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Autonomy and hypersets
Available online at www.sciencedirect.com
BioSystems 91 (2008) 320–330
Autonomy and hypersets Anthony Chemero a,∗ , Michael T. Turvey b a b
Scientific and Philosophical Studies of Mind Program, Franklin & Marshall College, United States Center for the Ecological Study of Perception and Action, University of Connecticut, United States Received 19 December 2006; received in revised form 2 April 2007; accepted 25 May 2007
Abstract This paper has two primary aims. The first is to provide an introductory discussion of hyperset theory and its usefulness for modeling complex systems. The second aim is to provide a hyperset analysis of several perspectives on autonomy: Robert Rosen’s metabolism-repair systems and his claim that living things are closed to efficient cause, Maturana and Varela’s autopoietic systems, and Kauffman’s cataytically closed systems. Consequences of the hyperset models for Rosen’s claim that autonomous systems have non-computable models are discussed. © 2007 Elsevier Ireland Ltd. All rights reserved. Keywords: Autonomy; Autopoiesis; Catalytic closure; Francisco Varela; Hyperset; Robert Rosen
1. Introduction Toward the end of his book Life Itself, Robert Rosen (Rosen, 1991; see also Rosen, 2000) claimed that the defining feature of living systems is that they are “closed to efficient cause”. The idea of closure to efficient cause is wrapped up with a series of other concepts (metabolism-repair systems, computatibility, impredicativity, complexity) in a somewhat obscure way. (See Letelier et al., 2006 for a recent attempt at clarification.) One aim of this paper is to set out exactly how these concepts are related. Rosen’s view was that closure to efficient causation is a variety of complexity, where complex systems are systems whose models contain impredicativities, and, therefore, are not computable. We will see in this paper that the relationship between complexity and computability is not exactly the one that
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Corresponding author. E-mail addresses: [email protected] (A. Chemero), [email protected] (M.T. Turvey).
Rosen imagined, but that his overall approach is still vindicated. In particular, we use non-well founded set theory to find fault with Rosen’s claim that, because of this complexity of living systems, the mathematical and computational tools of the mainstream cognitive sciences, derived from computability theory and designed for simple systems, are inappropriate as tools for analyzing systems that are closed to efficient causation. So, a further, and more important aim of this paper, is to introduce non-well founded set theory or hyperset theory (Aczel, 1988; Barwise and Etchemendy, 1987; Barwise and Moss, 1996; Kercel, 2003), and show that it is a useful tool to make sense of the kind of complexity that is characteristic of living systems. Our plan is as follows. We will begin by introducing non-well founded set theory or hyperset theory. Hyperset theory comes with an intuitive, easy-to-understand system of graphs that make it possible to test systems for complexity. In the next section, we will discuss hypersets and introduce the graphing system. We will show that complex systems all have graphs with a similar structure—they all have loops. In the follow-
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ing section, we will use these tools to analyze Robert Rosen’s metabolism-repair systems (1991). The purpose of this will be to show that, as suggested by Kercel (2003), the diagrams do indeed capture the complexity of Rosen’s model. Furthermore, it will allow us to say what distinguishes systems that are closed to efficient cause from other complex systems. This will allow us to discuss briefly the relationship among closure to efficient cause and three other proposed base cases of autonomous system: operationally closed systems (Varela, 1979), autopoietic systems (Varela et al., 1974; Thompson, 2007), and collectively autocatalytic sets (Kauffman, 1995, 2000) Finally, we offer a short discussion of how hyperset theory might be applied to complex systems and the possibilities for future work. Preliminary application of hyperset theory to model the Belouszov–Zhabotinsky reaction and animal-environment systems as studied within ecological psychology (Chemero and Turvey, 2007) suggests a special role for hyperset graphs. Namely, they can function as “complexity detectors”. They can help determine – within science’s various branches – just which systems are complex and in what ways. Our impression is that hypersets will prove helpful in developing a general theory of autonomy and context dependence. Indeed, we close by suggesting that the term ‘autonomy’ be defined in terms of hyperset models, and outlining consequences of such a definition. 2. Introducing Hyperset Theory During the late 19th and early 20th century work that codified and established set theory, it became evident that certain paradoxes threatened the foundations of set theory. The most famous of these paradoxes was first discussed by Bertrand Russell, and is now known as Russell’s Paradox (1903). Imagine the barber who shaves all and only those who don’t shave themselves. Who shaves the barber? It turns out that if the barber shaves himself, he does not shave himself, and vice versa. Put in terms of sets, the paradox is as follows. Call all the sets that do not contain themselves as members normal sets; those that do contain themselves are abnormal sets. Consider S the set of all normal sets. Is S normal? If so, S is a member of the set of all normal sets. But since S is the set of all normal sets, S would then be abnormal. If S is abnormal, then S is a member of itself. But if S is a member of itself (the set of all normal sets), then S must be normal. So: S is a member of itself just in case it is not a member of itself, or, equivalently, S is normal just in case it is abnormal. This means that the sentence “S is a normal set” is simultaneously true and false.
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(We ask that readers convince themselves of this before continuing.) This was taken to be a serious problem with the theory of sets. In response to this problem, Russell introduced the vicious circle principle, which outlawed any sentence which was circular. This was later formalized as the theory of types, most of the details of which we do not need to be concerned with, but whose main effect was to make it illegal for sets to be members of themselves. According to the theory of types, that is, there are no abnormal sets, and there is also no set of all sets. Paradox solved. Poincar´e’s (1906) banishment of impredicative definitions from mathematics, which may be more familiar to some readers, serves a similar purpose. Roughly, a predicative definition applies to members of some domain so that its application is not altered by addition of new members to the domain; an impredicative definition, on the other hand, picks out different members of a domain should new individuals be added to the domain. The definition that picks out “the individuals in Ms. Riley’s class whose first names begins with P” is predicative, but the definition that picks out “the 4th tallest individual in Ms. Riley’s class” is impredicative. A new student’s arrival does not change the first letter of Peter’s name, but it may make him 5th tallest. Essentially, impredicative definitions pick out individuals or properties whose falling under that definition depend on other members of a set. Another way to put this is that impredicative definitions pick out individuals in a way that is context-dependent. “Abnormal set” and “the barber who shaves all and only those who do not shave themselves” are defined impredicatively, so outlawing impredicativities makes defining these types of set impossible. Again, paradox solved. The elimination of abnormal sets leaves us with standard set theory as taught in College-level logic and mathematics courses. This set theory is usually referred to as well-founded. There are two related senses in which standard set theory is well-founded. First, it is wellfounded in that its implementation does not lead to paradoxes; that is, it is logically well-founded. The second way in which it is well-founded comes from graph theory, a standard tool used to understand set membership. In graph theory, every set determines a tree-like graph in which the set itself is the upper-most node and each member of the set is a descendent node with an arrow pointing to it from the original set of which it is a member; this is applied recursively. Fig. 1 shows a graph for the set P = {a, b, c}. Notice that there is an arrow from the node representing the set P to each of its members. Fig. 2 shows a graph for the set Q = {a, b, {c}}. Notice that in this graph, there is an additional complexity in which the node for the member of Q, {c},
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Figs. 1–4. Graphs of (Hyper)sets described in text.
