The Life & Mind Seminar Network

Thoughts on trivial vs non-trivial closure

Posted in General by Ezequiel on February 24, 2007

Semi-random thoughts following Tom’s post.

1. I cannot recall where I’ve read it in the primary literature, but I’m almost certain that Varela has written somewhere that what at some point he referred to as operational closure he was now calling organizational closure (or vice versa). The point is that I have never actually distinguished the two phrases. But of course, formulating a refined distinction could be useful. But I didn’t quite understand the distinction that Tom is proposing (what would be an example that is one and not the other?).

2. Operational closure (because I’m working from memory I’ll use only this term) comes in trivial and non-trivial varieties. But this is never said explicitly anywhere as far as I know.

3. The trivial variety only requires that a network of “processes” be such that its operation results in the continuation of the processes that compose this network. This is another way of saying that the processes, whatever they are, form a closed system in the mathematical or group theory sense. For instance, the set of even numbers is closed under multiplication, the symmetries of a sphere are closed under all solid rotations, a glider is closed under the rules of the game of life. Notice that in all cases there is a distinction between operator and operant. Trivial closure is when this distinction remains unbroken. It may admit of very simple examples, e.g., the continuous self-identity of a static entity.

4. Some cases of trivial closure, however, are strong enough to support non-representationalist alternatives to cognitive explanation. Consider Bittorio, the CA wrapped around itself interacting with a sea of random binary numbers (Varela et al 1991) or a simple fully connected CTRNN showing some endogenous activity (e.g., an unstable centre-crossing network). Do all cases of trivial closure afford this? No. I propose that the key factor is when trivial closure involves dynamical processes (i.e., process of explicit transformation where states differ over time). Why do they support non-representationalism (support here used in the sense of allowing such interpretations, not demanding them)? Because, as long as we’re describing the system’s operation, we need not involve any relational element describing the system’s interactions. Or rather, such elements will appear in our descriptions as perturbations to its operation, as contextual changes to the system’s structure and hence to its particular form of realizing its closure. Therefore, a relational concept such as a representation (doubly relating itself to a content or referent and a consumer; R represents O to S) cannot be part of the operation of an operationally closed system. If it were, we would have a system that is not closed, or we would have to open up the system so that an extended enough new system becomes closed.

5. Life and other forms of biological autonomy go beyond trivial closure. How they do so is now a matter of intense debate and investigation. I can mention two ways, which I still don’t know exactly how related they are (one implies the other, at least, and this may be reciprocal too): i) contextual precariousness leading to a self-defined unity; ii) breaking of distinction between operator and components operated upon.

6. Precariousness. If and when an operationally closed system is established out of processes that not only sustain themselves as a network, but additionally, would not be able to sustain themselves unless they were organized as such a network, we are in the presence of a precarious operationally closed system. Fundamentally, component processes would not normally be present or durable enough, unless the conditions that are sustained by the rest of the network were in place. The operationally closed system not only maintains itself; it enables itself. It produces its own preconditions (cf. Zizek’s comments on autopoiesis in The Parallax View who focuses this crucial point). An isolated component process, under the same physical conditions — but in the absence of the network’s organization — would not be sustainable. There are two consequences of this. The operationally closed system (in general a system of relations distinguished by the observer) implies in addition a self-distinction; i.e., it produces itself as a unity, rather than sustaining itself as a set of invariant relations. An environment is thereby co-defined. In addition, a precarious closed system must engage in interactions to sustain itself. It must actively keep the natural tendencies of isolated component processes to disintegrate at bay. Hence a non-trivially closed system is by necessity interactionally open. This, and not internal operations, is the locus of mind.

