ppt

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!