has an additional member, the object c. The sets P and Q are well-founded sets; they are not abnormal and do not contain themselves as members. The graphs of P and Q show a property common to the graphs of all wellfounded sets. Graphs of well-founded sets contain no infinite cycles or loops, and each path from the original set terminates in some object (its foundation, as it were). In fact, Aczel, 1988 proves that a graph will contain no cycles or loops if and only if it is well founded. The qualifier ‘if and only if’ means that a graph containing loops or cycles is a picture of a non-well-founded set. The presence of cycles and/or loops, that is, would indicate that some set has itself as a member, or that the concept, system, or definition it models is impredicative. (See Appendix A.) Consider, as an example, the abnormal set A = {A, b, c}. This will have the graph in Fig. 3. Aczel also defines the special abnormal set W as the set that contains only itself, W = {W}. W is graphed in Fig. 4. The sets A and W are not allowed in well-founded set theory, but they do have coherent and useful graphs. Aczel’s Anti-Foundation Axiom that embraces nonwell-founded (i.e., abnormal) sets is as follows: every graph, with or without loops, pictures a genuine set. Hypersets are defined as graphable sets. So well founded sets and non-well-founded sets are both types of hyperset. Non-well-founded sets were banished from set theory in the interest of the logical soundness of the theory,
and with the explicit goal of reducing mathematics to set theory, and set theory to logic. Re-introducing non-wellfounded sets into our set theory makes such reduction impossible. But this loss (if it really is a loss) is offset by the increased ability to model real-world phenomena. Work by Gupta (1982), Barwise and Etchemendy (1987), Gupta and Belnap (1993) and Barwise and Moss (1996) makes very clear that many concepts and real-world systems are circular or otherwise not-well-founded, hence illegal according to standard set theory. The most well known example is the concept ‘truth’, and the predicate ‘is true’. Gupta (1982) argues that the semantic paradoxes involving truth, such as the liar (“This very sentence is false”) and the truth-teller (“This very sentence is true”) show that the predicate “is true” can only be defined circularly, and develops a logical technique (the revision theory) for dealing with circular concepts. Later, Barwise and Etchemendy (1987) use hypersets to understand the truth predicate. In each of these cases, some object in a system is defined impredicatively, and is therefore illegal in standard set theory. Of course, the illegality of these systems and concepts in standard set theory does not change their behavior, or cause them to cease to exist. Such systems do exist. Adequately modeling them requires a richer set theory, such as hyperset theory, that allows and accounts for the behavior of systems with impredicativities. The graph theory that goes with hyperset theory will be especially important in what follows. As Aczel has
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proved and Figs. 1–4 demonstrate, graphs of hypersets can be used to determine whether the set pictured is or is not well-founded, and if it is not well-founded exactly what it is that makes it non-well-founded. These graphs, that is, can be used to determine whether systems contain impredicativities, and so are complex. The graphs also allow us to determine the precise source and nature of the complexity in such systems. We will use this feature of hypersets and their graphs to look at Rosen’s model of living things as metabolism-repair systems. 3. Rosen on Life Itself According to Rosen (1991, 2000), a central feature of living things is complexity. Rosen, it should be noted, has an idiosyncratic understanding of complexity. First, according to Rosen, whether a system is complex depends only secondarily on the system itself; it depends primarily on the system’s models. A physical system is complex if and only if it has impredicative models; otherwise, the system is simple. (As mentioned in Section 1 above, Rosen typically described this in terms of Turing-computability. We discuss this issue in Section 7.) This connects in important ways with the above discussion of impredicative definitions and Russell’s theory of types. Put most simply, Russell introduced the theory of types and Poincar´e outlawed impredicative definitions specifically to keep logic, set theory and mathematics well-founded. Thus systems that have models that disobey the theory of types or contain impredicativities – models, that is, that are not well-founded – are complex in Rosen’s sense. If this is so, we can use hyperset diagrams as a means of modeling systems that are complex. Indeed, we can use hyperset diagrams of models as complexity detectors. We can see this by examining Rosen’s metabolismrepair model of living systems (1991, 2000). According to Rosen, it is a necessary condition on living systems that their models contain three functions: a metabolic function, a repair function, and a replication function. We will discuss these in order. The metabolic function hi is a mapping from raw materials xi ∈ X to behavior yi ∈ Y. That is, it determines the way the system reacts to environmental input. A living system must also have a repair function that repairs the metabolic function to keep the system reacting appropriately to the environment. That is, it must have a function which maps behaviors yi ∈ Y onto the members of set H, where metabolic hi ∈ H. Finally, the repair function must itself be repaired. This repair is done by the system’s behavior. That is, each member yi of the set of behaviors Y is itself a replication function that maintains the repair function, by
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mapping hi onto ai ∈ A. This is, admittedly, complicated. The idea, though, is rather simple. The living system must be metabolizing consistently, so must maintain the causal processes that it uses in metabolizing. This is the job of the repair function. But the repair function must also be maintained, and this is done by the system’s behavioral output. Rosen’s diagrammatic representation of his model of living systems is shown in Fig. 5. Note that in this diagram a function P from Q to R is represented as ‘P’ connected to ‘Q’ with a solid arrow, and ‘Q’ connected to ‘R’ with a dashed arrow. Fig. 6 is a hyperset diagram of Rosen’s model of living systems. In Fig. 6, two common set-theoretic conventions are followed. First, functions are represented as ordered pairs. That is, if P is a function from Q to R, P is represented as P = Q, R. Second, to preserve their ordering, ordered pairs are represented as sets containing the set containing their first member and the set containing both their members. Thus Q, R is represented as {{Q}, {Q, R}}. In Fig. 6, the sets of inputs X, behaviors Y, metabolic functions H, and repair functions A are represented as if they were individuals. This is done to make the already-complicated diagram simpler. The thing to notice about the hyperset diagram in Fig. 6 is that there are loops, indicating that the sets are not well-founded. Indeed, in the graph there is only one location where the system does not loop back on itself: the set X of inputs from the environment. This makes sense of two important aspects of Rosen’s model of life. First, the loops indicate that Rosen’s diagram is indeed complex: it involves irremovable impredicativities. Rosen took just this kind of complexity to be one of the most important features of living systems, and often wrote of the impredicativities found in living systems. Fig. 6 bears this out. Second, the fact that the only place the diagram “bottoms out” is with raw materials is also important. Rosen often wrote that living systems are “closed to efficient causation”. In Rosen’s Aristotelian language, an ‘efficient cause’ matches up with our common sense intuition about the cause of an event. Thus, the efficient cause of a billiard ball moving is that it was struck by another billiard ball. When Rosen says that living systems are ‘closed to efficient cause’, he means that
Fig. 5. Rosen’s model of living systems (1991). Note that Y = H (X) is represented by a solid line from H to X and a dashed line from X to Y.
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Fig. 6. Hyperset diagram of Rosen’s metabolism-repair system. Functions are represented as ordered pairs containing their domain and range. So f (a) = b is represented as f = a, b.
all of the efficient causes of the system are produced by the system. Entities and events outside the system cannot be efficient causes of its behavior, they can only be raw material (Aristotelian ‘material causes’) for the system. We can see this in Fig. 5, Rosen’s own representation of living systems. In that diagram, only X, the raw material input from the environment, is not both (a) itself a function in the system and (b) the output of a function of the system. To put this in Rosen’s (and Aristotle’s) language, all functions but X are both efficient and material causes in the system. X is only a material cause. The hyperset diagrams make clear how this is the case. With this analysis of Rosen’s closure to efficient cause in hand, we can briefly consider two other analyses of autonomy: Maturana and Varela on autopoiesis and operational closure and Kauffmann and collective autocatalysis and catalytic closure. 4. On Operational Closure and Autopoiesis As a way to spell out what is distinctive about autonomous systems, Maturana, Varela and their colleagues (Maturana and Varela, 1980, 1987; Varela, 1979; Varela et al., 1974; Varela et al., 1991) define the notions operational closure and autopoiesis, and offer each as a base case of automonous system. We discuss these in order.
A system is operationally closed if and only if every effect of the system is also a part of the system. The one-dimensional cellular automaton Bittorio, discussed in Varela et al., 1991, is perhaps the clearest example of an operationally closed system. Bittorio is a onedimensional cellular automata, shaped like a ring. (See Fig. 7.) At each time step, the states of Bittorio’s cells are determined by the states of neighboring cells. Thus, left alone, the cells of Bittorio will pass through a series of states that form a self-organizing pattern. Because of this, Varela, Thompson and Rosch claim, Bittorio is autonomous. This is not to say, of course, that Bittorio is immune to influence from the environment. The environment can perturb Bittorio by changing the state of one of its cells, altering Bittorio’s dynamics, either temporarily or permanently depending on the nature of the perturbation and the rules governing Bittorio’s cells. So operational closure, like closure to efficient cause, does not imply separation from the environment. Although operationally closed, Bittorio is structurally coupled to its environment.
Fig. 7. The one-dimensional cellular automaton Bittorio.
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Fig. 8. Hyperset diagram of three cells of the cellular automaton Bittorio. Each cell’s state is a function of that of its neighbors. So B = F (A, C}. This is represented as F = {A, C},{B}.
