Computing symmetrie functions with AND/OR circuits and a single MAJORITY gate

Zhi Li Zhang; David A.Mix Barrington; Jun Tarui

doi:10.1007/3-540-56503-5_53

Computing symmetrie functions with AND/OR circuits and a single MAJORITY gate

Zhi Li Zhang, David A.Mix Barrington, Jun Tarui

Computer Science and Engineering

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

8 Scopus citations

Abstract

Fagin et al. characterized those symmetric Boolean functions which can be computed by small AND/OR circuits of constant depth and unbounded fan-in. Here we provide a similar characterization for d-perceptrons-AND/OR circuits of constant depth and unbounded fan-in with a single MAJORITY gate at the output. We show that a symmetric function has small (quasipolynomial, or 2^{log O(1) n} size) d-perceptrons iff it has only poly-log many sign changes (i.e., it changes value log^O(1) n times as the number of positive inputs varies from zero to n). A consequence of the lower bound is that a recent construction of Beigel is optimal. He showed how to convert a constant-depth unbounded fan-in AND/OR circuit with poly-log many MAJORITY gates into an equivalent d-perceptron-we show that more than poly-log MAJORITY gates cannot in general be converted to one.

Original language	English (US)
Title of host publication	STACS 1993 - 10th Annual Symposium on Theoretical Aspects of Computer Science, Proceedings
Editors	Patrice Enjalbert, Alain Finkel, Klaus W. Wagner
Publisher	Springer Verlag
Pages	535-544
Number of pages	10
ISBN (Print)	9783540565031
DOIs	https://doi.org/10.1007/3-540-56503-5_53
State	Published - 1993
Event	10th Annual Symposium on Theoretical Aspects of Computer Science, STACS 1993 - Wurzburg, Germany Duration: Feb 25 1993 → Feb 27 1993

Publication series

Name	Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume	665 LNCS
ISSN (Print)	0302-9743
ISSN (Electronic)	1611-3349

Other

Other	10th Annual Symposium on Theoretical Aspects of Computer Science, STACS 1993
Country/Territory	Germany
City	Wurzburg
Period	2/25/93 → 2/27/93

Bibliographical note

Funding Information:
The first author was supported by grants CCR-8812567 and CCR-9008416. The second author was supported by NSF Computer and Computation Theory grants CCR-8922098 and CCR-9207829. The third author was supported in part by the ESPRIT IIBRA Programme of the EC under contract 7141 (ALCOM II).

Publisher Copyright:
© Springer-Verlag Berlin Heidelberg 1993.

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Zhang, Z. L., Barrington, D. A. M., & Tarui, J. (1993). Computing symmetrie functions with AND/OR circuits and a single MAJORITY gate. In P. Enjalbert, A. Finkel, & K. W. Wagner (Eds.), STACS 1993 - 10th Annual Symposium on Theoretical Aspects of Computer Science, Proceedings (pp. 535-544). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 665 LNCS). Springer Verlag. https://doi.org/10.1007/3-540-56503-5_53

Computing symmetrie functions with AND/OR circuits and a single MAJORITY gate. / Zhang, Zhi Li; Barrington, David A.Mix; Tarui, Jun.
STACS 1993 - 10th Annual Symposium on Theoretical Aspects of Computer Science, Proceedings. ed. / Patrice Enjalbert; Alain Finkel; Klaus W. Wagner. Springer Verlag, 1993. p. 535-544 (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 665 LNCS).

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

Zhang, ZL, Barrington, DAM & Tarui, J 1993, Computing symmetrie functions with AND/OR circuits and a single MAJORITY gate. in P Enjalbert, A Finkel & KW Wagner (eds), STACS 1993 - 10th Annual Symposium on Theoretical Aspects of Computer Science, Proceedings. Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), vol. 665 LNCS, Springer Verlag, pp. 535-544, 10th Annual Symposium on Theoretical Aspects of Computer Science, STACS 1993, Wurzburg, Germany, 2/25/93. https://doi.org/10.1007/3-540-56503-5_53

Zhang ZL, Barrington DAM, Tarui J. Computing symmetrie functions with AND/OR circuits and a single MAJORITY gate. In Enjalbert P, Finkel A, Wagner KW, editors, STACS 1993 - 10th Annual Symposium on Theoretical Aspects of Computer Science, Proceedings. Springer Verlag. 1993. p. 535-544. (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)). doi: 10.1007/3-540-56503-5_53

Zhang, Zhi Li ; Barrington, David A.Mix ; Tarui, Jun. / Computing symmetrie functions with AND/OR circuits and a single MAJORITY gate. STACS 1993 - 10th Annual Symposium on Theoretical Aspects of Computer Science, Proceedings. editor / Patrice Enjalbert ; Alain Finkel ; Klaus W. Wagner. Springer Verlag, 1993. pp. 535-544 (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)).

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Computing symmetrie functions with AND/OR circuits and a single MAJORITY gate

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