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We are concerned with the behaviour of stochastic systems with inputs and outputs, and how this might relate to the pursuit of a goal. We model this using what we term transducers, which are a mathematical object that captures only the external behaviour of such a system and not its internal state. We present a framework for reasoning about the optimality of such a process, when it is coupled to a 'teleo-environment' consisting of another transducer that also embodies a success criterion. We find that (globally) optimal transducers have a property closely related to Bellman's theorem: a transducer that is optimal in one time step will again be optimal in the next time step, but with respect to a different environment (obtained from the original one by a modified version of Bayesian filtering). We also consider bounded rationality and its relationship to constrained optimality, which in our framework means optimal within some subset of all transducers. We describe a condition that is sufficient for such a subset to have this Bellman property. Additionally, we show that a policy is deterministic if and only if there exists a teleo-envionment for which it is uniquely optimal among the set of all transducers; this is at least conceptually related to classical representation theorems from decision theory. This need not hold for constrained subsets; we give an example of this related to the so-called absent-minded driver problem. All of the formalism is defined using coinduction, following the style proposed by Czajka [9]., Comment: 54 pages, 1 figure

We elucidate the mathematical structure of Bayesian filtering, and Bayesian inference more broadly, by applying recent work on category theoretical probability, specifically the concept of a strongly representable Markov category. We show that filtering, along with related concepts such as conjugate priors, arise from an adjunction: the process of taking a hidden Markov process is right adjoint to a forgetful functor. This has an interesting consequence. In practice, filtering is usually implemented using parametrised families of distributions. The Kalman filter is a particularly important example, which uses Gaussians. Rather than calculating a new posterior each time, the implementation only needs to udpate the parameters. This structure arises naturally from our adjunction; the correctness of such a model is witnessed by a map from the model into the system being modelled. Conjugate priors arise from this construction as a special case. In showing this we define a notion of unifilar machine, which has its origins in the literature on epsilon-machines. Unifilar machines are useful as models of the "observable behaviour" of stochastic systems; we show additionally that in the Kleisli category of the distribution monad there is a terminal unifilar machine, and its elements are controlled stochastic processes, mapping sequences of the input alphabet probabilistically to sequences of the output alphabet., Comment: In Proceedings ACT 2023, arXiv:2312.08138

Under what circumstances can a system be said to have beliefs and goals, and how do such agency-related features relate to its physical state? Recent work has proposed a notion of interpretation map, a function that maps the state of a system to a probability distribution representing its beliefs about an external world. Such a map is not completely arbitrary, as the beliefs it attributes to the system must evolve over time in a manner that is consistent with Bayes' theorem, and consequently the dynamics of a system constrain its possible interpretations. Here we build on this approach, proposing a notion of interpretation not just in terms of beliefs but in terms of goals and actions. To do this we make use of the existing theory of partially observable Markov processes (POMDPs): we say that a system can be interpreted as a solution to a POMDP if it not only admits an interpretation map describing its beliefs about the hidden state of a POMDP but also takes actions that are optimal according to its belief state. An agent is then a system together with an interpretation of this system as a POMDP solution. Although POMDPs are not the only possible formulation of what it means to have a goal, this nevertheless represents a step towards a more general formal definition of what it means for a system to be an agent., Comment: 17 pages, no figures, to be presented at 3rd International Workshop on Active Inference 2022

A central concept in active inference is that the internal states of a physical system parametrise probability measures over states of the external world. These can be seen as an agent's beliefs, expressed as a Bayesian prior or posterior. Here we begin the development of a general theory that would tell us when it is appropriate to interpret states as representing beliefs in this way. We focus on the case in which a system can be interpreted as performing either Bayesian filtering or Bayesian inference. We provide formal definitions of what it means for such an interpretation to exist, using techniques from category theory., Comment: 11 pages + 26 pages appendix, to be published in the proceedings of the 2nd International Workshop on Active Inference 2021

We offer a new approach to the information decomposition problem in information theory: given a 'target' random variable co-distributed with multiple 'source' variables, how can we decompose the mutual information into a sum of non-negative terms that quantify the contributions of each random variable, not only individually but also in combination? We derive our composition from cooperative game theory. It can be seen as assigning a "fair share" of the mutual information to each combination of the source variables. Our decomposition is based on a different lattice from the usual 'partial information decomposition' (PID) approach, and as a consequence our decomposition has a smaller number of terms: it has analogs of the synergy and unique information terms, but lacks terms corresponding to redundancy. Because of this, it is able to obey equivalents of the axioms known as 'local positivity' and 'identity', which cannot be simultaneously satisfied by a PID measure., Comment: under review by Kybernetika journal

Natural evolution gives the impression of leading to an open-ended process of increasing diversity and complexity. If our goal is to produce such open-endedness artificially, this suggests an approach driven by evolutionary metaphor. On the other hand, techniques from machine learning and artificial intelligence are often considered too narrow to provide the sort of exploratory dynamics associated with evolution. In this paper, we hope to bridge that gap by reviewing common barriers to open-endedness in the evolution-inspired approach and how they are dealt with in the evolutionary case - collapse of diversity, saturation of complexity, and failure to form new kinds of individuality. We then show how these problems map onto similar issues in the machine learning approach, and discuss how the same insights and solutions which alleviated those barriers in evolutionary approaches can be ported over. At the same time, the form these issues take in the machine learning formulation suggests new ways to analyze and resolve barriers to open-endedness. Ultimately, we hope to inspire researchers to be able to interchangeably use evolutionary and gradient-descent-based machine learning methods to approach the design and creation of open-ended systems., Comment: 17 pages, 6 figures; Final author version accepted for publication in Artificial Life

