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Two recursion theoretic characterizations of proof speed-ups

  • James S. Royer (a1)

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Smullyan in [Smu61] identified the recursion theoretic essence of incompleteness results such as Gödel's first incompleteness theorem and Rosser's theorem. Smullyan (improving upon [Kle50] and [Kle52]) showed that, for sufficiently complex theories, the collection of provable formulae and the collection of refutable formulae are effectively inseparable—where formulae and their Gödel numbers are identified. This paper gives a similar treatment for proof speed-up. We say that a formal system S1 is speedable over another system S0 on a set of formulae A iff, for each recursive function h, there is a formula α in A such that the length of the shortest proof of α in S0 is larger than h of the shortest proof of α in S1. (Here we equate the length of a proof with something like the number of characters making it up, not its number of lines.) We characterize speedability in terms of the inseparability by r.e. sets of the collection of formulae which are provable in S1 but unprovable in S0 from the collection A–where again formulae and their Gödel numbers are identified. We provide precise definitions of proof length, speedability and r.e. inseparability below.

We follow the terminology and notation of [Rog87] with borrowings from [Soa87]. Below, ϕ is an acceptable numbering of the partial recursive functions [Rog87] and Φ a (Blum) complexity measure associated with ϕ [Blu67], [DW83].

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[CEF83]Case, J., K. Ebcioglu, and Fulk, M., R.e. inseparable general and subrecursive index sets, this Journal, vol. 48 (1983), 12351236.
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Two recursion theoretic characterizations of proof speed-ups

  • James S. Royer (a1)

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