# Hilbert’s thirteenth problem and circuit complexity

Hansen, K.A. and Lachish, Oded and Miltersen, P.B.
(2009)
Hilbert’s thirteenth problem and circuit complexity.
In:
Dong, Y.F. and Du, D.-Z. and Ibarra, O.H. (eds.)
*Algorithms and Computation.*
Lecture Notes in Computer Science 5878.
Berlin, Germany:
Springer Verlag, pp. 153-162.
ISBN 9783642106316.

## Abstract

We study the following question, communicated to us by Miklós Ajtai: Can all explicit (e.g., polynomial time computable) functions f: ({0,1} w )3 →{0,1} w be computed by word circuits of constant size? A word circuit is an acyclic circuit where each wire holds a word (i.e., an element of {0,1} w ) and each gate G computes some binary operation gG:({0,1}w)2→{0,1}w , defined for all word lengths w. We present an explicit function so that its w’th slice for any w ≥ 8 cannot be computed by word circuits with at most 4 gates. Also, we formally relate Ajtai’s question to open problems concerning ACC0 circuits.

## Metadata

Item Type: | Book Section |
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School: | Birkbeck Faculties and Schools > Faculty of Science > School of Computing and Mathematical Sciences |

Depositing User: | Sarah Hall |

Date Deposited: | 30 May 2013 15:40 |

Last Modified: | 09 Aug 2023 12:33 |

URI: | https://eprints.bbk.ac.uk/id/eprint/7134 |

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