Computational Language $ \beta $ based on Orthomodular Lattices with the Non-distributivity of Quantum Logic

TL;DR

This paper introduces a computational language based on orthomodular lattices, emphasizing non-distributivity in quantum logic, to model transformation processes with deep algorithmic complexity in universal computing systems.

Contribution

It proposes a new computational framework using orthomodular lattices and non-distributive quantum logic to analyze complex transformation processes in universal machines.

Findings

Defines the class of functions with quasi-periodic structure, $\\Omega$, linked to irrational eigenvector components.

Models universal computation using a parallel processing framework on event-state systems.

Highlights the role of non-distributivity in Hilbert spaces for logical deduction in quantum-inspired computation.

Abstract

It is argued that transformation processes (generation rules) showing evidence of a long evolutionary history in universal computing systems can be generalized. The explicit function class $ \Omega $ is defined as follows: "Operators whose eigenvectors (or eigenvalues) have an irrational number in their components constitute a class of functions with quasi-periodic structure, $ \Omega $, and the class $ \Omega $ shows evidence of a long evolutionary history." In order to empirically prove this theorem by examining physical systems carrying out life activities or intellectual outputs of developed intelligence, the basic framework of the universal machine model C and the computational language $ \beta $ is presented as a model for general computational methods, which allow transformation processes (generation rules) with deep algorithmic complexity to be derived from generation results. C and $ \beta $ perform massively parallel computations on event-state systems consisting of exponential combinations of propositional elements expressed in terms of correlations between subsystems. The logical structure of the computational language relies on a non-distributivity in Hilbert spaces or orthogonal modular lattices, allowing for the manipulation and deduction of simultaneous propositions. In this logical local structure, the propositions implying certain consequences are not uniquely determined.