%!PS-Adobe-2.0 %%Creator: dvips(k) 5.86 Copyright 1999 Radical Eye Software %%Title: iso4.dvi %%Pages: 20 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips iso4.dvi -o iso4.ps %DVIPSParameters: dpi=600, compressed %DVIPSSource: TeX output 2005.05.09:0829 %%BeginProcSet: texc.pro %! /TeXDict 300 dict def TeXDict begin/N{def}def/B{bind def}N/S{exch}N/X{S N}B/A{dup}B/TR{translate}N/isls false N/vsize 11 72 mul N/hsize 8.5 72 mul N/landplus90{false}def/@rigin{isls{[0 landplus90{1 -1}{-1 1}ifelse 0 0 0]concat}if 72 Resolution div 72 VResolution div neg scale isls{ landplus90{VResolution 72 div vsize mul 0 exch}{Resolution -72 div hsize mul 0}ifelse TR}if Resolution VResolution vsize -72 div 1 add mul TR[ matrix currentmatrix{A A round sub abs 0.00001 lt{round}if}forall round exch round exch]setmatrix}N/@landscape{/isls true N}B/@manualfeed{ statusdict/manualfeed true put}B/@copies{/#copies X}B/FMat[1 0 0 -1 0 0] N/FBB[0 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M}B/k{4 M}B/w{0 rmoveto}B/l{p -4 w}B/m{p -3 w}B/n{ p -2 w}B/o{p -1 w}B/q{p 1 w}B/r{p 2 w}B/s{p 3 w}B/t{p 4 w}B/x{0 S rmoveto}B/y{3 2 roll p a}B/bos{/SS save N}B/eos{SS restore}B end %%EndProcSet %%BeginProcSet: pstricks.pro %! % PostScript prologue for pstricks.tex. % Version 97 patch 3, 98/06/01 % For distribution, see pstricks.tex. % /tx@Dict 200 dict def tx@Dict begin /ADict 25 dict def /CM { matrix currentmatrix } bind def /SLW /setlinewidth load def /CLW /currentlinewidth load def /CP /currentpoint load def /ED { exch def } bind def /L /lineto load def /T /translate load def /TMatrix { } def /RAngle { 0 } def /Atan { /atan load stopped { pop pop 0 } if } def /Div { dup 0 eq { pop } { div } ifelse } def /NET { neg exch neg exch T } def /Pyth { dup mul exch dup mul add sqrt } def /PtoC { 2 copy cos mul 3 1 roll sin mul } def /PathLength@ { /z z y y1 sub x x1 sub Pyth add def /y1 y def /x1 x def } def /PathLength { flattenpath /z 0 def { /y1 ED /x1 ED /y2 y1 def /x2 x1 def } { /y ED /x ED PathLength@ } {} { /y y2 def /x x2 def PathLength@ } /pathforall load stopped { pop pop pop pop } if z } def /STP { .996264 dup scale } def /STV { SDict begin normalscale end STP } def /DashLine { dup 0 gt { /a .5 def PathLength exch div } { pop /a 1 def PathLength } ifelse /b ED /x ED /y ED /z y x add def b a .5 sub 2 mul y mul sub z Div round z mul a .5 sub 2 mul y mul add b exch Div dup y mul /y ED x mul /x ED x 0 gt y 0 gt and { [ y x ] 1 a sub y mul } { [ 1 0 ] 0 } ifelse setdash stroke } def /DotLine { /b PathLength def /a ED /z ED /y CLW def /z y z add def a 0 gt { /b b a div def } { a 0 eq { /b b y sub def } { a -3 eq { /b b y add def } if } ifelse } ifelse [ 0 b b z Div round Div dup 0 le { pop 1 } if ] a 0 gt { 0 } { y 2 div a -2 gt { neg } if } ifelse setdash 1 setlinecap stroke } def /LineFill { gsave abs CLW add /a ED a 0 dtransform round exch round exch 2 copy idtransform exch Atan rotate idtransform pop /a ED .25 .25 % DG/SR modification begin - Dec. 12, 1997 - Patch 2 %itransform translate pathbbox /y2 ED a Div ceiling cvi /x2 ED /y1 ED a itransform pathbbox /y2 ED a Div ceiling cvi /x2 ED /y1 ED a % DG/SR modification end Div cvi /x1 ED /y2 y2 y1 sub def clip newpath 2 setlinecap systemdict /setstrokeadjust known { true setstrokeadjust } if x2 x1 sub 1 add { x1 % DG/SR modification begin - Jun. 1, 1998 - Patch 3 (from Michael Vulis) % a mul y1 moveto 0 y2 rlineto stroke /x1 x1 1 add def } repeat grestore } % def a mul y1 moveto 0 y2 rlineto stroke /x1 x1 1 add def } repeat grestore pop pop } def % DG/SR modification end /BeginArrow { ADict begin /@mtrx CM def gsave 2 copy T 2 index sub neg exch 3 index sub exch Atan rotate newpath } def /EndArrow { @mtrx setmatrix CP grestore end } def /Arrow { CLW mul add dup 2 div /w ED mul dup /h ED mul /a ED { 0 h T 1 -1 scale } if w neg h moveto 0 0 L w h L w neg a neg rlineto gsave fill grestore } def /Tbar { CLW mul add /z ED z -2 div CLW 2 div moveto z 0 rlineto stroke 0 CLW moveto } def /Bracket { CLW mul add dup CLW sub 2 div /x ED mul CLW add /y ED /z CLW 2 div def x neg y moveto x neg CLW 2 div L x CLW 2 div L x y L stroke 0 CLW moveto } def /RoundBracket { CLW mul add dup 2 div /x ED mul /y ED /mtrx CM def 0 CLW 2 div T x y mul 0 ne { x y scale } if 1 1 moveto .85 .5 .35 0 0 0 curveto -.35 0 -.85 .5 -1 1 curveto mtrx setmatrix stroke 0 CLW moveto } def /SD { 0 360 arc fill } def /EndDot { { /z DS def } { /z 0 def } ifelse /b ED 0 z DS SD b { 0 z DS CLW sub SD } if 0 DS z add CLW 4 div sub moveto } def /Shadow { [ { /moveto load } { /lineto load } { /curveto load } { /closepath load } /pathforall load stopped { pop pop pop pop CP /moveto load } if ] cvx newpath 3 1 roll T exec } def /NArray { aload length 2 div dup dup cvi eq not { exch pop } if /n exch cvi def } def /NArray { /f ED counttomark 2 div dup cvi /n ED n eq not { exch pop } if f { ] aload /Points ED } { n 2 mul 1 add -1 roll pop } ifelse } def /Line { NArray n 0 eq not { n 1 eq { 0 0 /n 2 def } if ArrowA /n n 2 sub def n { Lineto } repeat CP 4 2 roll ArrowB L pop pop } if } def /Arcto { /a [ 6 -2 roll ] cvx def a r /arcto load stopped { 5 } { 4 } ifelse { pop } repeat a } def /CheckClosed { dup n 2 mul 1 sub index eq 2 index n 2 mul 1 add index eq and { pop pop /n n 1 sub def } if } def /Polygon { NArray n 2 eq { 0 0 /n 3 def } if n 3 lt { n { pop pop } repeat } { n 3 gt { CheckClosed } if n 2 mul -2 roll /y0 ED /x0 ED /y1 ED /x1 ED x1 y1 /x1 x0 x1 add 2 div def /y1 y0 y1 add 2 div def x1 y1 moveto /n n 2 sub def n { Lineto } repeat x1 y1 x0 y0 6 4 roll Lineto Lineto pop pop closepath } ifelse } def /Diamond { /mtrx CM def T rotate /h ED /w ED dup 0 eq { pop } { CLW mul neg /d ED /a w h Atan def /h d a sin Div h add def /w d a cos Div w add def } ifelse mark w 2 div h 2 div w 0 0 h neg w neg 0 0 h w 2 div h 2 div /ArrowA { moveto } def /ArrowB { } def false Line closepath mtrx setmatrix } def % DG modification begin - Jan. 15, 1997 %/Triangle { /mtrx CM def translate rotate /h ED 2 div /w ED dup 0 eq { %pop } { CLW mul /d ED /h h d w h Atan sin Div sub def /w w d h w Atan 2 %div dup cos exch sin Div mul sub def } ifelse mark 0 d w neg d 0 h w d 0 %d /ArrowA { moveto } def /ArrowB { } def false Line closepath mtrx %setmatrix } def /Triangle { /mtrx CM def translate rotate /h ED 2 div /w ED dup CLW mul /d ED /h h d w h Atan sin Div sub def /w w d h w Atan 2 div dup cos exch sin Div mul sub def mark 0 d w neg d 0 h w d 0 d /ArrowA { moveto } def /ArrowB { } def false Line closepath mtrx % DG/SR modification begin - Jun. 1, 1998 - Patch 3 (from Michael Vulis) % setmatrix } def setmatrix pop } def % DG/SR modification end /CCA { /y ED /x ED 2 copy y sub /dy1 ED x sub /dx1 ED /l1 dx1 dy1 Pyth def } def /CCA { /y ED /x ED 2 copy y sub /dy1 ED x sub /dx1 ED /l1 dx1 dy1 Pyth def } def /CC { /l0 l1 def /x1 x dx sub def /y1 y dy sub def /dx0 dx1 def /dy0 dy1 def CCA /dx dx0 l1 c exp mul dx1 l0 c exp mul add def /dy dy0 l1 c exp mul dy1 l0 c exp mul add def /m dx0 dy0 Atan dx1 dy1 Atan sub 2 div cos abs b exp a mul dx dy Pyth Div 2 div def /x2 x l0 dx mul m mul sub def /y2 y l0 dy mul m mul sub def /dx l1 dx mul m mul neg def /dy l1 dy mul m mul neg def } def /IC { /c c 1 add def c 0 lt { /c 0 def } { c 3 gt { /c 3 def } if } ifelse /a a 2 mul 3 div 45 cos b exp div def CCA /dx 0 def /dy 0 def } def /BOC { IC CC x2 y2 x1 y1 ArrowA CP 4 2 roll x y curveto } def /NC { CC x1 y1 x2 y2 x y curveto } def /EOC { x dx sub y dy sub 4 2 roll ArrowB 2 copy curveto } def /BAC { IC CC x y moveto CC x1 y1 CP ArrowA } def /NAC { x2 y2 x y curveto CC x1 y1 } def /EAC { x2 y2 x y ArrowB curveto pop pop } def /OpenCurve { NArray n 3 lt { n { pop pop } repeat } { BOC /n n 3 sub def n { NC } repeat EOC } ifelse } def /AltCurve { { false NArray n 2 mul 2 roll [ n 2 mul 3 sub 1 roll ] aload /Points ED n 2 mul -2 roll } { false NArray } ifelse n 4 lt { n { pop pop } repeat } { BAC /n n 4 sub def n { NAC } repeat EAC } ifelse } def /ClosedCurve { NArray n 3 lt { n { pop pop } repeat } { n 3 gt { CheckClosed } if 6 copy n 2 mul 6 add 6 roll IC CC x y moveto n { NC } repeat closepath pop pop } ifelse } def /SQ { /r ED r r moveto r r neg L r neg r neg L r neg r L fill } def /ST { /y ED /x ED x y moveto x neg y L 0 x L fill } def /SP { /r ED gsave 0 r moveto 4 { 72 rotate 0 r L } repeat fill grestore } def /FontDot { DS 2 mul dup matrix scale matrix concatmatrix exch matrix rotate matrix concatmatrix exch findfont exch makefont setfont } def /Rect { x1 y1 y2 add 2 div moveto x1 y2 lineto x2 y2 lineto x2 y1 lineto x1 y1 lineto closepath } def /OvalFrame { x1 x2 eq y1 y2 eq or { pop pop x1 y1 moveto x2 y2 L } { y1 y2 sub abs x1 x2 sub abs 2 copy gt { exch pop } { pop } ifelse 2 div exch { dup 3 1 roll mul exch } if 2 copy lt { pop } { exch pop } ifelse /b ED x1 y1 y2 add 2 div moveto x1 y2 x2 y2 b arcto x2 y2 x2 y1 b arcto x2 y1 x1 y1 b arcto x1 y1 x1 y2 b arcto 16 { pop } repeat closepath } ifelse } def /Frame { CLW mul /a ED 3 -1 roll 2 copy gt { exch } if a sub /y2 ED a add /y1 ED 2 copy gt { exch } if a sub /x2 ED a add /x1 ED 1 index 0 eq { pop pop Rect } { OvalFrame } ifelse } def /BezierNArray { /f ED counttomark 2 div dup cvi /n ED n eq not { exch pop } if n 1 sub neg 3 mod 3 add 3 mod { 0 0 /n n 1 add def } repeat f { ] aload /Points ED } { n 2 mul 1 add -1 roll pop } ifelse } def /OpenBezier { BezierNArray n 1 eq { pop pop } { ArrowA n 4 sub 3 idiv { 6 2 roll 4 2 roll curveto } repeat 6 2 roll 4 2 roll ArrowB curveto } ifelse } def /ClosedBezier { BezierNArray n 1 eq { pop pop } { moveto n 1 sub 3 idiv { 6 2 roll 4 2 roll curveto } repeat closepath } ifelse } def /BezierShowPoints { gsave Points aload length 2 div cvi /n ED moveto n 1 sub { lineto } repeat CLW 2 div SLW [ 4 4 ] 0 setdash stroke grestore } def /Parab { /y0 exch def /x0 exch def /y1 exch def /x1 exch def /dx x0 x1 sub 3 div def /dy y0 y1 sub 3 div def x0 dx sub y0 dy add x1 y1 ArrowA x0 dx add y0 dy add x0 2 mul x1 sub y1 ArrowB curveto /Points [ x1 y1 x0 y0 x0 2 mul x1 sub y1 ] def } def /Grid { newpath /a 4 string def /b ED /c ED /n ED cvi dup 1 lt { pop 1 } if /s ED s div dup 0 eq { pop 1 } if /dy ED s div dup 0 eq { pop 1 } if /dx ED dy div round dy mul /y0 ED dx div round dx mul /x0 ED dy div round cvi /y2 ED dx div round cvi /x2 ED dy div round cvi /y1 ED dx div round cvi /x1 ED /h y2 y1 sub 0 gt { 1 } { -1 } ifelse def /w x2 x1 sub 0 gt { 1 } { -1 } ifelse def b 0 gt { /z1 b 4 div CLW 2 div add def /Helvetica findfont b scalefont setfont /b b .95 mul CLW 2 div add def } if systemdict /setstrokeadjust known { true setstrokeadjust /t { } def } { /t { transform 0.25 sub round 0.25 add exch 0.25 sub round 0.25 add exch itransform } bind def } ifelse gsave n 0 gt { 1 setlinecap [ 0 dy n div ] dy n div 2 div setdash } { 2 setlinecap } ifelse /i x1 def /f y1 dy mul n 0 gt { dy n div 2 div h mul sub } if def /g y2 dy mul n 0 gt { dy n div 2 div h mul add } if def x2 x1 sub w mul 1 add dup 1000 gt { pop 1000 } if { i dx mul dup y0 moveto b 0 gt { gsave c i a cvs dup stringwidth pop /z2 ED w 0 gt {z1} {z1 z2 add neg} ifelse h 0 gt {b neg} {z1} ifelse rmoveto show grestore } if dup t f moveto g t L stroke /i i w add def } repeat grestore gsave n 0 gt % DG/SR modification begin - Nov. 7, 1997 - Patch 1 %{ 1 setlinecap [ 0 dx n div ] dy n div 2 div setdash } { 1 setlinecap [ 0 dx n div ] dx n div 2 div setdash } % DG/SR modification end { 2 setlinecap } ifelse /i y1 def /f x1 dx mul n 0 gt { dx n div 2 div w mul sub } if def /g x2 dx mul n 0 gt { dx n div 2 div w mul add } if def y2 y1 sub h mul 1 add dup 1000 gt { pop 1000 } if { newpath i dy mul dup x0 exch moveto b 0 gt { gsave c i a cvs dup stringwidth pop /z2 ED w 0 gt {z1 z2 add neg} {z1} ifelse h 0 gt {z1} {b neg} ifelse rmoveto show grestore } if dup f exch t moveto g exch t L stroke /i i h add def } repeat grestore } def /ArcArrow { /d ED /b ED /a ED gsave newpath 0 -1000 moveto clip newpath 0 1 0 0 b grestore c mul /e ED pop pop pop r a e d PtoC y add exch x add exch r a PtoC y add exch x add exch b pop pop pop pop a e d CLW 8 div c mul neg d } def /Ellipse { /mtrx CM def T scale 0 0 1 5 3 roll arc mtrx setmatrix } def /Rot { CP CP translate 3 -1 roll neg rotate NET } def /RotBegin { tx@Dict /TMatrix known not { /TMatrix { } def /RAngle { 0 } def } if /TMatrix [ TMatrix CM ] cvx def /a ED a Rot /RAngle [ RAngle dup a add ] cvx def } def /RotEnd { /TMatrix [ TMatrix setmatrix ] cvx def /RAngle [ RAngle pop ] cvx def } def /PutCoor { gsave CP T CM STV exch exec moveto setmatrix CP grestore } def /PutBegin { /TMatrix [ TMatrix CM ] cvx def CP 4 2 roll T moveto } def /PutEnd { CP /TMatrix [ TMatrix setmatrix ] cvx def moveto } def /Uput { /a ED add 2 div /h ED 2 div /w ED /s a sin def /c a cos def /b s abs c abs 2 copy gt dup /q ED { pop } { exch pop } ifelse def /w1 c b div w mul def /h1 s b div h mul def q { w1 abs w sub dup c mul abs } { h1 abs h sub dup s mul abs } ifelse } def /UUput { /z ED abs /y ED /x ED q { x s div c mul abs y gt } { x c div s mul abs y gt } ifelse { x x mul y y mul sub z z mul add sqrt z add } { q { x s div } { x c div } ifelse abs } ifelse a PtoC h1 add exch w1 add exch } def /BeginOL { dup (all) eq exch TheOL eq or { IfVisible not { Visible /IfVisible true def } if } { IfVisible { Invisible /IfVisible false def } if } ifelse } def /InitOL { /OLUnit [ 3000 3000 matrix defaultmatrix dtransform ] cvx def /Visible { CP OLUnit idtransform T moveto } def /Invisible { CP OLUnit neg exch neg exch idtransform T moveto } def /BOL { BeginOL } def /IfVisible true def } def end % END pstricks.pro %%EndProcSet %%BeginProcSet: pst-dots.pro %!PS-Adobe-2.0 %%Title: Dot Font for PSTricks 97 - Version 97, 93/05/07. %%Creator: Timothy Van Zandt %%Creation Date: May 7, 1993 10 dict dup begin /FontType 3 def /FontMatrix [ .001 0 0 .001 0 0 ] def /FontBBox [ 0 0 0 0 ] def /Encoding 256 array def 0 1 255 { Encoding exch /.notdef put } for Encoding dup (b) 0 get /Bullet put dup (c) 0 get /Circle put dup (C) 0 get /BoldCircle put dup (u) 0 get /SolidTriangle put dup (t) 0 get /Triangle put dup (T) 0 get /BoldTriangle put dup (r) 0 get /SolidSquare put dup (s) 0 get /Square put dup (S) 0 get /BoldSquare put dup (q) 0 get /SolidPentagon put dup (p) 0 get /Pentagon put (P) 0 get /BoldPentagon put /Metrics 13 dict def Metrics begin /Bullet 1000 def /Circle 1000 def /BoldCircle 1000 def /SolidTriangle 1344 def /Triangle 1344 def /BoldTriangle 1344 def /SolidSquare 886 def /Square 886 def /BoldSquare 886 def /SolidPentagon 1093.2 def /Pentagon 1093.2 def /BoldPentagon 1093.2 def /.notdef 0 def end /BBoxes 13 dict def BBoxes begin /Circle { -550 -550 550 550 } def /BoldCircle /Circle load def /Bullet /Circle load def /Triangle { -571.5 -330 571.5 660 } def /BoldTriangle /Triangle load def /SolidTriangle /Triangle load def /Square { -450 -450 450 450 } def /BoldSquare /Square load def /SolidSquare /Square load def /Pentagon { -546.6 -465 546.6 574.7 } def /BoldPentagon /Pentagon load def /SolidPentagon /Pentagon load def /.notdef { 0 0 0 0 } def end /CharProcs 20 dict def CharProcs begin /Adjust { 2 copy dtransform floor .5 add exch floor .5 add exch idtransform 3 -1 roll div 3 1 roll exch div exch scale } def /CirclePath { 0 0 500 0 360 arc closepath } def /Bullet { 500 500 Adjust CirclePath fill } def /Circle { 500 500 Adjust CirclePath .9 .9 scale CirclePath eofill } def /BoldCircle { 500 500 Adjust CirclePath .8 .8 scale CirclePath eofill } def /BoldCircle { CirclePath .8 .8 scale CirclePath eofill } def /TrianglePath { 0 660 moveto -571.5 -330 lineto 571.5 -330 lineto closepath } def /SolidTriangle { TrianglePath fill } def /Triangle { TrianglePath .85 .85 scale TrianglePath eofill } def /BoldTriangle { TrianglePath .7 .7 scale TrianglePath eofill } def /SquarePath { -450 450 moveto 450 450 lineto 450 -450 lineto -450 -450 lineto closepath } def /SolidSquare { SquarePath fill } def /Square { SquarePath .89 .89 scale SquarePath eofill } def /BoldSquare { SquarePath .78 .78 scale SquarePath eofill } def /PentagonPath { -337.8 -465 moveto 337.8 -465 lineto 546.6 177.6 lineto 0 574.7 lineto -546.6 177.6 lineto closepath } def /SolidPentagon { PentagonPath fill } def /Pentagon { PentagonPath .89 .89 scale PentagonPath eofill } def /BoldPentagon { PentagonPath .78 .78 scale PentagonPath eofill } def /.notdef { } def end /BuildGlyph { exch begin Metrics 1 index get exec 0 BBoxes 3 index get exec setcachedevice CharProcs begin load exec end end } def /BuildChar { 1 index /Encoding get exch get 1 index /BuildGlyph get exec } bind def end /PSTricksDotFont exch definefont pop % END pst-dots.pro %%EndProcSet %%BeginProcSet: pst-text.pro %! % PostScript header file pst-text.pro % Version 97, 94/04/20 % For distribution, see pstricks.tex. /tx@TextPathDict 40 dict def tx@TextPathDict begin % Syntax: PathPosition - % Function: Searches for position of currentpath distance from % beginning. Sets (X,Y)=position, and Angle=tangent. /PathPosition { /targetdist exch def /pathdist 0 def /continue true def /X { newx } def /Y { newy } def /Angle 0 def gsave flattenpath { movetoproc } { linetoproc } { } { firstx firsty linetoproc } /pathforall load stopped { pop pop pop pop /X 0 def /Y 0 def } if grestore } def /movetoproc { continue { @movetoproc } { pop pop } ifelse } def /@movetoproc { /newy exch def /newx exch def /firstx newx def /firsty newy def } def /linetoproc { continue { @linetoproc } { pop pop } ifelse } def /@linetoproc { /oldx newx def /oldy newy def /newy exch def /newx exch def /dx newx oldx sub def /dy newy oldy sub def /dist dx dup mul dy dup mul add sqrt def /pathdist pathdist dist add def pathdist targetdist ge { pathdist targetdist sub dist div dup dy mul neg newy add /Y exch def dx mul neg newx add /X exch def /Angle dy dx atan def /continue false def } if } def /TextPathShow { /String exch def /CharCount 0 def String length { String CharCount 1 getinterval ShowChar /CharCount CharCount 1 add def } repeat } def % Syntax: InitTextPath - /InitTextPath { gsave currentpoint /Y exch def /X exch def exch X Hoffset sub sub mul Voffset Hoffset sub add neg X add /Hoffset exch def /Voffset Y def grestore } def /Transform { PathPosition dup Angle cos mul Y add exch Angle sin mul neg X add exch translate Angle rotate } def /ShowChar { /Char exch def gsave Char end stringwidth tx@TextPathDict begin 2 div /Sy exch def 2 div /Sx exch def currentpoint Voffset sub Sy add exch Hoffset sub Sx add Transform Sx neg Sy neg moveto Char end tx@TextPathSavedShow tx@TextPathDict begin grestore Sx 2 mul Sy 2 mul rmoveto } def end % END pst-text.pro %%EndProcSet %%BeginProcSet: pst-node.pro %! % PostScript prologue for pst-node.tex. % Version 97 patch 1, 97/05/09. % For distribution, see pstricks.tex. % /tx@NodeDict 400 dict def tx@NodeDict begin tx@Dict begin /T /translate load def end /NewNode { gsave /next ED dict dup 3 1 roll def exch { dup 3 1 roll def } if begin tx@Dict begin STV CP T exec end /NodeMtrx CM def next end grestore } def /InitPnode { /Y ED /X ED /NodePos { NodeSep Cos mul NodeSep Sin mul } def } def /InitCnode { /r ED /Y ED /X ED /NodePos { NodeSep r add dup Cos mul exch Sin mul } def } def /GetRnodePos { Cos 0 gt { /dx r NodeSep add def } { /dx l NodeSep sub def } ifelse Sin 0 gt { /dy u NodeSep add def } { /dy d NodeSep sub def } ifelse dx Sin mul abs dy Cos mul abs gt { dy Cos mul Sin div dy } { dx dup Sin mul Cos Div } ifelse } def /InitRnode { /Y ED /X ED X sub /r ED /l X neg def Y add neg /d ED Y sub /u ED /NodePos { GetRnodePos } def } def /DiaNodePos { w h mul w Sin mul abs h Cos mul abs add Div NodeSep add dup Cos mul exch Sin mul } def /TriNodePos { Sin s lt { d NodeSep sub dup Cos mul Sin Div exch } { w h mul w Sin mul h Cos abs mul add Div NodeSep add dup Cos mul exch Sin mul } ifelse } def /InitTriNode { sub 2 div exch 2 div exch 2 copy T 2 copy 4 index index /d ED pop pop pop pop -90 mul rotate /NodeMtrx CM def /X 0 def /Y 0 def d sub abs neg /d ED d add /h ED 2 div h mul h d sub Div /w ED /s d w Atan sin def /NodePos { TriNodePos } def } def /OvalNodePos { /ww w NodeSep add def /hh h NodeSep add def Sin ww mul Cos hh mul Atan dup cos ww mul exch sin hh mul } def /GetCenter { begin X Y NodeMtrx transform CM itransform end } def /XYPos { dup sin exch cos Do /Cos ED /Sin ED /Dist ED Cos 0 gt { Dist Dist Sin mul Cos div } { Cos 0 lt { Dist neg Dist Sin mul Cos div neg } { 0 Dist Sin mul } ifelse } ifelse Do } def /GetEdge { dup 0 eq { pop begin 1 0 NodeMtrx dtransform CM idtransform exch atan sub dup sin /Sin ED cos /Cos