Construction and
Theoretical Properties of
Hierarchical Complex
Systems.
Boris Mitavskiy,
University of Birmingham,
School of Computer Science.
Many structures in biology and in computer
science are modeled by digraphs with
extra structure:
Many structures in biology and in computer
science are modeled by digraphs with
extra structure:
Many structures in biology and in computer
science are modeled by digraphs with
extra structure:
Artificial neural nets form a “multi

sorted
algebraic theory” (i.e. a category with
finite products):
Artificial neural nets form a “multi

sorted
algebraic theory” (i.e. a category with
finite products):
Many structures in biology and in computer
science are modeled by digraphs with
extra structure:
Many structures in biology and in computer
science are modeled by digraphs with
extra structure:
Many structures in biology and in computer
science are modeled by digraphs with
extra structure:
Another category associated to neural nets is
is one where neurons are the objects and
synapses are the morphism (more
Common view among the mathematical
neuroscientists: Andree Ehresmann et. al.)
Another category associated to neural nets is
is one where neurons are the objects and
synapses are the morphism (more
Common view among the mathematical
neuroscientists: Andree Ehresmann et. al.)
1.
The leading idea behind the creation of complicated objects is to construct them
from simpler “building block” objects. Indeed, a single neuron can not accomplish
very much.
Modularization
is a
crucial
component of cognition and other
complicated processes.
2.
Directed graphs with extra structure (usually categories) model a vast number of
artificial and biological complex systems. The extra structure (such as composition,
for instance) allows us to build higher layers from the lower ones. A descriptive
algebraic framework for construction of efficient higher layers (modules) in the
biological systems has been investigated by Andree Ehresmann.
3.
A number of already existing computational models benefit significantly from a
hierarchical approach in terms of a dynamical system of directed graphs where the
higher layers represent efficient computational structures for a more specialized
(narrower) set of problems (ex.: higher and lower level programming languages,
object oriented programming)
4.
What’s missing then?
5.
The algorithms for the construction of higher layers from the
lower ones have not been studied in the general and systematic framework
I offer. On the other hand, many known AI techniques fit into
this framework quite well. These include, for instance,
reinforcement learning methods, evolutionary computing
techniques and Hebbian learning methods.
6.
To the best of my knowledge, such a general algebraic framework
(category theory etc.) has not been used in conjunction with probability theory before.
The main goals of the project I propose are to investigate how
the “missing aspects” (5 and 6) can help us
* to produce systematic methods for the construction of the
artificial hierarchical complex systems and study their theoretical
properties (complexity of construction and efficiency of performance, etc.)
* to understand better and also to enhance the already existing
AI techniques using the language and tools from category theory
(mainly the notion of a structure

preserving morphism) and from
probability theory
Probability space of all possible situations,
Probability space of all possible situations,
Probability space of all possible situations,
Probability space of all possible situations,
Probability space of all possible situations,
Probability space of all possible situations,
Probability space of all possible situations,
Probability space of all possible situations,
Probability space of all possible situations,
Probability space of all possible situations,
Probability space of all possible situations,
Probability space of all possible situations,
Probability space of all possible situations,
Probability space of all possible situations,
Probability space of all possible situations,
Morphisms of digraphs, functors and abstract level comparison:
Probability space of all possible situations,
Morphisms of digraphs, functors and abstract level comparison:
Probability space of all possible situations,
Morphisms of digraphs, functors and abstract level comparison:
Probability space of all possible situations,
Morphisms of digraphs, functors and abstract level comparison:
Probability space of all possible situations,
Morphisms of digraphs, functors and abstract level comparison:
Probability space of all possible situations,
Morphisms of digraphs, functors and abstract level comparison:
Probability space of all possible situations,
Morphisms of digraphs, functors and abstract level comparison:
Probability space of all possible situations,
Morphisms of digraphs, functors and abstract level comparison:
Questions?
Questions?
Please don’t be too harsh with me on my
30
th
birthday!
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