6,798 research outputs found
An Analysis of Tennenbaum\u27s Theorem in Constructive Type Theory
No full textTennenbaum\u27s theorem states that the only countable model of Peano arithmetic (PA) with computable arithmetical operations is the standard model of natural numbers. In this paper, we use constructive type theory as a framework to revisit, analyze and generalize this result. The chosen framework allows for a synthetic approach to computability theory, exploiting that, externally, all functions definable in constructive type theory can be shown computable. We then build on this viewpoint, and furthermore internalize it by assuming a version of Church\u27s thesis, which expresses that any function on natural numbers is representable by a formula in PA. This assumption provides for a conveniently abstract setup to carry out rigorous computability arguments, even in the theorem\u27s mechanization. Concretely, we constructivize several classical proofs and present one inherently constructive rendering of Tennenbaum\u27s theorem, all following arguments from the literature. Concerning the classical proofs in particular, the constructive setting allows us to highlight differences in their assumptions and conclusions which are not visible classically. All versions are accompanied by a unified mechanization in the Coq proof assistant
The Future of Canadian Climate Policy — with Marc Lee
No full textMarc Lee is a Senior Economist at the Canadian Centre for Policy Alternatives\u27 BC Office. In addition to tracking federal and provincial budgets and economic trends, Marc has published on a range of topics from poverty and inequality to globalization and international trade to public services and regulation. Marc is the Co-Director of the Climate Justice Project, a research partnership with UBC\u27s School of Community and Regional Planning that examines the links between climate change policies and social justice.Resources:Climate Justice Project: www.policyalternatives.ca/projects/cli…tice-projectMarc Lee\u27s Posts on Policy Note: www.policynote.ca/author/marclee/Canadian Centre for Policy Alternatives: www.policyalternatives.ca/Marc\u27s Twitter: twitter.com/MarcLeeCCPA International Panel on Climate Change, 2021 report: www.ipcc.ch/report/ar6/wg1
Synthetic Undecidability and Incompleteness of First-Order Axiom Systems in Coq
No full textWe mechanise the undecidability of various first-order axiom systems in Coq, employing the synthetic approach to computability underlying the growing Coq Library of Undecidability Proofs. Concretely, we cover both semantic and deductive entailment in fragments of Peano arithmetic (PA) and Zermelo-Fraenkel set theory (ZF), with their undecidability established by many-one reductions from solvability of Diophantine equations, i.e. Hilbert’s tenth problem (H10), and the Post correspondence problem (PCP), respectively. In the synthetic setting based on the computability of all functions definable in a constructive foundation, such as Coq’s type theory, it suffices to define these reductions as meta-level functions with no need for further encoding in a formalised model of computation.
The concrete cases of PA and ZF are prepared by a general synthetic theory of undecidable axiomatisations, focusing on well-known connections to consistency and incompleteness. Specifically, our reductions rely on the existence of standard models, necessitating additional assumptions in the case of full ZF, and all axiomatic extensions still justified by such standard models are shown incomplete. As a by-product of the undecidability of ZF formulated using only membership and no equality symbol, we obtain the undecidability of first-order logic with a single binary relation
An Analysis of Tennenbaum’s Theorem in Constructive Type Theory
No full textTennenbaum’s theorem states that the only countable model of Peano arithmetic (PA) with computable arithmetical operations is the standard model of natural numbers. In this paper, we use constructive type theory as a framework to revisit and generalize this result.
The chosen framework allows for a synthetic approach to computability theory, by exploiting the fact that, externally, all functions definable in constructive type theory can be shown computable. We internalize this fact by assuming a version of Church’s thesis expressing that any function on natural numbers is representable by a formula in PA. This assumption allows for a conveniently abstract setup to carry out rigorous computability arguments and feasible mechanization.
