8,713 research outputs found
Sharing a Library between Proof Assistants: Reaching out to the HOL Family
We observe today a large diversity of proof systems. This diversity has the
negative consequence that a lot of theorems are proved many times. Unlike
programming languages, it is difficult for these systems to co-operate because
they do not implement the same logic. Logical frameworks are a class of theorem
provers that overcome this issue by their capacity of implementing various
logics. In this work, we study the STTforall logic, an extension of Simple Type
Theory that has been encoded in the logical framework Dedukti. We present a
translation from this logic to OpenTheory, a proof system and interoperability
tool between provers of the HOL family. We have used this translation to export
an arithmetic library containing Fermat's little theorem to OpenTheory and to
two other proof systems that are Coq and Matita.Comment: In Proceedings LFMTP 2018, arXiv:1807.0135
Analytic Torsion on Manifolds with Edges
Let (M,g) be an odd-dimensional incomplete compact Riemannian singular space
with a simple edge singularity. We study the analytic torsion on M, and in
particular consider how it depends on the metric g. If g is an admissible edge
metric, we prove that the torsion zeta function is holomorphic near s = 0,
hence the torsion is well-defined, but possibly depends on g. In general
dimensions, we prove that the analytic torsion depends only on the asymptotic
structure of g near the singular stratum of M; when the dimension of the edge
is odd, we prove that the analytic torsion is independent of the choice of
admissible edge metric. The main tool is the construction, via the methodology
of geometric microlocal analysis, of the heat kernel for the Friedrichs
extension of the Hodge Laplacian in all degrees. In this way we obtain detailed
asymptotics of this heat kernel and its trace.Comment: 36 pages, 5 figures, v2: minor improvement
Symmetries and Paraparticles as a Motivation for Structuralism
This paper develops an analogy proposed by Stachel between general relativity
(GR) and quantum mechanics (QM) as regards permutation invariance. Our main
idea is to overcome Pooley's criticism of the analogy by appeal to
paraparticles.
In GR the equations are (the solution space is) invariant under
diffeomorphisms permuting spacetime points. Similarly, in QM the equations are
invariant under particle permutations. Stachel argued that this feature--a
theory's `not caring which point, or particle, is which'--supported a
structuralist ontology.
Pooley criticizes this analogy: in QM the (anti-)symmetrization of fermions
and bosons implies that each individual state (solution) is fixed by each
permutation, while in GR a diffeomorphism yields in general a distinct, albeit
isomorphic, solution.
We define various versions of structuralism, and go on to formulate Stachel's
and Pooley's positions, admittedly in our own terms. We then reply to Pooley.
Though he is right about fermions and bosons, QM equally allows more general
types of symmetry, in which states (vectors, rays or density operators) are not
fixed by all permutations (called `paraparticle states'). Thus Stachel's
analogy is revived.Comment: 45 pages, Latex, 3 Figures; forthcoming in British Journal for the
Philosophy of Scienc
- …