ED /NodeSep ED NodePos NodeMtrx dtransform CM idtransform end } { 1 eq {{exch}} {{}} ifelse /Do ED pop XYPos } ifelse } def /AddOffset { 1 index 0 eq { pop pop } { 2 copy 5 2 roll cos mul add 4 1 roll sin mul sub exch } ifelse } def /GetEdgeA { NodeSepA AngleA NodeA NodeSepTypeA GetEdge OffsetA AngleA AddOffset yA add /yA1 ED xA add /xA1 ED } def /GetEdgeB { NodeSepB AngleB NodeB NodeSepTypeB GetEdge OffsetB AngleB AddOffset yB add /yB1 ED xB add /xB1 ED } def /GetArmA { ArmTypeA 0 eq { /xA2 ArmA AngleA cos mul xA1 add def /yA2 ArmA AngleA sin mul yA1 add def } { ArmTypeA 1 eq {{exch}} {{}} ifelse /Do ED ArmA AngleA XYPos OffsetA AngleA AddOffset yA add /yA2 ED xA add /xA2 ED } ifelse } def /GetArmB { ArmTypeB 0 eq { /xB2 ArmB AngleB cos mul xB1 add def /yB2 ArmB AngleB sin mul yB1 add def } { ArmTypeB 1 eq {{exch}} {{}} ifelse /Do ED ArmB AngleB XYPos OffsetB AngleB AddOffset yB add /yB2 ED xB add /xB2 ED } ifelse } def /InitNC { /b ED /a ED /NodeSepTypeB ED /NodeSepTypeA ED /NodeSepB ED /NodeSepA ED /OffsetB ED /OffsetA ED tx@NodeDict a known tx@NodeDict b known and dup { /NodeA a load def /NodeB b load def NodeA GetCenter /yA ED /xA ED NodeB GetCenter /yB ED /xB ED } if } def /LPutLine { 4 copy 3 -1 roll sub neg 3 1 roll sub Atan /NAngle ED 1 t sub mul 3 1 roll 1 t sub mul 4 1 roll t mul add /Y ED t mul add /X ED } def /LPutLines { mark LPutVar counttomark 2 div 1 sub /n ED t floor dup n gt { pop n 1 sub /t 1 def } { dup t sub neg /t ED } ifelse cvi 2 mul { pop } repeat LPutLine cleartomark } def /BezierMidpoint { /y3 ED /x3 ED /y2 ED /x2 ED /y1 ED /x1 ED /y0 ED /x0 ED /t ED /cx x1 x0 sub 3 mul def /cy y1 y0 sub 3 mul def /bx x2 x1 sub 3 mul cx sub def /by y2 y1 sub 3 mul cy sub def /ax x3 x0 sub cx sub bx sub def /ay y3 y0 sub cy sub by sub def ax t 3 exp mul bx t t mul mul add cx t mul add x0 add ay t 3 exp mul by t t mul mul add cy t mul add y0 add 3 ay t t mul mul mul 2 by t mul mul add cy add 3 ax t t mul mul mul 2 bx t mul mul add cx add atan /NAngle ED /Y ED /X ED } def /HPosBegin { yB yA ge { /t 1 t sub def } if /Y yB yA sub t mul yA add def } def /HPosEnd { /X Y yyA sub yyB yyA sub Div xxB xxA sub mul xxA add def /NAngle yyB yyA sub xxB xxA sub Atan def } def /HPutLine { HPosBegin /yyA ED /xxA ED /yyB ED /xxB ED HPosEnd } def /HPutLines { HPosBegin yB yA ge { /check { le } def } { /check { ge } def } ifelse /xxA xA def /yyA yA def mark xB yB LPutVar { dup Y check { exit } { /yyA ED /xxA ED } ifelse } loop /yyB ED /xxB ED cleartomark HPosEnd } def /VPosBegin { xB xA lt { /t 1 t sub def } if /X xB xA sub t mul xA add def } def /VPosEnd { /Y X xxA sub xxB xxA sub Div yyB yyA sub mul yyA add def /NAngle yyB yyA sub xxB xxA sub Atan def } def /VPutLine { VPosBegin /yyA ED /xxA ED /yyB ED /xxB ED VPosEnd } def /VPutLines { VPosBegin xB xA ge { /check { le } def } { /check { ge } def } ifelse /xxA xA def /yyA yA def mark xB yB LPutVar { 1 index X check { exit } { /yyA ED /xxA ED } ifelse } loop /yyB ED /xxB ED cleartomark VPosEnd } def /HPutCurve { gsave newpath /SaveLPutVar /LPutVar load def LPutVar 8 -2 roll moveto curveto flattenpath /LPutVar [ {} {} {} {} pathforall ] cvx def grestore exec /LPutVar /SaveLPutVar load def } def /NCCoor { /AngleA yB yA sub xB xA sub Atan def /AngleB AngleA 180 add def GetEdgeA GetEdgeB /LPutVar [ xB1 yB1 xA1 yA1 ] cvx def /LPutPos { LPutVar LPutLine } def /HPutPos { LPutVar HPutLine } def /VPutPos { LPutVar VPutLine } def LPutVar } def /NCLine { NCCoor tx@Dict begin ArrowA CP 4 2 roll ArrowB lineto pop pop end } def /NCLines { false NArray n 0 eq { NCLine } { 2 copy yA sub exch xA sub Atan /AngleA ED n 2 mul dup index exch index yB sub exch xB sub Atan /AngleB ED GetEdgeA GetEdgeB /LPutVar [ xB1 yB1 n 2 mul 4 add 4 roll xA1 yA1 ] cvx def mark LPutVar tx@Dict begin false Line end /LPutPos { LPutLines } def /HPutPos { HPutLines } def /VPutPos { VPutLines } def } ifelse } def /NCCurve { GetEdgeA GetEdgeB xA1 xB1 sub yA1 yB1 sub Pyth 2 div dup 3 -1 roll mul /ArmA ED mul /ArmB ED /ArmTypeA 0 def /ArmTypeB 0 def GetArmA GetArmB xA2 yA2 xA1 yA1 tx@Dict begin ArrowA end xB2 yB2 xB1 yB1 tx@Dict begin ArrowB end curveto /LPutVar [ xA1 yA1 xA2 yA2 xB2 yB2 xB1 yB1 ] cvx def /LPutPos { t LPutVar BezierMidpoint } def /HPutPos { { HPutLines } HPutCurve } def /VPutPos { { VPutLines } HPutCurve } def } def /NCAngles { GetEdgeA GetEdgeB GetArmA GetArmB /mtrx AngleA matrix rotate def xA2 yA2 mtrx transform pop xB2 yB2 mtrx transform exch pop mtrx itransform /y0 ED /x0 ED mark ArmB 0 ne { xB1 yB1 } if xB2 yB2 x0 y0 xA2 yA2 ArmA 0 ne { xA1 yA1 } if tx@Dict begin false Line end /LPutVar [ xB1 yB1 xB2 yB2 x0 y0 xA2 yA2 xA1 yA1 ] cvx def /LPutPos { LPutLines } def /HPutPos { HPutLines } def /VPutPos { VPutLines } def } def /NCAngle { GetEdgeA GetEdgeB GetArmB /mtrx AngleA matrix rotate def xB2 yB2 mtrx itransform pop xA1 yA1 mtrx itransform exch pop mtrx transform /y0 ED /x0 ED mark ArmB 0 ne { xB1 yB1 } if xB2 yB2 x0 y0 xA1 yA1 tx@Dict begin false Line end /LPutVar [ xB1 yB1 xB2 yB2 x0 y0 xA1 yA1 ] cvx def /LPutPos { LPutLines } def /HPutPos { HPutLines } def /VPutPos { VPutLines } def } def /NCBar { GetEdgeA GetEdgeB GetArmA GetArmB /mtrx AngleA matrix rotate def xA2 yA2 mtrx itransform pop xB2 yB2 mtrx itransform pop sub dup 0 mtrx transform 3 -1 roll 0 gt { /yB2 exch yB2 add def /xB2 exch xB2 add def } { /yA2 exch neg yA2 add def /xA2 exch neg xA2 add def } ifelse mark ArmB 0 ne { xB1 yB1 } if xB2 yB2 xA2 yA2 ArmA 0 ne { xA1 yA1 } if tx@Dict begin false Line end /LPutVar [ xB1 yB1 xB2 yB2 xA2 yA2 xA1 yA1 ] cvx def /LPutPos { LPutLines } def /HPutPos { HPutLines } def /VPutPos { VPutLines } def } def /NCDiag { GetEdgeA GetEdgeB GetArmA GetArmB mark ArmB 0 ne { xB1 yB1 } if xB2 yB2 xA2 yA2 ArmA 0 ne { xA1 yA1 } if tx@Dict begin false Line end /LPutVar [ xB1 yB1 xB2 yB2 xA2 yA2 xA1 yA1 ] cvx def /LPutPos { LPutLines } def /HPutPos { HPutLines } def /VPutPos { VPutLines } def } def /NCDiagg { GetEdgeA GetArmA yB yA2 sub xB xA2 sub Atan 180 add /AngleB ED GetEdgeB mark xB1 yB1 xA2 yA2 ArmA 0 ne { xA1 yA1 } if tx@Dict begin false Line end /LPutVar [ xB1 yB1 xA2 yA2 xA1 yA1 ] cvx def /LPutPos { LPutLines } def /HPutPos { HPutLines } def /VPutPos { VPutLines } def } def /NCLoop { GetEdgeA GetEdgeB GetArmA GetArmB /mtrx AngleA matrix rotate def xA2 yA2 mtrx transform loopsize add /yA3 ED /xA3 ED /xB3 xB2 yB2 mtrx transform pop def xB3 yA3 mtrx itransform /yB3 ED /xB3 ED xA3 yA3 mtrx itransform /yA3 ED /xA3 ED mark ArmB 0 ne { xB1 yB1 } if xB2 yB2 xB3 yB3 xA3 yA3 xA2 yA2 ArmA 0 ne { xA1 yA1 } if tx@Dict begin false Line end /LPutVar [ xB1 yB1 xB2 yB2 xB3 yB3 xA3 yA3 xA2 yA2 xA1 yA1 ] cvx def /LPutPos { LPutLines } def /HPutPos { HPutLines } def /VPutPos { VPutLines } def } def % DG/SR modification begin - May 9, 1997 - Patch 1 %/NCCircle { 0 0 NodesepA nodeA \tx@GetEdge pop xA sub 2 div dup 2 exp r %r mul sub abs sqrt atan 2 mul /a ED r AngleA 90 add PtoC yA add exch xA add %exch 2 copy /LPutVar [ 4 2 roll r AngleA ] cvx def /LPutPos { LPutVar t 360 %mul add dup 5 1 roll 90 sub \tx@PtoC 3 -1 roll add /Y ED add /X ED /NAngle ED /NCCircle { NodeSepA 0 NodeA 0 GetEdge pop 2 div dup 2 exp r r mul sub abs sqrt atan 2 mul /a ED r AngleA 90 add PtoC yA add exch xA add exch 2 copy /LPutVar [ 4 2 roll r AngleA ] cvx def /LPutPos { LPutVar t 360 mul add dup 5 1 roll 90 sub PtoC 3 -1 roll add /Y ED add /X ED /NAngle ED % DG/SR modification end } def /HPutPos { LPutPos } def /VPutPos { LPutPos } def r AngleA 90 sub a add AngleA 270 add a sub tx@Dict begin /angleB ED /angleA ED /r ED /c 57.2957 r Div def /y ED /x ED } def /NCBox { /d ED /h ED /AngleB yB yA sub xB xA sub Atan def /AngleA AngleB 180 add def GetEdgeA GetEdgeB /dx d AngleB sin mul def /dy d AngleB cos mul neg def /hx h AngleB sin mul neg def /hy h AngleB cos mul def /LPutVar [ xA1 hx add yA1 hy add xB1 hx add yB1 hy add xB1 dx add yB1 dy add xA1 dx add yA1 dy add ] cvx def /LPutPos { LPutLines } def /HPutPos { xB yB xA yA LPutLine } def /VPutPos { HPutPos } def mark LPutVar tx@Dict begin false Polygon end } def /NCArcBox { /l ED neg /d ED /h ED /a ED /AngleA yB yA sub xB xA sub Atan def /AngleB AngleA 180 add def /tA AngleA a sub 90 add def /tB tA a 2 mul add def /r xB xA sub tA cos tB cos sub Div dup 0 eq { pop 1 } if def /x0 xA r tA cos mul add def /y0 yA r tA sin mul add def /c 57.2958 r div def /AngleA AngleA a sub 180 add def /AngleB AngleB a add 180 add def GetEdgeA GetEdgeB /AngleA tA 180 add yA yA1 sub xA xA1 sub Pyth c mul sub def /AngleB tB 180 add yB yB1 sub xB xB1 sub Pyth c mul add def l 0 eq { x0 y0 r h add AngleA AngleB arc x0 y0 r d add AngleB AngleA arcn } { x0 y0 translate /tA AngleA l c mul add def /tB AngleB l c mul sub def 0 0 r h add tA tB arc r h add AngleB PtoC r d add AngleB PtoC 2 copy 6 2 roll l arcto 4 { pop } repeat r d add tB PtoC l arcto 4 { pop } repeat 0 0 r d add tB tA arcn r d add AngleA PtoC r h add AngleA PtoC 2 copy 6 2 roll l arcto 4 { pop } repeat r h add tA PtoC l arcto 4 { pop } repeat } ifelse closepath /LPutVar [ x0 y0 r AngleA AngleB h d ] cvx def /LPutPos { LPutVar /d ED /h ED /AngleB ED /AngleA ED /r ED /y0 ED /x0 ED t 1 le { r h add AngleA 1 t sub mul AngleB t mul add dup 90 add /NAngle ED PtoC } { t 2 lt { /NAngle AngleB 180 add def r 2 t sub h mul t 1 sub d mul add add AngleB PtoC } { t 3 lt { r d add AngleB 3 t sub mul AngleA 2 t sub mul add dup 90 sub /NAngle ED PtoC } { /NAngle AngleA 180 add def r 4 t sub d mul t 3 sub h mul add add AngleA PtoC } ifelse } ifelse } ifelse y0 add /Y ED x0 add /X ED } def /HPutPos { LPutPos } def /VPutPos { LPutPos } def } def /Tfan { /AngleA yB yA sub xB xA sub Atan def GetEdgeA w xA1 xB sub yA1 yB sub Pyth Pyth w Div CLW 2 div mul 2 div dup AngleA sin mul yA1 add /yA1 ED AngleA cos mul xA1 add /xA1 ED /LPutVar [ xA1 yA1 m { xB w add yB xB w sub yB } { xB yB w sub xB yB w add } ifelse xA1 yA1 ] cvx def /LPutPos { LPutLines } def /VPutPos@ { LPutVar flag { 8 4 roll pop pop pop pop } { pop pop pop pop 4 2 roll } ifelse } def /VPutPos { VPutPos@ VPutLine } def /HPutPos { VPutPos@ HPutLine } def mark LPutVar tx@Dict begin /ArrowA { moveto } def /ArrowB { } def false Line closepath end } def /LPutCoor { NAngle tx@Dict begin /NAngle ED end gsave CM STV CP Y sub neg exch X sub neg exch moveto setmatrix CP grestore } def /LPut { tx@NodeDict /LPutPos known { LPutPos } { CP /Y ED /X ED /NAngle 0 def } ifelse LPutCoor } def /HPutAdjust { Sin Cos mul 0 eq { 0 } { d Cos mul Sin div flag not { neg } if h Cos mul Sin div flag { neg } if 2 copy gt { pop } { exch pop } ifelse } ifelse s add flag { r add neg } { l add } ifelse X add /X ED } def /VPutAdjust { Sin Cos mul 0 eq { 0 } { l Sin mul Cos div flag { neg } if r Sin mul Cos div flag not { neg } if 2 copy gt { pop } { exch pop } ifelse } ifelse s add flag { d add } { h add neg } ifelse Y add /Y ED } def end % END pst-node.pro %%EndProcSet %%BeginProcSet: special.pro %! 