Artificial Life VII
August 1-6, 2000, Portland, Oregon.
... This result is obviously substantial, but to express its real force we must formulate it in such a way that it cannot be trivialized...--Burks (1970a, p. 49)
Arthur Burks makes this comment following a 46 page recapitulation of John von Neumann's design for a self-reproducing machine (realised as a configuration embedded in a 29-state, two dimensional, cellular automaton or CA). The comment can be fairly said to have initiated an extended debate about the significance of this work of von Neumann's, which has waxed and waned over time, but still persists to the present day.
Von Neumann's design is large and complex, and relies for its operation on exact and intricate interactions between the many relatively simple parts. In that sense, it is certainly substantial; but Burks is absolutely accurate in pointing out that this intricacy, in itself, is not necessarily interesting or significant. In particular, if the same ``results'' could be achieved, or the same ``problem'' solved, with drastically simpler machinery, then the interest of von Neumann's design would be critically undermined.
This is no idle concern on Burks' part. As he himself points out, within the CA framework, one can easily formulate a simple rule whereby a cell in a distinct state (labelled, say, 1) will cause adjacent quiescent cells (labelled, say, 0) to transit to state 1 also. By essentially the same definition or criterion as was applied to von Neumann's system, such a single cell, state 1, configuration, would qualify as a self-reproducing machine--and would seem to render von Neumann's fabulously baroque design completely redundant.
Burks concluded that ``... what is needed is a requirement that the self-reproducing automaton have some minimal complexity.'' And at first sight, this does seem eminently reasonable. Presumably (?) it is relatively easy to construct a ``simple'' machine, by whatever means; therefore it need not surprise us unduly if we manage to concoct a ``simple'' machine which can construct other ``simple'' machines, including ones ``like'' itself, and which therefore qualifies as self-reproducing. Whereas, it is relatively difficult to construct a ``complex'' machine, by any means; and therefore it may well be a challenging problem to exhibit a ``complex'' machine that is capable of self-reproduction. Von Neumann's machine certainly appears ``complex'', and certainly succeeds in constructing other machines like itself (i.e., in reproducing itself). So if we could just more formally express the precise sense in which von Neumann's machine is ``complex'', then we might indeed be able to clarify the ``real force'' of his achievement.
However, even at this point, we should be at least somewhat wary--because, while von Neumann himself certainly did introduce and discuss the notion of ``complexity'' in relation to this work, he did not attempt any formalisation of it. Indeed, he described his own concept of complexity as ``vague, unscientific and imperfect'' (von Neumann, 1949, p. 78). It would therefore seem unlikely that the significance of his eventual results should actually rely on such a formalisation.
Nonetheless, Burks went on to propose just such a formal criterion of complexity--namely the ability to carry out universal computation. And by this criterion, von Neumann's design (or, at least, a straightforward derivative of it) would qualify as a ``complex'' self-reproducing machine, and thus be clearly distinguished from the single cell, ``1 state self-reproducer,'' which would remain merely a ``simple'' (and thus trivial) self-reproducing machine.
This seems a reasonable enough suggestion by Burks; though I would emphasise again that, as far as I have been able to establish, such a thing was never proposed by von Neumann himself--and, indeed, it jars seriously with von Neumann's calculated refusal to formalise complexity.
In any case, it turns out that Burks' suggestion is unsatisfactory and unsustainable. While it is true that von Neumann's machine (suitably formulated) can satisfy Burks' criterion for ``complex'' self-reproduction, this still represents an interesting result only if this criterion cannot be satisfied by very much simpler means. But in fact--and with hindsight, this now seems unsurprising--Burks' criterion can be satisfied by much simpler means than those deployed by von Neumann. This is because universal computation, per se, does not actually require particularly complex machinery (see, for example, Minsky, 1967).
This fact was formally demonstrated by Herman (1973), when he essentially showed how the basic single cell, 1 state, self-reproducer described earlier, could be combined with a single cell capable of universal computation. This results in a CA system in which the individual cells are ``simpler'' than in von Neumann's CA (i.e., have fewer states), and yet there are single cell configurations capable of both self-reproduction and universal computation. Granted, the universal computation ability relies on an adjacent, indefinitely long, ``tape'' configuration; but that was equally true of the universal computation ability of von Neumann's design, and is not a relevant distinguishing factor.