Bittorio is operationally closed because the state of each of its cells has an effect only on the state of some of Bittorio’s other cells—all these effects are within Bittorio. But note that Bittorio is not closed to efficient cause. Bittorio’s operations (i.e., the rules that generate its behavior) are not produced by Bittorio, but are instead determined from outside, by a computer programmer. Fig. 8 is a hyperset diagram of three of Bittorio’s cells, which follows the same conventions as Fig. 6. Fig. 8 looks quite complicated, but it contains no impredicative loops: each of the system’s functions (F, G, and H) is independent of the others. This is not to say, however, that operational closure is not interestingly related to closure to efficient cause. A sub-class of operationally closed systems, the autopoietic systems (Maturana and Varela, 1973, Varela et al., 1974; Varela, 1979) are closed to efficient cause. Autopoietic systems are systems that are self-producing. They are operationally closed because
their effects are system-internal; in addition, though, the effects are system-constituting. Maturana and Varela (1973, 1980) claim that autopoiesis is necessary and sufficient for life. Thus, from their point of view, all operationally closed systems are autonomous and a subset of the operational closed systems, namely, the autopoietic systems, are living. The autopoietic system most commonly discussed is the cell. In a cell, the metabolism produces a cell membrane and the membrane provides a boundary, which plays a crucial role in the cell’s metabolism. This is depicted in Fig. 9, which follows the same set theoretic convention as Figs. 6 and 8. Note that the diagram has a form very similar to Rosen’s metabolism-repair system: every function in the system (the metabolism, the membrane) is produced by the system itself and only raw materials are not produced by the system. In other words, the diagram shows that autopoietic systems like the cell are closed to efficient cause.
Fig. 9. Hyperset diagram of the cell and cell membrane, a canonical autopoietic system.
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5. Collective Autocatalysis and Catalytic Closure Now consider another proposed minimal case of autonomy: catalytically closed systems (Kauffman, 1993, 1995, 2000). Kauffman introduces catalytic closure to argue that the origin of autonomous and, ultimately, living systems was not, as has been previously argued, improbable. Start by considering living systems as networks of chemical reactions. In a dilute solution (e.g., a pond), most chemical reactions proceed very slowly, so for reactions to proceed fast enough to keep a collection of them going, they must be catalyzed. But the likelihood that the catalyst for any particular chemical reaction just happens to be around is very small. For a network of chemical reactions to be autonomous, then, the network must produce its own catalysts. Kauffman shows, statistically, that for large enough networks of reactions it is overwhelmingly likely that reactions in the network will be catalyzed by the products of other reactions in the network. When every product in a network of chemical reactions has a catalyst that is also a product in the network, the network of reactions is a collectively autocatalytic system. Collectively autocatalytic systems are catalytically closed, and hence, autonomous. We can see this in Fig. 10, which is a hyperset diagram of the following (imaginary) catalytically closed set of reactions: A
R−→B B
S −→A In this set of reactions, chemicals R and S are raw materials. If they are in steady supply, these reactions will form a self-sustaining network. In the hyperset diagram, we represent this system by equating each product with the set reactants and catalysts that produce it. Thus the product A is {S, B}; B is {R, A}. In the figure, there is an impredicative loop involving catalysts A and B. The impredicativity indicates that collectively autocatalytic systems are complex systems. Notice also that in the
Fig. 10. Hyperset diagram of a collectively autocatalytic network of reactions. System products are represented as the set of reactants that produce them.
diagram only “bottoms out” at S and T, the raw materials. This is just as is the case with the metabolism-repair system and the autocatalytic systems described above. Like those systems, systems that are catalytically closed are closed to efficient cause. 6. Conclusions Concerning Hypersets and Autonomy The preceding leads to two different kinds of conclusions, those regarding hypersets as an analytic tool for understanding complex systems and those regarding artificial autonomy. We discuss the implications of our results for Rosen’s claims concerning the relationship among life, complexity and computability in Section 7. We take it that the hyperset analysis of Rosen’s model of living systems is useful for three reasons. First, it demonstrates some of Rosen’s claims about his model, viz., that it contains impredicativities and is closed to efficient causation. It also shows exactly what the relationship among parts of a system must be for it to be closed to efficient causation. We think, therefore, that the hyperset analysis is enlightening. Second, it allows us to see in some detail how closure to efficient cause differs from operational closure and catalytic closure. Third, and most importantly for our purposes, the hyperset diagrams show that such an analysis is useful and does in fact capture important properties concerning the complexity of models. We must be clear that hypersets are in no way intended as a replacement for analysis of complex systems with coupled differential equations. Because hyperset theory is a new mathematical tool, the mathematical relationship between hypersets and differential equations is still to be worked out by mathematicians. So whether it will be possible to use hypersets to model ordinary differential equations is an open question. Although we must wait for mathematicians to sort out these details of the relationship, we can say now that hyperset analyses are most useful in situations in which analysis with differential equations is inappropriate for one reason or another. Examples of this include the following: systems that are intractable using differential equations such as the animal-environment systems as studied in ecological psychology; systems about which not enough is known to develop differential equations such as the cell; and model systems intended as conceptual tools such as the Brusselator and the systems discussed above (metabolism-repair systems, autopoietic systems, catalytically closed systems). Elsewhere we have presented hyperset analyses of the Belouzov–Zhabotinsky reaction, the Brusselator
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and animal-environment systems (Chemero and Turvey, 2007). There are also several conclusions regarding autonomy and artificial life. First, if we assume that closure to efficient cause is the most basic kind of autonomy (note that we have given no particular reason to do so), then autopoietic and catalytically closed systems are autonomous. Second, we have seen that metabolism-repair systems that are closed to efficient cause, autopoiesis and catalytic closure are structurally similar when viewed in hyperset diagrams. The hyperset diagrams show that for each, every causally important part of a system is also a product of the system. From our point of view, Rosen’s metabolism-repair systems and Maturana and Varela’s autopoietic systems are valuable as characterizations of an abstract property, while Kauffman’s work is valuable for connecting this abstract property to chemical processes. So each characterization is useful for certain purposes, and one might develop an argument that one description of this abstract property is better than others. The hyperset diagrams show unequivocally that these systems share an important structural feature: having every function be a product of a system, having loopy hyperset diagrams that terminate only in raw materials. We propose that this feature be called ‘autonomy’. Finally, note that if, as we suggest, metabolismrepair, autopoietic, and catalytically closed systems are autonomous, there is reason to be skeptical that autonomy is sufficient for life. Rosen eventually (2000) admitted that being a metabolism-repair systems is merely necessary, but not sufficient for life. Many authors have made similar claims about autopoiesis. Recently, for example, Bourgine and Stewart (2004) claimed that cognition is also necessary; Ruiz-Mirazo and Moreno (2004), Ruiz-Mirazo and Mavelli (this issue) and Suzuki and Ikegama (this issue) claim that thermodynamically-realistic metabolism is also necessary. If any of these authors are right, artificial autonomy has already been achieved, but artificial autonomy is not the same thing as artificial life. 7. Conclusions Concerning Computability and Life The hyperset analysis of metabolism-repair and autopoietic systems is germane to the debate over the possibility of artificial life. Rosen’s central argument is that organisms will have impredicative models, which he takes to say that they have models that are not Turing computable, which is to say that they are not machines. We have demonstrated using hypersets that autopoietic
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and metabolism-repair systems have impredicative models. Following Rosen, one might conclude that these systems are not Turing-computable. Considering the nature of impredicativity (and non-well-founded sets), it is not hard to see why Rosen made this inference. A set S is Turing-computable if and only if a Turing machine given any input n from S’s domain halts with output 1 if n is in S and halts with 0 if n is not in S. Now consider the case of circularly defined sets. Suppose that S = {. . . A . . .} and A = {. . . S . . .}. A Turing machine T that can determine whether some n is a member of S will need to check the members of S to determine whether n is among them. In doing so, T will need to check the membership of each set that is a member of S. In this case, it will check the membership of A. To determine the membership of A, it will need to determine the membership of all the sets that are members of A, including S in this case. This, it seems, leads to an infinite loop, which would mean that T does not halt. So it would seem that S and A are not Turing-computable. Another reason to believe that circularly defined sets are not Turing-computable comes from experience writing computer programs. Any decent compiler will give you error messages if you write a program with circularities. For these reasons, in earlier papers (Chemero and Turvey, 2007; Turvey, 2004), we followed Rosen in taking impredicatively defined sets to be non-Turing-computable. We have since learned that there is no necessary connection between impredicative definitions and nonTuring-computability. For example, work by Martin-L¨of (1970) and Girard (1972) on impredicative types has been extremely valuable in computer science. More importantly, hyperset membership has been shown to be Turing-computable in polynomial time (Lisitsa and Sazonov, 1999; Sazonov, 2006), a result anticipated in the pioneering work of Barwise and Moss (1996). Deciding on hyperset membership therefore does not necessarily involve infinite loops with their unwelcome consequences of adding infinitely many copies of elements to sets and an inability to terminate. Thus Rosen’s claim that “there is no algorithm for building something that is impredicative” (2000, p. 294) is incorrect (to the extent that ‘to build’ and ‘to compute’ are overlapping conceptions). The question at this point is to what extent the Turing-computability of impredicative sets, whether modeled as hypersets or not, affects Rosen’s most important contributions. Rosen claimed that impredicative systems are not computable. We have shown that impredicative systems are usefully modeled as hypersets, and hypersets are Turing-computable. In sum, we have shown that Rosen’s (2000) claim that impredicative systems are not Turing-
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computable is no longer tenable, which undercuts the justification for his conclusion that models of living systems are not computable. This agrees with the arguments of Chu and Ho (2006), who dispute Rosen’s purported proof that artificial life is impossible, and Wells (2006), who argued that Rosen’s metabolism-repair systems are computable. Nonetheless, we believe that much of the spirit of Rosen’s work can be maintained. We can see this by realizing that Rosen’s characterization of the distinction between simple and complex systems is multi-dimensional. (Gwinn (2006) lists eight ways in which Rosen makes this distinction.) In distinguishing between simple and complex systems, Rosen argued that simple systems are impoverished when compared to complex systems. Among these ways in which simple systems are impoverished are the following: (i) complex systems have models that are computable and non-computable, while simple systems only have models that are computable; (ii) complex systems have models that are impredicative and models that are predicative, while simple systems have only predicative models; (iii) complex systems can have fractionable and nonfractionable structures, while simple systems can have only fractionable structures; (iv) complex systems can have both linear chains and closed loops of entailment, while simple systems can only have linear chains of entailment; (v) complex systems have both syntactic and semantic elements, while simple systems have only syntactic elements. (See Gwinn, 2006.)
