**Kinematic
Self-Replicating Machines**

**©
2004 Robert A. Freitas Jr. and Ralph C. Merkle. All Rights Reserved.**

Robert A. Freitas Jr., Ralph C. Merkle, *Kinematic Self-Replicating Machines*,
Landes Bioscience, Georgetown, TX, 2004.

**5.9.2 Strategies for
Exponential Kinematic Self-Replication**

Fibonacci’s rabbits must wait one time period to mature before they can begin replicating, markedly slowing the population growth rate. But consider a replicator that is a fully functional adult device from birth and can immediately begin the construction of its own daughter without waiting. Such entities may be called “birth-mature replicators.” At the end of the first cycle (n = 1), there are 2 adult devices, the first having constructed the second. At the end of the second cycle (n = 2), there are 4 adult devices, each of the first-cycle devices having constructed their own copies. The series progression, starting from the end of the first replication cycle, is therefore 2, 4, 8, 16, ..., and so the population of birth-mature replicators follows the simple exponential law:

after n cycles have elapsed; P(n) > F_{end}(n)
for all n > 0. Note that the replicator population can be restated as P(t)
in terms of t, the total elapsed time since replication began, and the replication
time t, the time a replicator requires to complete
a replication cycle.

Even faster population growth P_{fast}(n) > P(n)
may be obtained by placing each completed subunit of the daughter that is currently
being built into full productive use immediately, instead of waiting for the
final completion of the daughter device. There are many design configurations
where something close to this can be achieved efficiently, most notably the
factory models (e.g., Sections 3.11, 3.13.2.2,
3.15, 4.9.3, 4.14,
and 5.9.6) wherein more specialized subunits within
the factory (e.g., an individual lathe) could be pressed into immediate service
prior to completion of the entire factory (Freitas and Gilbreath [2],
Section 5.3.5 at p. 218; see also “Operating-Subsystem-in-Process Replication”,
Section 5.1.8), and there has been at least one experimental
demonstration of this by the Chirikjian group (see “Robot 4” in
Section 3.23.2 and Figure
3.75).

Let us consider that the replication cycle is divided into
k time segments, such that at the end of each time segment all components built
during that time segment can begin fully contributing to replicative activities.
For k = 1, the base population of fast replicators p_{fast}(1) = 1 +
1 (replicator plus daughter) = 2 devices, at the end of the n = 1 cycle. For
k = 2, at the end of half a cycle, the original replicator has built half a
daughter, and that half now enters service. By the end of the replication cycle,
the second half of the daughter has been built by the original replicator and
the daughter has also had half a cycle to begin construction of a granddaughter
device, which is now one-quarter completed; hence p_{fast}(1) = 1 +
2·(1/k) + 1·(1/k)^{2} = 2.25 devices. For k = 3, p_{fast}(1)
= 1 + 3·(1/k) + 3·(1/k)^{2} + 1·(1/k)^{3}
= 2.37 devices; for k = 4, p_{fast}(1) = 1 + 4·(1/k) + 6·(1/k)^{2}
+ 4·(1/k)^{3} + 1·(1/k)^{4} = 2.44 devices produced
during the n = 1 replication cycle. Since the coefficients of each term are
horizontal elements of Pascal’s triangle,*
which is a two-dimensional number array constructed by sequential addition much
like Fibonacci numbers, it can be shown that p_{fast}(n) = p_{fast}(n,
1) + p_{fast}(n, 2) + ... is an infinite series with the individual
terms:

whose sum converges to p_{fast}(n) = 1 + 1/(1!) +
1/(2!) + 1/(3!) + 1/(4!) + ... = e = 2.71828...., for n = 1 cycle. Hence:

in the limit as k → ∞, providing another example
of exponential growth. Table
5.4 shows the replicator populations resulting from each of the four modes
of replication discussed so far: P_{fast}(n) > P(n) > F_{end}(n)
> F_{mat}(n, t_{mat}/t)
for n > 3, and after n = 20 replication cycles, [P_{fast}(20) ~ 10^{9}]
>> [P(20) ~ 10^{6}] >> [F_{end}(20) ~ 10^{4}]
> [F_{mat}(20, t_{mat}/t
= 2) ~ 10^{3}]. Clearly, the choice of replication strategy produces
significantly different population growth rates.

* The French mathematician Blaise Pascal described a triangular arrangement of numbers – now called Pascal’s Triangle [2843] – that are useful in algebra and probability theory, in which every number in the interior of the triangle is the sum of the two numbers on either side directly above it:

So far we have considered only replicators that produce a single offspring during each replication cycle. If each replicator can produce D daughters per generation – i.e., a “litter” of D offspring per pregnancy, rather than just a single daughter per pregnancy – then for birth-mature replicators:*

* Star Trek’s fictional “Tribbles” are furry earless rabbits that are said to be “born pregnant” and to produce a litter of D = 10 offspring every 12 hours [2844], so three days of unconstrained replication (n = 6) produces a total population of 1,771,561 animals, using Eqn. 12 for birth-mature replicators.

