On the logical complexity of cyclic arithmetic

TL;DR

This paper analyzes the logical complexity of cyclic arithmetic proofs, establishing equivalences with fragments of Peano Arithmetic and providing bounds on proof sizes, thereby advancing understanding of proof complexity and logical strength.

Contribution

It proves the equivalence of cyclic arithmetic fragments with certain arithmetic theories and improves bounds on proof and logical complexity.

Findings

$Ioldsymbol{ extSigma}_{n+1}$ and $Coldsymbol{ extSigma}_n$ prove the same $oldsymbol{ extPi}_{n+1}$ theorems.

Proofs in cyclic arithmetic and Peano Arithmetic differ only exponentially in size.

Certain fragments of cyclic arithmetic have their consistency provable in $Ioldsymbol{ extSigma}_{n+2}$ but not in $Ioldsymbol{ extSigma}_{n+1}$.

Abstract

We study the logical complexity of proofs in cyclic arithmetic ($\mathsf{CA}$), as introduced in Simpson '17, in terms of quantifier alternations of formulae occurring. Writing $C\Sigma_n$ for (the logical consequences of) cyclic proofs containing only $\Sigma_n$ formulae, our main result is that $I\Sigma_{n+1}$ and $C\Sigma_n$ prove the same $\Pi_{n+1}$ theorems, for all $n\geq 0$. Furthermore, due to the 'uniformity' of our method, we also show that $\mathsf{CA}$ and Peano Arithmetic ($\mathsf{PA}$) proofs of the same theorem differ only exponentially in size. The inclusion $I\Sigma_{n+1} \subseteq C\Sigma_n$ is obtained by proof theoretic techniques, relying on normal forms and structural manipulations of $\mathsf{PA}$ proofs. It improves upon the natural result that $I\Sigma_n$ is contained in $C\Sigma_n$. The converse inclusion, $C\Sigma_n \subseteq I\Sigma_{n+1}$, is obtained by calibrating the approach of Simpson '17 with recent results on the reverse mathematics of B\"uchi's theorem in Ko{\l}odziejczyk, Michalewski, Pradic & Skrzypczak '16 (KMPS'16), and specialising to the case of cyclic proofs. These results improve upon the bounds on proof complexity and logical complexity implicit in Simpson '17 and also an alternative approach due to Berardi & Tatsuta '17. The uniformity of our method also allows us to recover a metamathematical account of fragments of $\mathsf{CA}$; in particular we show that, for $n\geq 0$, the consistency of $C\Sigma_n$ is provable in $I\Sigma_{n+2}$ but not $I\Sigma_{n+1}$. As a result, we show that certain versions of McNaughton's theorem (the determinisation of $\omega$-word automata) are not provable in $\mathsf{RCA}_0$, partially resolving an open problem from KMPS '16.