Logics for Action

Abstract. Logics of action, for reasoning about the effects of state change, and logics of belief, accounting for belief revision and update, have much in common. Furthermore, we may undertake an action because we hold a particular belief, and revise our beliefs in the light of observed consequences of an action. So studies of these two aspects are inevitably intertwined. However, we argue, a clear separation of the two is helpful in understanding their interactions. We give a semantic presentation of such a separation, introducing a semantic setting that supports one logic for describing the effects of actions, which are modeled as changing the values of particular atomic properties, or fluents, and another for expressing more complex facts or beliefs about the world. We use a simple state-logic, to account for state change, and show how it can be integrated with a variety of domain-logics, of fact or belief, for reasoning about the world. State- and domain-logics are linked, syntactically and semantically; but separate. The state-logic, our logic for action, is quantified propositional logic. Bounded existential propositional quantification is used to specify which literals may b

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. 1 Introduction Building on work on the semantics of programming languages in realizability models, in particular that of Wesley Phoa [Pho90] and John Longley [Lon95], we sketch a simple approach to elements of the ML modules system, such as signatures, structures and functors. Once the basic machinery is set up, we will need only quite basic category theory. This paper is an updated and completely revised version of an earlier paper by Michael Fourman and Wesley Phoa [PF92]. The construction of "generic" (in a sense to be defined below) elements and types presented here is essentially the same as in that paper. However, our presentation is ..

The Gelfand spectrum of a noncommutative C*-algebra: a topos-theoretic approach

We compare two influential ways of defining a generalized notion of space. The first, inspired by Gelfand duality, states that the category of 'noncommutative spaces' is the opposite of the category of C*-algebras. The second, loosely generalizing Stone duality, maintains that the category of 'pointfree spaces' is the opposite of the category of frames (i.e., complete lattices in which the meet distributes over arbitrary joins). One possible relationship between these two notions of space was unearthed by Banaschewski and Mulvey, who proved a constructive version of Gelfand duality in which the Gelfand spectrum of a commutative C*-algebra comes out as a pointfree space. Being constructive, this result applies in arbitrary toposes (with natural numbers objects, so that internal C*-algebras can be defined). Earlier work by the first three authors, shows how a noncommutative C*-algebra gives rise to a commutative one internal to a certain sheaf topos. The latter, then, has a constructive Gelfand spectrum, also internal to the topos in question. After a brief review of this work, we compute the so-called external description of this internal spectrum, which in principle is a fibered pointfree space in the familiar topos Sets of sets and functions. However, we obtain the external spectrum as a fibered topological space in the usual sense. This leads to an explicit Gelfand transform, as well as to a topological reinterpretation of the Kochen-Specker Theorem of quantum mechanics, which supplements the remarkable topos-theoretic version of this theorem due to Butterfield and Isham.Comment: 12 page

Transdisciplinary working to shape systematic reviews and interpret the findings: Commentary

This is the final version. Available from BMC via the DOI in this record. Important policy questions tend to span a range of academic disciplines, and the relevant research is often carried out in a variety of social, economic and geographic contexts. In efforts to synthesise research to help inform decisions arising from the policy questions, systematic reviews need conceptual frameworks and ways of thinking that combine knowledge drawn from different academic traditions and contexts; in other words, transdisciplinary research. This paper considers how transdisciplinary working can be achieved with: conceptual frameworks that span traditional academic boundaries; methods for shaping review questions and conceptual frameworks; and methods for interpreting the relevance of findings to different contexts. It also discusses the practical challenges and ultimate benefits of transdisciplinary working for systematic reviews.World Health OrganizationUK Department for International DevelopmentUK aidNational Institute for Health Research (NIHR

Geometric Logic in Computer Science

We present an introduction to geometric logic and the mathematical structures associated with it, such as categorical logic and toposes. We also describe some of its applications in computer science including its potential as a logic for spec-i cation languages.

Continuous Truth II: Reflections

Abstract. In the late 1960s, Dana Scott first showed how the Stone-Tarski topological interpretation of Heyting’s calculus could be extended to model intuitionistic analysis; in particular Brouwer’s continuity prin-ciple. In the early ’80s we and others outlined a general treatment of non-constructive objects, using sheaf models—constructions from topos theory—to model not only Brouwer’s non-classical conclusions, but also his creation of “new mathematical entities”. These categorical models are intimately related to, but more general than Scott’s topological model. The primary goal of this paper is to consider the question of iterated extensions. Can we derive new insights by repeating the second act? In Continuous Truth I, presented at Logic Colloquium ’82 in Florence, we showed that general principles of continuity, local choice and local com-pactness hold in the gros topos of sheaves over the category of separable locales equipped with the open cover topology. We touched on the question of iteration. Here we develop a more gen-eral analysis of iterated categorical extensions, that leads to a reflection schema for statements of predicative analysis. We also take the opportunity to revisit some aspects of both Continuous Truth I and Formal Spaces (Fourman & Grayson 1982), and correct two long-standing errors therein

A topos for algebraic quantum theory

The aim of this paper is to relate algebraic quantum mechanics to topos theory, so as to construct new foundations for quantum logic and quantum spaces. Motivated by Bohr's idea that the empirical content of quantum physics is accessible only through classical physics, we show how a C*-algebra of observables A induces a topos T(A) in which the amalgamation of all of its commutative subalgebras comprises a single commutative C*-algebra. According to the constructive Gelfand duality theorem of Banaschewski and Mulvey, the latter has an internal spectrum S(A) in T(A), which in our approach plays the role of a quantum phase space of the system. Thus we associate a locale (which is the topos-theoretical notion of a space and which intrinsically carries the intuitionistic logical structure of a Heyting algebra) to a C*-algebra (which is the noncommutative notion of a space). In this setting, states on A become probability measures (more precisely, valuations) on S(A), and self-adjoint elements of A define continuous functions (more precisely, locale maps) from S(A) to Scott's interval domain. Noting that open subsets of S(A) correspond to propositions about the system, the pairing map that assigns a (generalized) truth value to a state and a proposition assumes an extremely simple categorical form. Formulated in this way, the quantum theory defined by A is essentially turned into a classical theory, internal to the topos T(A).Comment: 52 pages, final version, to appear in Communications in Mathematical Physic

An overview of population-based algorithms for multi-objective optimisation

In this work we present an overview of the most prominent population-based algorithms and the methodologies used to extend them to multiple objective problems. Although not exact in the mathematical sense, it has long been recognised that population-based multi-objective optimisation techniques for real-world applications are immensely valuable and versatile. These techniques are usually employed when exact optimisation methods are not easily applicable or simply when, due to sheer complexity, such techniques could potentially be very costly. Another advantage is that since a population of decision vectors is considered in each generation these algorithms are implicitly parallelisable and can generate an approximation of the entire Pareto front at each iteration. A critique of their capabilities is also provided