This article revisits the concept of autopoiesis and examines its relation to cognition and life. We present a mathematical model of a 3D tesselation automaton, considered as a minimal example of autopoiesis. This leads us to a thesis T1: “An autopoietic system can be described as a random dynamical system, which is defined only within its organized autopoietic domain.” We propose a modified definition of autopoiesis: “An autopoietic system is a network of processes that produces the components that reproduce the network, and that also regulates the boundary conditions necessary for its ongoing existence as a network.” We also propose a definition of cognition: “A system is cognitive if and only if sensory inputs serve to trigger actions in a specific way, so as to satisfy a viability constraint.” It follows from these definitions that the concepts of autopoiesis and cognition, although deeply related in their connection with the regulation of the boundary conditions of the system, are not immediately identical: a system can be autopoietic without being cognitive, and cognitive without being autopoietic. Finally, we propose a thesis T2: “A system that is both autopoietic and cognitive is a living system.”

This article focuses on an artificial life approach to some important problems in machine learning such as statistical discrimination, curve approximation, and pattern recognition. We describe a family of models, collectively referred to as semi-algebraic networks (SAN). These models are strongly inspired by two complementary lines of thought: the biological concept of autopoiesis and morphodynamical notions in mathematics. Mathematically defined as semi-algebraic sets, SANs involve geometric components that are submitted to two coupled processes: (a) the adjustment of the components (under the action of the learning examples), and (b) the regeneration of new components. Several examples of SANs are described, using different types of components. The geometric nature of SANs gives new possibilities for solving the bias/variance dilemma in discrimination or curve approximation problems. The question of building multilevel semi-algebraic networks is also addressed, as they are related to cognitive problems such as memory and morphological categorization. We describe an example of such multilevel models.