On the logical complexity of cyclic arithmetic

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 $CΣ_n$ for (the logical consequences of) cyclic proofs containing only $Σ_n$ formulae, our main result is that $IΣ_{n+1}$ and $CΣ_n$ prove the same $Π_{n+1}$ 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 $IΣ_{n+1} \subseteq CΣ_n$ is obtained by proof theoretic techniques, relying on normal forms and structural manipulations of $\mathsf{PA}$ proofs. It improves upon the natural result that $IΣ_n$ is contained in $CΣ_n$. The converse inclusion, $CΣ_n \subseteq IΣ_{n+1}$, is obtained by calibrating the approach of Simpson '17 with recent results on the reverse mathematics of Büchi's theorem in Kołodziejczyk, 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 $CΣ_n$ is provable in $IΣ_{n+2}$ but not $IΣ_{n+1}$. As a result, we show that certain versions of McNaughton's theorem (the determinisation of $ω$-word automata) are not provable in $\mathsf{RCA}_0$, partially resolving an open problem from KMPS '16.