39 research outputs found
Minimal lambda-theories by ultraproducts
A longstanding open problem in lambda calculus is whether there exist
continuous models of the untyped lambda calculus whose theory is exactly the
least lambda-theory lambda-beta or the least sensible lambda-theory H
(generated by equating all the unsolvable terms). A related question is
whether, given a class of lambda models, there is a minimal lambda-theory
represented by it. In this paper, we give a general tool to answer positively
to this question and we apply it to a wide class of webbed models: the
i-models. The method then applies also to graph models, Krivine models,
coherent models and filter models. In particular, we build an i-model whose
theory is the set of equations satisfied in all i-models.Comment: In Proceedings LSFA 2012, arXiv:1303.713
Graph easy sets of mute lambda terms
Among the unsolvable terms of the lambda calculus, the mute ones are those having the highest degree of undefinedness. In this paper, we define for each natural number n, an infinite and recursive set M-n of mute terms, and show that it is graph-easy: for any closed term t of the lambda calculus there exists a graph model equating all the terms of M-n to t. Alongside, we provide a brief survey of the notion of undefinedness in the lambda calculus. (C) 2015 Elsevier B.V. All rights reserved
Synthetic Philosophy of Mathematics and Natural Sciences Conceptual analyses from a Grothendieckian Perspective
ISBN-13: 978-0692593974. Giuseppe Longo. Synthetic Philosophy of Mathematics and Natural Sciences, Conceptual analyses from a Grothendieckian Perspective, Reflections on “Synthetic Philosophy of Contemporary Mathematics” by F. Zalamea, Urbanomic (UK) and Sequence Press (USA), 2012. Invited Paper, in Speculations: Journal of Speculative Realism, Published: 12/12/2015, followed by an answer by F. Zalamea.International audienceZalamea’s book is as original as it is belated. It is indeed surprising, if we give it a moment’s thought, just how greatly behind schedule philosophical reflection on contemporary mathematics lags, especially considering the momentous changes that took place in the second half of the twentieth century. Zalamea compares this situation with that of the philosophy of physics: he mentions D’Espagnat’s work on quantum mechanics, but we could add several others who, in the last few decades, have elaborated an extremely timely philosophy of contemporary physics (see for example Bitbol 2000; Bitbol et al. 2009). As was the case in biology, philosophy – since Kant’s crucial observations in the Critique of Judgment, at least – has often “run ahead” of life sciences, exploring and opening up a space for reflections that are not derived from or integrated with its contemporary scientific practice. Some of these reflections are still very much auspicious today. And indeed, some philosophers today are saying something truly new about biology..
Realizability with Stateful Computations for Nonstandard Analysis
In this paper we propose a new approach to realizability interpretations for nonstandard arithmetic. We deal with nonstandard analysis in the context of intuitionistic realizability, focusing on the Lightstone-Robinson construction of a model for nonstandard analysis through an ultrapower. In particular, we consider an extension of the ?-calculus with a memory cell, that contains an integer (the state), in order to indicate in which slice of the ultrapower ?^{?} the computation is being done. We shall pay attention to the nonstandard principles (and their computational content) obtainable in this setting. We then discuss how this product could be quotiented to mimic the Lightstone-Robinson construction
Mathematical Logic: Proof theory, Constructive Mathematics
The workshop “Mathematical Logic: Proof Theory, Constructive Mathematics” was centered around proof-theoretic aspects of current mathematics, constructive mathematics and logical aspects of computational complexit
Analysis and construction of logical systems: a category-theoretic approach
The aim of this dissertation is to develop categorical foundations for studying lambda calculi and their logics formed into logical systems. We show how internal models for polymorphic lambda calculi arise in any 2-category with a notion of discreteness. We generalise to a 2-categorical setting the famous theorem of Peter Freyd saying that there are no sufficiently (co)complete non-degenerate categories. As a simple corollary, we obtain a variant of Freyd theorem for categories internal to any tensored category. We introduce the concept of an associated category, and relying on it, provide a representation theorem relating our internal models with well-studied fibrational models for polymorphism. Finally, we define Yoneda triangles as relativisations of internal adjunctions, and use them to characterise universes that admit a notion of convolution. We show that such universes induce semantics for lambda calculi. We prove that a construction analogical to enriched Day convolution works for categories internal to a locally cartesian closed category with finite colimits.Celem niniejszej rozprawy jest zbudowanie teoriokategoryjnych fundamentów umożliwiających studiowanie rachunków lambda i ich logik opisanych za pomocą systemów logicznych. Pokazujemy w jaki sposób 2-kategorie z notacją dyskretności pozawalają mówić o modelach dla polimorficznych rachunków lambda. Uogólniamy i internalizujemy w 2-kategoriach klasyczne twierdzenia Petera Freyda o nieistnieniu dostatecznie (ko)zupełnych niezdegenerowanych kategorii. Jako prosty wniosek otrzymujemy wariant twierdzenia Freyda dla kategorii wewnętrznych względem dowolnej kategorii z tensorami. Wprowadzamy pojęcie kategorii stowarzyszonej i bazując na nim dowodzimy twierdzenie o reprezentacji wiążące wprowadzone w rozprawie wewnętrzne modele z dobrze znanymi modelami rozwłóknieniowymi dla polimorfizmu. Definiujemy pojęcie trójkąta Yonedy jako relatywizację wewnętrznych sprzężeń i używamy go do charakteryzacji uniwersów posiadających notację splotu. Pokazujemy, że takie uniwersa indukują semantykę dla rachunków lambda. Dowodzimy, że konstrukcja analogiczna do splotu Day'a dla kategorii wzbogacanych strukturą monoidalną zachodzi także dla kategorii wewnętrznych względem dowolnej lokalnie kartezjańsko domkniętej kategorii ze skończonymi kogranicami
Stateful Realizers for Nonstandard Analysis
In this paper we propose a new approach to realizability interpretations for
nonstandard arithmetic. We deal with nonstandard analysis in the context of
(semi)intuitionistic realizability, focusing on the Lightstone-Robinson
construction of a model for nonstandard analysis through an ultrapower. In
particular, we consider an extension of the -calculus with a memory
cell, that contains an integer (the state), in order to indicate in which slice
of the ultrapower the computation is being done. We pay
attention to the nonstandard principles (and their computational content)
obtainable in this setting. In particular, we give non-trivial realizers to
Idealization and a non-standard version of the LLPO principle. We then discuss
how to quotient this product to mimic the Lightstone-Robinson construction