Algorithmic Information Dynamics (AID) is an algorithmic probabilistic framework for causal discovery and causal analysis. It enables a numerical solution to inverse problems based or motivated on principles of algorithmic probability. AID studies dynamical systems in software space where all possible computable models can be found or approximated under the assumption that discrete longitudinal data such as particle orbits in state and phase space can approximate continuous systems by Turing-computable means. AID combines perturbation analysis and algorithmic information theory to guide a search for sets of models compatible with observations and to precompute and exploit those models as testable generative mechanisms and causal first principles underlying data and systems. AID is an alternative or a complement to other approaches and methods of experimental inference, such as statistical machine learning and classical information theory. AID connects with and across other parallel fields of active research such as logical inference, causal reasoning, and neuro-symbolic computation. AID studies how candidate discrete computable equations as generating mechanisms are affected by changes in observed phenomena over time as a result of a system evolving (e.g. under the influence of noise) or being externally perturbed. AID is related to other areas such as computational mechanics and program synthesis. However, unlike methods such as Bayesian networks, AID does not rely on graphical models or the (often inaccessible) empirical estimation of mass probability distributions. AID encompasses the foundations and methods that make the area of algorithmic information and algorithmic complexity more relevant to scientific discovery and causal analysis.