Document: John von Neumann and the Evolutionary Growth of Complexity

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5 Evolvability: A Closer Look

The set of von Neumann self-reproducers anchored on a single $u_0$ have precisely this in common: they process the same formal ``genetic language'' for describing machines. In biological terms we may say that this set incorporates a fixed, or absolute mapping between genotype (description tape) and phenotype (self-reproducing automaton).

Thus, in committing ourselves (following von Neumann) to solving the problem of the evolutionary growth of complexity purely within the resources of a single such set, we are also committing ourselves to the equivalent of what I call Genetic Absolutism [13, Section 5.3]. I should note that, in the latter paper, I argue at length against the idea of Genetic Absolutism; but not in the sense that it is ``bad'' in itself--it just is not a tenable theory of biological evolution. Now von Neumann is not yet trying to capture all the complications of biological evolution: he is merely trying to establish that some key features, at least, can be recreated in a formal, or artificial, system. If this can be done within what is, in effect, a framework of Genetic Absolutism, and if there is some advantage to doing this in that particular way, then the fact that it is still ``unbiological'' (in this specific respect) should not be held too severely against it. (Indeed, there are arguably much more severe discrepancies than this in any case.)

Now, as it happens, adopting Genetic Absolutism does have a significant advantage for von Neumann. Working within such a framework it is necessary (for the solution of von Neumann's problem) to exhibit one core general constuctive automaton, $u_0$; and it is necessary to establish that this is sufficiently powerful to satisfy the informal requirements of the evolutionary growth of complexity; and it is finally necessary to show that, based on the formal genetic language processed by $u_0$, there is a reasonable likelihood that most, if not all, of the corresponding self-reproducers will be directly or indirectly connected under mutation. But if all this can be done, then the problem immediately at issue for von Neumann can, indeed be solved.

But the key point is that even though this may suffice for von Neumann's immediate purposes, nonetheless his framework is actually capable of going well beyond this; and I will claim that there may be advantages in doing so.

As I indicated in the previous section, the alternative to Genetic Absolutism is Genetic Relativism [13, Section 5.4], which envisages that the mapping between genotype (description tape) and phenotype (self-reproducing automaton) is not fixed or absolute but may vary from one organism (automaton) to another.

If we tackle von Neumann's problem in a framework of Genetic Relativism, we do not restrict attention to a single $u_0$, giving rise to an ``homogenous'' set of self-reproducers, all sharing the same genetic language. Instead we introduce the possibility of having many different core automata--$u_0^1$, $u_0^2$, etc. Each of these will process a more or less different genetic language, and will thus give rise to its own unique set of related self-reproducers. We must still establish that most if not all self-reproducers in each such set are connected under mutation; but, in addition, we must try to show that there are at least some mutational connections between the different such sets (in order to establish pathways for evolution of the mapping itself).

The latter is, of course, a much more difficult task, because the mutations in question are now associated with changes in the very languages used to decode the description tapes. But, if such connections could be established, then, for the purposes of solving von Neumann's problem we are no longer restricted to considering the range of complexities of any single von Neumman set of self-reproducers (i.e., anchored on a single $u_0$, with a common description language), but can instead consider the union of many--indeed a potential infinity--of such sets.

Now clearly, in terms simply of solving von Neumann's problem, Genetic Relativism introduces severe complications which are not necessary, or even strictly useful. For now we have to exhibit not one, but multiple core general constructive automata, processing not one, but multiple genetic languages; and we have to characterise the range of complexity, and mutational connectivity, of not one but multiple sets of self-reproducers; and finally, we still have to establish the existence of mutational links between these different sets of self-reproducers. At face value, the only benefit in this approach seems to be the rather weak one that maybe--just maybe--the distinct general constructive automata can be, individually, significantly simpler or less powerful than the single one required under Genetic Absolutism; but it seems quite unlikely that this could outweigh the additional complications.

Let me say then that I actually accept all this: that for the solution of von Neumann's problem, as I have stated it, adopting the framework of Genetic Absolutism seems to be quite the simplest and most efficacious approach, and I endorse it as such. Nonetheless, I think it worthwhile to point out the possibility of working in the alternative framework of Genetic Relativism for a number of distinct reasons.

Firstly, it would be easy, otherwise, to mistake what is merely a pragmatic preference for using Genetic Absolutism in solving von Neumann's problem with the minimum of effort, for a claim that Genetic Absolutism is, in some sense, necessary for the solution of this problem. It is not. More generally, our chosen problem is only concerned with what may be possible, or sufficient--not what is necessary.

