Turing Machines Equipped with CTC in Physical Universes

TL;DR

This paper examines the physical plausibility of Turing machines with closed time-like curves (CTCs), proposing axioms to ensure consistency and analyzing their implications for computation and information transfer.

Contribution

It introduces strong and weak axioms for CTCs, discusses their physical implications, and proposes the data transferring hypothesis to address computational limitations.

Findings

CTCs imply the destruction of particles before completing a full loop.

Under the weak axiom, Turing machines cannot transmit information around CTCs.

The data transferring hypothesis offers a potential solution for information transfer in CTC-containing universes.

Abstract

We study the paradoxical aspects of closed time-like curves and their impact on the theory of computation. After introducing the $\text{TM}_\text{CTC}$, a classical Turing machine benefiting CTCs for backward time travel, Aaronson et al. proved that $\text{P} = \text{PSPACE}$ and the $\Delta_2$ sets, such as the halting problem, are computable within this computational model. Our critical view is the physical consistency of this model, which leads to proposing the strong axiom, explaining that every particle rounding on a CTC will be destroyed before returning to its starting time, and the weak axiom, describing the same notion, particularly for Turing machines. We claim that in a universe containing CTCs, the two axioms must be true; otherwise, there will be an infinite number of any particle rounding on a CTC in the universe. An immediate result of the weak axiom is the incapability of Turing machines to convey information for a full round on a CTC, leading to the proposed $\text{TM}_\text{CTC}$ programs for the aforementioned corollaries failing to function. We suggest our solution for this problem as the data transferring hypothesis, which applies another $\text{TM}_\text{CTC}$ as a means for storing data. A prerequisite for it is the existence of the concept of Turing machines throughout time, which makes it appear infeasible in our universe. Then, we discuss possible physical conditions that can be held for a universe containing CTCs and conclude that if returning to an approximately equivalent universe by a CTC was conceivable, the above corollaries would be valid.