Closure to efficient causation, computability and artificial life☆
Introduction
Studies of artificial life depend heavily on efforts to set up and simulate models of living organisms in the computer. According to Rosen (1991), however, a living organism is not a machine, and so it cannot have a computer-simulable model. Not surprisingly, his conclusion has stimulated an intense argument among computer scientists, mathematicians and biologists (Landauer and Bellman, 2002, McMullin, 2004, Wells, 2006, Chu and Ho, 2006; Chu and Ho, 2007a, Chu and Ho, 2007b; Louie, 2007, Wolkenhauer, 2007, Wolkenhauer and Hofmeyr, 2007, Mossio et al., 2009), because if it is valid it imposes a formidable barrier to modern theories of computation in a topic that is as central to our scientific endeavour as it is to the nature of living systems. It is important to emphasize at the outset that Rosen did not argue that artificial life was impossible,1 but only that organisms are “closed to efficient causation” and that this essential property excludes any possibility of simulable models.
The numerous new papers cited above that deal with the issue of computability make it necessary to examine the controversy. We are convinced that full understanding of Rosen's work requires study of more than just the well known closure diagram in Life Itself (Rosen, 1991, Fig. 10C.6 ; Fig. 1a). His argument against computability requires a detailed analysis of metabolic closure as set out in a series of papers spanning 15 years (Rosen, 1958a, Rosen, 1958b, Rosen, 1959, Rosen, 1966, Rosen, 1971, Rosen, 1973). Progress in the matter of computability requires a thorough knowledge of the conceptual steps that Rosen took in this regard (for example the distinction that he made between simulation and modelling), and not just the summary encapsulated by the diagram.
A recent analysis in terms of and the theory of computer programming (Mossio et al., 2009) led to the opposite conclusion, that a system closed to efficient causation can certainly have computable models. The authors pointed out the apparent contradiction that autopoiesis (Maturana and Varela, 1980), which has strong underlying similarities with Rosen's theory (Letelier et al., 2003), including closure to efficient causation, is claimed to have computable models (McMullin, 2004). Moreover, it is not obvious that the examples of (M, R)-systems that we have proposed (Letelier et al., 2004, Letelier et al., 2006, Cornish-Bowden et al., 2007; Cornish-Bowden and Cárdenas, 2007, Cornish-Bowden and Cárdenas, 2008) cannot be simulated. Other criticisms of Rosen's analysis also deserve to be answered, as they affect his insights in relation to closure, which we regard as essential for understanding living systems.
How does the existence, or not, of computable models of living organisms affect other aspects of the study of life? According to Chemero and Turvey (2008), “little rides on whether genuine artificial life is possible”, and we agree: the existence of simulable models of living organisms has less importance than Rosen's essential insight, that organisms must be closed to efficient cause and hence metabolically closed. However, the argument about simulability will certainly continue: the work of many groups, including those attempting to develop life in silico, depends on the assumption that computer simulation of living systems is in principle possible, and any claims that it is not possible can expect to meet vehement opposition.
We believe that the way forward, both for getting a better understanding of life and for deciding whether it can have simulable models, will require Rosen's abstract and mathematical ideas to be brought into much closer correspondence with biological reality. Future models need not only to reflect the mathematics accurately but they must also be biochemically reasonable.
Section snippets
Closure to efficient cause
Closure to efficient cause, illustrated in Fig. 1, is central to Rosen's view of life, and we shall briefly resume what it means. Rosen drew the diagram, for example in Fig. 10C.6 of Rosen (1991), as in Fig. 1a. Cottam et al. (2007) have argued that the underlying logic is symmetrical and can be better illustrated with a symmetrical figure-of-eight layout, as in Fig. 1b. Their arrangement is visually appealing but it suggests a misleading parallelism between efficient and material causation,
Analysis of closure in terms of hypersets
Hypersets are generalized sets in which the restriction that sets cannot be members of themselves is relaxed. This restriction was made at the beginning of the 20th century as a way of resolving Russell's paradox and the problems of ambiguity that arise when impredicative definitions are permitted, i.e. definitions that allow the entity being defined to participate in its own definition. However, the impredicativity that is central to Rosen's view of an organism does not prevent it from being
Autocatalytic sets and autopoiesis
There is a fundamental difference between autocatalytic sets (Kauffman, 1993, Kauffman, 1986) and autopoiesis (Maturana and Varela, 1980), and it is necessary to understand this because of the implications of these approaches to the origin of life. For Kauffman an autocatalytic set is inevitably a large set, with, as a minimum, thousands of elements based on amino acids or RNA bases, because only large systems can have the statistical properties needed for closure to become virtually
Are autopoietic systems really computable?
As Letelier et al. (2003) pointed out, autopoietic systems share many features of (M,R)-systems and can be regarded as a subset of (M,R)-systems. They are thus by implication closed to efficient causation, and should inherit the property of not having computable models.5 However, it has been claimed that autopoietic systems have been modelled (McMullin, 2004),6
(M,R)-systems considered in terms of Cartesian closed categories
In developing their abstract cell model of a living organism, Wolkenhauer and Hofmeyr (2007) state, but do not prove, that the category needed for a mathematical model of a self-organizing cell must be Cartesian closed. In essence this means that the category behaves like the category of sets and mappings regarding the relationship between functions of two variables and functions of one variable. It is well known, indeed, that the graph of a function of two variables, such as , can be
Simulation and modelling
Simulating an organism and creating a model of an organism may appear to be the same thing, and so it is important for discussing Rosen's work to emphasize that he attached clearly different meanings to these two ideas. There are two key issues in the notion of simulation, the first of which is the possibility of developing by means of a computer program a sequence of steps that behave identically to the phenomenon to be simulated. In general this simulation could not give any information about
Rosen's analysis
The conclusion of Mossio et al. (2009) that (M,R)-systems can have computable models is based on an analysis of the fundamental equations of (M,R)-systems in terms of the theory of computer programming, specifically in terms of . Their analysis omits an essential part of the argument, however, and arrives in consequence at a result that we contest. As we discussed previously (Letelier et al., 2006) the summary of Rosen's system shown in Fig. 1 can be expressed in mathematically much
(M,R)-systems
Although there has been a considerable resurgence of interest in Rosen's view of organisms in recent years, a large part of the discussion has focussed on his diagram in Fig. 10C.6 of his book (Rosen, 1991), corresponding to Fig. 1 a of the present paper. However, as we have emphasized in the Introduction, full understanding of Rosen's work cannot be obtained from a single diagram, and we, following an idea of Morán et al. (1996), have been exploring the characteristics of a very small
Conclusions
Efforts to mathematically disprove Rosen's contention that an organism cannot have simulable models have not resolved the question. Louie (2007) has been highly critical of some of the arguments (Chu and Ho, 2006), and, as we have discussed in Section 3, there are problems also with some of the others. Other supposed contradictions can be attributed to the use of loose definitions in place of Rosen's very precise ones. As noted above, for example, the definition of computability used by Mossio
Acknowledgements
This work was supported by Fondecyt 1030371 (JCL), Fondecyt 1070246 (JSA) and the CNRS (AC-B, MLC).
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2020, BioSystemsCitation Excerpt :To understand this it is important to realize that he made a crucial distinction between modelling, which he considered impossible, and simulation, which he considered possible. He did not regard simulation as the same as modelling, and to understand his theoretical ideas it is important to keep the two concepts separate, as we have tried to explain elsewhere (Cárdenas et al., 2010; Cornish-Bowden et al., 2013). For him a model of, for example, a machine incorporates understanding of how the machine works.
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All authors contributed equally to the work.