271,354 research outputs found
The prospects for mathematical logic in the twenty-first century
The four authors present their speculations about the future developments of
mathematical logic in the twenty-first century. The areas of recursion theory,
proof theory and logic for computer science, model theory, and set theory are
discussed independently.Comment: Association for Symbolic Logi
Fourier spectra of measures associated with algorithmically random Brownian motion
In this paper we study the behaviour at infinity of the Fourier transform of
Radon measures supported by the images of fractal sets under an algorithmically
random Brownian motion. We show that, under some computability conditions on
these sets, the Fourier transform of the associated measures have, relative to
the Hausdorff dimensions of these sets, optimal asymptotic decay at infinity.
The argument relies heavily on a direct characterisation, due to Asarin and
Pokrovskii, of algorithmically random Brownian motion in terms of the prefix
free Kolmogorov complexity of finite binary sequences. The study also
necessitates a closer look at the potential theory over fractals from a
computable point of view.Comment: 24 page
On Martin's Pointed Tree Theorem
We investigate the reverse mathematics strength of Martin's pointed tree
theorem (MPT) and one of its variants, weak Martin's pointed tree theorem
(wMPT)
A Logical Characterization of Constant-Depth Circuits over the Reals
In this paper we give an Immerman's Theorem for real-valued computation. We
define circuits operating over real numbers and show that families of such
circuits of polynomial size and constant depth decide exactly those sets of
vectors of reals that can be defined in first-order logic on R-structures in
the sense of Cucker and Meer. Our characterization holds both non-uniformily as
well as for many natural uniformity conditions.Comment: 24 pages, submitted to WoLLIC 202
Changing a semantics: opportunism or courage?
The generalized models for higher-order logics introduced by Leon Henkin, and
their multiple offspring over the years, have become a standard tool in many
areas of logic. Even so, discussion has persisted about their technical status,
and perhaps even their conceptual legitimacy. This paper gives a systematic
view of generalized model techniques, discusses what they mean in mathematical
and philosophical terms, and presents a few technical themes and results about
their role in algebraic representation, calibrating provability, lowering
complexity, understanding fixed-point logics, and achieving set-theoretic
absoluteness. We also show how thinking about Henkin's approach to semantics of
logical systems in this generality can yield new results, dispelling the
impression of adhocness. This paper is dedicated to Leon Henkin, a deep
logician who has changed the way we all work, while also being an always open,
modest, and encouraging colleague and friend.Comment: 27 pages. To appear in: The life and work of Leon Henkin: Essays on
his contributions (Studies in Universal Logic) eds: Manzano, M., Sain, I. and
Alonso, E., 201
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