Herman draws the following conclusion (Herman, 1973, p. 62):
...the existence of a self-reproducing universal computer-constructor in itself is not relevant to the problem of biological and machine self-reproduction. Hence, there is a need for new mathematical conditions to insure non-trivial self-reproduction.
So we see that Herman rejects Burks' specific criterion, while still continuing to accept Burks' formulation of the issue at stake--namely the identification of a suitable criterion for distinguishing ``non-trivial'' self-reproduction, albeit in Herman's version this is no longer explicitly tied to the notion of ``complexity''.
The discussion was taken up again by Langton (1984) (though apparently without reference to Herman's work). He presented a rather different analysis of Burks' criterion, but with ultimately complementary results. Langton pointed out that, as a general principle, there is little evidence to suggest that living organisms contain universal computation devices embedded within them. Since the self-reproduction of living organisms is presumably to be regarded as non-trivial (by definition-at least in this context), we should not, therefore, adopt universal computation as a criterion. So in this respect, albeit for different reasons, Langton concurs with Herman.
More importantly, Langton goes on to suggest a specific alternative criterion. He points out that self-reproduction in living organisms relies on a decomposition of the organism into two parts or components playing very distinct roles in the reproductive process:
Langton does not explain just why such a decomposition may be important, but rather seems to accept its pervasiveness among biological organisms as reason enough to adopt it as a criterion. And, in some sense it ``works'', because, indeed, von Neumann's self-reproducing design does have this architecture, whereas (say) Herman's self-declared ``trivial'' design does not. So it seems like this may be a satisfactory or even illuminating demarcation.
However: Langton did not stop there. With this new criterion in hand, he went on to consider whether it could be satisfied with a design significantly simpler than von Neumann's--and it transpires that it can. In fact, Langton was able to present a design for a CA space which itself is rather simpler than von Neumann's (i.e., having fewer states per cell), into which he could embed a self-reproducing automaton which, like von Neumann's, has an explicit decomposition into genotype and phenotype, but which is vastly smaller and simpler, occupying a region of just 150 cells--compared to the several hundred thousand cells required for von Neumann's device!
Langton's automaton is certainly still quite intricate, and its design involved a subtle interplay between designing the CA itself and designing the automaton to be embedded within it. In this sense it is a significant and substantive achievement. But is remains very unsatisfactory from the point of view of evaluating von Neumann's work. If Langton's criterion for non-trivial self-reproduction is accepted as an appropriate measure to judge von Neumann's work by, then we must conclude that the latter's design is vastly more complex than necessary. While this may be true, I suggest that we should be reluctant to accept it without some much more substantive rationale for Langton's criterion. Or to put it another way, there may still be more ``real force'' to von Neumann's achievement than is captured or implied by Langton's criterion.
I propose to resolve this enigma in a rather different way.
Firstly, I fully agree with Burks that to appreciate the full force of von Neumann's work, we must understand what problem he was attempting to solve. In particular, if it should turn out that this problem can be solved by trivial (Herman) or at least much simpler (Langton) means then we should have to conclude that it was not such a substantial achievement after all. Where I differ from these, and indeed, most other, commentators, is that I think it is a mistake to view von Neumann's problem as having been wholly, or even largely, concerned with self-reproduction!
Of course, this is not to deny that von Neumann did, indeed, present a design for a self-reproducing automaton. I do not dispute that at all. Rather, my claim is that this self-reproducing capability, far from being the object of the design, is actually an incidental--indeed, trivial, though highly serendipitous--corollary of von Neumann's having solved at least some aspects of a far deeper problem.
This deeper problem is what I call the evolutionary growth of complexity. More specifically, the problem of how, in a general and open-ended way, machines can manage to construct other machines more ``complex'' that themselves. For if our best theories of biological evolution are correct, and assuming that biological organisms are, in some sense, ``machines'', then we must hold that such a constructive increase in complexity has happened not just once, but innumerable times in the course of phylogenetic evolution. Note that this claim does not rely on any sophisticated, much less formal, definition of complexity; it requires merely the crudest of qualitative rankings. Nor does it imply any necessary or consistent growth in complexity through evolution, but merely an acceptance that complexity has grown dramatically in some lineages.