resented in their models. More precisely, autonomous systems are closed to efficient cause and so must have impredicative models. We have shown in this paper that Rosen’s metabolism-repair systems and two other candidates for minimal autonomous systems (autopoietic and catalytically closed systems) are in fact closed to efficient cause and, so, must have impredicative models. We have also seen that these impredicative models are Turing-computable. Indeed, for one of these candidates, autopoietic systems, there has been a long and fruitful history of computational modeling (Varela et al., 1974; McMullin and Varela, 1997, and Bourgine and Stewart, 2004). We believe that those interested in pursuing research along the lines suggested by Rosen should view this as an opportunity. Computational modeling using hypersets could become an important tool for studying non-fractionable, circularly causal, partly semantic systems such as Rosen’s metabilism-repair system. Acknowledgements Larry Moss provided invaluable advice on hypersets. John Stewart, Xabier Barandiaran, Abhey Shah and Michael Silberstein gave helpful advice on earlier drafts, as did an anonymous referee. Chemero’s work was supported by a National Science Foundation grant SBR 00-04097 and the Mellon Foundation/Central Pennsylvania Consortium. Turvey was supported by National Science Foundation grants SBR 00-04097 and SBR 0423036. Appendix A. Using Hyperset Models
As we have seen, Rosen’s thinking that (ii) entails (i) is no longer justified. But Rosen’s great insights were in seeing that complex systems have non-fractionable structures, loopy causal entailment, and built-in semantics, and in seeing that this leads to models of complex systems with impredicativities. Future research on Rosen-related themes must focus on the contrasts between simple and complex systems in points (ii)–(v) above. We contend that the central idea of Rosen’s work is not that artificial life or valuable computational models of autonomous systems are impossible. Very little rides on whether genuine artificial life is possible: we agree with Moreno, Extebarria and Umerez (this issue) that the real value of computational research is in what it teaches us about biological autonomy. Instead, the most important point of Rosen’s work is that autonomous systems have a structure that is very different from non-living things and that this structure must be rep-
Scattered throughout this paper are comments that, taken together, should make it possible for interested readers to use hypersets to model systems of interest to them. The purpose of this appendix is to provide explicit instructions for using hypersets, all in one place. Step 1: Think of the system of interest S in terms of a group of functions, mapping inputs to outputs. In most cases, there is no algorithm for coming with the right set of functions for your system. It will also typically be the case that there are many different sets of functions that capture the causal structure of the system. Step 2: Represent each of these functions as an ordered pair containing the input as the first member and the output as the second. Thus, f (x) = y
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Fig. 11. Graph of sample set.
becomes: f = x, y. Step 3: Represent each ordered pair as a Kuratowska pair, a set including the singleton set of the first member of the ordered pair and the set of both elements of the ordered pair. Thus, f = x, y becomes f = {{x}, {x, y}}. Representing the ordered pair this way makes it into a graphable set, yet marks the ordering of the ordered pair. Step 4: Graph the set. To do so, draw an arrow from each set to each of its members. Thus, one draws lines from the function f = {{x}, {x, y}} to {x} and {x, y}. Then one draws and line from {x} to x and lines from {x, y} to x and y. The result is Fig. 11. Step 5: Analyze the graph. If for any function g, chasing arrows from the function to its members, then to their members and so on leads back to g, the model is impredicative and the system modeled is complex. If every function of the system is the output of another function, the system modeled is closed to efficient cause. We have argued in this paper that this also makes the system autonomous. References Aczel, P., 1988. Non-Well-Founded Sets. CSLI, Stanford. Barwise, J., Etchemendy, J., 1987. The Liar: An Essay on Truth and Circularity. Oxford University Press, New York.
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Barwise, J., Moss, L. (Eds.), 1996. Vicious Circles. CSLI,Stanford. Bourgine, P., Stewart, J., 2004. Autopoiesis and cognition. Artif. Life 10, 327–345. Chemero, A., Turvey, M.T., 2007. Hypersets, complexity and ecological approach to perception-action. Biol. Theory 2 (1), 23–36. Chu, D., Ho, W., 2006. A category theoretical argument against the possibility of artificial life: Robert Rosen’s central proof revisited. Artif. Life 12 (1), 117–134. Girard, J.Y., 1972, Interpretation fonctionelle et eleimination des coupures dans l’arithmetique d’ordre superieure, These d’Etat, Universit´e Paris 7. Gupta, A., 1982. Truth and paradox. J. Philos. Logic 11, 1– 60. Gupta, A., Belnap, N., 1993. The Revision Theory of Truth. MIT Press, Cambridge. Gwinn, T. 2006. Rosennean Complexity. Robert Rosen: Complexity and Life Web Page. URL=. Kauffman, S., 1993. The Origins of Order: Self-Organization and Selection in Evolution. Oxford University Press, New York. Kauffman, S., 1995. At Home in the Universe. Viking, New York. Kauffman, S., 2000. Investigations. Oxford University Press, New York. Kercel, S.W., 2003. Endogenous causes-bizarre effects. Evol. Cogn. 8, 130–144. Letelier, J.-C., Soto-Andrade, J., Gu´ın˜ ez Abarz´ua, F., Cornish-Bowden, A., C´ardenas, M.L., 2006. Organizational invariance and metabolic closure: analysis in terms of (M,R) systems. J. Theor. Biol. 236, 949–961. Lisitsa, A.P., Sazonov, V.Yu., 1999. Linear ordering on graphs, antifounded sets and polynomial time computability. Theor. Comput. Sci. 224, 173–213. Maturana, H., Varela, F., 1980. Autopoiesis and Cognition: The Realization of the Living. D Reidel Publishing, Boston. Maturana, H., Varela, F., 1987. The Tree of Knowledge: The Biological Roots of Human Understanding. Shambhala, Boston. Martin-L¨of, P., 1970. Notes on Constructive Mathematics. Almqvist & Wiksell, Stockholm. McMullin, B., Varela, F., 1997. Rediscovering computational autopoiesis. In: Husbands, P., Harvey, I. (Eds.), Proceeedings of the Fourth European Conference on Artificial Life. MIT Press, Cambridge. Poincar´e, H., 1906. Les math´ematiques et la logique. Revue de m’etaphysique et de morale 14, 294–317. Rosen, R., 1991. Life Itself. Columbia University Press, New York. Rosen, R., 2000. Essays on Life Itself. Columbia University Press, New York. Ruiz-Mirazo, K., Moreno, A., 2004. Basic Autonomy as a Fundamental Step in the Synthesis of Life. Artif. Life 10, 235– 259. Russell, B., 1903. Principles of Mathematics. Cambridge University Press, Cambridge, UK. Sazonov, V.Yu., 2006. Querying hyperset/web-like databases. Logic J.l IGPL 14 (5), 785–814. Thompson, E., 2007. Life in Mind. Harvard University Press, New York. Turvey, M.T., 2004. Impredicativity, dynamics, and the perceptionaction divide. In: Jirsa, V.K., Kelso, J.A.S. (Eds.), Coordination Dynamics: Issues and Trends. Vol. 1 Applied Complex Systems. Springer Verlag, New York, pp. 1–20.
