On the logical complexity of cyclic arithmetic
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We study the logical complexity of proofs in cyclic arithmetic (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≥ 0. Furthermore, due to the 'uniformity' of our method, we also show that CA and Peano Arithmetic (PA) proofs of the same theorem differ only exponentially in size. The inclusion IΣ_n+1⊆ CΣ_n is obtained by proof theoretic techniques, relying on normal forms and structural manipulations of PA proofs. It improves upon the natural result that IΣ_n is contained in CΣ_n. The converse inclusion, CΣ_n ⊆ 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, 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 CA; in particular we show that, for n≥ 0, the consistency of CΣ_n is provable in IΣ_n+2 but not IΣ_n+1.
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