Abstract: Cognitive Reflectivity

[mjgeddes]

I tripped over my own boot-laces and stumbled – whacking my head.
Bent over to rub my head and retie the damn laces, all a sudden I had
the following quick rough thoughts for a paper:

'Cognitive Reflectivity'
Marc Geddes
Melbourne, Australia
18th June, 2009

Abstract


"A change in the goal-system of an agent is equivalent to a change in
the way in which knowledge is represented by the agent.  It follows
that it is equivalent to a change in the complexity of the program
representing the agent.  Thus we require a method of comparing the
complexity of strings in order to ensure that relevant program
structure is preserved with state transitions over time.  Standard
probability theory cannot be used because; (1) Consistent probability
calculations require implicit universal generalizations, but a
universal measure of the complexity of  finite strings is a logical
impossibility (fromGodel, Lob theorems); and (2) Standard measures of
complexity (e.g Kolmogorov complexity) from information theory deal
only with one aspect of information (i.e.  Shannon information), and
fail to consider semantic content. The solution must resolve both
these problems.

Regarding (2) the solution is as follows:, information theory is
generalized to deal with the actual meaning of information  (i.e . the
semantics of Shannon information) .The generalized definition of the
complexity of a finite string is based on the conceptual clustering of
semantic categories specifying the knowledge a string represents.  The
generation of hierarchical category structures representing the
knowledge in a string is also associated with a generalization of
Occam’s razor.   The justification for Occam’s razor and the problem
of priors in induction is resolved by defining ‘utility’ in terms of
‘aesthetic goodness’, which is the degree of integration of different
concept hierarchies.  This considers the process through which a
theory is generated;  it is a form of process-oriented evaluation.

Regarding (1); The Godel limitation is bypassed by using relative
complexity measures of pairs of strings .  This requires generalizing
standard Bayesian induction ; in fact induction is merely a special
case of a new form of case-based reasoning (analogical reasoning) .
Analogical reasoning can be formalized by utilizing concepts from
category theory to implement prototype theory, where mathematical
categories are regarded as semantic categories. Semantic concepts
representing the knowledge encoded in strings can be considered to
reside in multi-dimensional feature space, and this enables mappings
between concepts; such mappings are defined by functors representing
conceptual distance; this gives a formal definition of an analogy.
The reason this overcomes the Godel limitation and is more general
than induction is because it always enables relative comparisons of
the complexity of pairs of strings. This is because case-based
reasoning depends only on the specific details on the strings being
compared, whereas induction makes implicit universal generalizations,
and thus fails.

To summarize:  Induction is shown to be merely a special case of a new
type of generalized case-based (analogical) reasoning. Concepts from
category theory enable a formal definition of an analogy, which is
based on the notion of conceptual distance between concepts. The
notion of complexity is generalized to deal with semantics, where the
information in a string is considered to be a concept hierarchy. This
enables comparisons between pairs of strings; relations between
strings are defined in terms of the mappings between concepts, and the
mapping is evaluated in terms of its aesthetic goodness.  Godelian
limitations are overcome, since analogical reasoning always enables a
comparison of the relative complexity between any two finite strings.
Further the new metric of aesthetic goodness ensures that the relevant
program structure is preserved between state transitions and thus
maintains a stable goal system."

Cheers

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[Abram Demski]

Marc,

This looks fun! Do you plan on actually writing the paper? I
appreciate seeing that you are attempting a unified assault on the
problem of cognition from an abstract vantage point.

However, I am unsure about the bit concerning overcoming Godelian
limitations. Switching to relative complexity comparisons does not
obviously solve anything, at least to me. You will have to spell this
out in the paper! :) Along with everything else, like your notion of
analogy and so on.

