March 2011 Double-exponential inseparability of Robinson subsystem Q+
Lavinia Egidi, Giovanni Faglia
J. Symbolic Logic 76(1): 94-124 (March 2011). DOI: 10.2178/jsl/1294170991


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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Lavinia Egidi. Giovanni Faglia. "Double-exponential inseparability of Robinson subsystem Q+." J. Symbolic Logic 76 (1) 94 - 124, March 2011.


Published: March 2011
First available in Project Euclid: 4 January 2011

zbMATH: 1222.03033
MathSciNet: MR2791339
Digital Object Identifier: 10.2178/jsl/1294170991

Rights: Copyright © 2011 Association for Symbolic Logic


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Vol.76 • No. 1 • March 2011
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