Concretely, we constructivize several classical proofs and present one inherently constructive rendering of Tennenbaum’s theorem, all following arguments from the literature. Concerning the classical proofs in particular, the constructive setting allows us to highlight differences in their assumptions and conclusions which are not visible classically. All versions are accompanied by a unified mechanization in the Coq proof assistant
Modular Verification of Intrusive List and Tree Data Structures in Separation Logic
No full textIntrusive linked data structures are commonly used in low-level programming languages such as C for efficiency and to enable a form of generic types. Notably, intrusive versions of linked lists and search trees are used in the Linux kernel and the Boost C++ library. These data structures differ from ordinary data structures in the way that nodes contain only the meta data (i.e. pointers to other nodes), but not the data itself. Instead the programmer needs to embed nodes into the data, thereby avoiding pointer indirections, and allowing data to be part of several data structures.
In this paper we address the challenge of specifying and verifying intrusive data structures using separation logic. We aim for modular verification, where we first specify and verify the operations on the nodes (without the data) and then use these specifications to verify clients that attach data. We achieve this by employing a representation predicate that separates the data structure’s node structure from the data that is attached to it. We apply our methodology to singly-linked lists - from which we build cyclic and doubly-linked lists - and binary trees - from which we build binary search trees. All verifications are conducted using the Coq proof assistant, making use of the Iris framework for separation logic
Climate Justice & Inequality: The Future of Canadian Climate Policy — with Marc Lee
No full textMarc Lee is a Senior Economist at the Canadian Centre for Policy Alternatives\u27 BC Office. In addition to tracking federal and provincial budgets and economic trends, Marc has published on a range of topics from poverty and inequality to globalization and international trade to public services and regulation. Marc is the Co-Director of the Climate Justice Project, a research partnership with UBC\u27s School of Community and Regional Planning that examines the links between climate change policies and social justice.Resources: Climate Justice Project: https://www.policyalternatives.ca/projects/climate-justice-projectMarc Lee\u27s Posts on Policy Note: https://www.policynote.ca/author/marclee/Canadian Centre for Policy Alternatives: https://www.policyalternatives.ca/Marc\u27s Twitter: https://twitter.com/MarcLeeCCPA International Panel on Climate Change, 2021 report: https://www.ipcc.ch/report/ar6/wg1
D. Jones, Enjoinder and argument in Ovid's Remedia amoris, 1997. (Hermes : H. 77)
No full textVergé-Borderolle Jean-Marc. D. Jones, Enjoinder and argument in Ovid's Remedia amoris, 1997. (Hermes : H. 77). In: Revue des Études Anciennes. Tome 99, 1997, n°3-4. Mélanges dédiés à la mémoire de J. Coupry. pp. 581-582
UKMARC AMC: Draft Rev 4.0: UK MARC format for archives and manuscripts control (UK MARC AMC)
No full textThis draft is the first attempt to establish a UK MARC specifically for Archives and Manuscripts Control since the British Library indicated that it would countenance such extensions to the national UK MARC format. In order to keep consistency with the general UK MARC format, standard UK MARC subject fields are not included in this document, since they should be taken from the latest version of the UK MARC manual. {A note of them should perhaps be included in UK MARC AMC.} {NB Text in braces is intended to be explanatory material for readers of this draft}. Certain other fields have not been included that might occasionally be used in the cataloguing of archival materials but would generally only be used for such materials in organizations which were combining archive
databases with library databases. This MARC version is intended for use with descriptions of archive or anuscript material that follow, or fit, the traditional style of cataloguing: we assume that these will normally relate
to paper or parchment originals. It is not intended for use with descriptions of other kinds of material. For these, fields may be drawn from the appropriate UK MARC document. MARC versions for use with archives in special formats should be developed, in order to complete the full range of facilities available to archivists and curators
MARC 21 para recursos contínuos
No full textTranslation and adaptation of the MARC 21 Format for Bibliographic Data, and MARC 21 Format for Holdings Data, Network Development and MARC Standards Office, Library of Congress, USA, by Angela Salles. Rio de Janeiro, 2010. 2 v. V.1 MARC 21 format for bibliographic data (updated until October 2010). V.2 MARC 21 format for data collection (Holdings) (updated until October 2008)
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