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Varela, F., 1979. Principles of Biological Autonomy. Elsevier, North Holland. Varela, F., Maturana, H., Uribe, R., 1974. Autopoiesis: The organization of living systems, its characterization and a model. Biosystems 5, 187–196.
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BioSystems 91 (2008) 320–330
Autonomy and hypersets Anthony Chemero a,∗ , Michael T. Turvey b a b
Scientific and Philosophical Studies of Mind Program, Franklin & Marshall College, United States Center for the Ecological Study of Perception and Action, University of Connecticut, United States Received 19 December 2006; received in revised form 2 April 2007; accepted 25 May 2007
Abstract This paper has two primary aims. The first is to provide an introductory discussion of hyperset theory and its usefulness for modeling complex systems. The second aim is to provide a hyperset analysis of several perspectives on autonomy: Robert Rosen’s metabolism-repair systems and his claim that living things are closed to efficient cause, Maturana and Varela’s autopoietic systems, and Kauffman’s cataytically closed systems. Consequences of the hyperset models for Rosen’s claim that autonomous systems have non-computable models are discussed. © 2007 Elsevier Ireland Ltd. All rights reserved. Keywords: Autonomy; Autopoiesis; Catalytic closure; Francisco Varela; Hyperset; Robert Rosen
1. Introduction Toward the end of his book Life Itself, Robert Rosen (Rosen, 1991; see also Rosen, 2000) claimed that the defining feature of living systems is that they are “closed to efficient cause”. The idea of closure to efficient cause is wrapped up with a series of other concepts (metabolism-repair systems, computatibility, impredicativity, complexity) in a somewhat obscure way. (See Letelier et al., 2006 for a recent attempt at clarification.) One aim of this paper is to set out exactly how these concepts are related. Rosen’s view was that closure to efficient causation is a variety of complexity, where complex systems are systems whose models contain impredicativities, and, therefore, are not computable. We will see in this paper that the relationship between complexity and computability is not exactly the one that
∗
Corresponding author. E-mail addresses: [email protected] (A. Chemero), [email protected] (M.T. Turvey).
Rosen imagined, but that his overall approach is still vindicated. In particular, we use non-well founded set theory to find fault with Rosen’s claim that, because of this complexity of living systems, the mathematical and computational tools of the mainstream cognitive sciences, derived from computability theory and designed for simple systems, are inappropriate as tools for analyzing systems that are closed to efficient causation. So, a further, and more important aim of this paper, is to introduce non-well founded set theory or hyperset theory (Aczel, 1988; Barwise and Etchemendy, 1987; Barwise and Moss, 1996; Kercel, 2003), and show that it is a useful tool to make sense of the kind of complexity that is characteristic of living systems. Our plan is as follows. We will begin by introducing non-well founded set theory or hyperset theory. Hyperset theory comes with an intuitive, easy-to-understand system of graphs that make it possible to test systems for complexity. In the next section, we will discuss hypersets and introduce the graphing system. We will show that complex systems all have graphs with a similar structure—they all have loops. In the follow-
0303-2647/$ – see front matter © 2007 Elsevier Ireland Ltd. All rights reserved. doi:10.1016/j.biosystems.2007.05.010
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ing section, we will use these tools to analyze Robert Rosen’s metabolism-repair systems (1991). The purpose of this will be to show that, as suggested by Kercel (2003), the diagrams do indeed capture the complexity of Rosen’s model. Furthermore, it will allow us to say what distinguishes systems that are closed to efficient cause from other complex systems. This will allow us to discuss briefly the relationship among closure to efficient cause and three other proposed base cases of autonomous system: operationally closed systems (Varela, 1979), autopoietic systems (Varela et al., 1974; Thompson, 2007), and collectively autocatalytic sets (Kauffman, 1995, 2000) Finally, we offer a short discussion of how hyperset theory might be applied to complex systems and the possibilities for future work. Preliminary application of hyperset theory to model the Belouszov–Zhabotinsky reaction and animal-environment systems as studied within ecological psychology (Chemero and Turvey, 2007) suggests a special role for hyperset graphs. Namely, they can function as “complexity detectors”. They can help determine – within science’s various branches – just which systems are complex and in what ways. Our impression is that hypersets will prove helpful in developing a general theory of autonomy and context dependence. Indeed, we close by suggesting that the term ‘autonomy’ be defined in terms of hyperset models, and outlining consequences of such a definition. 2. Introducing Hyperset Theory During the late 19th and early 20th century work that codified and established set theory, it became evident that certain paradoxes threatened the foundations of set theory. The most famous of these paradoxes was first discussed by Bertrand Russell, and is now known as Russell’s Paradox (1903). Imagine the barber who shaves all and only those who don’t shave themselves. Who shaves the barber? It turns out that if the barber shaves himself, he does not shave himself, and vice versa. Put in terms of sets, the paradox is as follows. Call all the sets that do not contain themselves as members normal sets; those that do contain themselves are abnormal sets. Consider S the set of all normal sets. Is S normal? If so, S is a member of the set of all normal sets. But since S is the set of all normal sets, S would then be abnormal. If S is abnormal, then S is a member of itself. But if S is a member of itself (the set of all normal sets), then S must be normal. So: S is a member of itself just in case it is not a member of itself, or, equivalently, S is normal just in case it is abnormal. This means that the sentence “S is a normal set” is simultaneously true and false.
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(We ask that readers convince themselves of this before continuing.) This was taken to be a serious problem with the theory of sets. In response to this problem, Russell introduced the vicious circle principle, which outlawed any sentence which was circular. This was later formalized as the theory of types, most of the details of which we do not need to be concerned with, but whose main effect was to make it illegal for sets to be members of themselves. According to the theory of types, that is, there are no abnormal sets, and there is also no set of all sets. Paradox solved. Poincar´e’s (1906) banishment of impredicative definitions from mathematics, which may be more familiar to some readers, serves a similar purpose. Roughly, a predicative definition applies to members of some domain so that its application is not altered by addition of new members to the domain; an impredicative definition, on the other hand, picks out different members of a domain should new individuals be added to the domain. The definition that picks out “the individuals in Ms. Riley’s class whose first names begins with P” is predicative, but the definition that picks out “the 4th tallest individual in Ms. Riley’s class” is impredicative. A new student’s arrival does not change the first letter of Peter’s name, but it may make him 5th tallest. Essentially, impredicative definitions pick out individuals or properties whose falling under that definition depend on other members of a set. Another way to put this is that impredicative definitions pick out individuals in a way that is context-dependent. “Abnormal set” and “the barber who shaves all and only those who do not shave themselves” are defined impredicatively, so outlawing impredicativities makes defining these types of set impossible. Again, paradox solved. The elimination of abnormal sets leaves us with standard set theory as taught in College-level logic and mathematics courses. This set theory is usually referred to as well-founded. There are two related senses in which standard set theory is well-founded. First, it is wellfounded in that its implementation does not lead to paradoxes; that is, it is logically well-founded. The second way in which it is well-founded comes from graph theory, a standard tool used to understand set membership. In graph theory, every set determines a tree-like graph in which the set itself is the upper-most node and each member of the set is a descendent node with an arrow pointing to it from the original set of which it is a member; this is applied recursively. Fig. 1 shows a graph for the set P = {a, b, c}. Notice that there is an arrow from the node representing the set P to each of its members. Fig. 2 shows a graph for the set Q = {a, b, {c}}. Notice that in this graph, there is an additional complexity in which the node for the member of Q, {c},
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Figs. 1–4. Graphs of (Hyper)sets described in text.
