Journal of Symbolic Logic

Double-exponential inseparability of Robinson subsystem Q+

Lavinia Egidi and Giovanni Faglia

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In this work a double exponential time inseparability result is proven for a finitely axiomatizable first order theory Q+. The theory, subset of Presburger theory of addition S+, is the additive fragment of Robinson system Q. We prove that every set that separates Q+ from the logically false sentences of addition is not recognizable by any Turing machine working in double exponential time. The lower bound is given both in the non-deterministic and in the linear alternating time models.

The result implies also that any theory of addition that is consistent with Q+—in particular any theory contained in S+—is at least double exponential time difficult. Our inseparability result is an improvement on the known lower bounds for arithmetic theories.

Our proof uses a refinement and adaptation of the technique that Fischer and Rabin used to prove the difficulty of S+. Our version of the technique can be applied to any incomplete finitely axiomatizable system in which all of the necessary properties of addition are provable.

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J. Symbolic Logic, Volume 76, Issue 1 (2011), 94-124.

First available in Project Euclid: 4 January 2011

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Egidi, Lavinia; Faglia, Giovanni. Double-exponential inseparability of Robinson subsystem Q +. J. Symbolic Logic 76 (2011), no. 1, 94--124. doi:10.2178/jsl/1294170991.

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