7. Tail-biting. I think that a consequence of a precarious system with a self-generated identity (as described in 6) is that the processes that compose it must blur the distinction between operator and that which is operated upon. This is hard to deal with with linear causality, but trivially true in natural, self-organized systems (proteins are products of genetic regulation and regulators themselves, language can be used to talk about language, etc.). This is what makes autopoiesis difficult to capture in a model. A simulation model, for instance, must keep at some level a strict separation between operations of transformation and those elements that are transformed. There is a way around this, which is to model the system at an underlying level (e.g., chemical interactions on a spatial grid) and observe tail-biting effects on the properties that are distinguished at a higher level (the classic example in biological modelling in the 90s was spiral-wave patterns in Pauline Hogeweg’s work on autocatalysis, but there are many others). I think eventually, a limitation is found in such models that maybe it’s not present in real living processes and this is what Robert Rosen’s book Life Itself attempts to demonstrate, the non-computability of life. This issue is badly in need of clearer formalization.

8. A note on material non-trivial closure (autopoiesis). Physical self-construction implies both of the above conditions (precariousness and tail-biting). Does it imply others? I copy a footnote from my Autopoiesis, Adaptivity, etc. paper on this point. Talking about the requirement of materiality in defining autopoiesis: “The demand for physicality seems at first ad hoc and arbitrary in a definition that is proposed as strictly formal and systemic. This is an unresolved issue at the core of continuing debates about the possibility of computational, or other non-physical kinds of autopoietic processes (Boden, 1999; McMullin, 2004; Moreno & Ruiz-Mirazo, 1999). The existence of computer realizations of algorithmic processes showing sometimes surprisingly lifelike properties such as growth, membrane construction, and self-repair (e.g., Ono & Ikegami, 2000) belies the fact that removing physicality from autopoiesis elevates trivial examples such as attractor states in cellular automata to the category of the living. Varela (1979) attempts to resolve the issue by defining autonomy or generalized operational closure as the class of self-producing processes occurring in arbitrary domains. Autopoiesis is the restriction of this class to the physical, molecular domain. But this merely defers the problem which is now that of trivial cases of autonomy. It is very likely that physicality brings more to the interpretation of autopoiesis than initially anticipated and that these elements also need to be spelled out for an improved definition of life. This worry had led some researchers like Alvaro Moreno and colleagues to explore the implications of thermodynamic constraints on autopoiesis, (see Ruiz-Mirazo & Moreno, 2004). The important result that this line of research is achieving is precisely determining what it is about physical constraints that make real life non-trivial. Interestingly, anyone who believes that life is a formal, substrate-independent, property, as opposed to formal and material, should follow these developments closely because any chance of substantiating their beliefs lies in being able to formalize the restrictions brought by physical constraints that make life a non-trivial affair.” In view of things said in 7, I’m inclined to say that the formalist view is flawed. Life is both form and matter. Or rather, it trascends that distinction. Hence, computational models of life can never be instantiations of life.



6 Responses

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  1. tomfroese said, on February 26, 2007 at 4:57 pm

    Thanks for this Ezequiel.

    I think the key distinction I was trying to make is that organizational closure refers to a network of processes that is self-producing, whereas operational closure refers to a network of processes which is (merely) self-referential.

    From this it would follow that all organizationally closed systems are necessarily also operationally closed, but not vice versa.

    For example, when we think of the closure of the sensorimotor loop of an organism in its environment it is clear that an explanation of anything which happens (sensation or motion) must make reference to another process within this system (sensation or motion). But I’m not sure in what sense we could say, for example, that the processes of sensation and motion actually produce each other. The sensorimotor loop might therefore be better described as operationally rather than organizationally closed.

    You provide an even clearer example, namely that the set of even numbers is closed under multiplication. This would be a case of operational, but not organizational, closure. Perhaps my distinction between operational closure and organizational closure coincides with your distinction between trivial and non-trivial closure, but I’m not sure.

    It might also be worth pointing out that I’ve read a recent Thompson paper in which he said that an autonomous system is defined by organizational AND operational closure, which implies that he might think that there is a difference between the two, but he does not elaborate on it.


