Sabrina Alexandra Gaube’s research while affiliated with Leibniz Universität Hannover and other places

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Publications (2)


Figure 1. Example graphs that satisfy the imposed conditions of Example 28.
The sequence d(8, 2, 2)
Logical characterizations of algebraic circuit classes over integral domains
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May 2024

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15 Reads

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1 Citation

Mathematical Structures in Computer Science

Timon Barlag

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Sabrina A. Gaube

We present an adapted construction of algebraic circuits over the reals introduced by Cucker and Meer to arbitrary infinite integral domains and generalize the ACR\mathrm{AC}_{\mathbb{R}} and NCR\mathrm{NC}_{\mathbb{R}}^{} classes for this setting. We give a theorem in the style of Immerman’s theorem which shows that for these adapted formalisms, sets decided by circuits of constant depth and polynomial size are the same as sets definable by a suitable adaptation of first-order logic. Additionally, we discuss a generalization of the guarded predicative logic by Durand, Haak and Vollmer, and we show characterizations for the ACR\mathrm{AC}_{R} and NCR\mathrm{NC}_R^{} hierarchy. Those generalizations apply to the Boolean AC\mathrm{AC} and NC\mathrm{NC} hierarchies as well. Furthermore, we introduce a formalism to be able to compare some of the aforementioned complexity classes with different underlying integral domains.

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Logical Characterization of Algebraic Circuit Classes over Integral Domains

February 2023

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14 Reads

We present an adapted construction of algebraic circuits over the reals introduced by Cucker and Meer to arbitrary infinite integral domains and generalize the ACR\mathrm{AC}_{\mathbb{R}} and NCR\mathrm{NC}_{\mathbb{R}}-classes for this setting. We give a theorem in the style of Immerman's theorem which shows that for these adapted formalisms, sets decided by circuits of constant depth and polynomial size are the same as sets definable by a suitable adaptation of first-order logic. Additionally, we discuss a generalization of the guarded predicative logic by Durand, Haak and Vollmer and we show characterizations for the ACR\mathrm{AC}_{R} and NCR\mathrm{NC}_{R} hierarchy. Those generalizations apply to the Boolean AC\mathrm{AC} and NC\mathrm{NC} hierarchies as well. Furthermore, we introduce a formalism to be able to compare some of the aforementioned complexity classes with different underlying integral domains.