Interplay Between Vertical and Horizontal Schemes of Computation: From Bayesian Inference to Quantum Logic via Gluing Boolean Algebras.

Artificial intelligence is typically formulated as an information-processing system composed of artificial neurons, where computation is understood as recursive operations connecting inputs and outputs. However, real neural systems are materially embodied and continuously reconfigured by metabolic and physical processes, suggesting that computation cannot be reduced to fixed causal structures. In this paper, we propose a theoretical framework that captures the interplay between informational and material processes as the interaction between two computational schemes: a vertical scheme, representing fixed cause-effect relations, and a horizontal scheme, representing transformations between such relations. We show that the vertical scheme corresponds to Bayesian inference, which updates probability distributions over a fixed hypothesis space, and is consistent with the free-energy minimization principle. In contrast, the horizontal scheme is formalized as inverse Bayesian inference, which modifies the hypothesis space itself by updating likelihood structures based on experienced data. We further demonstrate that the interplay between these schemes can be expressed algebraically as a process of continuously gluing Boolean algebras. This construction yields a non-distributive orthomodular lattice, i.e., quantum logic, without invoking Hilbert space formalism. In this view, quantum logic emerges not as a static logical system but as a structural consequence of dynamically reconfiguring causal contexts. This framework provides a unified perspective in which inference is understood not only as optimization within a fixed model but also as a process that generates and transforms the model itself. It offers a formal basis for describing open-ended computation and suggests a connection to approaches such as unconventional computing and Natural Born Intelligence, where computational structures evolve through interaction with material processes. Unlike existing approaches, this framework derives quantum-logic-like structure from the continual reconfiguration of causal contexts rather than from Hilbert-space assumptions or optimization within a fixed hypothesis space.