Navigating Complexity: How Resource-Limited Agents Derive Probability and Generate Emergence

In the Kolmogorov Theory of consciousness (KT), an algorithmic agent is an information-processing system that compresses sensory data into simpler representations to plan actions that optimize its objective function. Algorithmic agents operate under limited data access, finite computational resources, and fundamental limits from the algorithmic information theory (AIT). In this paper, we demonstrate how these limitations naturally give rise to the principles of probability, Bayesian inference, and the concept of emergence within this framework.Using a toy example of an agent compressing data from a large library, we demonstrate how limitations in data access naturally lead to a multi-model strategy, where probabilistic reasoning and Occam's razor emerge as the agent navigates between different models. This process is naturally extended by allowing it to accommodate objective functions beyond compression. Next, we propose a formal definition of emergence from the notions of coarse-graining and Kolmogorov complexity. Due to its limited resources, the agent must employ coarse-graining to transform data that initially appears incompressible into a compressible, aggregate form while retaining a non-trivial structure. The agent must find patterns and operate at some coarse-graining level to ensure survival. Coarse-graining may include various forms of data reorganization, such as spatiotemporal averaging, compressive sensing, and dimensionality reduction techniques.We discuss the connections to other theoretical approaches, such as Jaynes' robot, the Free Energy Principle, and Active Inference. By addressing how an ideal agent copes with the inherent limitations of data access and computational capacity, we provide a unified framework for understanding both probabilistic reasoning and emergence in algorithmic agents.