Computational tameness of classical non-causal models

TL;DR

This paper characterizes the computational power of non-causal circuit models with logical consistency, showing they are limited to problems in UP∩coUP and cannot efficiently solve NP-complete problems, unlike other CTC models.

Contribution

It provides a complete complexity-theoretic characterization of non-causal circuit models, linking their power to UP∩coUP and contrasting them with other CTC models.

Findings

Non-causal circuit models are equivalent to UP∩coUP.

Classical deterministic CTCs cannot efficiently solve NP-complete problems.

The results relate fixed points in non-causal models to complexity classes.

Abstract

We show that the computational power of the non-causal circuit model, i.e., the circuit model where the assumption of a global causal order is replaced by the assumption of logical consistency, is completely characterized by the complexity class~$\operatorname{\mathsf{UP}}\cap\operatorname{\mathsf{coUP}}$. An example of a problem in that class is factorization. Our result implies that classical deterministic closed timelike curves (CTCs) cannot efficiently solve problems that lie outside of that class. Thus, in stark contrast to other CTC models, these CTCs cannot efficiently solve~$\operatorname{\mathsf{NP-complete}}$ problems, unless~$\operatorname{\mathsf{NP}}=\operatorname{\mathsf{UP}}\cap\operatorname{\mathsf{coUP}}=\operatorname{\mathsf{coNP}}$, which lets their existence in nature appear less implausible. This result gives a new characterization of~$\operatorname{\mathsf{UP}}\cap\operatorname{\mathsf{coUP}}$ in terms of fixed points.