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b(our)f(goal)e(is)h(to)h(fo)s(cus)g(on)f(a)h(few)g (topics)257 3339 y(and)e(to)g(iden)m(tify)f(in)m(teresting)h(op)s(en)g (questions)h(for)e(whic)m(h,)i(hop)s(efully)-8 b(,)36 b(the)g(answ)m(ers)257 3460 y(do)c(not)f(lie)e(to)s(o)i(far)f(b)s(ey)m (ond)j(reac)m(h.)44 b(In)31 b(a)g(brief)g(surv)m(ey)i(of)e(this)g (nature)g(it)f(is)h(di\036cult)257 3580 y(to)26 b(touc)m(h)g(up)s(on)g (rami\034cations)d(of)j(the)g(area)f(in)g(computational)e(group)j (theory)-8 b(,)28 b(whic)m(h)257 3700 y(is)41 b(a)f(sub)5 b(ject)42 b(b)m(y)g(itself.)66 b(Also,)42 b(a)f(certain)f(bias)g(due)h (to)g(our)f(researc)m(h)i(in)m(terests)g(in)257 3821 y(complexit)m(y)32 b(theory)h(is)f(una)m(v)m(oidable.)257 4193 y Fv(2)156 b(Preliminaries)257 4425 y Fu(By)43 b(graphs)f(w)m(e)h (mean)e(\034nite)g(simple)g(graphs,)j(usually)d(denoted)i(b)m(y)g Ft(X)51 b Fs(=)43 b(\()p Ft(V)5 b(;)17 b(E)6 b Fs(\))p Fu(,)257 4545 y(where)45 b Ft(V)65 b Fu(is)43 b(the)h(v)m(ertex)i(set)e (and)g Ft(E)52 b Fr(\022)1879 4465 y Fq(\000)1925 4501 y Fp(V)1935 4580 y Fo(2)1981 4465 y Fq(\001)2027 4545 y Fu(.)77 b(W)-8 b(e)44 b(sa)m(y)g(t)m(w)m(o)g(graphs)g Ft(X)3091 4560 y Fo(1)3174 4545 y Fu(and)g Ft(X)3456 4560 y Fo(2)257 4666 y Fu(are)36 b Fn(isomorphic)j Fu(if)c(there)h(is)f (a)g(bijection)f Ft(')f Fs(:)g Ft(V)2073 4681 y Fo(1)2145 4666 y Fr(\000)-17 b(!)33 b Ft(V)2395 4681 y Fo(2)2469 4666 y Fu(suc)m(h)k(that)f Fs(\()p Ft(u;)17 b(v)t Fs(\))31 b Fr(2)i Ft(E)3336 4681 y Fo(1)3411 4666 y Fu(i\033)257 4786 y Fs(\()p Ft(')p Fs(\()p Ft(u)p Fs(\))p Ft(;)17 b(')p Fs(\()p Ft(v)t Fs(\)\))35 b Fr(2)h Ft(E)973 4801 y Fo(2)1013 4786 y Fu(.)57 b(W)-8 b(e)38 b(write)f Ft(X)1605 4801 y Fo(1)1681 4759 y Fr(\030)1681 4790 y Fs(=)1794 4786 y Ft(X)1875 4801 y Fo(2)1952 4786 y Fu(and)g(call)f Ft(')h Fu(an)g(isomorphism.)56 b(An)37 b Fn(au-)257 4907 y(tomorphism)i Fu(of)32 b(a)g(graph)g Ft(X)40 b Fu(is)32 b(an)g(isomorphism)d(from)i Ft(X)41 b Fu(to)31 b Ft(X)8 b Fu(.)44 b(Automorphisms)257 5027 y(are)33 b(p)s(erm)m(utations)e(on)h (the)h(set)g Ft(V)22 b Fu(,)32 b(and)g(the)h(set)g(of)f(automorphisms)e Fs(Aut\(X\))j Fu(forms)257 5147 y(a)43 b(group)h(under)g(p)s(erm)m (utation)e(comp)s(osition.)73 b(More)44 b(precisely)-8 b(,)46 b(if)c Fr(j)p Ft(V)21 b Fr(j)46 b Fs(=)g Ft(n)e Fu(then)257 5268 y Fs(Aut\(X\))39 b Fu(is)e(a)h(subgroup)g(of)g Ft(S)1405 5283 y Fp(n)1490 5268 y Fu(the)g(symmetric)f(group)h(on)g Ft(n)g Fu(elemen)m(ts.)60 b(It)38 b(is)g(w)m(ell)257 5388 y(kno)m(wn)29 b(that)e(graph)h(isomorphism)d(testing)i(is)g(p)s (olynomial)d(time)i(equiv)-5 b(alen)m(t)27 b(to)g(\034nd-)p Black Black eop %%Page: 3 3 3 2 bop Black Black 257 573 a Fu(ing)34 b(a)f(p)s(olynomial-size)e (generator)j(set)h(for)e(the)i(automorphism)d(group)i(of)f(a)h(graph.) 257 693 y(W)-8 b(e)33 b(no)m(w)g(recall)f(some)g(relev)-5 b(an)m(t)32 b(p)s(erm)m(utation)f(group)i(theory)-8 b(.)404 814 y(In)47 b(general,)k Fs(Sym\(\012\))d Fu(denotes)g(the)g(symmetric) e(group)i(on)f(the)h Fn(\034nite)f Fu(set)h Fs(\012)p Fu(.)257 934 y(A)42 b Fn(p)-5 b(ermutation)42 b(gr)-5 b(oup)42 b Fu(on)f Fs(\012)g Fu(is)g(a)g(subgroup)h(of)f Fs(Sym\(\012\))p Fu(.)70 b(F)-8 b(or)41 b Fr(j)p Fs(\012)p Fr(j)h Fs(=)h Ft(n)p Fu(,)g(w)m(e)g(let)257 1054 y Fs(\012)j(=)e([)p Ft(n)p Fs(])f Fu(and)g(simply)e(write)h Ft(S)1489 1069 y Fp(n)1579 1054 y Fu(of)g(all)f(p)s(erm)m(utations)g(on)i Fs([)p Ft(n)p Fs(])i(=)f Fr(f)p Fs(1)p Ft(;)17 b Fs(2)g Ft(:)g(:)g(:)e(;)i(n)p Fr(g)43 b Fu(to)257 1175 y(denote)f Fs(Sym\(\012\))p Fu(.)70 b(Giv)m(en)41 b Ft(g)46 b Fr(2)c Ft(S)1562 1190 y Fp(n)1650 1175 y Fu(and)g Ft(i)g Fr(2)h Fs([)p Ft(n)p Fs(])p Fu(,)h(w)m(e)e(denote)g(b)m(y)g Ft(i)2868 1139 y Fp(g)2950 1175 y Fu(the)f(image)f(of)257 1295 y Ft(i)k Fu(under)h(p)s(erm)m(utation)d Ft(g)t Fu(.)76 b(This)44 b(a)g(con)m(v)m(enien)m(t)h(notation)d(to)i(express)i(the)e (left)f(to)257 1416 y(righ)m(t)37 b(comp)s(osition)e Ft(g)1095 1431 y Fo(1)1134 1416 y Ft(g)1181 1431 y Fo(2)1257 1416 y Fu(of)i(p)s(erm)m(utations)f Ft(g)2021 1431 y Fo(1)2061 1416 y Ft(;)17 b(g)2152 1431 y Fo(2)2226 1416 y Fr(2)36 b Ft(S)2388 1431 y Fp(n)2435 1416 y Fu(.)58 b(More)38 b(precisely)-8 b(,)39 b(w)m(e)f(can)257 1536 y(write)j Ft(i)548 1500 y Fp(g)582 1509 y Fm(1)617 1500 y Fp(g)651 1509 y Fm(2)731 1536 y Fs(=)h(\()p Ft(i)920 1500 y Fp(g)954 1509 y Fm(1)993 1536 y Fs(\))1031 1500 y Fp(g)1065 1509 y Fm(2)1144 1536 y Fu(for)e(all)f Ft(i)k Fr(2)f Fs([)p Ft(n)p Fs(])p Fu(.)69 b(F)-8 b(or)41 b Fs(\001)h Fr(\022)h Fs([)p Ft(n)p Fs(])e Fu(and)g Ft(g)46 b Fr(2)c Ft(S)2876 1551 y Fp(n)2964 1536 y Fu(w)m(e)g(write)f Fs(\001)3455 1500 y Fp(g)257 1656 y Fu(for)f(its)f(image)f(under)j Ft(g)t Fu(:)57 b Fs(\001)1346 1620 y Fp(g)1426 1656 y Fs(=)40 b Fr(f)p Ft(j)46 b Fr(j)40 b Ft(j)46 b Fs(=)40 b Ft(i)1981 1620 y Fp(g)2021 1656 y Fr(g)p Fu(.)65 b(F)-8 b(or)39 b Fs(\001)i Fr(\022)f Fs([)p Ft(n)p Fs(])p Fu(,)i Ft(G)2842 1620 y Fo(\(\001\))3000 1656 y Fu(denotes)f(the)257 1777 y(subgroup)d(of)e Ft(G)h Fu(that)g(\034xes)h(eac)m(h)g(elemen)m(t) f(of)f Fs(\001)p Fu(,)i(and)f Ft(G)2475 1792 y Fo(\001)2575 1777 y Fu(denotes)h(the)g(subgroup)257 1897 y Fr(f)p Ft(g)31 b Fr(2)d Ft(G)g Fr(j)f Fs(\001)720 1861 y Fp(g)788 1897 y Fs(=)h(\001)p Fr(g)p Fu(.)404 2017 y(The)45 b(p)s(erm)m(utation) f(group)g Fn(gener)-5 b(ate)g(d)44 b Fu(b)m(y)i(a)e(subset)j Ft(A)d Fu(of)h Ft(S)2771 2032 y Fp(n)2862 2017 y Fu(is)g(the)g (smallest)257 2138 y(subgroup)30 b(of)f Ft(S)845 2153 y Fp(n)921 2138 y Fu(con)m(taining)f Ft(A)h Fu(and)g(is)g(denoted)h Fr(h)p Ft(A)p Fr(i)p Fu(.)42 b(W)-8 b(e)30 b(assume)f(that)g(subgroups) 257 2258 y(of)22 b Ft(S)418 2273 y Fp(n)487 2258 y Fu(are)g(presen)m (ted)i(b)m(y)f(generator)f(sets.)41 b(Since)23 b(an)m(y)f(\034nite)g (group)g Ft(G)g Fu(has)g(a)g(generator)257 2379 y(set)35 b(of)e(size)g Fs(log)17 b Fr(j)p Ft(G)p Fr(j)p Fu(,)33 b(subgroups)i(of)e Ft(S)1679 2394 y Fp(n)1759 2379 y Fu(ha)m(v)m(e)i(generator)f(sets)g(of)f(size)h(p)s(olynomial)c(in)257 2499 y Ft(n)p Fu(.)43 b(The)29 b(iden)m(tit)m(y)f(p)s(erm)m(utation)g (is)g(denoted)h(b)m(y)g Fs(1)g Fu(\(w)m(e)g(use)h Fs(1)e Fu(to)g(denote)h(the)g(iden)m(tit)m(y)257 2619 y(of)j(all)f(groups\).) 404 2740 y(F)-8 b(or)34 b(a)h(subgroup)g Ft(G)g Fu(of)g Ft(S)1376 2755 y Fp(n)1458 2740 y Fu(\(denoted)h Ft(G)31 b Fr(\024)i Ft(S)2145 2755 y Fp(n)2192 2740 y Fu(\))i(the)g(set)h Ft(i)2623 2704 y Fp(G)2714 2740 y Fs(=)c Fr(f)p Ft(i)2905 2704 y Fp(g)2977 2740 y Fr(j)g Ft(g)j Fr(2)d Ft(G)p Fr(g)j Fu(for)257 2860 y Ft(i)g Fr(2)g Fs([)p Ft(n)p Fs(])i Fu(is)f(the)h Ft(G)p Fn(-orbit)46 b Fu(of)36 b Ft(i)p Fu(,)i(and)f Ft(G)f Fu(is)g Fn(tr)-5 b(ansitive)44 b Fu(on)36 b Fs([)p Ft(n)p Fs(])h Fu(if)f Ft(i)2681 2824 y Fp(G)2775 2860 y Fs(=)e([)p Ft(n)p Fs(])j Fu(for)f Ft(i)f Fr(2)g Fs([)p Ft(n)p Fs(])p Fu(.)257 2980 y(Let)41 b Ft(G)f Fr(\024)h Fs(Sym\(\012\))f Fu(b)s(e)g(transitiv)m(e)f(on)h Fs(\012)p Fu(.)66 b(A)40 b Ft(G)p Fn(-blo)-5 b(ck)50 b Fu(is)40 b(a)f(subset)j Fs(\001)e Fu(of)f Fs([)p Ft(n)p Fs(])i Fu(suc)m(h)257 3101 y(that)29 b(for)f(ev)m(ery)j Ft(g)g Fr(2)d Ft(G)g Fu(either)h Fs(\001)1495 3065 y Fp(g)1563 3101 y Fs(=)f(\001)h Fu(or)f Fs(\001)1973 3065 y Fp(g)2028 3101 y Fr(\\)14 b Fs(\001)29 b(=)e Fr(;)p Fu(.)42 b(F)-8 b(or)28 b(a)g(transitiv)m(e)h(group)f Ft(G)p Fu(,)257 3221 y(the)34 b(set)h Fs([)p Ft(n)p Fs(])f Fu(and)g(the)g(singleton)e(sets)j Fr(f)p Ft(i)p Fr(g)p Fu(,)f Ft(i)c Fr(2)g Fs([)p Ft(n)p Fs(])k Fu(are)f Fn(trivial)44 b Fu(blo)s(c)m(ks.)j(A)34 b(transitiv)m(e)257 3342 y(group)f Ft(G)g Fu(is)f Fn(primitive)40 b Fu(if)32 b(it)g(do)s(es)h(not)g(ha)m (v)m(e)h(an)m(y)g(non)m(trivial)c(blo)s(c)m(ks)j(otherwise)h(it)e(is) 257 3462 y(called)g Fn(imprimitive)p Fu(.)404 3582 y(Let)22 b Ft(G)645 3597 y Fo(1)707 3582 y Fu(and)g Ft(G)963 3597 y Fo(2)1024 3582 y Fu(b)s(e)h(t)m(w)m(o)g(\034nite)f(groups.)40 b(W)-8 b(e)22 b(sa)m(y)i(that)e Ft(G)2503 3597 y Fo(1)2564 3582 y Fu(and)g Ft(G)2820 3597 y Fo(2)2882 3582 y Fu(are)g(isomorphic) 257 3703 y(if)43 b(there)i(is)e(a)g(bijection)g Ft(')k Fs(:)g Ft(G)1492 3718 y Fo(1)1578 3703 y Fr(\000)-16 b(!)46 b Ft(G)1862 3718 y Fo(2)1945 3703 y Fu(that)e(preserv)m(es)j (the)d(group)f(op)s(eration.)257 3823 y(Lik)m(ewise,)36 b(for)e(t)m(w)m(o)i(\034nite)e(rings)h Ft(R)1584 3838 y Fo(1)1658 3823 y Fu(and)g Ft(R)1924 3838 y Fo(2)1964 3823 y Fu(,)g(w)m(e)h(sa)m(y)g(that)e(they)i(are)f(isomorphic)d(if)257 3944 y(there)42 b(is)f(a)h(bijection)e Ft(')i Fs(:)h Ft(R)1370 3959 y Fo(1)1453 3944 y Fr(\000)-16 b(!)42 b Ft(R)1730 3959 y Fo(2)1811 3944 y Fu(that)f(preserv)m(es)j(the)e (ring)e(op)s(erations.)70 b(As)257 4064 y(for)33 b(graphs,)i (automorphisms)d(are)h(isomorphisms)f(from)g(an)i(algebraic)e (structure)j(to)257 4184 y(itself,)26 b(and)f(the)g(automorphisms)e (form)h(a)g(group)h(under)h(the)f(comp)s(osition)d(op)s(eration.)404 4305 y(W)-8 b(e)46 b(brie\035y)h(recall)e(the)h(de\034nitions)g(and)g (notation)f(for)h(some)g(standard)g(com-)257 4425 y(plexit)m(y)38 b(classes.)60 b(Details)37 b(can)h(b)s(e)g(found)g(in)f(a)g(textb)s(o)s (ok)h(lik)m(e)g([14)o(].)59 b(Let)38 b Fs(P)g Fu(denote)257 4545 y(the)29 b(class)g(of)f(languages)g(\(decision)f(problems\))h (that)g(are)h(accepted)h(b)m(y)f(deterministic)257 4666 y(T)-8 b(uring)30 b(mac)m(hines)g(in)f(time)g(b)s(ounded)i(b)m(y)g(a)e (p)s(olynomial)e(in)i(input)h(size,)h(and)f Fs(NP)g Fu(de-)257 4786 y(note)g(the)g(class)g(of)f(languages)g(accepted)i(b)m(y)f (nondeterministic)e(T)-8 b(uring)29 b(mac)m(hines)h(in)257 4907 y(p)s(olynomial)24 b(time.)41 b(W)-8 b(e)27 b(denote)i(the)f (class)f(of)g(functions)h(computable)e(in)h(p)s(olynomial)257 5027 y(time)32 b(b)m(y)h(FP)-8 b(.)404 5147 y(A)31 b(function)f Ft(f)38 b Fs(:)28 b Fr(f)p Fs(0)p