39 research outputs found
Cohomology in Grothendieck Topologies and Lower Bounds in Boolean Complexity
This paper is motivated by questions such as P vs. NP and other questions in
Boolean complexity theory. We describe an approach to attacking such questions
with cohomology, and we show that using Grothendieck topologies and other ideas
from the Grothendieck school gives new hope for such an attack.
We focus on circuit depth complexity, and consider only finite topological
spaces or Grothendieck topologies based on finite categories; as such, we do
not use algebraic geometry or manifolds.
Given two sheaves on a Grothendieck topology, their "cohomological
complexity" is the sum of the dimensions of their Ext groups. We seek to model
the depth complexity of Boolean functions by the cohomological complexity of
sheaves on a Grothendieck topology. We propose that the logical AND of two
Boolean functions will have its corresponding cohomological complexity bounded
in terms of those of the two functions using ``virtual zero extensions.'' We
propose that the logical negation of a function will have its corresponding
cohomological complexity equal to that of the original function using duality
theory. We explain these approaches and show that they are stable under
pullbacks and base change. It is the subject of ongoing work to achieve AND and
negation bounds simultaneously in a way that yields an interesting depth lower
bound.Comment: 70 pages, abstract corrected and modifie
Sheaves and Duality in the Two-Vertex Graph Riemann-Roch Theorem
For each graph on two vertices, and each divisor on the graph in the sense of
Baker-Norine, we describe a sheaf of vector spaces on a finite category whose
zeroth Betti number is the Baker-Norine "Graph Riemann-Roch" rank of the
divisor plus one. We prove duality theorems that generalize the Baker-Norine
"Graph Riemann-Roch" Theorem
Extreme Rays of AND-Measures in Circuit Complexity
This paper is motivated by the problem of proving lower bounds on the formula size of boolean functions, which leads to lower bounds on circuit depth. We know that formula size is bounded from below by all formal complexity measures. Thus, we study formula size by investigating AND-measures, which are weakened forms of formal complexity measures. The collection of all AND-measures is a pointed polyhedral cone; we study the extreme rays of this cone in order to better understand AND-measures. From the extreme rays, we attempt to discover useful properties of AND-measures that may help in proving new lower bounds on formula size and circuit depth. This paper focuses on describing some of the properties of AND-measures, especially those that are extreme rays. Furthermore, it describes some algorithhms for finding the extreme rays
Natural Communication
In Natural Communication, the author criticizes the current paradigm of specific goal orientation in the complexity sciences. His model of "natural communication" encapsulates modern theoretical concepts from mathematics and physics, in particular category theory and quantum theory. The author is convinced that only by looking to the past is it possible to establish continuity and coherence in the complexity science
Topological Foundations of Cognitive Science
A collection of papers presented at the First International Summer Institute in Cognitive Science, University at Buffalo, July 1994, including the following papers:
** Topological Foundations of Cognitive Science, Barry Smith
** The Bounds of Axiomatisation, Graham White
** Rethinking Boundaries, Wojciech Zelaniec
** Sheaf Mereology and Space Cognition, Jean Petitot
** A Mereotopological Definition of 'Point', Carola Eschenbach
** Discreteness, Finiteness, and the Structure of Topological Spaces, Christopher Habel
** Mass Reference and the Geometry of Solids, Almerindo E. Ojeda
** Defining a 'Doughnut' Made Difficult, N .M. Gotts
** A Theory of Spatial Regions with Indeterminate Boundaries, A.G. Cohn and N.M. Gotts
** Mereotopological Construction of Time from Events, Fabio Pianesi and Achille C. Varzi
** Computational Mereology: A Study of Part-of Relations for Multi-media Indexing, Wlodek Zadrozny and Michelle Ki
Topos and Stacks of Deep Neural Networks
Every known artificial deep neural network (DNN) corresponds to an object in
a canonical Grothendieck's topos; its learning dynamic corresponds to a flow of
morphisms in this topos. Invariance structures in the layers (like CNNs or
LSTMs) correspond to Giraud's stacks. This invariance is supposed to be
responsible of the generalization property, that is extrapolation from learning
data under constraints. The fibers represent pre-semantic categories (Culioli,
Thom), over which artificial languages are defined, with internal logics,
intuitionist, classical or linear (Girard). Semantic functioning of a network
is its ability to express theories in such a language for answering questions
in output about input data. Quantities and spaces of semantic information are
defined by analogy with the homological interpretation of Shannon's entropy
(P.Baudot and D.B. 2015). They generalize the measures found by Carnap and
Bar-Hillel (1952). Amazingly, the above semantical structures are classified by
geometric fibrant objects in a closed model category of Quillen, then they give
rise to homotopical invariants of DNNs and of their semantic functioning.
Intentional type theories (Martin-Loef) organize these objects and fibrations
between them. Information contents and exchanges are analyzed by Grothendieck's
derivators
International Congress of Mathematicians: 2022 July 6–14: Proceedings of the ICM 2022
Following the long and illustrious tradition of the International Congress of Mathematicians, these proceedings include contributions based on the invited talks that were presented at the Congress in 2022.
Published with the support of the International Mathematical Union and edited by Dmitry Beliaev and Stanislav Smirnov, these seven volumes present the most important developments in all fields of mathematics and its applications in the past four years. In particular, they include laudations and presentations of the 2022 Fields Medal winners and of the other prestigious prizes awarded at the Congress.
The proceedings of the International Congress of Mathematicians provide an authoritative documentation of contemporary research in all branches of mathematics, and are an indispensable part of every mathematical library