A second closely related point is this: prima facie, our solution based on Genetic Absolutism may seem to imply that a general constructive automaton (i.e., capable of constructing a very wide range of target machines) is a pre-requisite to any evolutionary growth of complexity. It is not. Indeed, we may say that, if such an implication were present, we should probably have to regard our solution as defective, for it would entirely beg the question of how such a relatively complex entity as $u_0$ (or something fairly close to it) could arise in the first place. Conversely, once we recognise the possibility of evolution within the framework of Genetic Relativism, we can at least see how such prior elaboration of the powers of the constructive automata could occur ``in principle''; this insight remains valid, at least as a coherent conjecture, even if we have not demonstrated it in operation. This has a possible advantage in relation to the solution of von Neumann's problem in that it may permit us to work, initially at least, with significantly more primitive constructive automata as the bases of our self-reproducers.

Thirdly, Genetic Absolutism views all the self-reproducers under investigation as connected by a single ``genetic network'' of mutational changes. This is sufficient to solve von Neumann's problem, as stated, which called only for exhibiting the possibility of mutational growth of complexity. In practice, however, we are interested in this as a basis for a Darwinian growth of complexity. Roughly speaking, this can only occur, if at all, along paths in the genetic network which lead ``uphill'' in terms of ``fitness''. If the genetic network is fixed then this may impose severe limits on the practical paths of Darwinian evolution (and thus on the practical growth of complexity). Again, once we recognise the possibility of evolution within a framework of Genetic Relativism--which offers the possibility, in effect, of changing, or jumping between, different genetic networks--the practical possibilities for the (Darwinian) growth of complexity are evidently greatly increased.

This last point represents a quite different reason for favouring the framework (or perhaps we may now say ``research programme'') of Genetic Relativism, and it is independent of the ``power'' of particular core constructive automata. In particular, even if we can exhibit a single full blown general constructive automaton, which yields a mutationally connected set of self-reproducers spanning (virtually) every possible behavior supported in the system, there could still be advantages, from the point of view of supporting Darwinian evolution, in identifying alternative constructive automata, defining alternative genetic networks (viewed now as evolutionarily accessible pathways through the space of possible automaton behaviors).

Indeed, this need not be all that difficult to do: it provides a particular reason to consider combining a basic constructive automaton with a turing machine (or something of similar computational powers): the latter is arranged so that it ``pre-processes'' the description tape in some (turing computable) fashion. The program of the turing machine could then effectively encode a space of alternative genetic languages (subject to the primitive constructional abilities of the original constructive automaton); with moderately careful design, it should be possible to open up an essentially infinite set of constructive automata, which are themselves connected under mutation (of the program for the embedded turing machine--another tape of some sort), thus permitting a multitude of different genetic networks for potential exploitation by a Darwinian evolutionary process. This should greatly enhance the possibilities for Darwinian evolution of any sort, and thus, in turn, for evolution involving the growth of complexity.

This particular idea seems to have been anticipated by Codd:

A further special case of interest is that in which both a universal computer and a universal constructor [sic] exist and the set of all tapes required by the universal constructor is included in the Turing domain $T$. For in this case it is possible to present in coded form the specifications of configurations to be constructed and have the universal computer decode these specifications ...Then the universal constructor can implement the decoded specifications. Codd [6, pp. 13-14]

While Codd did not elaborate on why such flexibility in ``coding'' should be of any special interest, it seems plausible that he had in mind precisely the possibility of opening up alternative genetic networks.

A final consideration here is the ``compositionality'' of the genetic mapping or language. When tackling von Neumann's problem within the framework of Genetic Absolutism, it was necessary to assume a degree of compositionality in the genetic language, to assure that there would exist a range of mutations not affecting the core constructive automaton in a self-reproducer; without this assumption it would be difficult, if not impossible, to argue that the set of self-reproducers anchored on this single core general constructive automaton would be connected under mutation. This compositionality assumption is more or less equivalent to the biological hypothesis of Genetic Atomism, which holds that genomes may be systematically decomposed into distinct genes which, individually, have absolute effects on phenotypic characteristics (see McMullin [13, p. 11], Dawkins [7, p. 271]). This again represents a divergence between von Neumann's pragmatically convenient solution schema for his particular problem, and the realities of the biological world (where any simple Genetic Atomism is quite untenable). I conjecture therefore that, should we wish to move away from a strict Genetic Absolutism in our formal or artificial systems we might well find it useful, if not essential, to abandon simple compositionality in our genetic language(s) (i.e., Genetic Atomism) also. This, in turn, would ultimately lead away from self-reproducer architectures in which there is any simple or neat division between the core constructive automaton and the rest of the automaton (though there might still be a fairly strict separation of the description tape from the rest of the machine--i.e., a genotype/phenotype division).

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Document: John von Neumann and the Evolutionary Growth of Complexity

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Timestamp: 2002-11-07