We investigate the use of attentional neural network layers in order to learn a `behavior characterization' which can be used to drive novelty search and curiosity-based policies. The space is structured towards answering a particular distribution of questions, which are used in a supervised way to train the attentional neural network. We find that in a 2d exploration task, the structure of the space successfully encodes local sensory-motor contingencies such that even a greedy local `do the most novel action' policy with no reinforcement learning or evolution can explore the space quickly. We also apply this to a high/low number guessing game task, and find that guessing according to the learned attention profile performs active inference and can discover the correct number more quickly than an exact but passive approach., Comment: 8 pages, 7 figures, ALife 2018

Many examples of cooperation exist in biology. In chemical systems however, which can sometimes be quite complex, we do not appear to observe intricate cooperative interactions. A key question for the origin of life, is then how can molecular cooperation first arise in an abiotic system prior to the emergence of biological replication. We postulate that selection at the molecular level is a driving force behind the complexification of chemical systems, particularly during the origins of life. In the theory of multilevel selection the two selective forces are: within-group and between-group, where the former tends to favor "selfish" replication of individuals and the latter favor cooperation between individuals enhancing the replication of the group as a whole. These forces can be quantified using the Price equation, which is a standard tool used in evolutionary biology to quantify evolutionary change. Our central claim is that replication and heredity in chemical systems are subject to selection, and quantifiable using the multilevel Price equation. We demonstrate this using the Graded Autocatalysis Replication Domain computer model, describing simple protocell composed out of molecules and its replication, which respectively analogue to the group and the individuals. In contrast to previous treatments of this model, we treat the lipid molecules themselves as replicating individuals and the protocells they form as groups of individuals. Our goal is to demonstrate how evolutionary biology tools and concepts can be applied in chemistry and we suggest that molecular cooperation may arise as a result of group selection. Further, the biological relation of parent-progeny is proposed to be analogue to the reactant-product relation in chemistry, thus allowing for tools from evolutionary biology to be applied to chemistry and would deepen the connection between chemistry and biology., Comment: Keywords: Composomes, Evolution, Cooperation, Price equation, Group selection, Compositional information, Population dynamics

We propose an alternative to the prevailing two origin of life narratives, one based on a replicator first hypothesis, and one based on a metabolism first hypothesis. Both hypotheses have known difficulties: All known evolvable molecular replicators such as RNA require complex chemical (enzymatic) machinery for the replication process. Likewise, contemporary cellular metabolisms require several enzymatically catalyzed steps, and it is difficult to identify a non-enzymatic path to their realization. We propose that there must have been precursors to both replication and metabolism that enable a form of selection to take place through action of simple chemical and physical processes. We model a concrete example of such a process, repeated sequestration of binary molecular combinations after exposure to an environment with a broad distribution of chemical components, as might be realized experimentally in in a repeated wet-dry cycle. We show that the repeated sequestration dynamics results in a selective amplification of a very small subset of molecular species present in the environment, thus providing a candidate primordial selection process., Comment: 7 pages, 6 figures

The introduction of convolutional layers greatly advanced the performance of neural networks on image tasks due to innately capturing a way of encoding and learning translation-invariant operations, matching one of the underlying symmetries of the image domain. In comparison, there are a number of problems in which there are a number of different inputs which are all 'of the same type' --- multiple particles, multiple agents, multiple stock prices, etc. The corresponding symmetry to this is permutation symmetry, in that the algorithm should not depend on the specific ordering of the input data. We discuss a permutation-invariant neural network layer in analogy to convolutional layers, and show the ability of this architecture to learn to predict the motion of a variable number of interacting hard discs in 2D. In the same way that convolutional layers can generalize to different image sizes, the permutation layer we describe generalizes to different numbers of objects., Comment: 7 pages, 4 figures

Due to recent advances in synthetic biology and artificial life, the origin of life is currently a hot topic of research. We review the literature and argue that the two traditionally competing "replicator-first" and "metabolism-first" approaches are merging into one integrated theory of individuation and evolution. We contribute to the maturation of this more inclusive approach by highlighting some problematic assumptions that still lead to an impoverished conception of the phenomenon of life. In particular, we argue that the new consensus has so far failed to consider the relevance of intermediate timescales. We propose that an adequate theory of life must account for the fact that all living beings are situated in at least four distinct timescales, which are typically associated with metabolism, motility, development, and evolution. On this view, self-movement, adaptive behavior and morphological changes could have already been present at the origin of life. In order to illustrate this possibility we analyze a minimal model of life-like phenomena, namely of precarious, individuated, dissipative structures that can be found in simple reaction-diffusion systems. Based on our analysis we suggest that processes in intermediate timescales could have already been operative in prebiotic systems. They may have facilitated and constrained changes occurring in the faster- and slower-paced timescales of chemical self-individuation and evolution by natural selection, respectively., Comment: 29 pages, 5 figures, Artificial Life