Why is this growth of complexity a problem? Well, put simply, all our pragmatic experience of machines and engineering points in the opposite direction. In general, if we want to construct a machine of any given degree of complexity, we use even more complex machinery in its construction. While this is not definitive, it is surely suggestive of a difficulty.
To make all this a little more concrete, imagine that we could exhibit the following:
Now consider the two (highly schematic) graphs shown in Figure 1. In both cases I have shown the putative set partitioned more or less coarsely by the complexity measure . That is, the inner rings or subsets are those of small (low complexity), while the outer, more inclusive rings or subsets include machines of progressively greater (higher complexity). The graph on the left indicates our ``typical'' engineering experience: all constructional pathways lead inward (from more complex to less complex). As a result, complexity will always, and unconditionally, degenerate in time. Conversely, the graph on the right indicates the abstract situation posited by our best current theories of biological evolution: at least some edges of the graph (constructional pathways) lead from the inner rings to their outer neighbors. Provided there are sufficient such pathways, then, starting from only the very simplest machines (or organisms), there will be potential constructional pathways whereby even the most complex of machines can eventually be constructed (in time). Thus complexity might grow in time.1
The problem which this presents is to show that the experience and intuition underlying the graph on the left is mistaken; to show, in other words, how the situation on the right might in fact be realised.
What would it take to solve this problem, even in principle? Well, one would need to exhibit a concrete class of machines , in sufficient detail to satisfy ourselves that they are purely mechanistic; one would need to show that they span a significant range of complexity; and finally, one would have to demonstrate that there are constructional pathways leading from the simplest to the most complex (there may, or may not, also be pathways in the other direction, but that is not the point at issue--or at least, not immediately).
I believe that this is precisely the problem which von Neumann set out to solve. Furthermore, it seems to me that he did, indeed, solve it; and that it is only by seeing his work in this light that its ``real force'' can be understood.
As to the first claim, that this--rather than ``self-reproduction'' per se--was von Neumann's problem, it is certainly relevant that he introduced his work in essentially these terms in both his recorded public presentations of it--at the Hixon Symposium in September 1948 (von Neumann, 1951, p. 312) and at the University of Illinois in December 1949 (von Neumann, 1949, pp. 78-79). Granted, in both cases he did also refer to the issue of self-reproduction. Specifically, he pointed out that self-reproduction is an example of a constructing machine where the output or offspring is just precisely matched in complexity to the parent--complexity neither increasing nor decreasing. But the critical point here--the significance of self-reproduction--is clearly as a special or ``watershed'' case of the more general situation: it is interesting precisely, and only, because it may mark a transition from strictly-degenerating to potentially-growing complexity. Conversely, if we encounter a form of ``self-reproduction'' that is not associated with such a transition, then it will not be of relevance to von Neumann's problem at all.
Indeed, von Neumann (1949, p. 86) made this point even more explicitly, when he actually noted the triviality of self-reproduction in ``growing crystals'' (essentially Burks' 1-state reproducer in a CA). But pace Burks, von Neumann's resolution of this was not at all to impose a criterion of complexity, but rather to stipulate that the reproductive process should be such as to support ``inheritable mutations''. Taken in context, I think this must be interpreted as supporting at least some ``mutations'' where the offspring is of increased complexity--which again returns us squarely to the problem of the evolutionary growth of complexity, rather than self-reproduction.
My second claim--that von Neumann actually solved this problem--may seem more far fetched; but let us take it in steps.
First note that while von Neumann exhibited the design and operation of only one machine in detail--his self-reproducing machine--he consistently pointed out how his CA space could support an indefinite variety of quite arbitrary machine configurations and behaviours. In this sense, he certainly exhibited not just a single particular machine, but a whole class of machines (which he identified via the infinite set of arbitrary, finite, ``initially quiescent'' configurations in his CA). This can correspond to the class in my problem formulation above (though I will refine this somewhat below).
