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Threshold Circuits for Iterated Matrix Product and Powering

Published online by Cambridge University Press:  15 April 2002

Carlo Mereghetti  and
Beatrice Palano
Carlo Mereghetti
Affiliation:
Dipartimento di Informatica, Sist. e Com., Università degli Studi di Milano -Bicocca, Via Bicocca degli Arcimboldi 8, 20126 Milano, Italy; e-mail: mereghetti@disco.unimib.it
Beatrice Palano
Affiliation:
Dipartimento di Informatica, Università degli Studi di Torino, Corso Svizzera 185, 10149 Torino, Italy; e-mail: beatrice@di.unito.it
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Abstract

The complexity of computing, via threshold circuits, the iteratedproduct and powering of fixed-dimension $k\times k$ matriceswith integer or rational entries is studied. We call these twoproblems $\sf IMP_k$ and $\sf MPOW_k$ , respectively, for short. We prove that: (i) For $k\geq2$ , $\sf IMP_k$ does not belong to ${\rm TC}^0$ , unless ${\rm TC}^0={\rm NC}^1$ .newline(ii) For stochastic matrices : $\sf IMP_2$ belongs to ${\rm TC}^0$ while, for $k\geq3$ , $\sf IMP_k$ does not belong to ${\rm TC}^0$ , unless ${\rm TC}^0={\rm NC}^1$ . (iii) For any k, $\sf MPOW_k$ belongs to ${\rm TC}^0$ .

Keywords

Threshold circuitsiterated matrix productmatrix powering.

Information

Type
Research Article
Information
RAIRO - Theoretical Informatics and Applications , Volume 34 , Issue 1 , January 2000 , pp. 39 - 46
Copyright
© EDP Sciences, 2000

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