Ft(;)17 b Fs(1)p Fr(g)1271 5111 y Fl(\003)1337 5147 y Fr(!)27 b Fk(N)46 b Fu(is)30 b(said)g(to)h(b)s(e)g(in)f(the)h(coun)m(ting)g(class)f Fs(#)p Fu(P)i(if)d(there)257 5268 y(is)36 b(a)f(p)s(olynomial)d(time)i (nondeterministic)g(T)-8 b(uring)35 b(mac)m(hine)g Ft(M)47 b Fu(suc)m(h)37 b(that)e Ft(f)11 b Fs(\()p Ft(x)p Fs(\))36 b Fu(is)257 5388 y(the)d(n)m(um)m(b)s(er)g(of)f(accepting)h(paths)g(of) f Ft(M)43 b Fu(on)33 b(input)f Ft(x)p Fu(.)p Black Black eop %%Page: 4 4 4 3 bop Black Black 404 573 a Fu(A)37 b(function)f Ft(f)48 b Fu(in)36 b(the)h(class)g Fs(FP)1649 531 y Fp(A)1743 573 y Fu(is)g(computable)f(b)m(y)i(p)s(olynomial-time)31 b(deter-)257 693 y(ministic)44 b Fn(or)-5 b(acle)52 b Fu(T)-8 b(uring)45 b(mac)m(hine)h Ft(M)56 b Fu(whic)m(h)47 b(has)f(access)i(to)d(oracle)g Ft(A)p Fu(:)70 b Ft(M)57 b Fu(can)257 814 y(en)m(ter)44 b(a)e(sp)s(ecial)g(query)h(state)g(and)g (query)h(the)f(mem)m(b)s(ership)f(of)g(a)g(string)g Ft(y)j Fu(in)d Ft(A)p Fu(.)257 934 y(W)-8 b(e)32 b(can)f(similarly)d(de\034ne) k Fs(FP)1409 892 y Fp(f)1486 934 y Fu(for)f(a)g(function)f(oracle)h Ft(f)11 b Fu(.)43 b(Let)31 b Fr(C)37 b Fu(b)s(e)32 b(a)f(relativizable) 257 1054 y(complexit)m(y)h(class.)44 b(A)32 b(language)g Ft(A)g Fu(is)g(said)g(to)h(b)s(e)f Fn(low)43 b Fu(for)32 b Fr(C)39 b Fu(if)31 b Fr(C)2758 1018 y Fp(A)2843 1054 y Fs(=)d Fr(C)6 b Fu(.)257 1389 y Fv(3)156 b(Hardness)257 1608 y Fu(GI)33 b(has)g(sev)m(eral)g(prop)s(erties)g(that)f(are)h(not)g (kno)m(wn)h(to)e(hold)g(b)m(y)i(NP-complete)e(prob-)257 1729 y(lems.)71 b(F)-8 b(or)41 b(example,)j(the)f(coun)m(ting)e(v)m (ersion)i(of)e(GI)h(is)f(reducible)h(to)f(its)h(decision)257 1849 y(v)m(ersion)e([28)o(].)62 b(Moreo)m(v)m(er,)42 b(it)c(is)g(kno)m(wn)i(that)e(graph)h(non-isomorphism,)e(the)i(com-)257 1969 y(plemen)m(t)44 b(of)f(GI,)g(b)s(elongs)h(to)f(the)h(class)g(AM)g (of)g(decision)f(problems)g(whose)i(\020y)m(es\021)257 2090 y(instances)g(ha)m(v)m(e)g(short)f(mem)m(b)s(ership)g(pro)s(ofs)f (in)g(a)h(probabilistic)d(sense)46 b([7)o(].)78 b(This)257 2210 y(implies)30 b(that)h(if)g(the)i(problem)d(w)m(ere)j(NP-complete,) f(then)g(the)h(p)s(olynomial)28 b(time)i(hi-)257 2330 y(erarc)m(h)m(y)i(w)m(ould)e(collapse)f(to)h(its)g(second)h(lev)m(el)f ([15,)g(34].)42 b(Because)32 b(of)e(these)h(facts,)g(w)m(e)257 2451 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5147 y(function)32 b(construction)h(for)f(GI:) h(the)g(OR)f(function)g(doubles)h(the)g(size)g(of)f(its)g(inputs.)257 5268 y(Therefore,)40 b(in)d(order)h(to)f(k)m(eep)j(the)e(output)f(of)g (the)i(reduction)e(p)s(olynomial)d(in)j(size,)257 5388 y(the)j(ab)s(o)m(v)m(e)h(metho)s(d)e(can)g(only)g(b)s(e)h(applied)e(to) i(for)f(circuits)f(ha)m(ving)i(a)f(logarithmic)p Black Black eop %%Page: 5 5 5 4 bop Black Black 257 573 a Fu(n)m(um)m(b)s(er)27 b(of)f(OR-gates)g (in)g(an)m(y)h(path)g(from)e(an)i(input)f(to)g(the)h(output)g(gate.)41 b(A)26 b(natural)257 693 y(question)33 b(in)f(this)g(con)m(text)i(is)e (the)h(follo)m(wing.)p Black 257 873 a Fj(Problem)53 b(2.)p Black 49 w Fn(Do)-5 b(es)48 b(gr)-5 b(aph)48 b(isomorphism)e (have)i(an)g(e\036ciently)g(c)-5 b(omputable)48 b Fs(OR)257 994 y Fn(function)35 b Ft(f)45 b Fn(such)35 b(that)g Ft(f)11 b Fs(\()p Ft(x;)17 b(y)t Fs(\))34 b Fn(has)g(size)h(at)g(most)f Ft(c)p Fs(\()p Fr(j)p Ft(x)p Fr(j)22 b Fs(+)g Fr(j)p Ft(y)t Fr(j)p Fs(\))p Fn(,)33 b(wher)-5 b(e)34 b Ft(c)28 b(<)f Fs(2)p Fn(?)404 1174 y Fu(The)48 b(hardness)i(results)e(for)f(GI) g(from)g([20)o(])h(w)m(ere)h(impro)m(v)m(ed)e(in)g([35])h(to)f(other) 257 1294 y(complexit)m(y)37 b(classes)h(using)f(a)g(di\033eren)m(t)g (metho)s(d.)57 b(In)37 b(order)g(to)g(sim)m(ulate)f(a)h(certain)257 1415 y(kind)42 b(of)g(circuit)f(gate)h Ft(g)k Fu(with)41 b(inputs)i Ft(x)f Fu(and)g Ft(y)t Fu(,)i(a)e(graph)g(gadget)g(is)g (constructed)257 1535 y(ha)m(ving)34 b(some)h(v)m(ertices)g(related)f (to)g(the)h(inputs)f(of)g Ft(g)k Fu(and)c(some)g(v)m(ertices)i(related) e(to)257 1656 y(the)d(outputs.)43 b(An)30 b(automorphism)e(in)h(the)i (gadget)f(graph)g(with)f(certain)h(restrictions)257 1776 y(enco)s(ding)45 b(the)g(input)f(v)-5 b(alues)45 b(of)f Ft(g)k Fu(is)c(forced)h(to)g(map)f(the)h(no)s(des)g(related)f(to)h(the) 257 1896 y(output)36 b(in)f(a)g(w)m(a)m(y)h(enco)s(ding)f Ft(g)t Fs(\()p Ft(x;)17 b(y)t Fs(\))p Fu(.)51 b(An)36 b(example)e(of)h(suc)m(h)i(a)e(gadget)h(enco)s(ding)e(a)257 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tx@Dict begin gsave STV newpath 0.8 SLW 0 setgray /ArrowA { moveto } def /ArrowB { } def /NCLW CLW def tx@NodeDict begin 0.0 0.0 neg 0.0 0.0 0 0 /N@D /N@H InitNC { NCLine } if end gsave 0.8 SLW 0 setgray 0 setlinecap stroke grestore grestore end 1168 4735 a 1168 4735 a tx@Dict begin gsave STV newpath 0.8 SLW 0 setgray /ArrowA { moveto } def /ArrowB { } def /NCLW CLW def tx@NodeDict begin 0.0 0.0 neg 0.0 0.0 0 0 /N@F /N@I InitNC { NCLine } if end gsave 0.8 SLW 0 setgray 0 setlinecap stroke grestore grestore end 1168 4735 a 1168 4735 a tx@Dict begin gsave STV newpath 0.8 SLW 0 setgray /ArrowA { moveto } def /ArrowB { } def /NCLW CLW def tx@NodeDict begin 0.0 0.0 neg 0.0 0.0 0 0 /N@G /N@I InitNC { NCLine } if end gsave 0.8 SLW 0 setgray 0 setlinecap stroke grestore grestore end 1168 4735 a 1168 4735 a tx@Dict begin gsave STV newpath 0.8 SLW 0 setgray /ArrowA { moveto } def /ArrowB { } def /NCLW CLW def tx@NodeDict begin 0.0 0.0 neg 0.0 0.0 0 0 /N@E /N@J InitNC { NCLine } if end gsave 0.8 SLW 0 setgray 0 setlinecap stroke grestore grestore end 1168 4735 a 1168 4735 a tx@Dict begin gsave STV newpath 0.8 SLW 0 setgray /ArrowA { moveto } def /ArrowB { } def /NCLW CLW def tx@NodeDict begin 0.0 0.0 neg 0.0 0.0 0 0 /N@H /N@J InitNC { NCLine } if end gsave 0.8 SLW 0 setgray 0 setlinecap stroke grestore grestore end 1168 4735 a 1238 2316 a Ft(x)1293 2331 y Fo(0)1238 2671 y Ft(x)1293 2686 y Fo(1)1242 3519 y Ft(y)1290 3534 y Fo(0)1242 3873 y Ft(y)1290 3888 y Fo(1)1801 3633 y Ft(u)1857 3648 y Fo(1)p Fp(;)p Fo(1)1801 3326 y Ft(u)1857 3341 y Fo(1)p Fp(;)p Fo(0)1801 2854 y Ft(u)1857 2869 y Fo(0)p Fp(;)p Fo(1)1801 2546 y Ft(u)1857 2561 y Fo(0)p Fp(;)p Fo(0)2424 2789 y Ft(z)2469 2804 y Fo(0)2424 3403 y Ft(z)2469 3418 y Fo(1)p Black 790 4939 a Fu(Figure)g(1:)43 b(A)33 b(graph)f(gadget)g(sim)m(ulating)e(a)i (parit)m(y)g(gate.)p Black Black 404 5268 a(F)-8 b(or)22 b(the)i Fr(\010)p Fu(-gate)f(considered)h(in)f(the)h(\034gure,)h(the)f (input)f(v)-5 b(alues)23 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b(series)h Ft(G)27 b Fs(=)h Ft(G)1180 4692 y Fo(0)1237 4677 y Fi(\003)18 b Ft(G)1409 4692 y Fo(1)1465 4677 y Ft(:)f(:)g(:)g Fi(\003)i Ft(G)1770 4692 y Fp(t)1827 4677 y Fs(=)28 b(1)k Fn(e)-5 b(ach)32 b(c)-5 b(omp)g(osition)32 b(factor)g Ft(G)3120 4692 y Fp(i)3149 4677 y Ft(=G)3275 4692 y Fp(i)p Fo(+1)3426 4677 y Fn(is)257 4797 y(either)j(ab)-5 b(elian)34 b(or)g(is)h (isomorphic)e(to)i(a)g(sub)-5 b(gr)g(oup)35 b(of)f Ft(S)2369 4812 y Fp(d)2410 4797 y Fn(.)404 5027 y Fu(The)i(class)f Fs(\000)899 5042 y Fp(d)975 5027 y Fu(of)f(\034nite)h(groups)h(is)f (algorithmically)29 b(imp)s(ortan)m(t.)50 b(It)35 b(has)h(pla)m(y)m(ed) 257 5147 y(an)48 b(imp)s(ortan)m(t)e(role)h(in)g(pro)m(ving)h(time)e(b) s(ounds)j(for)e(sev)m(eral)i(p)s(erm)m(utation)d(group)257 5268 y(algorithms,)32 b(including)g(the)j(curren)m(t)g(b)s(est)g (algorithm)c(for)i(the)i(graph)f(isomorphism)257 5388 y(problem)e(\(e.g.)g(see)i([27]\).)p Black Black eop %%Page: 8 8 8 7 bop Black Black 404 573 a Fu(Giv)m(en)28 b(a)f(p)s(erm)m(utation)g (group)h Ft(G)f Fr(\024)i Ft(S)1854 588 y Fp(n)1929 573 y Fu(b)m(y)f(a)g(generating)g(set)g Ft(A)g Fu(and)h(a)e(subset)j Fs(\001)257 693 y Fu(of)i Fr(f)p Fs(1)p Ft(;)17 b Fs(2)p Ft(;)g Fr(\001)g(\001)g(\001)30 b Ft(;)17 b(n)p Fr(g)p Fu(,)31 b(the)h Fn(set)j(stabilizer)e(pr)-5 b(oblem)38 b Fu(is)31 b(to)g(compute)h(a)f(generating)g(set)i(for)257 814 y(the)j(stabilizer)d(group)h Ft(G)1203 829 y Fo(\001)1266 814 y Fu(.)51 b(Set)35 b(stabilizer)e(is)i(of)f(in)m(terest)h(b)s (ecause)i(GI)d(is)h(reducible)257 934 y(to)h(it.)51 b(T)-8 b(o)36 b(see)h(it)d(note)i(that)g(it)e(su\036ces)k(to)d(sho)m(w)i(that) e(\034nding)g(the)h(automorphism)257 1054 y(group)e(is)f(reducible)h (to)g(set)g(stabilizer.)46 b(F)-8 b(or)33 b(a)h(graph)f Ft(X)38 b Fs(=)30 b(\()p Ft(V)5 b(;)17 b(E)6 b Fs(\))p Fu(,)34 b(let)g Ft(G)29 b Fs(=)h Ft(S)3284 1069 y Fp(n)3365 1054 y Fu(act)257 1175 y(on)j(the)g(pairs)800 1094 y Fq(\000)845 1131 y Fp(V)856 1209 y Fo(2)902 1094 y Fq(\001)980 1175 y Fu(where)h Fr(j)p Ft(V)21 b Fr(j)28 b Fs(=)f Ft(n)p Fu(.)44 b(Clearly)-8 b(,)31 b(for)h Fs(\001)c(=)g Ft(E)39 b Fu(w)m(e)33 b(ha)m(v)m(e)h Ft(G)2931 1190 y Fo(\001)3026 1175 y Fu(is)e Fs(Aut\(X\))p Fu(.)p Black 257 1377 a Fj(Prop)s(osition)i(2.)p Black 41 w Fn(Gr)-5 b(aph)34 b(isomorphism)d(is)j(p)-5 b(olynomial-time)32 b(r)-5 b(e)g(ducible)33 b(to)h(set)f(sta-)257 1497 y(bilizer.)404 1699 y Fu(A)38 b(more)g(in)m(v)m(olv)m(ed)h(reduction)f(in)g([26])g (sho)m(ws)i(that)f(BDGI)f(is)g(NC)h(reducible)g(to)257 1819 y(p)s(erm)m(utation)29 b(groups)i(in)f Fs(\000)1299 1834 y Fp(d)1339 1819 y Fu(.)43 b(Therefore,)32 b(one)e(w)m(a)m(y)i(to) e(put)h(BDGI)f(in)g(NC)h(w)m(ould)f(b)s(e)257 1939 y(to)j(sho)m(w)g (that)g(the)f(set)i(stabilizer)c(problem)i(is)g(in)g(NC.)p Black 257 2141 a Fj(Problem)d(5.)p Black 37 w Fn(Pr)-5 b(e)g(cisely)29 b(classify)g(the)h(c)-5 b(omplexity)28 b(of)h(the)h(set)f(stabilizer)g(pr)-5 b(oblem)28 b(for)257 2262 y(gr)-5 b(oups)35 b(in)f Fs(\000)745 2277 y Fp(d)821 2262 y Fn(\(or)g(even)g(solvable)g(gr)-5 b(oups\).)404 2463 y Fu(It)36 b(follo)m(ws)f(from)g(the)h(results)h(of)e(Babai,)h (Luks,)j(and)d(S\351ress)i([11)o(,)f(8)o(])f(that)g(graph)257 2584 y(isomorphism)30 b(for)i(the)h(b)s(ounded)h(eigen)m(v)-5 b(alue)32 b(m)m(ultiplicit)m(y)c(case)34 b(is)e(in)g(NC.)p Black 257 2786 a Fj(Problem)43 b(6.)p Black 45 w Fn(Classify)d(the)h(c) -5 b(omplexity)40 b(of)h(gr)-5 b(aph)40 b(isomorphism)f(for)h(gr)-5 b(aphs)40 b(with)257 2906 y(b)-5 b(ounde)g(d)34 b(eigenvalue)g (multiplicity.)257 3238 y Fv(5)156 b(Graph)52 b(Canonization)257 3457 y Fu(Let)36 b Fr(G)494 3472 y Fp(n)576 3457 y Fu(denote)g(the)f (set)h(of)f(all)e(simple)g(undirected)j(graphs)f(on)g Ft(n)h Fu(v)m(ertices.)52 b(A)35 b Fn(c)-5 b(an-)257 3578 y(onizing)34 b(function)40 b Fu(for)32 b Fr(G)1193 3593 y Fp(n)1272 3578 y Fu(is)g(a)h(function)f Ft(f)43 b