Computational Processes and Incompleteness

TL;DR

This paper formalizes Wolfram's concept of computational processes using cellular automata, classifies them into decidable, intermediate, and complete, and demonstrates the non-existence of intermediate processes within a specific computational framework.

Contribution

It introduces a formal definition of cellular automata-based computational processes and proves the non-existence of intermediate processes in finite injury priority arguments.

Findings

Classifies cellular automata processes into decidable, intermediate, and complete.

Shows that intermediate processes cannot be established within standard finite injury priority arguments.

Provides a formal framework connecting Wolfram's notions with classical computability theory.

Abstract

We introduce a formal definition of Wolfram's notion of computational process based on cellular automata, a physics-like model of computation. There is a natural classification of these processes into decidable, intermediate and complete. It is shown that in the context of standard finite injury priority arguments one cannot establish the existence of an intermediate computational process.