The next question is whether this class of machines spans a significant range of complexity. Given that we are using only the most vague definition of ``complexity'' here, the answer can be at best informal and qualitative. To my knowledge, von Neumann did not quite comment on this explicitly. He did explain at length that in setting up this sort of ``axiomatic'' theory of automata, there is necessarily a degree of arbitrariness in selecting the primitive or atomic parts from which the composite automata will be constructed. And indeed, between 1948 and 1953 he considered a number of alternative approaches before settling on a particular CA formulation (the latter concept having been suggested by Ulam--see Burks, 1966, p. 94). But given the highly complicated (co-ordinated, systematic, purposeful) behaviour of the one machine he did design in detail, it certainly seems to me that if we allow, as von Neumann did, for the construction of machines with indefinitely more parts, in arbitrary configurations, then this set surely might span a sufficient range of complexity to meet our requirements.
This then leaves what seems by far the most intractable aspect of the problem: to show that there are constructional pathways leading from the simplest to the most complex machines. At first sight, it seems hopeless to demand a demonstration of this; firstly because the measure of complexity is so vague; but secondly because it would seem to demand separate analysis or demonstration of the constructional potentialities for most if not all of the elements of the infinite set . But this, it seems to me, was where von Neumann's crucial insight occurred.
Drawing on the capability for a single universal Turing machine to carry out the computations of any Turing machine at all, given a suitable description of that machine, von Neumann recognised that a single ``general constructive automaton'' (von Neumann, 1949, p. 85) might be able to construct ``any'' machine at all (i.e., any element of his specified ), given a ``description tape'' of that target machine. This immediately suggests the possibility that certain such special, ``programmable'', constructor (sub-)systems may enable enormous constructive potentialities--and also, incidentally (?) open up a powerful and generic route toward self-reproducing capability.
Von Neumann exploited this general idea by making a minor, but subtle, modification of his general constructive automaton (in a manner incidentally having no analog or significance in pure computation theory): as well as requiring it to decode a description to produce an offspring, he required that it must also copy the description, attaching this as part of the offspring. As Langton surmised, this combination of decoding and copying potentially has quite dramatic consequences--but the challenge is still to make those consequences explicit.
Let me denote a basic general constructive (decoding and copying) machine by . We require that .2 Let denote a ``description of '', (relative to ). That is, letting denote the composition of and a tape describing an arbitrary , this will result in the construction of an instance of , via decoding of the description, itself composed with a copy of the original tape, i.e., . We write this as:
Since this applies for all it applies to itself and we have:
This is the single self-reproducing automaton, which has been commonly identified as von Neumann's ``result'' or ``achievement''. Certainly at this point we have a case of construction where complexity has been fully preserved--but the question remains whether this gives us an avenue into the case where complexity can grow.
Now note that can be combined or augmented with fairly arbitrary ancillary machinery, while still retaining its general constructive ability. That is, we can say that is still a general constructive automaton for ``almost all'' . The exception would be where in some sense interferes with or disrupts the normal operation of . Discounting those cases, we can say that the existence of a single general constructive automaton, , actually implies the existence of a whole infinite set of related general constructive automata, , where has the form (i.e., all the members of this set share the same ``general constructive sub-system''). This, in turn, implies the existence of a whole infinite set of self-reproductive automata, where each has the form . Furthermore, given that is derived from just by excluding some exceptional or pathological cases (i.e., where the operation of would be disrupted), we can say that , and thus , will still span essentially the same significant range of complexity as itself.
Now this is clearly a much stronger result than merely demonstrating the existence of a single self-reproducing machine. In particular, it indicates the possibility of arbitrarily complex machines that are still capable of self-reproduction. This in itself certainly distinguishes von Neumann's result from that of Langton (1984). Although the operation of Langton's machine can be decomposed into copying and decoding activities, it does not incorporate anything like a ``general constructive automaton''. The decoding capability is extremely impoverished so that, in effect, there is only one description that can be effectively processed--that of the one self-reproducing configuration which Langton exhibited.
But in any case, there is still a further critical step in the analysis of von Neumann's work. The goal is not just to show a class of arbitrarily complex constructors capable of constructing machines of equal complexity (which is what self-reproduction illustrates), but to demonstrate constructional pathways whereby complexity can grow. And it turns out that this further result can also be demonstrated by the set .