Fu(from)31 b Fr(G)2214 3593 y Fp(n)2294 3578 y Fu(to)h Fr(G)2472 3593 y Fp(n)2552 3578 y Fu(suc)m(h)i(that)p Black 403 3780 a Fr(\017)p Black 48 w Fu(F)-8 b(or)32 b(an)m(y)h(graph)g Ft(G)27 b Fr(2)h(G)1394 3795 y Fp(n)1441 3780 y Fu(,)33 b Ft(f)11 b Fs(\()p Ft(G)p Fs(\))32 b Fu(is)g(isomorphic)f(to)h Ft(G)p Fu(.)p Black 403 3983 a Fr(\017)p Black 48 w Fu(F)-8 b(or)32 b Ft(G)753 3998 y Fo(1)792 3983 y Ft(;)17 b(G)913 3998 y Fo(2)980 3983 y Fr(2)28 b(G)1133 3998 y Fp(n)1180 3983 y Fu(,)33 b Ft(f)11 b Fs(\()p Ft(G)1414 3998 y Fo(1)1453 3983 y Fs(\))28 b(=)f Ft(f)11 b Fs(\()p Ft(G)1796 3998 y Fo(2)1835 3983 y Fs(\))33 b Fu(if)e(and)i(only)f(if)f Ft(G)2565 3998 y Fo(1)2637 3983 y Fu(is)h(isomorphic)f(to)h Ft(G)3425 3998 y Fo(2)3464 3983 y Fu(.)257 4184 y(In)37 b(other)g(w)m(ords,)h(a)e (canonizing)f(function)h(assigns)h(a)f Fn(c)-5 b(anonic)g(al)37 b(form)43 b Fu(to)36 b(eac)m(h)h(iso-)257 4305 y(morphism)31 b(class)i(of)f(graphs.)404 4425 y(F)-8 b(or)30 b(example,)h(the)h (function)f Ft(f)42 b Fu(suc)m(h)33 b(that)e Ft(f)11 b Fs(\()p Ft(G)p Fs(\))31 b Fu(is)g(the)h(lexicographically)c(least)257 4545 y(graph)k(in)f(the)i(isomorphism)c(class)j(con)m(taining)f Ft(G)h Fu(is)f(a)h(canonizing)e(function.)43 b(Ho)m(w-)257 4666 y(ev)m(er,)29 b(as)d(observ)m(ed)i(in)d([10)o(,)h(27],)h(this)f (canonizing)e(function)h(is)h(NP-hard.)41 b(Notice)25 b(that)257 4786 y(this)32 b(function)g(can)g(b)s(e)g(computed)g(in)g Fs(FP)1832 4750 y Fo(NP)1969 4786 y Fu(b)m(y)h(a)e(simple)g(pre\034x)i (searc)m(h)g(algorithm.)404 4907 y(The)38 b(in)m(triguing)d(op)s(en)j (question)g(is)f(whether)i(there)g(is)e Fn(some)44 b Fu(canonizing)36 b(func-)257 5027 y(tion)e(for)g(graphs)i(that)e(can)h (b)s(e)g(computed)g(in)g(p)s(olynomial)c(time.)49 b(No)35 b(b)s(etter)g(upp)s(er)257 5147 y(b)s(ound)30 b(than)g Fs(FP)910 5111 y Fo(NP)1044 5147 y Fu(is)f(kno)m(wn)i(for)e(general)g (graphs)h(\(for)f(an)m(y)h(canonizing)f(function\).)257 5268 y(Th)m(us,)i(it)c(is)h(a)g(basic)g(complexit)m(y-theoretic)f (question)i(to)f(classify)f(the)i(complexit)m(y)f(of)257 5388 y(canonization.)42 b(Is)33 b(this)g(problem)e(lo)m(w)h(for)g(an)m (y)h(lev)m(el)f(of)g(the)h(p)s(olynomial)c(hierarc)m(h)m(y?)p Black Black eop %%Page: 9 9 9 8 bop Black Black 404 573 a Fu(W)-8 b(e)31 b(note)g(that)g(obtaining) e(canonical)g(forms)h(for)h(algebraic)e(ob)5 b(jects)32 b(is)f(a)f(natural)257 693 y(and)23 b(fruitful)e(pursuit)i(in)f (mathematics.)38 b(F)-8 b(or)22 b(instance,)j(w)m(e)f(ha)m(v)m(e)g(the) g(Jordan)e(Canon-)257 814 y(ical)36 b(F)-8 b(orm)36 b(for)h(matrices)g (under)i(similarit)m(y)34 b(transformations.)57 b(Lik)m(ewise,)39 b(w)m(e)g(ha)m(v)m(e)257 934 y(the)47 b(Hermite)d(Normal)g(F)-8 b(orm)44 b(for)i(lattices)e(under)j(unimo)s(dular)c(transformations.) 257 1054 y(These)48 b(normal)c(forms)h(can)i(pla)m(y)e(an)h(imp)s (ortan)m(t)e(role)h(in)g(the)i(design)f(of)f(e\036cien)m(t)257 1175 y(algorithms)30 b(for)i(problems.)404 1296 y(On)j(\034rst)g(sigh)m (t)g(it)f(w)m(ould)h(app)s(ear)g(that)g(graph)g(canonization)e(is)i (closely)g(related)257 1417 y(to)45 b(the)g(problem)f(of)g(isomorphism) f(testing.)80 b(Indeed,)50 b(for)44 b(one)h(direction)f(w)m(e)i(can)257 1537 y(observ)m(e)34 b(that)e(isomorphism)e(testing)i(for)g(graphs)g (is)g(p)s(olynomial-time)27 b(reducible)32 b(to)257 1657 y(graph)h(canonization.)42 b(Ho)m(w)33 b(ab)s(out)f(the)h(con)m(v)m (erse?)p Black 257 1867 a Fj(Problem)43 b(7.)p Black 45 w Fn(Is)d(gr)-5 b(aph)40 b(c)-5 b(anonization)39 b(p)-5 b(olynomial)40 b(time)g(r)-5 b(e)g(ducible)40 b(to)h(gr)-5 b(aph)40 b(iso-)257 1987 y(morphism?)404 2197 y Fu(There)c(is)e(in)m (teresting)h(evidence)h(supp)s(orting)e(a)h(p)s(ositiv)m(e)f(answ)m(er) i(in)e(the)i(results)257 2317 y(of)i(Babai)f(and)h(Luks)h([10].)60 b(Building)36 b(on)i(the)g(earlier)f(seminal)f(w)m(ork)j(of)e(Luks)i ([25],)257 2438 y(Babai)26 b(and)h(Luks)h(tak)m(e)g(an)f(algebraic)e (approac)m(h)j(to)f(the)g(canonization)e(problem.)41 b(W)-8 b(e)257 2558 y(recall)31 b(some)i(de\034nitions)f(b)s(efore)g(w) m(e)i(explain)e(a)g(k)m(ey)i(result)e(in)g(their)g(pap)s(er.)404 2680 y(First)g(w)m(e)j(can)f(assume)g(b)m(y)g(enco)s(ding)g(that)f(w)m (e)i(are)e(w)m(orking)h(with)f(strings)g(\(o)m(v)m(er)257 2800 y(a)k(\034nite)g(alphab)s(et,)g(sa)m(y)h Fr(f)p Fs(0)p Ft(;)17 b Fs(1)p Fr(g)p Fu(\))36 b(as)h(our)g(ob)5 b(jects)38 b(instead)f(of)f(graphs.)57 b(Let)37 b Ft(G)f Fr(\024)f Ft(S)3448 2815 y Fp(n)257 2920 y Fu(b)s(e)44 b(a)e(subgroup)i(of)f(the)g(symmetric)f(group)h(acting)f(on)h Fr(f)p Fs(0)p Ft(;)17 b Fs(1)p Fr(g)2689 2884 y Fp(n)2778 2920 y Fu(as)43 b(follo)m(ws:)63 b(for)42 b(a)257 3041 y(p)s(erm)m(utation)37 b Ft(g)j Fr(2)e Ft(G)g Fu(and)g Ft(x)g Fs(=)f Ft(x)1583 3056 y Fo(1)1623 3041 y Ft(x)1678 3056 y Fo(2)1734 3041 y Fr(\001)17 b(\001)g(\001)e Ft(x)1922 3056 y Fp(n)2007 3041 y Fr(2)37 b(f)p Fs(0)p Ft(;)17 b Fs(1)p Fr(g)2352 3005 y Fp(n)2398 3041 y Fu(,)40 b Ft(g)h Fu(maps)d Ft(x)g Fu(to)g Ft(y)j Fu(\(denoted)257 3161 y Ft(x)312 3125 y Fp(g)381 3161 y Fs(=)27 b Ft(y)t Fu(\),)32 b(where)i Ft(y)c Fs(=)e Ft(x)1152 3176 y Fp(i)1176 3185 y Fm(1)1215 3161 y Ft(x)1270 3176 y Fp(i)1294 3185 y Fm(2)1350 3161 y Fr(\001)17 b(\001)g(\001)e Ft(x)1538 3176 y Fp(i)1562 3184 y Fh(n)1641 3161 y Fu(suc)m(h)34 b(that)f Ft(i)2106 3176 y Fp(k)2176 3161 y Fs(=)28 b Ft(k)2334 3125 y Fp(g)2407 3161 y Fu(for)k Fs(1)27 b Fr(\024)h Ft(k)j Fr(\024)d Ft(n)p Fu(.)404 3283 y(No)m(w,)50 b(the)c(general)f(problem)g(can)h(b)s(e)g(stated)h(as)f(follo)m(ws:)69 b(W)-8 b(e)46 b(sa)m(y)h(that)f(t)m(w)m(o)257 3403 y(strings)40 b Ft(x)h Fu(and)f Ft(y)k Fu(are)c Ft(G)p Fu(-isomorphic)e(if)h Ft(x)1898 3367 y Fp(g)1979 3403 y Fs(=)i Ft(y)i Fu(for)d(some)g Ft(g)k Fr(2)d Ft(G)p Fu(.)67 b(W)-8 b(e)40 b(sa)m(y)h(that)257 3523 y Ft(f)e Fs(:)28 b Fr(f)p Fs(0)p Ft(;)17 b Fs(1)p Fr(g)641 3487 y Fp(n)714 3523 y Fr(\000)-16 b(!)27 b(f)p Fs(0)p Ft(;)17 b Fs(1)p Fr(g)1144 3487 y Fp(n)1215 3523 y Fu(is)25 b(a)f(canonizing)g(function)h(w.r.t.)g(the)h(group)f Ft(G)p Fu(,)h(if)e Ft(f)11 b Fs(\()p Ft(x)p Fs(\))25 b Fu(and)257 3644 y Ft(x)36 b Fu(are)f Ft(G)p Fu(-isomorphic)e(for)i (all)e Ft(x)j Fu(and)f Ft(f)11 b Fs(\()p Ft(x)p Fs(\))32 b(=)h Ft(f)11 b Fs(\()p Ft(y)t Fs(\))34 b Fu(i\033)g Ft(x)i Fu(and)f Ft(y)j Fu(are)e Ft(G)p Fu(-isomorphic.)257 3764 y(In)42 b(an)m(y)h(case,)i(lexicographic)40 b(canonization)g (remains)g(hard)i(ev)m(en)h(for)f(v)m(ery)h(simple)257 3885 y(groups)33 b Ft(G)p Fu(.)p Black 257 4094 a Fj(Prop)s(osition)43 b(3.)p Black 45 w Fu([10])e Fn(The)f(lexic)-5 b(o)g(gr)g(aphic)39 b(c)-5 b(anonizing)40 b(function)g(w.r.t.)h(arbitr)-5 b(ary)257 4214 y(gr)g(oups)38 b Ft(G)g Fn(for)g(strings)g(is)f Fs(NP)q Fn(-har)-5 b(d)37 b(even)h(if)f Ft(G)h Fn(is)g(r)-5 b(estricte)g(d)38 b(to)g(b)-5 b(e)38 b(an)g(elementary)257 4335 y(ab)-5 b(elian)34 b Fs(2)p Fn(-gr)-5 b(oup.)404 4544 y Fu(The)44 b(idea)e(of)h(the)h(pro)s(of)e(is)g(to)h(giv)m(e)g(a)g (p)s(olynomial-time)38 b(reduction)43 b(from)f(the)257 4665 y(maxim)m(um)30 b(clique)j(problem)e(to)h(the)h(lexicographic)e (canonization)g(problem.)404 4786 y(More)i(imp)s(ortan)m(t,)f(on)h(the) h(p)s(ositiv)m(e)f(side,)h(Babai)e(and)h(Luks)i(giv)m(e)e(an)g (algorithm)257 4907 y(for)44 b(computing)e(a)i(canonizing)f(function)g Fn(that)j(dep)-5 b(ends)51 b Fu(on)44 b(the)g(structure)h(of)f(the)257 5027 y(group)27 b Ft(G)p Fu(.)42 b(This)27 b(algorithm)d(is)i(based)i (on)f(a)g(divide-and-conquer)g(strategy)g(along)f(the)257 5147 y(same)e(lines)f(as)h(dev)m(elop)s(ed)h(b)m(y)g(Luks)g(in)e([25)o (]:)40 b(the)24 b(divide)f(and)h(conquer)h(is)f(done)g(on)g(the)257 5268 y(group)36 b Ft(G)f Fu(based)i(on)e(its)h(in)m(ternal)e(structure) j(\(transitiv)m(e)e(constituen)m(ts,)i(primitivit)m(y)257 5388 y(and)c(imprimitivit)m(y)28 b(structure\).)p Black Black eop %%Page: 10 10 10 9 bop Black Black 404 573 a Fu(If)32 b Ft(G)f Fu(is)h(a)f(p)s(erm)m (utation)g(group)g(in)g(the)i(class)f Fs(\000)2190 588 y Fp(d)2230 573 y Fu(,)g(it)f(turns)i(out)e(that)h(this)f(canon-)257 693 y(ization)e(algorithm)f(runs)j(in)f(p)s(olynomial)d(time.)42 b(Crucially)-8 b(,)29 b(the)i(fact)g(that)g(primitiv)m(e)257 814 y(groups)d(in)f Fs(\000)737 829 y Fp(d)804 814 y Fu(are)h(of)f(size)g(at)g(most)g Ft(n)1653 777 y Fp(O)r Fo(\()p Fp(d)p Fo(\))1832 814 y Fu(is)g(used)h(for)f(this)g(analysis.) 41 b(T)-8 b(o)27 b(summarize:)p Black 257 1017 a Fj(Theorem)35 b(4.)p Black 40 w Fu([10])e Fn(Given)g(a)g(gr)-5 b(oup)33 b Ft(G)27 b Fr(\024)i Ft(S)1944 1032 y Fp(n)2024 1017 y Fn(such)k(that)g Ft(G)28 b Fr(2)g Fs(\000)2701 1032 y Fp(d)2741 1017 y Fn(,)34 b(ther)-5 b(e)33 b(is)f(an)h Ft(n)3344 981 y Fp(O)r Fo(\()p Fp(d)p Fo(\))257 1138 y Fn(algorithm)h(that)i(c)-5 b(omputes)34 b(a)h(c)-5 b(anonizing)33 b(function)h(for)h Fr(G)2483 1153 y Fp(n)2530 1138 y Fn(.)404 1341 y Fu(In)e([10)o(])g(it)f(is)g(also)g(sho)m(wn)i (that)f(for)f(general)g(graphs)h(there)h(is)e(a)g Ft(c)2881 1305 y Fp(n)2924 1282 y Fm(1)p Fh(=)p Fm(2+)p Fh(o)p Fm(\(1\))3213 1341 y Fu(canon-)257 1462 y(izing)38 b(algorithm)e(whic)m (h)k(closely)f(matc)m(hes)h(the)g(running)e(time)g(of)h(the)h(b)s(est)g (kno)m(wn)257 1582 y(isomorphism)34 b(test)j(for)f(general)g(graphs.)55 b(Ho)m(w)m(ev)m(er,)39 b(from)d(a)g(complexit)m(y-theoretic)257 1702 y(p)s(oin)m(t)i(of)g(view)h(the)g(relativ)m(e)e(di\036cult)m(y)i (is)f(not)g(clear)g(ev)m(en)i(for)e(the)h(problem)e(of)h Ft(G)p Fu(-)257 1823 y(isomorphism)24 b(testing)h(for)g(a)g(group)h Ft(G)h Fr(2)i Fs(\000)1880 1838 y Fp(d)1920 1823 y Fu(.)41 b(In)26 b(particular,)f(w)m(e)i(w)m(ould)f(lik)m(e)f(to)g(kno)m(w)257 1943 y(an)34 b(answ)m(er)g(for)f(the)h(follo)m(wing)c(question.)46 b(W)-8 b(e)34 b(conjecture)g(that)f(the)h(answ)m(er)g(should)257 2063 y(b)s(e)f(p)s(ositiv)m(e.)p Black 257 2267 a Fj(Problem)38 b(8.)p Black 42 w Fn(L)-5 b(et)37 b Ft(G)30 b Fr(\024)h Ft(S)1287 2282 y Fp(n)1370 2267 y Fn(b)-5 b(e)36 b(a)g(p)-5 b(ermutation)36 b(gr)-5 b(oup)37 b(that)f(is)g(in)g Fs(\000)2883 2282 y Fp(d)2924 2267 y Fn(.)49 b(Is)36 b(the)g(pr)-5 b(ob-)257 2387 y(lem)36 b(of)f(testing)h(if)g(two)g(strings)f Ft(x)i Fn(and)e Ft(y)k Fn(ar)-5 b(e)36 b Ft(G)p Fn(-isomorphic)e(NC)i (e)-5 b(quivalent)35 b(to)h(the)257 2508 y(c)-5 b(orr)g(esp)g(onding)33 b(c)-5 b(anonization)32 b(pr)-5 b(oblem?)43 b(W)-7 b(e)34 b(c)-5 b(an)33 b(also)h(ask)f(a)h(similar)f(question)g(for)257 2628 y(solvable)h(p)-5 