From: rh@smds.com (Richard Harter)
Newsgroups: talk.origins
Subject: Genome Space
Date: 28 Jan 93 08:08:24 GMT
Reply-To: rh@ishmael.UUCP (Richard Harter)
Organization: Software Maintenance & Development Systems, Inc.

Jon Livesey and I have been conducting an interesting interchange
here.  Before going through another series of multiple quotations
ad nauseum, I am going to do a background piece.

			GENONE SPACE

I am going to specify a simple mathematical model to make some of
the arguments less ambiguous.  Consider a genome as a sequence of
genes [simplification alert].  Assign to each possible gene a distinct
index number.  A particular genome is specified by the index numbers
of the genes in the genome.  We can treat this as a phase space where
each index is a different dimension.  Values in this space are binary
(genes are present or absent).  [Note: in this formulation alleles are
different genes.]  Any particular genome corresponds to a point in genome
space; the points of the space represent all possible genomes.

Now consider any particular genome.  Either (a) it specifies an
organism that is not viable in any environemnt, or one that is viable
in some environment.  Call the set V of all points in genome space that
correspond to potentially viable organisms the viability subspace.

For any point P in genome space there is a set of points which represent
possible mates for the organism specified by point P.  Call this set
M(P).  Consider a point Q in M(P).  For each point X in genome space
there is some probability p(P,Q,X) that the mating of P and Q will
produce the organism specified by X.  Define the accessibility a(P,X) of
X from P as the maximum value of p(P,Q,X) for Q in M(P).

All of these probabilities are quite small but the range is enormous.
We can specify a threshold T below which the likelihood of X being an
offspring of P corresponds to a miracle rather than to normal biological
processes.  Let A(P) be the set of X such that a(P,X)>T and X is in
the viability subspace.  Then A(P) is the set of all naturally possible
potentially viable offspring of P.

We are now ready to divvy up viability subspace.  Let P be any point
in V.  Form the closure D(P) of P as the smallest set of points such
that P is in D(P) and, for any point R in D(P), A(R) is a subset of
D(P).  In other words, D(P) is the set of all possible descendents of
P.  Now form the full closure E(P) of P as the smallest set of points
such that P is in E(P) and for any two points P1 and P2, if P2 is in
D(P1) and P2 is in E(P) then P1 is in E(P).  In other words E(P) is
all possible ancestors of P and all of their possible descendents.

Now I claim that the operation of performing full closures partitions
viability space, i.e. that viability space V is divided into disjoint
subsets, V0, V1,... [There may be only one such subspace.]  Call these
disjoint subsets islands of evolutionary feasibility.

Now the point of all of this rigamarole is simply to make it possible
to have some modest amount of rigor and clarity.  We are now ready
to state a couple of possibilities.

1.	(Common Descent)

a)	All currently extant organisms belong to the same island of
evolutionary feasibility.

b)	There is an ancestral organism a0 such that all extant organisms
are in D(a0), i.e common descent is feasible.

c)	Common descent is a fact.

Possibility (1) summarizes our current understanding of Evolution.

2.	(Doctrine of Pre-existent Kinds)

a)	Currently extant organisms belong to multiple islands of 
evolutionary feasibility.

b)	These islands are small.  That is the maximal distance between
points in a given island is small compared to the distance between
islands.

Possibility (2) summarizes a strong version of the doctrine of kinds.
Some creationists assert this doctrine; some others raise the
possibility.

3.	(Current disjunction)

a)	Currently extant organisms can be divided using D(P) as the
basis for partitioning.  In other words, current organisms can be
divided into kinds, such that organisms in different kinds can never
have common descendents.

I claim that current disjunction is consistent with either (1) or (2).

-- 
Richard Harter: SMDS Inc.  Net address: rh@smds.com Phone: 508-369-7398 
US Mail: SMDS Inc., PO Box 555, Concord MA 01742.    Fax: 508-369-8272
In the fields of Hell where the grass grows high
Are the graves of dreams allowed to die.