9 research outputs found
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 model of the untyped
-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.
Programming Languages and Systems
This open access book constitutes the proceedings of the 29th European Symposium on Programming, ESOP 2020, which was planned to take place in Dublin, Ireland, in April 2020, as Part of the European Joint Conferences on Theory and Practice of Software, ETAPS 2020. The actual ETAPS 2020 meeting was postponed due to the Corona pandemic. The papers deal with fundamental issues in the specification, design, analysis, and implementation of programming languages and systems
Analysis in univalent type theory
Some constructive real analysis is developed in univalent type theory (UTT). We develop various types of real numbers, and prove several equivalences between those types. We then study computation with real numbers. It is well known how to compute with real numbers in intensional formalizations of mathematics, where equality of real numbers is specified by an imposed equivalence relation on representations such as Cauchy sequences. However, because in UTT equality of real numbers is captured directly by identity types, we are prevented from making any nontrivial discrete observations of arbitrary real numbers. For instance, there is no function which converts real numbers to decimal expansions, as this would not be continuous. To avoid breaking extensionality, we thus restrict our attention to real numbers that have been equipped with a simple structure called a \emph{locator}. In order to compute, we modify existing constructions in analysis to work with locators, including Riemann integrals, intermediate value theorems and differential equations. Hence many of the proofs involving locators look familiar, showing that the use of locators is not a conceptual burden. We discuss the possibility of implementing the work in proof assistants and present a Haskell prototype
Cardinal Arithmetic: From Silver’s Theorem to Shelah’s PCF Theory
Treballs Finals del Màster de Lògica Pura i Aplicada, Facultat de Filosofia, Universitat de Barcelona, Curs: 2019-2020, Tutor: Joan Bagaria PigrauThe main goal of this master’s thesis is to give a detailed description of the major ZFC advances in cardinal arithmetic from Silver’s Theorem to Shelah’s pcf theory and his bound on 2אω. In our attempt to make this thesis as self-contained as possible, we have devoted the first chapter to review the most elementary concepts of set theory, which include all the classical results from the first period of developement of cardinal arithmetic, from 1870 to 1930, due to Cantor, Hausdorff, König, and Tarski
A Friendly Introduction to Mathematical Logic
At the intersection of mathematics, computer science, and philosophy, mathematical logic examines the power and limitations of formal mathematical thinking. In this expansion of Leary’s user-friendly 1st edition, readers with no previous study in the field are introduced to the basics of model theory, proof theory, and computability theory. The text is designed to be used either in an upper division undergraduate classroom, or for self study. Updating the 1st Edition’s treatment of languages, structures, and deductions, leading to rigorous proofs of Gödel’s First and Second Incompleteness Theorems, the expanded 2nd Edition includes a new introduction to incompleteness through computability as well as solutions to selected exercises. Available on Lulu.com, IndiBound.com, and Amazon.com, as well as wholesale through Ingram Content Group.https://knightscholar.geneseo.edu/geneseo-authors/1005/thumbnail.jp
Recommended from our members
Aspects of Qualitative Consciousness: A Computer Science Perspective
The domain of artificial intelligence (AI) has been characterised by John Searle [Sear84] by distinguishing between iveak AI, according to which computers are useful tools for studying mind, and strong AI, according to which an equivalence is made between mind and programs such that computers executing programs actually possess minds. This dissertation explores a third alternative, namely: the prospects and promise of m ild AI, according to which a suitable computer is capable of possessing species of mentality that may differ from or be weaker than ordinary human mentality, but qualify as “mentality” nonetheless. The purpose of this dissertation is to explore the prospects and promise of mild AI.
The approach adopted explores whether mind can be replicated, as opposed to merely simulated, in digital machines. This requires a definition of mind in order to judge success. James Fetzer [Fetz90] has suggested minds can be defined as sign using systems in the sense of Charles Peirce’s semiotic (theory of signs) and, on this basis, argues convincingly against strong AI. Determining if his negative conclusion applies to mild AI requires rejoining Fetzer’s analysis of the analogical argument for strong AI and redressing his laws of human beings and digital machines. This is tackled by focusing on the nature and form of the operational relationship between the physical machine and mind, and suggesting some operational requirements for a minimal semiotic system independently of any underlying physical implementation. This involves four steps.
Firstly, as a formal foundation, a characterisation of systems is developed in terms of the causal structure and ontological levels in the system, where an ontological level is individuated by the laws that are in effect. This is in contrast to levels of organisation, such as levels of software abstraction. This exploration suggests the necessity — as a matter of natural law — for a mediating level between the physical machine and mind that is or, at least, appears to be necessary for producing forms of mentality. The lawful structure that appears to be required within this level and between levels is examined with respect to the prospects for implementing a semiotic system.
Secondly, how a system can operate in terms of semiotic processes based on a network of instantiated dispositions is explored. These are modelled as the temporal counterparts of state-transitions and stationary-representations, which are termed causal-flows and temporal-representations, respectively. They highlight the varying interactive structure of temporal patterns of causal activity in time. For the purposes of replicating mind, preserving the causal-flow structure of mental processes arises as an important requirement.
Thirdly, the system structure sufficient for generating consciousness is explored — a necessary condition for a cognitive semiotic system. This suggests a requirement relating to the causal accessibility of the contents of consciousness. This structuring is driven by the system’s need to signify reality by categorising these aspects as operational entities upon which decisions can be made. Consciousness arises through the manner in which the signified reality is generated. This makes mind and consciousness the result of a co-ordinated occurrent system wide activity.
Fourthly, in a mathematical sense, brains and computers can be classified as types of numeric and symbolic systems, respectively. These systems are compared and conditions formulated under which they may give rise to equivalent ontological levels. Peirce’s triadic sign relation is analysed in terms of ontological levels and the results used to clarify the nature of the ground relation in machine forms of mentality.
According to the theorems developed, the introduction of a dispositional mediating level might effectively enable a suitable computer to replicate species of mentality. An important factor in determining whether a computer is suitable for this purpose is its performance capacity and thus some estimates are calculated in this respect. It is shown how these requirements, along with a number of others, can help in the development of semiotic systems and variants, such as the iconic state machine of Igor Aleksander [Alek96]