b(ermutation)35 b(gr)-5 b(oups,)34 b(which)g(is)g(a)h(sub)-5 b(class)34 b(of)h Fs(\000)2505 2643 y Fp(d)2545 2628 y Fn(.)404 2832 y Fu(A)d(more)g(general)g (problem)f(is)h(the)h(follo)m(wing.)p Black 257 3035 a Fj(Problem)44 b(9.)p Black 46 w Fn(F)-7 b(or)41 b(di\033er)-5 b(ent)41 b(r)-5 b(estricte)g(d)41 b(gr)-5 b(aph)41 b(classes)g(c)-5 b(onsider)g(e)g(d)40 b(in)i(Se)-5 b(ction)41 b(4,)257 3156 y(what)35 b(is)g(the)f(r)-5 b(elative)35 b(c)-5 b(omplexity)34 b(of)g(isomorphism)f(and)i(c)-5 b(anonization?)404 3359 y Fu(W)d(e)32 b(next)g(brie\035y)g(discuss)g(canonization)e(for)h (\034nite)g(groups)h(and)g(rings.)42 b(What)32 b(is)257 3480 y(the)26 b(appropriate)e(notion)g(for)h(groups?)41 b(F)-8 b(or)24 b(ab)s(elian)g(groups,)j(the)e(structure)h(theorem)257 3600 y(decomp)s(osing)35 b(an)m(y)g(\034nite)g(ab)s(elian)f(group)h(in) m(to)f(a)h(direct)g(pro)s(duct)g(of)g(cyclic)g(groups)257 3721 y(is)40 b(a)h(natural)e(canonical)g(form)g(and)i(it)e(can)i(b)s(e) g(used)g(to)g(test)g(the)g(isomorphism)d(of)257 3841 y(t)m(w)m(o)46 b(ab)s(elian)e(groups.)82 b(Th)m(us,)50 b(the)c(problem)e(of)h(canonization)e(for)i(\034nite)g(ab)s(elian)257 3961 y(groups)34 b(b)s(oils)e(do)m(wn)i(to)f(computing)f(the)h(cyclic)g (group)g(decomp)s(osition.)44 b(Of)33 b(course,)257 4082 y(the)41 b(question)f(arises)h(whether)g(there)g(could)f(b)s(e)g (canonical)f(forms)g(that)h(are)g Fn(e)-5 b(asier)257 4202 y Fu(to)43 b(compute.)75 b(F)-8 b(or)42 b(nonab)s(elian)f(groups,) 46 b(it)c(do)s(es)i(not)f(app)s(ear)f(that)h(there)h(is)f(an)m(y)257 4322 y(suc)m(h)f(in)m(trinsic)c(canonical)g(form.)64 b(It)40 b(is)f(tempting)g(to)g(use)i(the)f(comp)s(osition)e(series)257 4443 y(\(or)c(some)f(other)h(series)h(for)e(groups\))h(but)g(these)h (are)f(only)g(partial)d(isomorphism)g(in-)257 4563 y(v)-5 b(arian)m(ts.)50 b(Of)34 b(course,)i(the)f(lexicographic)e(canonical)g (form)h(can)h(alw)m(a)m(ys)g(b)s(e)g(de\034ned)257 4684 y(for)j(\034nite)g(groups)h(\(and)f(rings\).)60 b(But)39 b(one)g(w)m(ould)f(susp)s(ect)i(that)e(it)f(is)h(NP-hard)h(to)257 4804 y(compute.)44 b(T)-8 b(urning)31 b(to)h(\034nite)g(rings,)g(w)m(e) h(can)g(try)f(to)g(use)h(W)-8 b(edderburn's)34 b(decomp)s(o-)257 4924 y(sition)26 b(theorem)h(for)f(semisimple)f(rings)h(to)h(de\034ne)h (canonical)e(forms.)40 b(T)-8 b(o)27 b(summarize,)257 5045 y(w)m(e)34 b(ha)m(v)m(e)g(the)f(follo)m(wing)d(op)s(en-ended)j (question.)p Black 257 5248 a Fj(Problem)45 b(10.)p Black 46 w Fn(What)e(ar)-5 b(e)42 b(the)h(suitable)f(c)-5 b(anonic)g(al)41 b(forms)g(for)i(\034nite)f(gr)-5 b(oups)42 b(and)257 5369 y(rings)35 b(and)f(what)g(is)h(c)-5 b(omplexity)34 b(of)h(c)-5 b(omputing)34 b(these)g(c)-5 b(anonic)g(al)34 b(forms?)p Black Black eop %%Page: 11 11 11 10 bop Black Black 257 573 a Fv(6)156 b(Ring)52 b(and)g(Group)g (Isomorphism)257 797 y Fu(W)-8 b(e)32 b(lo)s(ok)e(no)m(w)i(at)f(the)g (complexit)m(y)g(of)g(isomorphism)d(testing)j(for)g(rings)g(and)g (groups.)257 918 y(These)40 b(questions)f(ha)m(v)m(e)g(ev)m(ok)m(ed)h (in)m(terest)e(due)h(to)e(the)h(recen)m(t)h(w)m(ork)g(b)m(y)g(Ka)m(y)m (al)e(and)257 1038 y(Saxena)h([21)o(])f(relating)e(the)i(complexit)m(y) f(of)g(ring)f(isomorphism)f(to)j(b)s(oth)f(graph)g(iso-)257 1159 y(morphism)41 b(and)h(in)m(teger)g(factoring.)70 b(More)43 b(recen)m(tly)-8 b(,)45 b(Agra)m(w)m(al)c(and)h(Saxena)h(in)e (a)257 1279 y(fascinating)46 b(article)f([1])i(ha)m(v)m(e)i(highligh)m (ted)c(the)j(imp)s(ortance)d(of)i(\034nite)g(rings)f(and)257 1399 y(their)29 b(automorphisms)f(for)h(computational)d(problems)j(in)f (algebra)g(with)h(v)-5 b(arious)29 b(ex-)257 1520 y(amples.)404 1643 y(W)-8 b(e)35 b(discuss)i(the)e(main)e(results)j(ab)s(out)f(ring)f (isomorphism)e(and)j(automorphism)257 1763 y(from)d([21].)44 b(Alongside,)32 b(w)m(e)h(mak)m(e)g(some)g(new)h(observ)-5 b(ations)32 b(for)h(the)g(group)f(isomor-)257 1884 y(phism)g(problem)f (to)i(dra)m(w)g(comparisons)e(and)i(form)m(ulate)e(op)s(en)i (questions.)404 2007 y(Recall)e(that)i(a)f(ring)g Fs(\()p Ft(R)q(;)17 b Fs(+)p Ft(;)g(:)p Fs(\))32 b Fu(with)h(unit)m(y)g(is)f(a) h(comm)m(utativ)m(e)e(group)i(under)h(the)257 2127 y(addition)h(op)s (eration)h(with)h Fs(0)f Fu(as)h(iden)m(tit)m(y)g(and)g(is)f(a)h (monoid)e(under)j(m)m(ultiplication)257 2247 y(with)43 b Fs(1)g Fu(as)h(m)m(ultiplicativ)m(e)c(iden)m(tit)m(y)-8 b(,)45 b(together)f(with)f(m)m(ultiplication)38 b(distributing)257 2368 y(o)m(v)m(er)c(addition.)404 2491 y(The)28 b(complexit)m(y)f(of)g (isomorphism)e(problems)i(migh)m(t)f(c)m(hange)j(dep)s(ending)e(on)h (the)257 2611 y(w)m(a)m(y)h(the)f(input)f(instances)i(are)e(represen)m (ted.)45 b(W)-8 b(e)28 b(\034rst)g(consider)g(the)g(represen)m(tation) 257 2732 y(of)39 b(a)g(\034nite)f(ring)g Ft(R)q Fu(.)63 b(One)39 b(w)m(a)m(y)h(is)f(to)f(describ)s(e)i Ft(R)g Fu(explicitly)d(b)m(y)j(its)e(addition)f(and)257 2852 y(m)m(ultiplication)i(tables.)75 b(This)43 b Fn(table)h(r)-5 b(epr)g(esentation)50 b Fu(is)42 b(of)h(size)g Ft(O)s Fs(\()p Ft(k)s Fr(j)p Ft(R)q Fr(j)3097 2816 y Fo(2)3135 2852 y Fs(\))p Fu(,)j(where)257 2972 y(elemen)m(ts)33 b(of)f Ft(R)i Fu(are)e(enco)s(ded)i(as)f(strings)f(of)g(length)g Ft(k)s Fu(.)404 3095 y(A)f(more)f(compact)h Fn(b)-5 b(asis)33 b(r)-5 b(epr)g(esentation)37 b Fu(w)m(ould)31 b(b)s(e)h(to)e(describ)s (e)i Ft(R)g Fu(b)m(y)g(giving)d(a)257 3216 y Fn(b)-5 b(asis)44 b Fu(for)35 b Ft(R)q Fu(.)53 b(The)37 b(basis)f(is)f(an)h Fn(indep)-5 b(endent)44 b Fu(generating)35 b(set)i Fr(f)p Ft(e)2777 3231 y Fo(1)2816 3216 y Ft(;)17 b(e)2905 3231 y Fo(2)2944 3216 y Ft(;)g Fr(\001)g(\001)g(\001)32 b Ft(;)17 b(e)3227 3231 y Fp(m)3293 3216 y Fr(g)36 b Fu(for)257 3336 y(the)j(additiv)m(e)f(group)h Fs(\()p Ft(R)q(;)17 b Fs(+\))p Fu(.)61 b(Clearly)-8 b(,)40 b Fs(\()p Ft(R)q(;)17 b Fs(+\))38 b Fu(has)h(generating)f(sets)i(of)e(size)h Ft(m)f Fs(=)257 3457 y Ft(O)s Fs(\(log)16 b Fr(j)p Ft(R)q Fr(j)p Fs(\))p Fu(.)42 b(A)m(dditionally)-8 b(,)28 b(to)i(describ)s(e)h (the)g(m)m(ultiplicativ)m(e)c(structure,)32 b(\020structural)257 3577 y(constan)m(ts\021)51 b(of)40 b(the)i(ring)e Ft(\013)1306 3592 y Fp(ij)t(k)1448 3577 y Fr(2)j Fk(Z)p Ft(;)17 b Fs(1)40 b Fr(\024)j Ft(i;)17 b(j;)g(k)46 b Fr(\024)d Ft(m)e Fu(are)h(giv)m(en,)h(where)g Ft(e)3183 3592 y Fp(i)3239 3577 y Fr(\001)28 b Ft(e)3340 3592 y Fp(j)3419 3577 y Fs(=)257 3623 y Fq(P)363 3726 y Fp(k)422 3697 y Ft(\013)484 3712 y Fp(ij)t(k)583 3697 y Ft(e)628 3712 y Fp(k)671 3697 y Fu(.)53 b(Since)37 b(the)f Fn(char)-5 b(acteristic)41 b Fu(of)35 b Ft(R)i Fu(is)f(b)s(ounded)g(b)m(y)h Fr(j)p Ft(R)q Fr(j)p Fu(,)g(eac)m(h)f(structural)257 3818 y(constan)m(t)i(is)f Ft(m)g Fu(bits)g(long.)56 b(No)m(w,)39 b(supp)s(ose)f(that)f(the)h(elemen)m(ts)f(of)g Ft(R)h Fu(are)f(enco)s(ded)257 3938 y(as)c(strings)f(of)h(length)f Ft(k)s Fu(.)43 b(Clearly)32 b(the)h(en)m(tire)f(represen)m(tation)h(is) f(of)h(size)f Ft(O)s Fs(\()p Ft(k)s(m)3292 3902 y Fo(4)3331 3938 y Fs(\))p Fu(.)404 4061 y(Lik)m(ewise,)g(consider)g(\034nite)f (groups)h Ft(G)g Fu(whose)g(elemen)m(ts)g(are)g(enco)s(ded)h(as)e (strings)257 4182 y(of)36 b(length)g Ft(k)s Fu(.)54 b(W)-8 b(e)37 b(could)e(describ)s(e)i Ft(G)f Fu(b)m(y)h(the)g Fn(table)h(r)-5 b(epr)g(esentation)42 b Fu(b)m(y)37 b(giving)e(the)257 4302 y(m)m(ultiplication)27 b(table)k(of)g(size)h Ft(O)s Fs(\()p Ft(k)s Fr(j)p Ft(G)p Fr(j)1722 4266 y Fo(2)1760 4302 y Fs(\))p Fu(.)43 b(Again,)30 b(a)i(more)e(compact)i(represen)m (tation)257 4422 y(for)45 b(\034nite)g(groups)g(w)m(ould)g(b)s(e)g(to)g (giv)m(e)g(a)g(generating)f(set)i Fr(f)p Ft(g)2640 4437 y Fo(1)2679 4422 y Ft(;)17 b(g)2770 4437 y Fo(2)2809 4422 y Ft(;)g Fr(\001)g(\001)g(\001)31 b Ft(;)17 b(g)3093 4437 y Fp(k)3135 4422 y Fr(g)45 b Fu(for)f Ft(G)p Fu(,)257 4543 y(where)d(the)f(m)m(ultiplication)34 b(op)s(eration)k(is)h (implicitly)c(describ)s(ed)41 b(b)m(y)f(a)f(\020blac)m(k-b)s(o)m(x\021) 257 4663 y([13,)26 b(9].)41 b(The)27 b(\020blac)m(k-b)s(o)m(x\021)33 b(mo)s(del)24 b(in)m(tro)s(duced)i(b)m(y)h([13,)f(9)o(])g(is)g(a)f(con) m(v)m(enien)m(t)j(setting)e(to)257 4784 y(study)i(the)e(complexit)m(y)f (of)h(group-theoretic)f(problems)g(that)h(do)g(not)f(tak)m(e)i(adv)-5 b(an)m(tage)257 4904 y(of)32 b(the)h(actual)f(group)g(op)s(eration)f (\(p)s(erm)m(utation)g(groups)i(or)f(matrix)f(groups)i(etc\).)404 5027 y(A)j(third)f(p)s(ossibilit)m(y)f(for)i(represen)m(ting)h (\034nite)f(ab)s(elian)e(groups)j(b)m(y)g(giving)d Fn(inde-)257 5147 y(p)-5 b(endent)43 b Fu(generating)34 b(sets.)49 b(Whether)36 b(an)e(arbitrary)f(generating)h(set)h(can)f(b)s(e)g (trans-)257 5268 y(formed)39 b(to)g(an)g Fn(indep)-5 b(endent)48 b Fu(generating)38 b(set)j(in)d(p)s(olynomial)e(time)i(is)h (op)s(en.)64 b(It)40 b(is)257 5388 y(related)d(to)g(mem)m(b)s(ership)g (testing)g(and)g(the)h(discrete)g(log)d(problem.)57 b(Ho)m(w)m(ev)m (er,)41 b(this)p Black Black eop %%Page: 12 12 12 11 bop Black Black 257 573 a Fu(transformation)37 b(can)h(b)s(e)h(done)g(b)m(y)g(a)f(p)s(olynomial)d(time)i(quan)m(tum)i (algorithm.)57 b(In-)257 693 y(deed,)36 b(isomorphism)c(testing)h(for)h (ab)s(elian)e(blac)m(k-b)s(o)m(x)i(groups)h(giv)m(en)f(b)m(y)h (generating)257 814 y(sets)f(can)f(b)s(e)g(done)g(in)e(quan)m(tum)i(p)s (olynomial)c(time)i(\(see)j([31)o(])f(for)f(example\).)p Black 257 994 a Fj(Problem)38 b(11.)p Black 43 w Fn(What)f(is)g(the)f (c)-5 b(omplexity)36 b(of)h(c)-5 b(onverting)35 b(a)i(gener)-5 b(ator)36 b(r)-5 b(epr)g(esenta-)257 1115 y(tion)35 b(to)g(a)g(b)-5 b(asis)34 b(r)-5 b(epr)g(esentation)34 b(for)g(\034nite)h(rings?)404 1295 y Fu(Notice)43 b(that)g(the)h(basis)g(represen)m(tation)g(for)f (\034nite)h(rings)f(is)g(more)g(structured)257 1416 y(than)30 b(the)g(generator)g(represen)m(tation)h(for)e(\034nite)g(groups.)43 b(The)31 b(nicer)f(represen)m(tation)257 1536 y(is)g(basically)f(due)i (to)f(the)h(fact)g(that)f(the)h(additiv)m(e)f(group)g(of)g(a)g (\034nite)g(ring)g(is)g(comm)m(u-)257 1656 y(tativ)m(e.)43 b(Indeed,)32 b(if)d(a)h(\034nite)g(group)g Ft(G)g Fu(is)g(giv)m(en)g(b) m(y)h(a)f(generator)g(set)h Fr(h)p Ft(g)2934 1671 y Fo(1)2973 1656 y Ft(;)17 b(g)3064 1671 y Fo(2)3103 1656 y Ft(;)g Fr(\001)g(\001)g(\001)31 b Ft(;)17 b(g)3387 1671 y Fp(k)3429 1656 y Fr(i)p Fu(,)257 1777 y(in)46 b(general)h(it)e(is)i(not)f(p)s (ossible)g(to)h(express)h(an)f(arbitrary)f(elemen)m(t)g Ft(g)56 b Fr(2)c Ft(G)47 b Fu(as)f(a)257 1897 y(p)s(olynomial-size)39 b(pro)s(duct)1332 1822 y Fq(Q)1426 1849 