The Sierpinski Object in the Scott Realizability Topos

We study the Sierpinski object $\Sigma$ in the realizability topos based on Scott's graph model of the $\lambda$-calculus. Our starting observation is that the object of realizers in this topos is the exponential $\Sigma ^N$, where $N$ is the natural numbers object. We define order-discrete objects by orthogonality to $\Sigma$. We show that the order-discrete objects form a reflective subcategory of the topos, and that many fundamental objects in higher-type arithmetic are order-discrete. Building on work by Lietz, we give some new results regarding the internal logic of the topos. Then we consider $\Sigma$ as a dominance; we explicitly construct the lift functor and characterize $\Sigma$-subobjects. Contrary to our expectations the dominance $\Sigma$ is not closed under unions. In the last section we build a model for homotopy theory, where the order-discrete objects are exactly those objects which only have constant paths

Domain Theory in Constructive and Predicative Univalent Foundations

We develop domain theory in constructive and predicative univalent foundations (also known as homotopy type theory). That we work predicatively means that we do not assume Voevodsky's propositional resizing axioms. Our work is constructive in the sense that we do not rely on excluded middle or the axiom of (countable) choice. Domain theory studies so-called directed complete posets (dcpos) and Scott continuous maps between them and has applications in programming language semantics, higher-type computability and topology. A common approach to deal with size issues in a predicative foundation is to work with information systems, abstract bases or formal topologies rather than dcpos, and approximable relations rather than Scott continuous functions. In our type-theoretic approach, we instead accept that dcpos may be large and work with type universes to account for this. A priori one might expect that complex constructions of dcpos result in a need for ever-increasing universes and are predicatively impossible. We show that such constructions can be carried out in a predicative setting. We illustrate the development with applications in the semantics of programming languages: the soundness and computational adequacy of the Scott model of PCF and Scott's $D_\infty$ model of the untyped $\lambda$-calculus. We also give a predicative account of continuous and algebraic dcpos, and of the related notions of a small basis and its rounded ideal completion. The fact that nontrivial dcpos have large carriers is in fact unavoidable and characteristic of our predicative setting, as we explain in a complementary chapter on the constructive and predicative limitations of univalent foundations. Our account of domain theory in univalent foundations is fully formalised with only a few minor exceptions. The ability of the proof assistant Agda to infer universe levels has been invaluable for our purposes.Comment: PhD thesis, extended abstract in the pdf. v5: Fixed minor typos in 6.2.18, 6.2.19 and 6.4.

Formal proofs supporting the thesis "Domain Theory in Constructive and Predicative Univalent Foundations"

This deposit consists of formal proofs in Agda and Coq, together with the software libraries that these proofs depend on. Moreover, the HTML renderings of these proofs, generated for presentation and reading, are also included. The readme file has additional details describing each file

Apartness, sharp elements, and the Scott topology of domains

Working constructively, we study continuous directed complete posets (dcpos) and the Scott topology. Our two primary novelties are a notion of intrinsic apartness and a notion of sharp elements. Being apart is a positive formulation of being unequal, similar to how inhabitedness is a positive formulation of nonemptiness. To exemplify sharpness, we note that a lower real is sharp if and only if it is located. Our first main result is that for a large class of continuous dcpos, the Bridges-Vîţǎ apartness topology and the Scott topology coincide. Although we cannot expect a tight or cotransitive apartness on nontrivial dcpos, we prove that the intrinsic apartness is both tight and cotransitive when restricted to the sharp elements of a continuous dcpo. These include the strongly maximal elements, as studied by Smyth and Heckmann. We develop the theory of strongly maximal elements highlighting its connection to sharpness and the Lawson topology. Finally, we illustrate the intrinsic apartness, sharpness, and strong maximality by considering several natural examples of continuous dcpos: the Cantor and Baire domains, the partial Dedekind reals, the lower reals and, finally, an embedding of Cantor space into an exponential of lifted sets

Road pricing from a geographical perspective: a literature review and implications for research into accessibility

Road pricing policies have been a subject of research for many decades. Even though until now examples of actual implication in the real world are limited, many different road-pricing measures have been considered, both in literature as well as in the political debate in several countries. However, most literature focuses on economic aspects, more or less ignoring spatial consequences. In this paper we will concentrate on the spatial effects of pricing policy and introduce the typical geographic concept of accessibility into the discussion about pricing policy. The paper firstly gives some backgrounds of pricing policies. Some objectives of road pricing in general are given. Furthermore some examples of already implemented pricing measures in countries all over the world are mentioned. General literature concerning pricing policies aims specifically on economic effects. This is mainly because of the typical economic aspects, which can be found in the theory of pricing policy such as the pricing of a scarce good as infrastructure capacity, related to time aspects. Also studies concerning acceptability of road pricing policies are discussed, because acceptance plays an important role in the implementation of pricing policies. The paper shortly addresses some of the economic and acceptability related literature. But the literature review of the paper focuses specifically on the geographical aspects of pricing policies. These geographical aspects have received much less attention so far although road-pricing measures may cause important spatial effects. Therefore the second part of the paper focuses on these geographical aspects. A specific research field in geography is accessibility. Accessibility is a concept that connects infrastructure and land-use. The research fields of accessibility and pricing policies in isolation are well elaborated. However, the link between road pricing policies and accessibility (measures) forms a new research field. The paper explains the importance of the concept of accessibility. In practice accessibility can be computed with accessibility measures. These measures form quantifications of accessibility. Different types of accessibility measures exist differing in concept as well as complexity. All these measures have in common that transport costs are not included at all or at least not in a realistic way. After explaining the concept of accessibility different categories of accessibility measures are explained and their general advantages and disadvantages are given. Furthermore possibilities to adapt or improve accessibility measures are discussed. After this discussion the actual link between road pricing policies and accessibility measures is explained. The discussion begins with the presentation of a conceptual model of the accessibility (and spatial) effects of road pricing. Subsequently an observation is made where current measures fall short to include pricing policy costs in a realistic way. This observation will lead to the determination of directions for improvement. Besides the general possibilities to adjust different accessibility measures, each measure is specifically evaluated on the ability to improve the way of describing accessibility effects of road pricing.

Predicative aspects of order theory in univalent foundations

We investigate predicative aspects of order theory in constructive univalent foundations. By predicative and constructive, we respectively mean that we do not assume Voevodsky’s propositional resizing axioms or excluded middle. Our work complements existing work on predicative mathematics by exploring what cannot be done predicatively in univalent foundations. Our first main result is that nontrivial (directed or bounded) complete posets are necessarily large. That is, if such a nontrivial poset is small, then weak propositional resizing holds. It is possible to derive full propositional resizing if we strengthen nontriviality to positivity. The distinction between nontriviality and positivity is analogous to the distinction between nonemptiness and inhabitedness. We prove our results for a general class of posets, which includes directed complete posets, bounded complete posets and sup-lattices, using a technical notion of a δ_V-complete poset. We also show that nontrivial locally small δ_V-complete posets necessarily lack decidable equality. Specifically, we derive weak excluded middle from assuming a nontrivial locally small δ_V-complete poset with decidable equality. Moreover, if we assume positivity instead of nontriviality, then we can derive full excluded middle. Secondly, we show that each of Zorn’s lemma, Tarski’s greatest fixed point theorem and Pataraia’s lemma implies propositional resizing. Hence, these principles are inherently impredicative and a predicative development of order theory must therefore do without them. Finally, we clarify, in our predicative setting, the relation between the traditional definition of sup-lattice that requires suprema for all subsets and our definition that asks for suprema of all small families