and for prenatal-active replicators:

after n cycles have elapsed, for D > 0, n > 0. While
D > 1 gives much higher populations at any given number of cycles than for
D = 1, the replication time t will correspondingly
be longer for D > 1 because more mass must be replicated in each cycle. In
the simplest case, making twice as many daughters requires replicating twice
as much mass, which takes twice as much time at a constant production rate,
hence t / t_{0}
~ D, where t_{0} is the replication time
for D = 1 daughter per litter. In this simple case, Eqns.
12 and 13 predict that the population-maximizing
strategy is to minimize the number of daughters, i.e., D = 1 yields the fastest
population growth rate. What if the manufacturing time per daughter can be made
smaller for additional daughters by some technical means (e.g., “cheaper
by the dozen”; see Section 5.9.6)? The population
growth-neutral replication time dependency is t_{neutral}
/ t_{0} = log_{2}(D + 1) for birth-mature
replicators and t_{neutral} / t_{0}
= 1 + log_{e}(D) for prenatal-active replicators. That is, if t
= t_{neutral}, then choosing any number of
daughters per litter gives the same population growth rate. If t
> t_{neutral}, as where t
/ t_{0} ~ D in the simplest case above, then
minimizing the number of daughters per litter maximizes the population growth
rate. If t < t_{neutral},
then maximizing the number of daughters per litter maximizes the population
growth rate. Tradeoffs among these variables should be more systematically investigated.

In the case of the Merkle Cased Hydrocarbon Assembler (Section
4.10.2) where the parental birth-mature replicator is destroyed after each
generation is completed, from Eqn. 12A the replicator
population is only P(n) = D^{n} after n generations with D daughters
per generation. If a constant time is assumed to be required to assemble each
daughter machine and if there is no lag time between completing one cycle and
beginning the next one, then the total number of replication cycles N = nD,
hence P(n) = D^{N/D}. Taking the derivative with respect to D using
the general power rule for differentiation, (∂/∂D)P(n) = (1 –
ln(D)) · D^{((1/D)-2)} which goes to zero when the replicator
population P(n) is maximized. For D > 0, then (∂/∂D)P(n) = 0
occurs when 1 – ln(D) = 0; that is, when D = e = 2.718... ~ 3, the optimum
number of daughters. (Thanks to C. Phoenix for inspiring this analysis.)

In the case of birth-immature replicators (e.g., Fibonacci
rabbits or human beings), Darbro and von Tiesenhausen [1090]
report that if these replicators are allowed to produce D = 2 pairs of offspring
per replication cycle instead of just D = 1 pair of offspring, and keeping t_{mat}/t
= 1, then the original “rabbit formula” (Eqn.
4) for total rabbit-pair population F_{end}(n, D) after n replicative
cycles becomes:

For birth-immature replicators with D > 2, “some
similar expressions may be derived [but] they are not so compact” [1090];
some formulations have previously been addressed by Hoggatt and Lind [2845].
The authors are unaware of any similar calculations for extended-maturation
replicators (t_{mat}/t
> 1) with D > 1 that have been published.

Interestingly, a similar set of formulas applies in the financial
world, wherein the growth of a given quantity of money is dependent upon the
interest rate, “discount” rate, or rate of return. For example,
if an initial quantity of money $_{0} earns an annual interest rate
of i and is allowed to compound m_{$} ≥ 1 times every year, then
the quantity of money after a period of Y years is given by $(n) = $_{0}
· (1 + i/m_{$})^{n} where n = m_{$}Y is the number
of compounding periods elapsed. In the case of a 100% annual interest rate,
the original money doubles or “replicates” itself in one year if
compounded only once (m_{$} = 1), so i = 1.00 and $(n) = $_{0}
· (1 + (1/m_{$}))^{n} = $_{0} · 2^{n}
which is equivalent to Eqn. 10 for birth-mature
replicators.* In the limit as compounding becomes continuous (m_{$}
→ ∞), then $(n) = $_{0} · e^{n} which is
equivalent to Eqn. 11 for prenatal-active replicators.
In a properly functioning capitalist economy, money acts much like a self-replicating
entity, with an initial invested amount producing more of itself over time.

* According to one version [2988-2990]
of an ancient parable on compound interest, after a wise man performs a useful
service for a king in India and the king insists that the man name a reward,
the wise man agrees to take one grain of rice for the first square on the king’s
chessboard on the first day, two grains for the second square on the second
day, and so on, doubling the amount daily for all 64 squares of the chessboard.
The king, too proud to admit his inability to calculate the sum total of the
gift, foolishly grants the wish, at least until it becomes clear that it will
wipe out the royal granary (he has committed to deliver 2^{64} = 18,446,744,073,709,551,616
grains of rice or ~1 trillion metric tons, or 2600 times the world production
of milled rice in 2002 [2991]). Finally,
despite his pride the king takes back his repayment, justly embarrassed because
of his stupidity and the wise man’s obvious generosity in not wanting
repayment in the first place. Other versions of this folktale have similar events
occurring in China [2992], Persia, or Japan,
and there are darker accounts in which the king is outsmarted by a peasant who
is ordered killed when the king discovers his own error.

Last updated on 1 August 2005