Journal of Symbolic Logic

Double-exponential inseparability of Robinson subsystem Q+

Lavinia Egidi and Giovanni Faglia

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Abstract

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.

Article information

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

Dates
First available in Project Euclid: 4 January 2011

Permanent link to this document
https://projecteuclid.org/euclid.jsl/1294170991

Digital Object Identifier
doi:10.2178/jsl/1294170991

Mathematical Reviews number (MathSciNet)
MR2791339

Zentralblatt MATH identifier
1222.03033

Citation

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. https://projecteuclid.org/euclid.jsl/1294170991


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