to illustrate how these measures can be used to assess progress and inform policies intended to reduce the digital divide.

RemIX: A Distributed Internet Exchange for Remote and Rural Networks

The concept of the \ac{IXP}, an Ethernet fabric central to the structure of the global Internet, is largely absent from the development of community-driven collaborative network infrastructure. The reasons for this are two-fold. \acp{IXP} exist in central, typically urban, environments where strong network infrastructure ensures high levels of connectivity. Between rural and remote regions, where networks are separated by distance and terrain, no such infrastructure exists. In this paper we present RemIX a distributed \acp{IXP} architecture designed for the community network environment. We examine this praxis using an implementation in Scotland, with suggestions for future development and research.