Self-organising behaviour in biological systems is a consequence of regularities of physical laws. The concept of regularity can be made more precise by relying on the mathematical concept of symmetry. Since the mathematical formalisation of invariance is expressed through the language of algebra, IC research has greatly benefited from algebraic automata theory. The computational interpretation of the algebraic structure of automata derived from various biochemical systems (for example the presence of simple non-abelian groups, or SNAGs, as functionally complete algebras (Maurer and Rhodes, 1965; Horváth, 2008)) is far from obvious. Therefore, one of the aims of BIOMICS is to compare well-understood features of dynamical systems, such as their Lie group structure (Olver, 1993), to features of the algebraic structure of the automata derived from the same dynamical systems (e.g. Chevalley, 1955) in the hope that the former will shed light on the latter, as shown schematically in** Figure 1**

**Figure 1. Algebraic and logic structure underpinning BIOMICS research**

In order to be able to specify the desired behavioural properties of biologically-inspired interaction computing software systems, we need to be able to identify the primitives of a specification language capable of expressing them. Category theory can help in this direction, in two steps. First, we will use category theory to develop a unifying mathematical framework for these phenomena, including the dynamical system and automaton structure, through the natural duality between algebras and coalgebras, and the adjunction relationships between behaviour and its realisation (Goguen, 1972; Rhodes, 2010; Plotkin,

2004). Second, the already well-established correspondence between coalgebras and temporal logics (Andreka et al., 2009) opens the possibility of developing an “environment specification” language, which can be seen as a higher-abstraction software engineering specification language addressing both the structure and content of bio-inspired digital systems. The right-hand side of Figure 1 shows how category theory provides a general framework that supports the mapping of results found in dynamical systems and automata

theory to the field of specification languages and logic.

The links between the above theoretical framework and applications in software engineering and systems biology are shown schematically in **Figure 2.**

**Figure 2: General framework for theoretical and applied Interaction Computing research**

**WP1-Algebraic Structure of Continuous and Discrete Models of Cellular Pathways**

**WP2-Category Theory Framework**

**WP3-Mathematical Model of Interaction Computing**

**WP4-Environment Specification Language**

**WP5-Proof-of-Concept Prototype of Interaction Computing Environment**

**WP6-Dissemination and Collaboration**

IC takes inspiration from cellular processes – regulatory and metabolic pathways – rather than from evolution, neural processes, or population dynamics. Since cellular processes have evolved to fulfil many requirements simultaneously and involve very large quantities of interacting components, they embody architectural and algorithmic features that are optimised for performing massively parallel asynchronous computation. At a fundamental level, this can be explained by the fact that proto-evolutionary processes in the primordial soup were able to harness *interactions *as the driver of the fundamental information processing architecture that underpins all life forms. Rather than reproducing origin-of-life scenarios as the phylogenetic basis of an evolutionary framework, however, BIOMICS aims to use the experience gained from millions of years of evolutionary trial and error by leveraging *existing *cell metabolic and regulatory mechanisms as the ontogenetic basis of a model for IC. However, because the knowledge to properly mimic, exploit and adapt these systems to computer science is lacking, BIOMICS will also advance the state of the art in the mathematics of bio-computing in significant and fundamental ways.

Thus, BIOMICS benefits from the merging of three research traditions:

- dynamical systems theory rooted in physics and informing much of modern-day systems biology;
- theoretical computer science rooted in algebraic automata theory;
- and formal specification languages rooted in logic.

The whole is held together by a category theory framework and leads to a model of interaction computing that will be implemented as a proof-of-concept prototype. On the one hand, BIOMICS’s conceptual framework for IC has been developed mainly from a statistical physics, systems biology, dynamical systems, logic, and security point of view in a number of EU projects (DBE, BIONETS, OPAALS, EINS) since 2003 (Dini and Schreckling, 2010; Horváth and Dini, 2010). On the other hand, the concept of interacting automata stretches back to the origins of algebraic automata theory in the 1960s (Krohn and Rhodes, 1965; Rhodes, 2010), and has greatly benefited from very recent work in the same mathematical tradition (Nehaniv and Rhodes, 2000; Dömösi and Nehaniv, 2005; Egri-Nagy and Nehaniv, 2008a; Egri-Nagy and Nehaniv 2008b; Egri-Nagy et al., 2008; Egri-Nagy and Nehaniv, 2010; Dini et al., 2012).There has been a recent renaissance in the study of metabolism and metabolic pathways. (Janes and Yaffe, 2006; McKnight, 2010; Bar-Even et al., 2012). The knowledge that is being acquired concerning the complex interactions and regulatory mechanisms involved in maintaining cellular homeostasis may usefully be transferred to a consideration of the principles involved in IC. Whereas the automata theory research will focus on the structural properties of self-organising systems, the BIOMICS specification language will instead focus on the specification of self-organising behaviour.

The Aim of BIOMICS is to develop the formal tools and frameworks from both points of view of the behavior realisation dichotomy (Goguen, 1972) to be able to effect their synthesis in the form of an environment which, through interactions, is capable of generating useful software systems that match the biological structure template – and are therefore themselves based on interactions. This foundational mathematical work of BIOMICS will be applicable to software systems of a radically new kind and to systems biology, creating a unified mathematical framework for understanding, predicting, manipulating, and dynamically synthesising algorithmic activity-in-context based on interactions (i.e. interaction computation) in both realms. This will be demonstrated not only by the application of the framework to the analysis of complex adaptive biological systems beyond those studied in the course of its development, but also by proof-of concept implementations of software systems (for example demonstrating security properties) as a potential new paradigm for Unconventional Computing.