The Modelwise Interpolation Property of Semantic Logics

In this paper we introduce the modelwise interpolation property of a logic that states that whenever $\models\phi\to\psi$ holds for two formulas $\phi$ and $\psi$, then for every model $\mathfrak{M}$ there is an interpolant formula $\chi$ formulated in the intersection of the vocabularies of $\phi$ and $\psi$, such that $\mathfrak{M}\models\phi\to\chi$ and $\mathfrak{M}\models\chi\to\psi$, that is, the interpolant formula in Craig interpolation may vary from model to model. We compare the modelwise interpolation property with the standard Craig interpolation and with the local interpolation property by discussing examples, most notably the finite variable fragments of first order logic, and difference logic. As an application we connect the modelwise interpolation property with the local Beth definability, and we prove that the modelwise interpolation property of an algebraizable logic can be characterized by a weak form of the superamalgamation property of the class of algebras corresponding to the models of the logic

Concrete sheaf models of higher-order recursion

This thesis studies denotational models, in the form of sheaf categories, of functional programming languages with higher-order functions and recursion. We give a general method for building such models and show how the method includes examples such as existing models of probabilistic and differentiable computation. Using our method, we build a new fully abstract sheaf model of higher-order recursion inspired by the fully abstract logical relations models of O’Hearn and Riecke. In this way, we show that our method for building sheaf models can be used both to unify existing models that have so far been studied separately and to discover new models. The models we build are in the style of Moggi, namely, a cartesian closed category with a monad for modelling non termination. More specifically, our general method builds sheaf categories by specifying a concrete site with a class of admissible monomorphisms, a concept which we define. We combine this approach with techniques from synthetic and axiomatic domain theory to obtain a lifting monad on the sheaf category and to model recursion. We then prove the models obtained in this way are computationally adequate

Event-B in the Institutional Framework: Defining a Semantics, Modularisation Constructs and Interoperability for a Specification Language

Event-B is an industrial-strength specification language for verifying the properties of a given system’s specification. It is supported by its Eclipse-based IDE, Rodin, and uses the process of refinement to model systems at different levels of abstraction. Although a mature formalism, Event-B has a number of limitations. In this thesis, we demonstrate that Event-B lacks formally defined modularisation constructs. Additionally, interoperability between Event-B and other formalisms has been achieved in an ad hoc manner. Moreover, although a formal language, Event-B does not have a formal semantics. We address each of these limitations in this thesis using the theory of institutions. The theory of institutions provides a category-theoretic way of representing a formalism. Formalisms that have been represented as institutions gain access to an array of generic specification-building operators that can be used to modularise specifications in a formalismindependent manner. In the theory of institutions, there are constructs (known as institution (co)morphisms) that provide us with the facility to create interoperability between formalisms in a mathematically sound way. The main contribution of this thesis is the definition of an institution for Event-B, EVT, which allows us to address its identified limitations. To this end, we formally define a translational semantics from Event- B to EVT. We show how specification-building operators can provide a unified set of modularisation constructs for Event-B. In fact, the institutional framework that we have incorporated Event-B into is more accommodating to modularisation than the current state-of-the-art for Rodin. Furthermore, we present institution morphisms that facilitate interoperability between the respective institutions for Event-B and UML. This approach is more generic than the current approach to interoperability for Event-B and in fact, allows access to any formalism or logic that has already been defined as an institution. Finally, by defining EVT, we have outlined the steps required in order to include similar formalisms into the institutional framework. Hence, this thesis acts as a template for defining an institution for a specification language

City-size and municipal effeciency: a study in the geography of city development

Many writers have attached considerable importance to the idea that there is an optimum size for cities. A cur¬ sory examination of these writings soon reveals that most represent little more than strongly held opinions. Very few individuals writing on the subiect have attempted to demon¬ strate the strength of their convictions. With an emphasis on the more quantitative aspects of the subject, this study undertakes an examination of the optimum size concept with specific attention to the criterion of municipal efficiency and the implications that it has within the institutional context of Great Britain.The analysis in the study is in part theoretical and in part empirical. The theoretical portion is carried out in general terms so that it could have application within almost any national institutional context. However, the empirical analysis is restricted to selected local authorities in Great Britain

The Universality Problem

The theme of this thesis is to explore the universality problem in set theory in connection to model theory, to present some methods for finding universality results, to analyse how these methods were applied, to mention some results and to emphasise some philosophical interrogations that these aspects entail. A fundamental aspect of the universality problem is to find what determines the existence of universal objects. That means that we have to take into consideration and examine the methods that we use in proving their existence or nonexistence, the role of cardinal arithmetic, combinatorics etc. The proof methods used in the mathematical part will be mostly set-theoretic, but some methods from model theory and category theory will also be present. A graph might be the simplest, but it is also one of the most useful notions in mathematics. We show that there is a faithful functor F from the category L of linear orders to the category G of graphs that preserves model theoretic-related universality results (classes of objects having universal models in exactly the same cardinals, and also having the same universality spectrum). Trees constitute combinatorial objects and have a central role in set theory. The universality of trees is connected to the universality of linear orders, but it also seems to present more challenges, which we survey and present some results. We show that there is no embedding between an ℵ2-Souslin tree and a non-special wide ℵ2 tree T with no cofinal branches. Furthermore, using the notion of ascent path, we prove that the class of non-special ℵ2-Souslin tree with an ω-ascent path a has maximal complexity number, 2ℵ2 = ℵ3. Within the general framework of the universality problem in set theory and model theory, while emphasising their approaches and their connections with regard to this topic, we examine the possibility of drawing some philosophical conclusions connected to, among others, the notions of mathematical knowledge, mathematical object and proof

Labor in politics : the state Labor parties in Australia, 1880-1920

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