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C00002 00002	parame[s84,jmc]		Parametrizing data bases
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parame[s84,jmc]		Parametrizing data bases

	We often need to conclude that one subject is irrelevant to others.
We need to do this in order to restrict what facts we take into account
in solving some problem.  Here is one way it might be done.

Propositional calculus -

	Suppose we have an axiom involving a fairly large number
of propositional variables which represents what we know about some
subject.  It can happen that these variables are tightly connected
to one another, so there is no way of changing the values of
one or a few of the variables
without changing most of the others and still keep the axiom true.
At the other extreme, it may be that the variables divide into
groups, and the axiom is a conjunction of formulas, each involving
the variables of only one group.  Another possibility is that we
we have a master group and a dependent group.  There is no restriction
on the variables of the master group, but then the values of the
dependent group are determined.  Less favorable: the master variables
can be chosen independently, but then the dependent variables have
some freedom left.  What freedom depends on the values of the
master variables?

	It is also possible that there are no independent sets of
variables as it stands, but a "change of co-ordinates" can be made
such that in the new co-ordinates, the states are more independently
parametrized.

	What is known about this?

	Is Belnap's relevance logic relevant to this?

1985 Feb 23

	Parametrization has a big potential role in AI.  The normal form
for monadic formulas ( monadi.tex[w85,jmc] ) gives a parametrization of all
monadic formulas as propositional combinations of the base formulas which
may be chosen independently.  It may be possible to develop moderately
general parametrization programs that will reduce search time by
parametrizing a space of possibilities that otherwise might have to be
searched.  Whenever we have a constraint, there is a hope for a
parametrization that will generate exactly those objects that satisfy the
constraint.

	A parametrizations may be the Cartesian of simpler parametrizations.
Basic parametrizations include propositional variables, variables ranging
over the natural numbers, the integers, the rationals, the reals or other
standard mathematical domain.  The set of sequences or the set of
S-expressions also occur.  The set of permutations.  The set of maps
from one space to another.

	Spaces that do not have good parametrizations need to be
identified.  The set of non-self-intersecting paths comes to mind.
It would be nice to have it for search graphs where previously
examined states are not of interest.  Here the breath first
search that successively generates all nodes of distance $n$ from
the origin plays an interesting mathematical role.  A pair consisting
of a non-self-intersecting path and a set of forbidden vertices
is more parametrizable.