To see this, we imagine the possibility of perturbations--``mutations''--to the description tape of a machine . With reasonable assumptions about the description language, it can be arranged that all, or almost all, tape configurations are ``legal'' (i.e., can be decoded to some target machine), and that the tape can be decomposed into a part, , coding for , and a part, , coding for the ancillary machinery, . That being the case, a more or less arbitrary modification of the portion of the tape will result in a legal description of some other machine, . The reproductive cycle will then result, not in self-reproduction, but in a ``mutant'' offspring ; but this is still an element of , and thus self-reproducing in its own right.3
Now these mutations essentially open up additional constructional pathways between the elements of . In particular it is now clear that this can allow incremental, bootstrapping, increases in complexity. In effect, the density of these mutational pathways reflects the combinatorics of the description code, so that we can be virtually guaranteed, without any detailed analysis, that there will be constructional pathways connecting the entire set . These will include, of course, degenerating pathways, where the offspring are less complex; but will also include vast numbers of pathways of increasing complexity.
In this way, finally, we see how von Neumann's detailed design of a single machine implies at least a schematic solution of the generic problem of the evolutionary growth of machine complexity.
While the designation of a distinct field known as Artificial Life is comparatively recent (Langton, 1989), I would argue that, looking backwards, it is clear that von Neumann's work properly defines its origin and inspiration. If the formulation above of von Neumann's problem--and his solution to it--is accepted, then a proper assessment of his contribution to the field he effectively founded presents at least three distinct questions:
My answer to question 1 is clearly an unambiguous ``yes''; and it stands as von Neumann's remarkable achievement both to have formulated this foundational problem and to have actually succeeded in solving it.
Question 2 is slightly more difficult. Given than von Neumann's problem situation has been poorly understood, it follows that subsequent contributors have not generally articulated clearly the relation between their work and this problem.
However, I do not hesitate to say that Thatcher (1970), Codd (1968) and Pesavento (1995) have all offered alternative, or at least significantly refined, solutions. Because these three systems are so very closely related to von Neumann's it follows that they are at least equally satisfactory in solving his problem.
Thatcher provided a somewhat simplified design for the basic general constructive automaton , in the same CA space as defined by von Neumann. Codd defined a significantly simpler CA space, which could still accommodate a functionally equivalent general constructive automaton. Much more recently, Pesavento has demonstrated a substantially simplified design for in a CA space marginally more complicated than von Neumann's; and, further, has actually demonstrated a functioning simulation of this complete .
Conversely, as already indicated, I think we can be clear that Langton's ``self-replicating loop'' system Langton (1984) does not qualify as an alternative solution to von Neumann's. Although it was derived from Codd's earlier system, and although he exhibits a self-reproducing configuration (involving ``copying'' and ``decoding'') this does not embody anything like a ``general constructive automaton'' and therefore has little or no evolutionary potential.
Another possible candidate solution to von Neumann's problem might be that of Berlekamp et al. (1982). There it is claimed that a general constructive automaton, of comparable functionality to von Neumann's , can be realized in Conway's well known ``Game of Life'' (GOL) CA. This arguably takes Codd's (and thus von Neumann's) result to some kind of limiting case, as the GOL cells have the absolute minimum of only 2 states. On the other hand, in contrast to the fully detailed designs for presented by Thatcher, Codd and Pesavento, Berlekamp et. al. provide only a very sketchy outline to indicate the possibility of an equivalent automaton in GOL. Moreover, because GOL is so radically simplified (as a CA) it would not be quite trivial to establish that it supports embedded automata of comparable range of ``complexity'' to those of Von Neumann etc.
This last issue affects the comparative evaluation of certain other systems even more seriously. I have in mind particularly Tierra (Ray, 1992) and related systems.
Tierra is, roughly, a shared memory parallel computer, with fairly simple ``von Neumann style'' processors.4 Ray has exhibited processes (``machines'' or ``automata'' in von Neumann's sense) in this framework which are capable of self-reproduction; and which, moreover, exhibit clear and significant evolutionary change. However, because the Tierra system is so very different in style from von Neumann's (or, indeed, any CA of 2 or more dimensions), it is very difficult to make even informal and qualitative comparisons of automata ``complexity'' between these systems.
