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Circular and non-wellfounded proofs have become an increasingly popular toolfor metalogical treatments of systems with forms of induction and/or recursion.In this work we investigate the expressivity of a variant CT of G\"odel'ssystem T where programs are circularly typed, rather than including an explicitrecursion combinator. In particular, we examine the abstraction complexity(i.e. type level) of C, and show that the G\"odel primitive recursivefunctionals may be typed more succinctly with circular derivations, using typesprecisely one level lower than in T. In fact we give a logical correspondencebetween the two settings, interpreting the quantifier-free type 1 theory oflevel n+1 T into that of level n C and vice-versa.  We also obtain some further results and perspectives on circular'derivations', namely strong normalisation and confluence, models based onhereditary computable functionals, continuity at type 2, and a translation toterms of $\T$ computing the same functional, at all types.

A circular version of G\"odel's T and its abstraction complexity