Parallel algorithms for power circuits and the word problem of the Baumslag group

Abstract

Power circuits have been introduced in 2012 by Myasnikov, Ushakov and Won as a data structure for non-elementarily compressed integers supporting the arithmetic operations addition and (x,y)↦x·2y\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(x,y) \\mapsto x\\cdot 2^y$$\\end{document}. The same authors applied power circuits to give a polynomial time solution to the word problem of the Baumslag group, which has a non-elementary Dehn function.In this work, we examine power circuits and the word problem of theBaumslag group under parallel complexity aspects. In particular, weestablish that the word problem of the Baumslag group can be solvedin NC—\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extemdash$$\\end{document}even though one of the essential steps is to compare twointegers given by power circuits and this, in general, is shown tobe P-complete. The key observation is that the depth of theoccurring power circuits is logarithmic and such power circuits canbe compared in NC.