  2. xabier said, on February 27, 2007 at 8:03 pm

    Thank you for your comments Ezequiel, I have never read such a clear exposition of operational/organizational closure. I was particularly happy to read your comment on the two level modeling approach to non-trivial operational closure (point 7). I though about it for long and I found it to be the most satisfying of the approaches (Rosen seems to me reaching too pessimistic conclusions to study and model life/mind). However I never read before about the reference to Hogeweg’s work. I found her website ( but couldn’t find the reference you mention. Could you please specify more about Pauline Hogeweg’s work on autocatalysis and any other work related to that (unless you are just referring to Alife models of “emergence”).



  3. ezequieldipaolo said, on February 27, 2007 at 9:23 pm

    Muchas gracias, Tom y Xabier.

    Just quickly…

    The work by Pauline Hogeweg is first described (I think) in this reference:

    Boerlijst, M.C. and Hogeweg, P. (1991) Spiral wave structure in prebiotic evolution: Hypercycles stable against parasites”, Physica D 48, 17-28

    But then they had other similar papers in Journal of Theoretical Biology and other places.

    Talking just from memory – so better check the paper! – they show that if you have a hypercycle (of autocatalytic chemical species) these can be unstable against invasion by parasites (other chemical species benefiting from catalysis but not “contributing anything back”; typical neo-Darwinian worry). Well, if you spread them out in space the form spiral wave patterns and somehow the same hypercycles are now robust to invasion by parasites. How? It turns out that the speed of rotation of the spiral waves is related to the integrity of the local cycle and that cheaters are regularly sweeped away by the waves (basically their benefits disappear from their locality regularly).

    This means that there is a dynamical systems (chemical interaction, diffusion in space) creating a global pattern (spiral waves) that affects the spatiotemporal constrains of the lower level dynamics (sweeps off parasites that would otherwise invade). You have now a nice example of a modelling of an emergent, maybe even tail-biting dynamical system on a computer simulation. So yes, in practice you can model these thigs. Rosen’s arguments I think attempt to show something else, but can easily be taken to mean you cannot even model such dynamics.

    I will get back to Tom’s points soon. Must have dinner…

  4. ezequieldipaolo said, on March 1, 2007 at 8:51 am

    Tom, the trouble I was having with your definition was that I understand how a network of processes can be self-producing, but it’s not so clear to me what it means to say it’s self-referential. From the examples you give, it does sound a bit like what I called trivial closure. Then, I agree that it may be a good distinction to propose.

    Logistically, I worry that having two terms that are so similar will just spread confusion. If, as you notice, some people already draw a distinction, like Thompson, we should then try to ask them what that distinction is.

  5. Mike Beaton said, on March 2, 2007 at 4:53 pm


    I also found your post to be extremely clear and helpful.

    Can I give a couple of comments?

    1. At the end of your point 8 you say that “Life is both form and matter” (and this is similar to comments you made during Joel’s seminar). Now, the specific comments towards the end of 8 sound compatible with a weak reading, something like this: life is a formally defined affair, but ‘life’ is always life with respect to a substrate. So a very good simulation of life will only be ‘alive’ with respect to its substrate. To be really alive, you need to meet the formal requirements with respect to the substrate of the physical Universe.

    But I see here that Varela has explicitly proposed this view, and that you are explicitly rejecting it.

    I’m still not sure why you reject this and are drawn to the ‘non-computable’ view. E.g. if Hogeweg’s self-sustaining, emergent chemical cycles were observed in real chemicals, not just in a simplified simulation, this would look like a striking potential step towards life. If it’s not yet life as such, why not add your own notion of ‘precariousness’, for instance, to the formal requirements?

    (David Deutsch, in the ‘Fabric of Reality’ gives an excellent overview of a modern, broader, physicists’ notion of computability, which is not just about symbol manipulation any more, but about whether one part of reality can model the behaviour of another part. E.g. there may be quantum behaviour which cannot be modelled by any classical, symbol manipulation system, but which can be modelled e.g. by a general purpose quantum computation system.

    I’m not sure why I think this is relevant – I suppose my thought is, that with such a broad, but plausible, notion of computability, it may be even harder to argue that life is non-computable.)