has an additional member, the object c. The sets P and Q are well-founded sets; they are not abnormal and do not contain themselves as members. The graphs of P and Q show a property common to the graphs of all wellfounded sets. Graphs of well-founded sets contain no infinite cycles or loops, and each path from the original set terminates in some object (its foundation, as it were). In fact, Aczel, 1988 proves that a graph will contain no cycles or loops if and only if it is well founded. The qualifier ‘if and only if’ means that a graph containing loops or cycles is a picture of a non-well-founded set. The presence of cycles and/or loops, that is, would indicate that some set has itself as a member, or that the concept, system, or definition it models is impredicative. (See Appendix A.) Consider, as an example, the abnormal set A = {A, b, c}. This will have the graph in Fig. 3. Aczel also defines the special abnormal set W as the set that contains only itself, W = {W}. W is graphed in Fig. 4. The sets A and W are not allowed in well-founded set theory, but they do have coherent and useful graphs. Aczel’s Anti-Foundation Axiom that embraces nonwell-founded (i.e., abnormal) sets is as follows: every graph, with or without loops, pictures a genuine set. Hypersets are defined as graphable sets. So well founded sets and non-well-founded sets are both types of hyperset. Non-well-founded sets were banished from set theory in the interest of the logical soundness of the theory,
and with the explicit goal of reducing mathematics to set theory, and set theory to logic. Re-introducing non-wellfounded sets into our set theory makes such reduction impossible. But this loss (if it really is a loss) is offset by the increased ability to model real-world phenomena. Work by Gupta (1982), Barwise and Etchemendy (1987), Gupta and Belnap (1993) and Barwise and Moss (1996) makes very clear that many concepts and real-world systems are circular or otherwise not-well-founded, hence illegal according to standard set theory. The most well known example is the concept ‘truth’, and the predicate ‘is true’. Gupta (1982) argues that the semantic paradoxes involving truth, such as the liar (“This very sentence is false”) and the truth-teller (“This very sentence is true”) show that the predicate “is true” can only be defined circularly, and develops a logical technique (the revision theory) for dealing with circular concepts. Later, Barwise and Etchemendy (1987) use hypersets to understand the truth predicate. In each of these cases, some object in a system is defined impredicatively, and is therefore illegal in standard set theory. Of course, the illegality of these systems and concepts in standard set theory does not change their behavior, or cause them to cease to exist. Such systems do exist. Adequately modeling them requires a richer set theory, such as hyperset theory, that allows and accounts for the behavior of systems with impredicativities. The graph theory that goes with hyperset theory will be especially important in what follows. As Aczel has
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proved and Figs. 1–4 demonstrate, graphs of hypersets can be used to determine whether the set pictured is or is not well-founded, and if it is not well-founded exactly what it is that makes it non-well-founded. These graphs, that is, can be used to determine whether systems contain impredicativities, and so are complex. The graphs also allow us to determine the precise source and nature of the complexity in such systems. We will use this feature of hypersets and their graphs to look at Rosen’s model of living things as metabolism-repair systems. 3. Rosen on Life Itself According to Rosen (1991, 2000), a central feature of living things is complexity. Rosen, it should be noted, has an idiosyncratic understanding of complexity. First, according to Rosen, whether a system is complex depends only secondarily on the system itself; it depends primarily on the system’s models. A physical system is complex if and only if it has impredicative models; otherwise, the system is simple. (As mentioned in Section 1 above, Rosen typically described this in terms of Turing-computability. We discuss this issue in Section 7.) This connects in important ways with the above discussion of impredicative definitions and Russell’s theory of types. Put most simply, Russell introduced the theory of types and Poincar´e outlawed impredicative definitions specifically to keep logic, set theory and mathematics well-founded. Thus systems that have models that disobey the theory of types or contain impredicativities – models, that is, that are not well-founded – are complex in Rosen’s sense. If this is so, we can use hyperset diagrams as a means of modeling systems that are complex. Indeed, we can use hyperset diagrams of models as complexity detectors. We can see this by examining Rosen’s metabolismrepair model of living systems (1991, 2000). According to Rosen, it is a necessary condition on living systems that their models contain three functions: a metabolic function, a repair function, and a replication function. We will discuss these in order. The metabolic function hi is a mapping from raw materials xi ∈ X to behavior yi ∈ Y. That is, it determines the way the system reacts to environmental input. A living system must also have a repair function that repairs the metabolic function to keep the system reacting appropriately to the environment. That is, it must have a function which maps behaviors yi ∈ Y onto the members of set H, where metabolic hi ∈ H. Finally, the repair function must itself be repaired. This repair is done by the system’s behavior. That is, each member yi of the set of behaviors Y is itself a replication function that maintains the repair function, by
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mapping hi onto ai ∈ A. This is, admittedly, complicated. The idea, though, is rather simple. The living system must be metabolizing consistently, so must maintain the causal processes that it uses in metabolizing. This is the job of the repair function. But the repair function must also be maintained, and this is done by the system’s behavioral output. Rosen’s diagrammatic representation of his model of living systems is shown in Fig. 5. Note that in this diagram a function P from Q to R is represented as ‘P’ connected to ‘Q’ with a solid arrow, and ‘Q’ connected to ‘R’ with a dashed arrow. Fig. 6 is a hyperset diagram of Rosen’s model of living systems. In Fig. 6, two common set-theoretic conventions are followed. First, functions are represented as ordered pairs. That is, if P is a function from Q to R, P is represented as P = Q, R. Second, to preserve their ordering, ordered pairs are represented as sets containing the set containing their first member and the set containing both their members. Thus Q, R is represented as {{Q}, {Q, R}}. In Fig. 6, the sets of inputs X, behaviors Y, metabolic functions H, and repair functions A are represented as if they were individuals. This is done to make the already-complicated diagram simpler. The thing to notice about the hyperset diagram in Fig. 6 is that there are loops, indicating that the sets are not well-founded. Indeed, in the graph there is only one location where the system does not loop back on itself: the set X of inputs from the environment. This makes sense of two important aspects of Rosen’s model of life. First, the loops indicate that Rosen’s diagram is indeed complex: it involves irremovable impredicativities. Rosen took just this kind of complexity to be one of the most important features of living systems, and often wrote of the impredicativities found in living systems. Fig. 6 bears this out. Second, the fact that the only place the diagram “bottoms out” is with raw materials is also important. Rosen often wrote that living systems are “closed to efficient causation”. In Rosen’s Aristotelian language, an ‘efficient cause’ matches up with our common sense intuition about the cause of an event. Thus, the efficient cause of a billiard ball moving is that it was struck by another billiard ball. When Rosen says that living systems are ‘closed to efficient cause’, he means that
Fig. 5. Rosen’s model of living systems (1991). Note that Y = H (X) is represented by a solid line from H to X and a dashed line from X to Y.
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Fig. 6. Hyperset diagram of Rosen’s metabolism-repair system. Functions are represented as ordered pairs containing their domain and range. So f (a) = b is represented as f = a, b.
all of the efficient causes of the system are produced by the system. Entities and events outside the system cannot be efficient causes of its behavior, they can only be raw material (Aristotelian ‘material causes’) for the system. We can see this in Fig. 5, Rosen’s own representation of living systems. In that diagram, only X, the raw material input from the environment, is not both (a) itself a function in the system and (b) the output of a function of the system. To put this in Rosen’s (and Aristotle’s) language, all functions but X are both efficient and material causes in the system. X is only a material cause. The hyperset diagrams make clear how this is the case. With this analysis of Rosen’s closure to efficient cause in hand, we can briefly consider two other analyses of autonomy: Maturana and Varela on autopoiesis and operational closure and Kauffmann and collective autocatalysis and catalytic closure. 4. On Operational Closure and Autopoiesis As a way to spell out what is distinctive about autonomous systems, Maturana, Varela and their colleagues (Maturana and Varela, 1980, 1987; Varela, 1979; Varela et al., 1974; Varela et al., 1991) define the notions operational closure and autopoiesis, and offer each as a base case of automonous system. We discuss these in order.
