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**Double-exponential inseparability of Robinson subsystem \(Q_{+}\).**
*(English)*
Zbl 1222.03033

Summary: In this work a double exponential time inseparability result is proven for a finitely axiomatizable first-order theory \(Q_{+}\). The theory, a subset of the Presburger theory of addition \(S_{+}\), is the additive fragment of Robinson’s 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 nondeterministic 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.

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.

### MSC:

03C07 | Basic properties of first-order languages and structures |

03D15 | Complexity of computation (including implicit computational complexity) |

68Q17 | Computational difficulty of problems (lower bounds, completeness, difficulty of approximation, etc.) |

### Keywords:

double exponential time inseparability; finitely axiomatizable first-order theory; additive fragment of Robinson’s system \(Q\); theory of addition
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\textit{L. Egidi} and \textit{G. Faglia}, J. Symb. Log. 76, No. 1, 94--124 (2011; Zbl 1222.03033)

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