> On an unrelated vein, I wonder what Kauffman is going to say about the
> number of human genes. Anyone has "Origins of Order" handy? I forgot the
> number he forecasts.
Do you recall the context in "Origins of Order"? On a cursory
examination, best I found was p 461, ch12, "Differentiation":
> Thus for yeast, the estimated number of cell types is 3 and the number
> of transcribed genes is 4000. In Drosophila, as in annelids, the
> number of cell types might be about 60 and the transcribed complexity
> about 17 000. In humans the number of cell types is 200 to 300 and the
> transcribed complexity is about 100 000. Thus on the basis of measured
> complexity of RNA sequences, the number of cell types increases roughly
> linearly with, or perhaps a bit faster than, the number of genes.
> It certainly does not increase much more rapidly than linearly.
> That is, the number of cell types does not increase as the number of
> genes squared or as an exponential function of the number of genes.
-Dave
To touch upon a vein related to other recent
FoRKposting, from p398, ch10, "Random Grammars":
> Fourth, several features of these models invite modifications of the
> core of neoclassical economics, which rests on general competetive
> equilibrium theory. This beautiful theory demands infinitely
> rational economic agents and what are called "complete markets."
> Such markets allow exchange of all possible dated contingent goods.
> An example would be a contract to deliver apples tomorrow if it
> rains in Nebraska today. In the absence of infinite rationality
> and complete markets, we see failure of the beautiful theorems
> guaranteeing an equilibrium such that all markets clear. If we take
> grammar models seriously as models of technological evolution, then
> the decidability problems in Filigreed Fogs and other objects imply
> that it may be logically impossible to deduce that a given good is
> ultimately producible from the current technologies. This implies
> that markets must be *incomplete*. The same failure of decidability
> may imply that economic agents must logically be bounded rational.
> Both these latter two points cut at the core of neoclassical economics
> and hence may invite its extension.
>
> Fifth, grammar models hold the hint of that extension, a hint which
> suggests the possibility of a theory of bounded rationality and a
> new nonequilibrium solution concept: markets may not clear. Rather,
> boundedly rational economic agents may achieve the edge of chaos,
> where markets come close to clearing. At the edge, bankruptcies, like
> extinctions, would propagate through the economy from time to time.
Not only does a Filigreed Fog (percolating neutral
filaments of large extension) mean that mutants can
go places at little cost, but neutral (and even
slightly deleterious) mutations can be ubiquitous
in a population. "All your base", "Kilroy", etc.,
like bankruptcies and extinctions, would propagate
through the memescape from time to time.
Appended are some old mails of mine in response to
a kragen-hacks post in which Kragen was generating
random lambda calculus terms, and wondering under
what conditions his recursion wouldn't terminate,
which turns out to be the same problem as above.
::::::::::::::::::::::::::::::::
> - I don't understand the conditions under which the random_expr code has
> a finite probability of infinite recursion.
Neither do I, quite, but I think I understand enough to point out
where your reasoning below fails. This is an interesting class of
problems, because even in cases where a generator terminates with
probability 1, it will generate arbitrarily long results --
effectively infinite in the short run.
> # The probability of the lambda production terminating is the same as
> # the probability of random_expr terminating.
> # The probability of the app production terminating is the square of the
> # probability of random_expr terminating.
These are the problematic statements, the correct model isn't quite
so simple and involves the limits of sequences which I don't
understand well enough yet to explain. Places to go look for
presentations include:
If you are a telecomms geek, treat it as a queueing theory problem:
an app production adds to the number of existing terms, a varref
diminishes.
If your inheritance is tied up in entail (is such a thing still a
legal form of land tenure?), treat it as the problem of "what is the
probability that a male line will go extinct?" and model varrefs as
daughters, lambda productions as heirs, and the extra terms from the
app productions as cadets.
If you are public health official, treat it as the problem of "given
an epidemic of lambda calculus, how likely is it to die out?".
If you are a population geneticist, it is similar to the problem of
"what are the chances that a neutral, or slightly deleterious,
genotype will become fixed in a population?".
If you are looking to IPO, or happen to be the sort of manager who
gets paid very well for positive outcomes, but does not lose in any
absolute sense for negative outcomes*, you may wish to look at the
other side of the problem: "how many dollars does the non-bankrupt
company have in the Nth quarter?" (the suprising answer here is
that when bankruptcy is common in a field, the surviving firms can
be awfully large)
A less cynical economic view can be taken of growth companies:
current growth produces the options for future growth, or so both
rational whigs and wild-eyed speculators tell us, but it obviously
can't go on forever, or present values would all be infinite and
sumerian investment bankers would have a corner on the market.
Anyway, treat your varrefs as failed speculations, lambda
productions as business as usual, and app productions as successful
investments, and you can model over how many years you may expect a
growth firm to exceed equilibrium. Of course, you can also use the
same expression to model over how many bets you may expect a winning
gambler's "hot streak" to last.
-Dave
* say I run a hedge fund, and I tell you, my client, not to worry --
I won't charge you a cent if I go negative, but I will take a 20%
"incentive fee" out of any profits. Why should you be worried that
I am set up for a fairly certain 1000 basis points?
::::::::::::::::::::::::::::::::::::
> # Say X for P(random_expr terminating), V for P(varref), L for
> # P(lambda), and A for P(app). Then
> # X = V + LX + AX^2
> # So it's just a plain old quadratic equation; result is (-b +- sqrt(b^2
> # - 4ac)) / 2a, or (-L +- sqrt(L^2 - 4 * A * V))/2A. In this case,
> # that's (-0.5 +- sqrt(0.25 - 4 * 0.2 * 0.3)/ (2 * 0.2)
> # Unfortunately, this comes out to -1.5 and -1, which are obviously
> # wrong. So I made a mistake somewhere in here.
My previous post was wrong; Kragen's quadratic term is correct. The
reason it doesn't have any roots between 0 and 1 is that (thankfully
for anyone executing the program) the expected number of recursions
to random_expr is .9 (0*.3 + 1*.5 + 2*.2), so the recursion should
terminate with probability 1.
If we boost the expectation for recursions, say by increasing the
probability of applications:
return new app (random_expr, random_expr) if ($r < 0.61);
Then we get the finite chance of non-termination; in this case a
solution for X = V + LX + AX^2 occurs somewhere around .97, and
there is ~3% chance of generating infinitely many terms.
Despite having that extra percentage edge which allows random_expr
to blow up in unusual cases, note that half of all calls to
random_expr would generate terms that are nested less than 3 deep.
We're a long way from bell curves when we play with long-tailed
distributions like these.
-Dave
Koerner, in The Pleasures of Counting (not, alas, The Joy of x), has
a chapter devoted to this topic, titled, Ecclesiastically enough,
"Time and Chance". In a sidebar, he mentions:
> Social Darwinism applies the Darwinian doctrine of the survival of
> the fittest to human society. Rich social darwinists take wealth as
> the best indication of fitness to survive, academic social
> darwinists take intellectual achievements as the best indication and
> so on. They are often haunted by the fear that the unfit do not
> understand this and may outbreed the fit.
This archive was generated by hypermail 2b29 : Sun Apr 29 2001 - 20:26:07 PDT