A system is operationally closed if and only if every effect of the system is also a part of the system. The one-dimensional cellular automaton Bittorio, discussed in Varela et al., 1991, is perhaps the clearest example of an operationally closed system. Bittorio is a onedimensional cellular automata, shaped like a ring. (See Fig. 7.) At each time step, the states of Bittorio’s cells are determined by the states of neighboring cells. Thus, left alone, the cells of Bittorio will pass through a series of states that form a self-organizing pattern. Because of this, Varela, Thompson and Rosch claim, Bittorio is autonomous. This is not to say, of course, that Bittorio is immune to influence from the environment. The environment can perturb Bittorio by changing the state of one of its cells, altering Bittorio’s dynamics, either temporarily or permanently depending on the nature of the perturbation and the rules governing Bittorio’s cells. So operational closure, like closure to efficient cause, does not imply separation from the environment. Although operationally closed, Bittorio is structurally coupled to its environment.
Fig. 7. The one-dimensional cellular automaton Bittorio.
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Fig. 8. Hyperset diagram of three cells of the cellular automaton Bittorio. Each cell’s state is a function of that of its neighbors. So B = F (A, C}. This is represented as F = {A, C},{B}.
Bittorio is operationally closed because the state of each of its cells has an effect only on the state of some of Bittorio’s other cells—all these effects are within Bittorio. But note that Bittorio is not closed to efficient cause. Bittorio’s operations (i.e., the rules that generate its behavior) are not produced by Bittorio, but are instead determined from outside, by a computer programmer. Fig. 8 is a hyperset diagram of three of Bittorio’s cells, which follows the same conventions as Fig. 6. Fig. 8 looks quite complicated, but it contains no impredicative loops: each of the system’s functions (F, G, and H) is independent of the others. This is not to say, however, that operational closure is not interestingly related to closure to efficient cause. A sub-class of operationally closed systems, the autopoietic systems (Maturana and Varela, 1973, Varela et al., 1974; Varela, 1979) are closed to efficient cause. Autopoietic systems are systems that are self-producing. They are operationally closed because
their effects are system-internal; in addition, though, the effects are system-constituting. Maturana and Varela (1973, 1980) claim that autopoiesis is necessary and sufficient for life. Thus, from their point of view, all operationally closed systems are autonomous and a subset of the operational closed systems, namely, the autopoietic systems, are living. The autopoietic system most commonly discussed is the cell. In a cell, the metabolism produces a cell membrane and the membrane provides a boundary, which plays a crucial role in the cell’s metabolism. This is depicted in Fig. 9, which follows the same set theoretic convention as Figs. 6 and 8. Note that the diagram has a form very similar to Rosen’s metabolism-repair system: every function in the system (the metabolism, the membrane) is produced by the system itself and only raw materials are not produced by the system. In other words, the diagram shows that autopoietic systems like the cell are closed to efficient cause.
Fig. 9. Hyperset diagram of the cell and cell membrane, a canonical autopoietic system.
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5. Collective Autocatalysis and Catalytic Closure Now consider another proposed minimal case of autonomy: catalytically closed systems (Kauffman, 1993, 1995, 2000). Kauffman introduces catalytic closure to argue that the origin of autonomous and, ultimately, living systems was not, as has been previously argued, improbable. Start by considering living systems as networks of chemical reactions. In a dilute solution (e.g., a pond), most chemical reactions proceed very slowly, so for reactions to proceed fast enough to keep a collection of them going, they must be catalyzed. But the likelihood that the catalyst for any particular chemical reaction just happens to be around is very small. For a network of chemical reactions to be autonomous, then, the network must produce its own catalysts. Kauffman shows, statistically, that for large enough networks of reactions it is overwhelmingly likely that reactions in the network will be catalyzed by the products of other reactions in the network. When every product in a network of chemical reactions has a catalyst that is also a product in the network, the network of reactions is a collectively autocatalytic system. Collectively autocatalytic systems are catalytically closed, and hence, autonomous. We can see this in Fig. 10, which is a hyperset diagram of the following (imaginary) catalytically closed set of reactions: A
R−→B B
S −→A In this set of reactions, chemicals R and S are raw materials. If they are in steady supply, these reactions will form a self-sustaining network. In the hyperset diagram, we represent this system by equating each product with the set reactants and catalysts that produce it. Thus the product A is {S, B}; B is {R, A}. In the figure, there is an impredicative loop involving catalysts A and B. The impredicativity indicates that collectively autocatalytic systems are complex systems. Notice also that in the
Fig. 10. Hyperset diagram of a collectively autocatalytic network of reactions. System products are represented as the set of reactants that produce them.
diagram only “bottoms out” at S and T, the raw materials. This is just as is the case with the metabolism-repair system and the autocatalytic systems described above. Like those systems, systems that are catalytically closed are closed to efficient cause. 6. Conclusions Concerning Hypersets and Autonomy The preceding leads to two different kinds of conclusions, those regarding hypersets as an analytic tool for understanding complex systems and those regarding artificial autonomy. We discuss the implications of our results for Rosen’s claims concerning the relationship among life, complexity and computability in Section 7. We take it that the hyperset analysis of Rosen’s model of living systems is useful for three reasons. First, it demonstrates some of Rosen’s claims about his model, viz., that it contains impredicativities and is closed to efficient causation. It also shows exactly what the relationship among parts of a system must be for it to be closed to efficient causation. We think, therefore, that the hyperset analysis is enlightening. Second, it allows us to see in some detail how closure to efficient cause differs from operational closure and catalytic closure. Third, and most importantly for our purposes, the hyperset diagrams show that such an analysis is useful and does in fact capture important properties concerning the complexity of models. We must be clear that hypersets are in no way intended as a replacement for analysis of complex systems with coupled differential equations. Because hyperset theory is a new mathematical tool, the mathematical relationship between hypersets and differential equations is still to be worked out by mathematicians. So whether it will be possible to use hypersets to model ordinary differential equations is an open question. Although we must wait for mathematicians to sort out these details of the relationship, we can say now that hyperset analyses are most useful in situations in which analysis with differential equations is inappropriate for one reason or another. Examples of this include the following: systems that are intractable using differential equations such as the animal-environment systems as studied in ecological psychology; systems about which not enough is known to develop differential equations such as the cell; and model systems intended as conceptual tools such as the Brusselator and the systems discussed above (metabolism-repair systems, autopoietic systems, catalytically closed systems). Elsewhere we have presented hyperset analyses of the Belouzov–Zhabotinsky reaction, the Brusselator
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and animal-environment systems (Chemero and Turvey, 2007). There are also several conclusions regarding autonomy and artificial life. First, if we assume that closure to efficient cause is the most basic kind of autonomy (note that we have given no particular reason to do so), then autopoietic and catalytically closed systems are autonomous. Second, we have seen that metabolism-repair systems that are closed to efficient cause, autopoiesis and catalytic closure are structurally similar when viewed in hyperset diagrams. The hyperset diagrams show that for each, every causally important part of a system is also a product of the system. From our point of view, Rosen’s metabolism-repair systems and Maturana and Varela’s autopoietic systems are valuable as characterizations of an abstract property, while Kauffman’s work is valuable for connecting this abstract property to chemical processes. So each characterization is useful for certain purposes, and one might develop an argument that one description of this abstract property is better than others. The hyperset diagrams show unequivocally that these systems share an important structural feature: having every function be a product of a system, having loopy hyperset diagrams that terminate only in raw materials. We propose that this feature be called ‘autonomy’. Finally, note that if, as we suggest, metabolismrepair, autopoietic, and catalytically closed systems are autonomous, there is reason to be skeptical that autonomy is sufficient for life. Rosen eventually (2000) admitted that being a metabolism-repair systems is merely necessary, but not sufficient for life. Many authors have made similar claims about autopoiesis. Recently, for example, Bourgine and Stewart (2004) claimed that cognition is also necessary; Ruiz-Mirazo and Moreno (2004), Ruiz-Mirazo and Mavelli (this issue) and Suzuki and Ikegama (this issue) claim that thermodynamically-realistic metabolism is also necessary. If any of these authors are right, artificial autonomy has already been achieved, but artificial autonomy is not the same thing as artificial life. 7. Conclusions Concerning Computability and Life The hyperset analysis of metabolism-repair and autopoietic systems is germane to the debate over the possibility of artificial life. Rosen’s central argument is that organisms will have impredicative models, which he takes to say that they have models that are not Turing computable, which is to say that they are not machines. We have demonstrated using hypersets that autopoietic
