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 2log O(1) n size) d-perceptrons iff it has only poly-log many sign changes (i.e., it changes value logO(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|
|Number of pages||10|
|State||Published - 1993|
|Event||10th Annual Symposium on Theoretical Aspects of Computer Science, STACS 1993 - Wurzburg, Germany|
Duration: Feb 25 1993 → Feb 27 1993
|Name||Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)|
|Other||10th Annual Symposium on Theoretical Aspects of Computer Science, STACS 1993|
|Period||2/25/93 → 2/27/93|
Bibliographical noteFunding 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).