I also have another quite different, and perhaps more important, misgiving about whether Tierra should be regarded as offering an equally satisfactory alternative solution to von Neumann's problem.
I originally expressed this reservation by stating that Tierra uses ``... a form of self-reproduction based on self-inspection (rather than a properly genetic system in the von Neumann sense)'' (McMullin, 1992, Chapter 4). The implication was that the self-reproducing entities in Tierra lack the important distinction between ``copying'' and ``decoding''--and that this perhaps affects evolutionary potential in the system. However Taylor (1999) has since persuaded me that this way of presenting the matter is, at best, unclear.
It might be more precise to say that Tierra does incorporate a distinction between copying and decoding processes: but that the ``decoding'' is represented by the execution of instructions by the Tierra processors. Thus, the decoding is hard-wired, whereas, in von Neumann's system, the decoding is itself a product of the specific design of , and, as such, is at least potentially mutable. In this way, von Neumann's system allows for what I have called Genetic Relativism (McMullin, 1992, Chapter 4), where the actual ``decoding'' map can itself evolve. It seems to me that this allows for a more profound form or degree of evolvability in von Neumann's system compared to Tierra. Having said that, I should also note that von Neumann himself seems to have discounted this idea. He stated explicitly that mutations affecting that part of a descriptor coding for would result in the production of ``sterile'' offspring (von Neumann, 1949, p. 86)--and would thus have no evolutionary potential at all. Clearly, on this specific point, I disagree with von Neumann, and consider that he actually underestimated the force of his own design in this particular respect.
As to my question 3 above--whether von Neumann's solution is the ``simplest''--this clearly does not admit of a clearcut answer, given the informality of our notions of simplicity and complexity. Certainly, the alternative solutions of Thatcher, Codd and Pesavento can all be said to be somewhat simpler in at least some respects. Similarly, if the outline design of Berlekamp et. al. is accepted then it is much simpler in one particular dimension (the number of cell states). Nonetheless, in considering the overall development of the field I would say that none of these ``simplifications'' either undermine, or significantly extend, von Neumann's seminal results.
By re-examining von Neumann's work in the light of his own description of the problem he was working on, I have tried to show that there is much more substance to it than has been generally recognised. However, as Popper (1976) has emphasised, the scientific enterprise is iterative: the solution of any given problem gives rise to a new problem situation, which is to say new problems to be addressed. It seems to me that von Neumann's work is particularly fruitful in this regard, as it poses a number of new and profound questions which need to be addressed by the (still) fledgling field of Artificial Life. Among these are:
In conclusion, it seems to me that von Neumann's work in Artificial Life--properly understood--is as profound and important today as it was half a century ago; and that it should continue to provide structure, insight and inspiration to the field for many years to come.
Many of the ideas presented here were first explored in my PhD thesis (McMullin, 1992); I was privileged to carry out that work under the supervision of the late John Kelly. A large number of people have since helped and challenged me in trying to refine the analysis and arguments. I am particularly indebted to Chris Langton, Glen Ropella, Noel Murphy, Tim Taylor, Mark Bedau and Moshe Sipper; the latter also provided critical encouragement to write this particular paper. I am grateful also to the ALife VII reviewers for comprehensive and constructive criticism. Financial support for the work has been provided by the Research Institute in Networks and Communications Engineering (RINCE) at Dublin City University.
First published in August 2000 as Technical Report No. bmcm-2000-01 of the Artificial Life Laboratory of Dublin City University. First presented at the conference Artificial Life VII August 1-6, 2000, Reed College, Portland, Oregon. Essentially the same text appears in Artificial Life VII: Proceedings of the Seventh International Conference, edited by Mark A. Bedau, John S. McCaskill, Norman H. Packard, and Steen Rasmussen (MIT Press, 2000), pp. 467-476.
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|Research Institute for Networks and Communications Engineering (RINCE)|
|Dublin City University|
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This report is copyright ©2000 by Barry McMullin. Certain rights have been assigned to The MIT Press in connection with its copyright in the edited volume Artificial Life VII: Proceedings of the Seventh International Conference, edited by Mark A. Bedau, John S. McCaskill, Norman H. Packard, and Steen Rasmussen (MIT Press, 2000).
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