    2. My second point is close to what I asked about in the seminar…

    The end of your 4 (“a relational concept … cannot be part of an operationally closed system”) and the end of your 6 (“a non-trivially closed system is by necessity interactionally open”) sound, to me, prima facie contradictory.

    Basically, an interaction *is* a relationship, so if we want to talk about a non-trivially closed system, in interaction with the world, we *have* to talk about its relationship to that world. Hence, while there might be valid arguments against using ‘representations’ when talking about operationally closed systems, working in their worlds, yours (based on the fact that they intrinsically involve relationships) seems not to be one of them.

    To me, the two sentences I quoted do sound more or less directly contradictory, but maybe that’s only based on my pre-existing point of view. If you can see (at least roughly) why these might sound contradictory to someone, perhaps you have enough time to briefly outline why they are not.

    But, many thanks again for such a useful post!


  6. ezequieldipaolo said, on March 3, 2007 at 11:37 am

    Thanks, Mike. These are useful thoughts too.

    1. I’m not terribly worried about the debate on the computability (or formalization) of the notion of life. It’s not a central interest of mine. Perhaps, as you suggest, adding constraints to current formal definitions, such as incorporating some notion of precariousness, could be a step towards a strictly formal definition. Life in this view would be a particular organization of a substrate (which couldn’t be just any possible substrate, but could possibly be a non-material one).

    My intuitions are a bit against this option. However, I do acknowledge that the debate is open. About computability, yes, perhaps other, broader notions of computation would change the face of the debate. There is on the other hand the argument by Rosen using category theory that “proves” the non-computability of life. I do think that there is something to it, but had always wanted to go carefully through the assumptions of the proof. Basically, life requires a sort of tail-biting that cannot be captured by any kind of recursive function.

    It’s an intriguing debate. But I don’t follow it closely.

    2. This point is more interesting to me. Yes, those two statements sound contradictory. But they are not. (They can, of course, be better expressed).

    By “a relational concept … cannot be part of an operationally closed system”, I mean it cannot describe any of the operations the make up that system as operationally closed. Say I am in relation R with respect to some aspect of my environment. Grammatically and logically speaking it makes no sense to use such a statement to describe some mechanisitic (operational) going-on inside me. I cannot use a relation of a whole to its surroundings to describe a part of that whole without breaching its integrity. (Imagine a molecular biologist saying that DNA is transcribed thanks to the presence of this and that protein and me looking at something threatening approaching towards me; that description, if it means anything, is not operational because it contains a relational statement; therefore my description has opened up the system and I’m not treating it as operationally closed).

    So that first statement says: Watch out when you mix your modes of discourse!

    (And it is not a trivial advice – I think that cognitive science since the 60’s has been about almost irremediably mixing these two modes of discourse – then at some point my looking at something threatening becomes a chemical reaction in itself).

    The other statement “a non-trivially closed system is by necessity interactionally open” says that if you have a precarious self-sustaining system, it MUST interact in order to survive. Not that its interactions will form part of its operations, but that in order to keep its operations going, the relations that the system enters into must provide it with the right boundary conditions. Since these are not already given (ex hypothesi because the system is precarious) then the system must perform some work as a whole to bring those conditions closer to satisfying its needs. So, this is a statement not about the operation of a system but about the implication that the operational conditions have for the system’s interactional conditions. I’m not contradicting the other statement.

    The upshot.

    i. At no point am I suggesting that we should abandon relational discourse. Absolutely not, I wouldn’t be able to figure out the simplest robot that I’ve evolved if I did this.

    ii. The interactive level is relational and central to cognitive science. I do even believe that ALL of cognition without exception happens only at this interactive level (cf my statement about the locus of mind in the original post).

    iii. Operational and relational/functional discourse interact in subtle ways (I wrote about this in my thesis; my ideas remain close to those expressed there). As you’ve seen it is possible for me to derive a relational implication from operational knowledge in my second statement. (The converse is not really so straightforward as I argued in my thesis).


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