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and metabolism-repair systems have impredicative models. Following Rosen, one might conclude that these systems are not Turing-computable. Considering the nature of impredicativity (and non-well-founded sets), it is not hard to see why Rosen made this inference. A set S is Turing-computable if and only if a Turing machine given any input n from S’s domain halts with output 1 if n is in S and halts with 0 if n is not in S. Now consider the case of circularly defined sets. Suppose that S = {. . . A . . .} and A = {. . . S . . .}. A Turing machine T that can determine whether some n is a member of S will need to check the members of S to determine whether n is among them. In doing so, T will need to check the membership of each set that is a member of S. In this case, it will check the membership of A. To determine the membership of A, it will need to determine the membership of all the sets that are members of A, including S in this case. This, it seems, leads to an infinite loop, which would mean that T does not halt. So it would seem that S and A are not Turing-computable. Another reason to believe that circularly defined sets are not Turing-computable comes from experience writing computer programs. Any decent compiler will give you error messages if you write a program with circularities. For these reasons, in earlier papers (Chemero and Turvey, 2007; Turvey, 2004), we followed Rosen in taking impredicatively defined sets to be non-Turing-computable. We have since learned that there is no necessary connection between impredicative definitions and nonTuring-computability. For example, work by Martin-L¨of (1970) and Girard (1972) on impredicative types has been extremely valuable in computer science. More importantly, hyperset membership has been shown to be Turing-computable in polynomial time (Lisitsa and Sazonov, 1999; Sazonov, 2006), a result anticipated in the pioneering work of Barwise and Moss (1996). Deciding on hyperset membership therefore does not necessarily involve infinite loops with their unwelcome consequences of adding infinitely many copies of elements to sets and an inability to terminate. Thus Rosen’s claim that “there is no algorithm for building something that is impredicative” (2000, p. 294) is incorrect (to the extent that ‘to build’ and ‘to compute’ are overlapping conceptions). The question at this point is to what extent the Turing-computability of impredicative sets, whether modeled as hypersets or not, affects Rosen’s most important contributions. Rosen claimed that impredicative systems are not computable. We have shown that impredicative systems are usefully modeled as hypersets, and hypersets are Turing-computable. In sum, we have shown that Rosen’s (2000) claim that impredicative systems are not Turing-
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computable is no longer tenable, which undercuts the justification for his conclusion that models of living systems are not computable. This agrees with the arguments of Chu and Ho (2006), who dispute Rosen’s purported proof that artificial life is impossible, and Wells (2006), who argued that Rosen’s metabolism-repair systems are computable. Nonetheless, we believe that much of the spirit of Rosen’s work can be maintained. We can see this by realizing that Rosen’s characterization of the distinction between simple and complex systems is multi-dimensional. (Gwinn (2006) lists eight ways in which Rosen makes this distinction.) In distinguishing between simple and complex systems, Rosen argued that simple systems are impoverished when compared to complex systems. Among these ways in which simple systems are impoverished are the following: (i) complex systems have models that are computable and non-computable, while simple systems only have models that are computable; (ii) complex systems have models that are impredicative and models that are predicative, while simple systems have only predicative models; (iii) complex systems can have fractionable and nonfractionable structures, while simple systems can have only fractionable structures; (iv) complex systems can have both linear chains and closed loops of entailment, while simple systems can only have linear chains of entailment; (v) complex systems have both syntactic and semantic elements, while simple systems have only syntactic elements. (See Gwinn, 2006.)
resented in their models. More precisely, autonomous systems are closed to efficient cause and so must have impredicative models. We have shown in this paper that Rosen’s metabolism-repair systems and two other candidates for minimal autonomous systems (autopoietic and catalytically closed systems) are in fact closed to efficient cause and, so, must have impredicative models. We have also seen that these impredicative models are Turing-computable. Indeed, for one of these candidates, autopoietic systems, there has been a long and fruitful history of computational modeling (Varela et al., 1974; McMullin and Varela, 1997, and Bourgine and Stewart, 2004). We believe that those interested in pursuing research along the lines suggested by Rosen should view this as an opportunity. Computational modeling using hypersets could become an important tool for studying non-fractionable, circularly causal, partly semantic systems such as Rosen’s metabilism-repair system. Acknowledgements Larry Moss provided invaluable advice on hypersets. John Stewart, Xabier Barandiaran, Abhey Shah and Michael Silberstein gave helpful advice on earlier drafts, as did an anonymous referee. Chemero’s work was supported by a National Science Foundation grant SBR 00-04097 and the Mellon Foundation/Central Pennsylvania Consortium. Turvey was supported by National Science Foundation grants SBR 00-04097 and SBR 0423036. Appendix A. Using Hyperset Models
As we have seen, Rosen’s thinking that (ii) entails (i) is no longer justified. But Rosen’s great insights were in seeing that complex systems have non-fractionable structures, loopy causal entailment, and built-in semantics, and in seeing that this leads to models of complex systems with impredicativities. Future research on Rosen-related themes must focus on the contrasts between simple and complex systems in points (ii)–(v) above. We contend that the central idea of Rosen’s work is not that artificial life or valuable computational models of autonomous systems are impossible. Very little rides on whether genuine artificial life is possible: we agree with Moreno, Extebarria and Umerez (this issue) that the real value of computational research is in what it teaches us about biological autonomy. Instead, the most important point of Rosen’s work is that autonomous systems have a structure that is very different from non-living things and that this structure must be rep-
Scattered throughout this paper are comments that, taken together, should make it possible for interested readers to use hypersets to model systems of interest to them. The purpose of this appendix is to provide explicit instructions for using hypersets, all in one place. Step 1: Think of the system of interest S in terms of a group of functions, mapping inputs to outputs. In most cases, there is no algorithm for coming with the right set of functions for your system. It will also typically be the case that there are many different sets of functions that capture the causal structure of the system. Step 2: Represent each of these functions as an ordered pair containing the input as the first member and the output as the second. Thus, f (x) = y
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Fig. 11. Graph of sample set.
becomes: f = x, y. Step 3: Represent each ordered pair as a Kuratowska pair, a set including the singleton set of the first member of the ordered pair and the set of both elements of the ordered pair. Thus, f = x, y becomes f = {{x}, {x, y}}. Representing the ordered pair this way makes it into a graphable set, yet marks the ordering of the ordered pair. Step 4: Graph the set. To do so, draw an arrow from each set to each of its members. Thus, one draws lines from the function f = {{x}, {x, y}} to {x} and {x, y}. Then one draws and line from {x} to x and lines from {x, y} to x and y. The result is Fig. 11. Step 5: Analyze the graph. If for any function g, chasing arrows from the function to its members, then to their members and so on leads back to g, the model is impredicative and the system modeled is complex. If every function of the system is the output of another function, the system modeled is closed to efficient cause. We have argued in this paper that this also makes the system autonomous. References Aczel, P., 1988. Non-Well-Founded Sets. CSLI, Stanford. Barwise, J., Etchemendy, J., 1987. The Liar: An Essay on Truth and Circularity. Oxford University Press, New York.
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Barwise, J., Moss, L. (Eds.), 1996. Vicious Circles. CSLI,Stanford. Bourgine, P., Stewart, J., 2004. Autopoiesis and cognition. Artif. Life 10, 327–345. Chemero, A., Turvey, M.T., 2007. Hypersets, complexity and ecological approach to perception-action. Biol. Theory 2 (1), 23–36. Chu, D., Ho, W., 2006. A category theoretical argument against the possibility of artificial life: Robert Rosen’s central proof revisited. Artif. Life 12 (1), 117–134. Girard, J.Y., 1972, Interpretation fonctionelle et eleimination des coupures dans l’arithmetique d’ordre superieure, These d’Etat, Universit´e Paris 7. Gupta, A., 1982. Truth and paradox. J. Philos. Logic 11, 1– 60. Gupta, A., Belnap, N., 1993. The Revision Theory of Truth. MIT Press, Cambridge. Gwinn, T. 2006. Rosennean Complexity. Robert Rosen: Complexity and Life Web Page. URL=
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