--abram

On Wed, Jun 17, 2009 at 8:51 PM, marc.geddes<marc.geddes@...> wrote:

>
> I tripped over my own boot-laces and stumbled – whacking my head.
> Bent over to rub my head and retie the damn laces, all a sudden I had
> the following quick rough thoughts for a paper:
>
> 'Cognitive Reflectivity'
> Marc Geddes
> Melbourne, Australia
> 18th June, 2009
>
> Abstract
>
>
> "A change in the goal-system of an agent is equivalent to a change in
> the way in which knowledge is represented by the agent.  It follows
> that it is equivalent to a change in the complexity of the program
> representing the agent.  Thus we require a method of comparing the
> complexity of strings in order to ensure that relevant program
> structure is preserved with state transitions over time.  Standard
> probability theory cannot be used because; (1) Consistent probability
> calculations require implicit universal generalizations, but a
> universal measure of the complexity of  finite strings is a logical
> impossibility (fromGodel, Lob theorems); and (2) Standard measures of
> complexity (e.g Kolmogorov complexity) from information theory deal
> only with one aspect of information (i.e.  Shannon information), and
> fail to consider semantic content. The solution must resolve both
> these problems.
>
> Regarding (2) the solution is as follows:, information theory is
> generalized to deal with the actual meaning of information  (i.e . the
> semantics of Shannon information) .The generalized definition of the
> complexity of a finite string is based on the conceptual clustering of
> semantic categories specifying the knowledge a string represents.  The
> generation of hierarchical category structures representing the
> knowledge in a string is also associated with a generalization of
> Occam’s razor.   The justification for Occam’s razor and the problem
> of priors in induction is resolved by defining ‘utility’ in terms of
> ‘aesthetic goodness’, which is the degree of integration of different
> concept hierarchies.  This considers the process through which a
> theory is generated;  it is a form of process-oriented evaluation.
>
> Regarding (1); The Godel limitation is bypassed by using relative
> complexity measures of pairs of strings .  This requires generalizing
> standard Bayesian induction ; in fact induction is merely a special
> case of a new form of case-based reasoning (analogical reasoning) .
> Analogical reasoning can be formalized by utilizing concepts from
> category theory to implement prototype theory, where mathematical
> categories are regarded as semantic categories. Semantic concepts
> representing the knowledge encoded in strings can be considered to
> reside in multi-dimensional feature space, and this enables mappings
> between concepts; such mappings are defined by functors representing
> conceptual distance; this gives a formal definition of an analogy.
> The reason this overcomes the Godel limitation and is more general
> than induction is because it always enables relative comparisons of
> the complexity of pairs of strings. This is because case-based
> reasoning depends only on the specific details on the strings being
> compared, whereas induction makes implicit universal generalizations,
> and thus fails.
>
> To summarize:  Induction is shown to be merely a special case of a new
> type of generalized case-based (analogical) reasoning. Concepts from
> category theory enable a formal definition of an analogy, which is
> based on the notion of conceptual distance between concepts. The
> notion of complexity is generalized to deal with semantics, where the
> information in a string is considered to be a concept hierarchy. This
> enables comparisons between pairs of strings; relations between
> strings are defined in terms of the mappings between concepts, and the
> mapping is evaluated in terms of its aesthetic goodness.  Godelian
> limitations are overcome, since analogical reasoning always enables a
> comparison of the relative complexity between any two finite strings.
> Further the new metric of aesthetic goodness ensures that the relevant
> program structure is preserved between state transitions and thus
> maintains a stable goal system."
>
> Cheers
>
> >
>

--
Abram Demski
http://dragonlogic-ai.blogspot.com/

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[mjgeddes]

On Jun 19, 2:44 am, Abram Demski <abramdem...@...> wrote:
> Marc,
>
> This looks fun! Do you plan on actually writing the paper? I
> appreciate seeing that you are attempting a unified assault on the
> problem of cognition from an abstract vantage point.

Er.. I'll test out the ideas first by implementing them.  If correct,
the results should be fairly dramatic and obvious.
>
> However, I am unsure about the bit concerning overcoming Godelian
> limitations. Switching to relative complexity comparisons does not
> obviously solve anything, at least to me. You will have to spell this
> out in the paper! :) Along with everything else, like your notion of
> analogy and so on.
>
> --abram
>

Well, my key idea is not relative complexity measures, but the idea
that analogical reasoning is actually more general than Bayesian
induction.  I think case-based reasoning can overcome Godel
limitations, since case-based reasoning is not trying to make
universal generalizations (which is where I think Bayes run into
trouble).


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