- Author:
- David A. M Barrington
WE SHOW THAT ANY LANGUAGE RECOGNIZED BY AN `NC(SUPERSCRIPT)1'' CIRCUIT (FAN-IN 2, DEPTH `O''(LOG `N'')) CAN BE RECOGNIZED BY A WIDTH-5 POLYNOMIAL- SIZE BRANCHING PROGRAM. AS ANY BOUNDED-WIDTH POLYNOMIAL-SIZE BRANCHING PROGRAM CAN BE SIMULATED BY AN `NC(SUPERSCRIPT)1 CIRCUIT, WE HAVE THAT THE CLASS OF LANGUAGES RECOGNIZED BY SUCH PROGRAMS IS EXACTLY NON-UNIFORM `NC(SUPERSCRIPT)1. FURTHER, FOLLOWING RUZZO [RU81] AND COOK [CO85], IF THE BRANCHING PROGRAMS ARE RESTRICTED TO BE `ATIME''(LOG`N'')-UNIFORM, THEY RECOG NIZE THE SAME LANGUAGES AS DO `ATIME''(LOG`N'')-UNIFORM `NC(SUPERSCRIPT)1'' CIRCUITS, THAT IS, THOSE LANGUAGES IN `ATIME''(LOG`N''). WE ALSO EXTEND THE METHOD OF PROOF TO INVESTIGATE THE COMPLEXITY OF THE WORD PROBLEM FOR A FIXED PERMUTATION GROUP AND SHOW THAT POLYNOMIAL SIZE `CIRCUITS'' OF WIDTH 4 ALSO RECOGNIZE EXACTLY NON-UNIFORM `NC(SUPERSCRIPT)1''.
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