The Complexity of Infinite Computations In Models of Set Theory

TL;DR

This paper demonstrates that the topological complexity of certain infinite computations varies across models of set theory, showing that ZFC axioms do not determine their complexity class.

Contribution

It establishes that the topological complexity of nguages accepted by B1uchi automata can differ between models of ZFC, revealing independence results in automata theory.

Findings

Existence of automata with complexity variance across models of ZFC

Topological complexity of nguages is not absolute but model-dependent

Improved lower bounds for decision problems related to these automata

Abstract

We prove the following surprising result: there exist a 1-counter B\"uchi automaton and a 2-tape B\"uchi automaton such that the \omega-language of the first and the infinitary rational relation of the second in one model of ZFC are \pi_2^0-sets, while in a different model of ZFC both are analytic but non Borel sets. This shows that the topological complexity of an \omega-language accepted by a 1-counter B\"uchi automaton or of an infinitary rational relation accepted by a 2-tape B\"uchi automaton is not determined by the axiomatic system ZFC. We show that a similar result holds for the class of languages of infinite pictures which are recognized by B\"uchi tiling systems. We infer from the proof of the above results an improvement of the lower bound of some decision problems recently studied by the author.