10 research outputs found
The extended predicative Mahlo universe in Martin-Lof type theory
This paper addresses the long-standing question of the predicativity of the Mahlo universe. A solution, called the extended predicative Mahlo universe, has been proposed by Kahle and Setzer in the context of explicit mathematics. It makes use of the collection of untyped terms (denoting partial functions) which are directly available in explicit mathematics but not in Martin-Lof type theory. In this paper, we overcome the obstacle of not having direct access to untyped terms in Martin-Lof type theory by formalizing explicit mathematics with an extended predicative Mahlo universe in Martin-Lof type theory with certain indexed inductive-recursive definitions. In this way, we can relate the predicativity question to the fundamental semantics of Martin-Lof type theory in terms of computation to canonical form. As a result, we get the first extended predicative definition of a Mahlo universe in Martin-Lof type theory. To this end, we first define an external variant of Kahle and Setzer\u27s internal extended predicative universe in explicit mathematics. This is then formalized in Martin-Lof type theory, where it becomes an internal extended predicative Mahlo universe. Although we make use of indexed inductive-recursive definitions that go beyond the type theory of indexed inductive-recursive definitions defined in previous work by the authors, we argue that they are constructive and predicative in Martin-Lof\u27s sense. The model construction has been type-checked in the proof assistant Agda
Generalized Universe Hierarchies and First-Class Universe Levels
In type theories, universe hierarchies are commonly used to increase the expressive power of the theory while avoiding inconsistencies arising from size issues. There are numerous ways to specify universe hierarchies, and theories may differ in details of cumulativity, choice of universe levels, specification of type formers and eliminators, and available internal operations on levels. In the current work, we aim to provide a framework which covers a large part of the design space. First, we develop syntax and semantics for cumulative universe hierarchies, where levels may come from any set equipped with a transitive well-founded ordering. In the semantics, we show that induction-recursion can be used to model transfinite hierarchies, and also support lifting operations on type codes which strictly preserve type formers. Then, we consider a setup where universe levels are first-class types and subject to arbitrary internal reasoning. This generalizes the bounded polymorphism features of Coq and at the same time the internal level computations in Agda
Broad Infinity and Generation Principles
This paper introduces Broad Infinity, a new and arguably intuitive axiom
scheme. It states that "broad numbers", which are three-dimensional trees whose
growth is controlled, form a set. If the Axiom of Choice is assumed, then Broad
Infinity is equivalent to the Ord-is-Mahlo scheme: every closed unbounded class
of ordinals contains a regular ordinal.
Whereas the axiom of Infinity leads to generation principles for sets and
families and ordinals, Broad Infinity leads to more advanced versions of these
principles. The paper relates these principles under various prior assumptions:
the Axiom of Choice, the Law of Excluded Middle, and weaker assumptions.Comment: 52 page
Partial functions and recursion in univalent type theory
We investigate partial functions and computability theory from within a constructive, univalent type theory. The focus is on placing computability into a larger mathematical context, rather than on a complete development of computability theory. We begin with a treatment of partial functions, using the notion of dominance, which is used in synthetic domain theory to discuss classes of partial maps. We relate this and other ideas from synthetic domain theory to other approaches to partiality in type theory. We show that the notion of dominance is difficult to apply in our setting: the set of ïżœ0 1 propositions investigated by Rosolini form a dominance precisely if a weak, but nevertheless unprovable, choice principle holds. To get around this problem, we suggest an alternative notion of partial function we call disciplined maps. In the presence of countable choice, this notion coincides with Rosoliniâs. Using a general notion of partial function,we take the first steps in constructive computability theory. We do this both with computability as structure, where we have direct access to programs; and with computability as property, where we must work in a program-invariant way. We demonstrate the difference between these two approaches by showing how these approaches relate to facts about computability theory arising from topos-theoretic and typetheoretic concerns. Finally, we tie the two threads together: assuming countable choice and that all total functions N - N are computable (both of which hold in the effective topos), the Rosolini partial functions, the disciplined maps, and the computable partial functions all coincide. We observe, however, that the class of all partial functions includes non-computable
partial functions
Fully Generic Programming Over Closed Universes of Inductive-Recursive Types
Dependently typed programming languages allow the type system to express arbitrary propositions of intuitionistic logic, thanks to the Curry-Howard isomorphism. Taking full advantage of this type system requires defining more types than usual, in order to encode logical correctness criteria into the definitions of datatypes. While an abundance of specialized types helps ensure correctness, it comes at the cost of needing to redefine common functions for each specialized type. This dissertation makes an effort to attack the problem of code reuse in dependently typed languages. Our solution is to write generic functions, which can be applied to any datatype.
