Hierarchical coordinate systems for understanding complexity and its evolution, with applications to genetic regulatory networks

Formalizing falsification for theories of consciousness across computational hierarchies

The scientific study of consciousness is currently undergoing a critical transition in the form of a rapidly evolving scientific debate regarding whether or not currently proposed theories can be assessed for their scientific validity. At the forefront of this debate is Integrated Information Theory (IIT), widely regarded as the preeminent theory of consciousness because it quantified subjective experience in a scalar mathematical measure called Φ that is in principle measurable. Epistemological issues in the form of the "unfolding argument" have provided a concrete refutation of IIT by demonstrating how it permits functionally identical systems to have differences in their predicted consciousness. The implication is that IIT and any other proposed theory based on a physical system's causal structure may already be falsified even in the absence of experimental refutation. However, so far many of these arguments surrounding the epistemological foundations of falsification arguments, such as the unfolding argument, are too abstract to determine the full scope of their implications. Here, we make these abstract arguments concrete, by providing a simple example of functionally equivalent machines realizable with table-top electronics that take the form of isomorphic digital circuits with and without feedback. This allows us to explicitly demonstrate the different levels of abstraction at which a theory of consciousness can be assessed. Within this computational hierarchy, we show how IIT is simultaneously falsified at the finite-state automaton level and unfalsifiable at the combinatorial-state automaton level. We use this example to illustrate a more general set of falsification criteria for theories of consciousness: to avoid being already falsified, or conversely unfalsifiable, scientific theories of consciousness must be invariant with respect to changes that leave the inference procedure fixed at a particular level in a computational hierarchy.

Cayley Graphs of Semigroups Applied to Atom Tracking in Chemistry

While atom tracking with isotope-labeled compounds is an essential and sophisticated wet-lab tool to, for example, illuminate reaction mechanisms, there exists only a limited amount of formal methods to approach the problem. Specifically, when large (bio-)chemical networks are considered where reactions are stereospecific, rigorous techniques are inevitable. We present an approach using the right Cayley graph of a monoid to track atoms concurrently through sequences of reactions and predict their potential location in product molecules. This can not only be used to systematically build hypothesis or reject reaction mechanisms (we will use the ANRORC mechanism "Addition of the Nucleophile, Ring Opening, and Ring Closure" as an example) but also to infer naturally occurring subsystems of (bio-)chemical systems. Our results include the analysis of the carbon traces within the tricarboxylic acid cycle and infer subsystems based on projections of the right Cayley graph onto a set of relevant atoms.

Combining Graph Transformations and Semigroups for Isotopic Labeling Design

The double pushout approach for graph transformation naturally allows an abstraction level of biochemical systems in which individual atoms of molecules can be traced automatically within chemical reaction networks. Aiming at a mathematical rigorous approach for isotope labeling design, we convert chemical reaction networks (represented as directed hypergraphs) into transformation semigroups. Symmetries within molecules correspond to permutations, whereas (not necessarily invertible) chemical reactions define the transformations of the semigroup. An approach for the automatic inference of informative labeling of atoms is presented, which allows to distinguish the activity of different pathway alternatives within reaction networks. To illustrate our approaches, we apply them to the reaction network of glycolysis, which is an important and well-understood process that allows for different alternatives to convert glucose into pyruvate.

Integrated Information Theory and Isomorphic Feed-Forward Philosophical Zombies

Any theory amenable to scientific inquiry must have testable consequences. This minimal criterion is uniquely challenging for the study of consciousness, as we do not know if it is possible to confirm via observation from the outside whether or not a physical system knows what it feels like to have an inside—a challenge referred to as the “hard problem” of consciousness. To arrive at a theory of consciousness, the hard problem has motivated development of phenomenological approaches that adopt assumptions of what properties consciousness has based on first-hand experience and, from these, derive the physical processes that give rise to these properties. A leading theory adopting this approach is Integrated Information Theory (IIT), which assumes our subjective experience is a “unified whole”, subsequently yielding a requirement for physical feedback as a necessary condition for consciousness. Here, we develop a mathematical framework to assess the validity of this assumption by testing it in the context of isomorphic physical systems with and without feedback. The isomorphism allows us to isolate changes in Φ without affecting the size or functionality of the original system. Indeed, the only mathematical difference between a “conscious” system with Φ > 0 and an isomorphic “philosophical zombie” with Φ = 0 is a permutation of the binary labels used to internally represent functional states. This implies Φ is sensitive to functionally arbitrary aspects of a particular labeling scheme, with no clear justification in terms of phenomenological differences. In light of this, we argue any quantitative theory of consciousness, including IIT, should be invariant under isomorphisms if it is to avoid the existence of isomorphic philosophical zombies and the epistemological problems they pose.

Symmetry structure in discrete models of biochemical systems: natural subsystems and the weak control hierarchy in a new model of computation driven by interactions

Interaction computing is inspired by the observation that cell metabolic/regulatory systems construct order dynamically, through constrained interactions between their components and based on a wide range of possible inputs and environmental conditions. The goals of this work are to (i) identify and understand mathematically the natural subsystems and hierarchical relations in natural systems enabling this and (ii) use the resulting insights to define a new model of computation based on interactions that is useful for both biology and computation. The dynamical characteristics of the cellular pathways studied in systems biology relate, mathematically, to the computational characteristics of automata derived from them, and their internal symmetry structures to computational power. Finite discrete automata models of biological systems such as the lac operon, the Krebs cycle and p53-mdm2 genetic regulation constructed from systems biology models have canonically associated algebraic structures (their transformation semigroups). These contain permutation groups (local substructures exhibiting symmetry) that correspond to 'pools of reversibility'. These natural subsystems are related to one another in a hierarchical manner by the notion of 'weak control'. We present natural subsystems arising from several biological examples and their weak control hierarchies in detail. Finite simple non-Abelian groups are found in biological examples and can be harnessed to realize finitary universal computation. This allows ensembles of cells to achieve any desired finitary computational transformation, depending on external inputs, via suitably constrained interactions. Based on this, interaction machines that grow and change their structure recursively are introduced and applied, providing a natural model of computation driven by interactions.

Effective theories for circuits and automata

Abstracting an effective theory from a complicated process is central to the study of complexity. Even when the underlying mechanisms are understood, or at least measurable, the presence of dissipation and irreversibility in biological, computational, and social systems makes the problem harder. Here, we demonstrate the construction of effective theories in the presence of both irreversibility and noise, in a dynamical model with underlying feedback. We use the Krohn-Rhodes theorem to show how the composition of underlying mechanisms can lead to innovations in the emergent effective theory. We show how dissipation and irreversibility fundamentally limit the lifetimes of these emergent structures, even though, on short timescales, the group properties may be enriched compared to their noiseless counterparts.

Automatic analysis of computation in biochemical reactions

We propose a modeling and analysis method for biochemical reactions based on finite state automata. This is a completely different approach compared to traditional modeling of reactions by differential equations. Our method aims to explore the algebraic structure behind chemical reactions using automatically generated coordinate systems. In this paper we briefly summarize the underlying mathematical theory (the algebraic hierarchical decomposition theory of finite state automata) and describe how such automata can be derived from the description of chemical reaction networks. We also outline techniques for the flexible manipulation of existing models. As a real-world example we use the Krebs citric acid cycle.