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Uniformity Within Parameterized Circuit Classes

Authors Steef Hegeman , Jan Martens , Alfons Laarman

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LIPIcs.IPEC.2025.27.pdf
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Subject Classification

ACM Subject Classification
  • Theory of computation → Complexity classes
  • Theory of computation → Circuit complexity
  • Theory of computation → Fixed parameter tractability
Keywords
  • Parameterized complexity
  • circuit complexity
  • uniformity
  • descriptive complexity

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Abstract

We study uniformity conditions for parameterized Boolean circuit families. Uniformity conditions require that the infinitely many circuits in a circuit family are in some sense easy to construct from one shared description. For shallow circuit families, logtime-uniformity is often desired but quite technical to prove. Despite that, proving it is often left as an exercise for the reader - even for recently introduced classes in parameterized circuit complexity, where uniformity conditions have not yet been explicitly studied. We formally define parameterized versions of linear-uniformity, logtime-uniformity, and FO-uniformity, and prove that these result in equivalent complexity classes when imposed on para-AC⁰ and para-AC^{0↑}. Overall, we provide a convenient way to verify uniformity for shallow parameterized circuit classes, and thereby substantiate claims of uniformity in the literature.

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Steef Hegeman, Jan Martens, and Alfons Laarman. Uniformity Within Parameterized Circuit Classes. In 20th International Symposium on Parameterized and Exact Computation (IPEC 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 358, pp. 27:1-27:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.IPEC.2025.27

Author Details

Steef Hegeman
  • Leiden University, The Netherlands
Jan Martens
  • Leiden University, The Netherlands
Alfons Laarman
  • Leiden University, The Netherlands

References

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