ALIFE XV Late Breaking Abstract

A set theoretic analysis shows different forms of closure in autopoietic theory, computational autopoiesis, and (M,R) systems

Dobromir Dotov and Tom Froese

There has been an enduring stance in artificial life that a notion of circular entailment is required in order to appropriately capture the self-maintenance of living systems. The detailed character of this circular entailment can differ across approaches. The recursive operation of dynamical systems, control feedback loops of cybernetics, operational closure of autopoiesis, top-down–bottom-up mutual constraint, and closure to efficient cause of Rosenian complexity are all forms of circular entailment. In the latter case, a living system is understood as a collection of biological processes such as metabolism and repair whereby the system contains within itself the efficient causes of the making of these processes. Each process is the produce of another process and these are contained in a closed set. In this sense a living system must exhibit closure to efficient cause.

How to determine the correspondence among these notions of circular cause and effect given that the different theoretical approaches have rarely been compared formally? A comparative approach is necessary in order to help consolidate the theory in artificial life. It has been proposed that set theory is both sufficiently abstract and rigorous to serve as framework for formalizing various domain-specific systems in nature such as particular social or biological systems as abstract cross-domain general systems (Mesarovic and Takahara, 1975). As a demonstration of what this could mean and how to achieve it, here we apply the suggested tools of set theory (Chemero and Turvey, 2008) in order to compare in a formal way autopoiesis, Randall Beer’s GoL computational model of autopoiesis (Beer, 2014), and Robert Rosen’s complexity.

The outcome of this analysis is that, unlike autopoiesis and complexity, the computational model appears to possess operational closure but not closure to efficient cause. This result is not surprising. Beer (2015, p.17) acknowledges that GoL is a model that reproduces certain properties of autopoiesis, not an instance of life. This result should also not be taken as diminishing the importance of simulation as a method of theoretical modeling. At a theoretical level, operational closure is a more encompassing condition than closure to efficient causation. In conclusion, the field of artificial life would benefit from the existence of comparative work synthesizing the various conditions (”closures”) necessary for living systems.

Beer, R. D. (2014). The cognitive domain of a glider in the game of life. Artificial Life, 20: 183–206.
Beer, R. D. (2015). Characterizing autopoiesis in the game of life. Artificial Life, 21(1):1–19.
Chemero, A. and Turvey, M. T. (2008). Autonomy and hypersets. BioSystems, 91(2):320–30.
Mesarovic, M. and Takahara, Y. (1975). General Systems Theory: Mathematical Foundations. Academic Press, New York, NY.


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