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:
18th June, 2009
"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
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."
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