Transformation semigroups as constructive dynamical spaces

The informal notion of constructive dynamical space, inspired by biochemical systems, gives the perspective from which a transformation semigroup can be considered as a programming language. This perspective complements a longer-term mathematical investigation into different understandings of the nature of computation that we see as fundamentally important for the realization of a formal framework for interaction computing based on algebraic concepts and inspired by cell metabolism. The interaction computing perspective generalizes further the individual transformation semigroup or automaton as a constructive dynamical space driven by programming language constructs, to a constructive dynamical 'meta-space' of interacting sequential machines that can be combined to realize various types of interaction structures. This view is motivated by the desire to map the self-organizing abilities of biological systems to abstract computational systems by importing the algebraic properties of cellular processes into computer science formalisms. After explaining how semigroups can be seen as constructive dynamical spaces we show how John Rhodes's formalism can be used to define an Interaction Machine and provide a conceptual discussion of its possible architecture based on Rhodes's analysis of cell metabolism. We close the paper with preliminary results from the holonomy decomposition of the semigroups associated with two automata derived from the same p53-mdm2 regulatory pathway being investigated in other papers at this same conference, at two different levels of discretization.