2,169 research outputs found
Bifinite Chu Spaces
This paper studies colimits of sequences of finite Chu spaces and their
ramifications. Besides generic Chu spaces, we consider extensional and
biextensional variants. In the corresponding categories we first characterize
the monics and then the existence (or the lack thereof) of the desired
colimits. In each case, we provide a characterization of the finite objects in
terms of monomorphisms/injections. Bifinite Chu spaces are then expressed with
respect to the monics of generic Chu spaces, and universal, homogeneous Chu
spaces are shown to exist in this category. Unanticipated results driving this
development include the fact that while for generic Chu spaces monics consist
of an injective first and a surjective second component, in the extensional and
biextensional cases the surjectivity requirement can be dropped. Furthermore,
the desired colimits are only guaranteed to exist in the extensional case.
Finally, not all finite Chu spaces (considered set-theoretically) are finite
objects in their categories. This study opens up opportunities for further
investigations into recursively defined Chu spaces, as well as constructive
models of linear logic
Linear logic for constructive mathematics
We show that numerous distinctive concepts of constructive mathematics arise
automatically from an interpretation of "linear higher-order logic" into
intuitionistic higher-order logic via a Chu construction. This includes
apartness relations, complemented subsets, anti-subgroups and anti-ideals,
strict and non-strict order pairs, cut-valued metrics, and apartness spaces. We
also explain the constructive bifurcation of classical concepts using the
choice between multiplicative and additive linear connectives. Linear logic
thus systematically "constructivizes" classical definitions and deals
automatically with the resulting bookkeeping, and could potentially be used
directly as a basis for constructive mathematics in place of intuitionistic
logic.Comment: 39 page
Hammering towards QED
This paper surveys the emerging methods to automate reasoning over large libraries developed with formal proof assistants. We call these methods hammers. They give the authors of formal proofs a strong “one-stroke” tool for discharging difficult lemmas without the need for careful and detailed manual programming of proof search. The main ingredients underlying this approach are efficient automatic theorem provers that can cope with hundreds of axioms, suitable translations of the proof assistant’s logic to the logic of the automatic provers, heuristic and learning methods that select relevant facts from large libraries, and methods that reconstruct the automatically found proofs inside the proof assistants. We outline the history of these methods, explain the main issues and techniques, and show their strength on several large benchmarks. We also discuss the relation of this technology to the QED Manifesto and consider its implications for QED-like efforts.Blanchette’s Sledgehammer research was supported by the Deutsche Forschungs-
gemeinschaft projects Quis Custodiet (grants NI 491/11-1 and NI 491/11-2) and
Hardening the Hammer (grant NI 491/14-1). Kaliszyk is supported by the Austrian
Science Fund (FWF) grant P26201. Sledgehammer was originally supported by the
UK’s Engineering and Physical Sciences Research Council (grant GR/S57198/01).
Urban’s work was supported by the Marie-Curie Outgoing International Fellowship
project AUTOKNOMATH (grant MOIF-CT-2005-21875) and by the Netherlands
Organisation for Scientific Research (NWO) project Knowledge-based Automated
Reasoning (grant 612.001.208).This is the final published version. It first appeared at http://jfr.unibo.it/article/view/4593/5730?acceptCookies=1
Formalising Mathematics in Simple Type Theory
Despite the considerable interest in new dependent type theories, simple type
theory (which dates from 1940) is sufficient to formalise serious topics in
mathematics. This point is seen by examining formal proofs of a theorem about
stereographic projections. A formalisation using the HOL Light proof assistant
is contrasted with one using Isabelle/HOL. Harrison's technique for formalising
Euclidean spaces is contrasted with an approach using Isabelle/HOL's axiomatic
type classes. However, every formal system can be outgrown, and mathematics
should be formalised with a view that it will eventually migrate to a new
formalism
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