Cosmic Logic: a Computational Model

TL;DR

This paper introduces a formal computational model of cosmic logic using Turing machines to analyze cosmological inference, revealing fundamental limits on computing probabilities in eternal inflation scenarios.

Contribution

It presents a novel Turing machine-based framework for cosmic logic, establishing the non-computability of certain cosmic measures and highlighting limitations in probability calculations.

Findings

CS machines are more fundamental than CM machines.

Non-computability of CS machines discriminating mortal and immortal CO machines.

Impossibility of computing probabilities over all CO machines in eternal inflation.

Abstract

We initiate a formal study of logical inferences in context of the measure problem in cosmology or what we call cosmic logic. We describe a simple computational model of cosmic logic suitable for analysis of, for example, discretized cosmological systems. The construction is based on a particular model of computation, developed by Alan Turing, with cosmic observers (CO), cosmic measures (CM) and cosmic symmetries (CS) described by Turing machines. CO machines always start with a blank tape and CM machines take CO's Turing number (also known as description number or G{\" o}del number) as input and output the corresponding probability. Similarly, CS machines take CO's Turing number as input, but output either one if the CO machines are in the same equivalence class or zero otherwise. We argue that CS machines are more fundamental than CM machines and, thus, should be used as building blocks in constructing CM machines. We prove the non-computability of a CS machine which discriminates between two classes of CO machines: mortal that halts in finite time and immortal that runs forever. In context of eternal inflation this result implies that it is impossible to construct CM machines to compute probabilities on the set of all CO machines using cut-off prescriptions. The cut-off measures can still be used if the set is reduced to include only machines which halt after a finite and predetermined number of steps.