Such a generic function can be applied to datatypes that are defined at the time the generic function was written, but they can also be applied to any datatype that is defined in the future. Our solution builds upon previous work on generic programming within dependently typed programming.
Type theory supports generic programming using a construction known as a universe. A universe can be considered the model of a programming language, such that writing functions over it models writing generic programs in the programming language. Historically, there has been a trade-off between the expressive power of the modeled programming language, and the kinds of generic functions that can be written in it. Our dissertation shows that no such trade-off is necessary, and that we can write future-proof generic functions in a model of a dependently typed programming language with a rich collection of types
A predicative variant of a realizability tripos for the Minimalist Foundation.
open2noHere we present a predicative variant of a realizability tripos validating
the intensional level of the Minimalist Foundation extended with Formal Church
thesis.the file attached contains the whole number of the journal including the mentioned pubblicationopenMaietti, Maria Emilia; Maschio, SamueleMaietti, MARIA EMILIA; Maschio, Samuel
Proof theory and Martin-Löf Type Theory
In this article an overview over the work of the author on developing proof theoretic strong extensions of Martin-Loef Type Theory including precise proof theoretic bounds is given. It presents the first publication of the proof theoretically strongest known extensions of Martin-Loef Type Theory, namely the hyper-Mahlo Universe, the hyper-alpha-Mahlo universe, the autononomous Mahlo universe and the Pi_3-reflecting universe. This is part of a proof theoretic program in developing proof theoretic as strong as possible constructive theories in order to obtain a constructive underpinning of strong classical theories with a full proof theoretic analysis
Proof Theory of Martin-Löf Type Theory. An overview
We give an overview over the historic development of proof theory and the main techniques used in ordinal theoretic proof theory. We argue, that in a revised Hilbertâs program, ordinal theoretic proof theory has to be supplemented by a second step, namely the development of strong equiconsistent constructive theories. Then we show, how, as part of such a programe, the proof theoretic analysis of Martin-Löf type theory with W-type and one microscopic universe containing only two finite sets is carried out. Then we look at the analysis Martin-Löf type theory with W-type and a universe closed under the W-type, and consider the extension of type theory by one Mahlo universe and its proof-theoretic analysis. Finally, we repeat the concept of inductive-recursive definitions, which extends the notion of inductive definitions substantially. We introduce a closed formalization, which can be used in generic programming, and explain, what is known about its strength.Nous donnons une vue dâensemble du dĂ©veloppement historique de la thĂ©orie de la preuve et des principales techniques utilisĂ©es dans la thĂ©orie ordinale de la preuve. Nous soutenons que, dans une forme rĂ©visĂ©e du programme dâHilbert, la thĂ©orie ordinale de la preuve doit ĂȘtre complĂ©tĂ©e par une seconde Ă©tape, Ă savoir le dĂ©veloppement de thĂ©ories constructives fortes et Ă©quiconsistantes. Comme partie dâun tel programme, nous prĂ©sentons ensuite lâanalyse, en thĂ©orie de la preuve, de la thĂ©orie des types de Martin-Löf avec un univers microscopique ne contenant que deux types finis. Nous examinons ensuite lâanalyse de la thĂ©orie des types de Martin-Löf avec type W et un univers clos pour ce type, puis nous Ă©tendons la thĂ©orie des types par un univers de Mahlo et considĂ©rons son analyse en thĂ©orie de la preuve. Enfin, nous prĂ©sentons le concept de dĂ©finition inductive-rĂ©cursive, qui Ă©tend de façon substantielle la notion de dĂ©finition inductive. Nous introduisons une formalisation close, qui peut ĂȘtre employĂ©e en programmation gĂ©nĂ©rique, et expliquons ce que nous savons de sa force ordinale
A model for a type theory with Mahlo universe
We present a type theory T T M, extending Martin-Löf Type Theory by adding one Mahlo universe V, a universe being the type theoretic analogue of one recursive Mahlo ordinal. A model, formulated in a Kripke-Platek style set theory KP M +, is given and we show that the proof theoretical strength of T T M is †|KP M + | = ÏâŠ1 (âŠM+Ï). By [Se96a], this bound is sharp.