y Fp(m)1426 1926 y(j)t Fo(=1)1570 1897 y Ft(g)1617 1912 y Fp(i)1641 1922 y Fh(j)1677 1897 y Fu(!)74 b(Ho)m(w)m(ev)m(er)45 b(the)e(reac)m (habilit)m(y)f(lemma)e(of)j([13)o(])257 2017 y(sho)m(ws)37 b(that)e(it)g(is)g(p)s(ossible)f(to)i(express)h Ft(g)i 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b(to)f(in)m(teger)g(factoring)f(\(via)h (randomized)g(reductions\).)404 4184 y(It)e(is)f(in)m(teresting)h(to)g (compare)f(these)i(results)g(with)e(the)i(situation)d(for)i(group)f (iso-)257 4305 y(morphism.)41 b(A)29 b(basic)g(di\033erence)h(b)s(et)m (w)m(een)h(the)f(t)m(w)m(o)g(problems)e(is)h(in)f(the)i(description)257 4425 y(of)43 b(an)g(isomorphism.)71 b(A)43 b(ring)f(isomorphism)e(b)s (et)m(w)m(een)45 b(t)m(w)m(o)f(rings)e Ft(R)2975 4440 y Fo(1)3058 4425 y Fu(and)g Ft(R)3331 4440 y Fo(2)3414 4425 y Fu(in)257 4545 y(basis)37 b(represen)m(tation)g(can)g(b)s(e)g (describ)s(ed)h(b)m(y)f(an)g(in)m(v)m(ertible)f(in)m(teger)g(matrix)g (\(after)257 4666 y(suitably)27 b(mo)s(difying)e(the)k(bases)g(in)e(p)s (olynomial)d(time\).)40 b(Ho)m(w)m(ev)m(er,)31 b(in)c(the)h(case)h(of)e (an)257 4786 y(isomorphism)32 b Ft(')i Fu(b)s(et)m(w)m(een)i(t)m(w)m(o) e(\034nite)g(groups)g Ft(G)g Fu(and)g Ft(H)42 b Fu(giv)m(en)34 b(b)m(y)h(generator)f(sets,)257 4907 y(there)26 b(seems)g(no)f 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y(Since)k(the)h(isomorphisms) d(\(or)i(automorphisms\))e(of)i(\034nite)g(groups)h(giv)m(en)f(b)m(y)h (gen-)257 935 y(erators)40 b(do)f(not)g(ha)m(v)m(e)i(explicit)d (mathematical)e(descriptions,)41 b(w)m(e)g(do)e(not)h(ha)m(v)m(e)g(an) 257 1055 y Fs(FP)387 1019 y Fo(AM)p Fl(\\)p Fo(coAM)771 1055 y Fu(b)s(ound)31 b(for)f(coun)m(ting)g(the)h(n)m(um)m(b)s(er)g(of) f(group)g(automorphisms)f(\(equiv-)257 1175 y(alen)m(tly)d (isomorphisms\).)39 b(Ho)m(w)m(ev)m(er,)30 b(supp)s(ose)d(a)f(mapping)f Ft(')j Fs(:)f Ft(G)h Fr(\000)-16 b(!)27 b Ft(H)34 b Fu(is)26 b(giv)m(en)g(b)m(y)257 1296 y Ft(')p Fs(\()p Ft(g)406 1311 y Fp(i)434 1296 y Fs(\))33 b Fu(as)g(a)g(straigh)m(t-line)d (program)i(o)m(v)m(er)i(the)f(generators)g(of)g Ft(H)40 b Fu(for)32 b(eac)m(h)i(generator)257 1416 y Ft(g)304 1431 y Fp(i)375 1416 y Fu(of)42 b Ft(G)p Fu(.)73 b(Then)43 b(testing)f(if)g Ft(')g Fu(de\034nes)i(an)e(isomorphism)e(is)i(in)g Fs(AM)29 b Fr(\\)g Fs(coAM)44 b Fu(due)257 1536 y(to)33 b(the)h(order-v)m(eri\034cation)f(in)m(teractiv)m(e)g(proto)s(col)f(of) h(Babai)f([9].)46 b(Using)33 b(this)g(w)m(e)i(can)257 1657 y(giv)m(e)41 b(a)f Fs(#)p Fu(P)703 1615 y Fo(NP)849 1657 y Fu(upp)s(er)h(b)s(ound:)60 b(Supp)s(ose)42 b(elemen)m(ts)f(of)f Ft(G)h Fu(and)f Ft(H)49 b Fu(are)40 b(enco)s(ded)i(as)257 1777 y(strings)29 b(of)f(length)h Ft(m)p Fu(.)42 b(F)-8 b(or)28 b(the)h(generators)g Ft(g)1975 1792 y Fo(1)2015 1777 y Ft(;)17 b(g)2106 1792 y Fo(2)2144 1777 y Ft(;)g Fr(\001)g(\001)g(\001)31 b Ft(;)17 b(g)2428 1792 y Fp(k)2471 1777 y Fu(,)29 b(the)g Fs(#)p Fu(P)h(oracle)e(mac)m(hine)257 1898 y(guesses)44 b(the)d(images)f Ft(')p Fs(\()p Ft(g)1258 1913 y Fp(i)1286 1898 y Fs(\))p Ft(;)17 b Fs(1)41 b Fr(\024)i Ft(i)g Fr(\024)g Ft(k)h Fu(as)d(strings)g(of)g(length)f Ft(m)p Fu(.)70 b(Using)41 b(an)g(NP)257 2018 y(oracle,)c(it)f(then)h (computes)g(the)g(straigh)m(t-line)d(programs)i(for)g(eac)m(h)i Ft(')p Fs(\()p Ft(g)3017 2033 y Fp(i)3044 2018 y Fs(\))p Fu(,)g(o)m(v)m(er)g(the)257 2138 y(generators)h(of)e Ft(H)8 b Fu(.)59 b(No)m(w,)40 b(a)e(new)g(group)g Ft(K)45 b Fu(is)38 b(formed,)g(that)g(is)f(generated)i(b)m(y)g(the)257 2259 y(pairs)33 b Fs(\()p Ft(g)582 2274 y Fp(i)610 2259 y Ft(;)17 b(')p Fs(\()p Ft(g)803 2274 y Fp(i)831 2259 y Fs(\)\))p Fu(.)46 b(Notice)33 b(that)g Ft(K)41 b Fu(is)33 b(a)g(subgroup)h(of)f Ft(G)23 b Fr(\002)g Ft(H)8 b Fu(.)46 b(The)35 b(order)e(v)m(eri\034ca-)257 2379 y(tion)f Fs(AM)i Fu(proto)s(col)d(of)h([9])h(can)g(no)m(w)h(b)s(e)f(used)h(to)f(compare) f(the)h(orders)h(of)e Ft(G)h Fu(and)g Ft(K)257 2500 y Fu(and)d(to)g(accept)g(if)f(and)h(only)f(if)g(their)g(orders)i(are)e (equal.)43 b(Clearly)-8 b(,)29 b(this)h(upp)s(er)g(b)s(ound)257 2620 y(also)h(holds)f(for)h(the)h(complexit)m(y)e(of)h(computing)e(the) j(n)m(um)m(b)s(er)g(of)e(automorphisms)g(of)257 2740 y Ft(G)p Fu(.)44 b(Since)32 b Fs(#)p Fu(P)807 2698 y Fl(9)p Fp(:)p Fo(AM)1015 2740 y Fs(=)c(#)p Fu(P)1266 2698 y Fo(AM)1416 2740 y Fs(=)f(#)p Fu(P)1667 2698 y Fo(NP)1804 2740 y Fu(w)m(e)34 b(ha)m(v)m(e)g(the)f(follo)m(wing:)p Black 257 2947 a Fj(Prop)s(osition)i(5.)p Black 42 w Fn(Computing)f(the)g(numb)-5 b(er)34 b(of)h(isomorphism)d(b)-5 b(etwe)g(en)34 b(two)h(gr)-5 b(oups)257 3067 y Ft(G)35 b Fn(and)f Ft(H)43 b Fn(given)34 b(by)h(gener)-5 b(ating)34 b(sets)g(is)h(in)f Fs(#)p Fn(P)2098 3025 y Fo(NP)2203 3067 y Fn(.)p Black 257 3273 a Fj(Problem)50 b(12.)p Black 48 w Fn(Tightly)d(classify)e(the)i(c)-5 b(omplexity)45 b(of)i(c)-5 b(omputing)45 b(the)i(numb)-5 b(er)46 b(of)257 3394 y(gr)-5 b(oup)46 b(isomorphisms,)g(when)e(the)i(gr)-5 b(oups)45 b(ar)-5 b(e)45 b(in)g(the)g(gener)-5 b(ator)45 b(r)-5 b(epr)g(esentation.)257 3514 y(Mor)g(e)50 b(pr)-5 b(e)g(cisely,)52 b(for)d(a)g(\034nite)g(gr)-5 b(oup)49 b Ft(G)h Fn(given)e(by)i(gener)-5 b(ators)48 b(in)h(the)h(black-b)-5 b(ox)257 3634 y(mo)g(del,)42 b(is)g(the)f(pr)-5 b(oblem)41 b(of)g(c)-5 b(omputing)41 b(the)g(numb)-5 b(er)41 b(of)h(automorphism)e (in)h Ft(G)h Fn(low)257 3755 y(for)35 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Fu(there)g(is)e(at)h(least)f(one)h (suc)m(h)i Ft(k)d(>)e Fs(1)i Fu(so)h(that)g Ft(a)d Fr(7!)f Ft(a)2190 1379 y Fp(k)2264 1416 y Fu(is)j(a)h(non)m(trivial)d (automorphism)257 1536 y(of)36 b Ft(G)449 1551 y Fp(i)477 1536 y Fu(.)53 b(This)36 b(can)g(b)s(e)g(extended)i(easily)d(to)g(a)h (non)m(trivial)d(automorphism)h(of)h Ft(G)p Fu(.)53 b(On)257 1656 y(the)40 b(other)g(hand,)h(if)d Fr(j)p Ft(G)1174 1671 y Fp(i)1202 1656 y Fr(j)h Fs(=)g(2)h Fu(for)e(eac)m(h)j Ft(i)p Fu(,)g(then)f Ft(G)g Fu(is)e(a)i(v)m(ector)g(space)h(o)m(v)m(er) f Fk(F)3332 1671 y Fo(2)3417 1656 y Fu(of)257 1777 y(dimension)27 b Ft(r)s Fu(.)42 b(Hence)30 b(an)m(y)f(nonsingular)d Ft(r)16 b Fr(\002)e Ft(r)32 b Fu(matrix)26 b(di\033eren)m(t)j(from)e (iden)m(tit)m(y)h(o)m(v)m(er)257 1897 y Fk(F)312 1912 y Fo(2)390 1897 y Fu(is)k(a)h(non)m(trivial)d(automorphism)g(of)i Ft(G)p Fu(.)p 1950 1897 49 49 v 404 2019 a(It)39 b(is)g(sho)m(wn)i(in)d ([21])h(that)g(all)f(rigid)f(rings)i(ha)m(v)m(e)h(a)g(simple)d 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Fs(1)e Fr(7!)f Ft(j)36 b Fu(de\034nes)31 b(an)e(automorphism)f(if)g(and)i(only)f(if) 257 3104 y Fs(gcd)q(\()p Ft(j;)17 b(n)p Fs(\))28 b(=)f(1)p Fu(,)j(since)g Ft(j)k Fr(2)28 b Fk(Z)1332 3119 y Fp(n)1406 3104 y Fu(is)h(a)h(generator)f(i\033)g(it)f(is)i(relativ)m(ely)e(prime) h(to)g Ft(n)p Fu(.)43 b(Th)m(us,)257 3224 y Fs(\()p Fk(Z)364 3239 y Fp(n)409 3224 y Ft(;)17 b Fs(+\))25 b Fu(has)i(precisely)f Ft(')p Fs(\()p Ft(n)p Fs(\))g Fu(generators,)i(where)f Ft(')f Fu(is)g(the)h(Euler)f Ft(')p Fu(-function.)40 b(It)26 b(fol-)257 3344 y(lo)m(ws)33 b(that)f(computing)f Fs(#Aut\()p Fk(Z)1518 3359 y Fo(n)1559 3344 y Fs(\))h Fu(implied)e(computing)h Ft(')p Fs(\()p Ft(n)p Fs(\))h Fu(whic)m(h)h(is)e(equiv)-5 b(alen)m(t)257 3465 y(to)33 b(in)m(teger)f(factoring)f(w.r.t.)i(randomized)f(p)s(olynomial-time)27 b(reductions.)p Black 257 3675 a Fj(Prop)s(osition)46 b(7.)p Black 46 w Fn(Inte)-5 b(ger)43 b(factoring)f(is)h(r)-5 b(e)g(ducible)42 b(to)h(c)-5 b(omputing)43 b(the)g(numb)-5 b(er)43 b(of)257 3795 y(automorphism)34 b(for)h(a)f(\034nite)h(gr)-5 b(oup)35 b Ft(G)f Fn(given)g(by)h(gener)-5 b(ators.)404 4005 y Fu(It)33 b(w)m(ould)h(b)s(e)f(in)m(teresting)h(to)f(kno)m(w)h (if)f(the)h(same)f(result)g(holds)h(for)f(p)s(erm)m(utation)257 4125 y(groups.)p Black 257 4335 a Fj(Problem)j(14.)p Black 42 w Fn(Is)f(inte)-5 b(ger)35 b(factoring)g(r)-5 b(e)g(ducible)34 b(to)i(c)-5 b(omputing)34 b(the)h(numb)-5 b(er)35 b(of)g(au-)257 4456 y(tomorphism)f(of)h(a)f(p)-5 b(ermutation)35 b(gr)-5 b(oup)34 b Ft(G)28 b Fr(\024)g Ft(S)2077 4471 y Fp(n)2159 4456 y Fn(given)34 b(by)h(gener)-5 b(ators?)404 4666 y Fu(In)35 b([21])f(it)g(is)h(sho)m(wn)h(that)f (\034nding)f(a)h(non)m(trivial)d(ring)i(automorphism)f(is)h(equiv-)257 4786 y(alen)m(t)h(to)f(in)m(teger)h(factoring.)49 b(In)m(terestingly)-8 b(,)36 b(for)e(the)h(case)h(of)e(groups)i(the)f(situation)257 4907 y(is)40 b(quite)f(di\033eren)m(t.)66 b(Let)40 b Ft(G)f Fu(b)s(e)h(a)g(group)f(giv)m(en)h(b)m(y)h(generators.)65 b(W)-8 b(e)40 b(can)g(c)m(hec)m(k)i(if)257 5027 y(it)g(is)g(nonab)s (elian)f(\(simply)g(b)m(y)i(c)m(hec)m(king)h(if)d(the)i(generators)g (comm)m(ute)f(with)g(eac)m(h)257 5147 y(other\).)i(If)32 b Ft(G)g Fu(is)g(nonab)s(elian,)f(w)m(e)i(will)d(\034nd)j(a)f (generator)g Ft(g)k Fu(suc)m(h)e(that)e Ft(g)t(g)3063 5162 y Fp(i)3118 5147 y Fr(6)p Fs(=)27 b Ft(g)3268 5162 y Fp(i)3296 5147 y Ft(g)36 b Fu(for)257 5268 y(some)30 b(other)f(generator)h Ft(g)1228 5283 y Fp(i)1285 5268 y Fu(of)f Ft(G)p Fu(.)42 b(Clearly)-8 b(,)29 b(the)h(inner)f (automorphism)f Ft(\034)2990 5283 y Fp(g)3059 5268 y Fu(de\034ned)j(b)m(y)257 5388 y Ft(\034)299 5403 y Fp(g)368 5388 y Fs(:)c Ft(x)h Fr(7!)g Ft(g)t(xg)790 5352 y Fl(\000)p Fo(1)916 5388 y Fu(is)k(a)g(non)m(trivial)e(automorphism.)p Black Black eop %%Page: 15 15 15 14 bop Black Black Black 257 573 a Fj(Prop)s(osition)24 b(8.)p Black 35 w Fn(Ther)-5 b(e)25 b(is)g(a)h(p)-5 b(olynomial-time)24 b(algorithm)h(for)h(\034nding)e(a)i(nontrivial)257 693 y(automorphism)34 b(of)h(a)f(nonab)-5 b(elian)33 b(gr)-5 b(oup)35 b(given)f(by)h(gener)-5 b(ator)34 b(set.)404 910 y Fu(Ab)s(elian)g(groups)j(dot)f(ha)m(v)m(e)i(inner)e (automorphisms.)53 b(On)36 b(the)h(other)f(hand)h(if)e Ft(G)257 1030 y Fu(is)h(an)h(ab)s(elian)e(group)h(giv)m(en)h(b)m(y)g (an)g Fn(indep)-5 b(endent)44 b Fu(generating)36 b(set)h Fr(h)p Ft(g)2934 1045 y Fo(1)2973 1030 y Ft(;)17 b(g)3064 1045 y Fo(2)3103 1030 y Ft(;)g Fr(\001)g(\001)g(\001)31 b Ft(;)17 b(g)3387 1045 y Fp(k)3429 1030 y Fr(i)p Fu(,)257 1151 y(and)28 b(a)f(m)m(ultiple)e Ft(n)j Fu(of)f Fr(j)p Ft(G)p Fr(j)g Fu(is)g(kno)m(wn,)j(then)f(it)d(is)h(easy)i(to)e(\034nd)h (a)g(non)m(trivial)d(automor-)257 1271 y(phism)35 b(applying)g(the)h (ideas)f(of)g(Prop)s(osition)f(6.)53 b(If)35 b(eac)m(h)i Ft(g)2509 1286 y Fp(i)2573 1271 y Fu(has)f(order)f Fs(2)h Fu(then)g Ft(G)g Fu(is)257 1391 y(v)m(ector)d(space)g(o)m(v)m(er)g Fk(F)1072 1406 y Fo(2)1149 1391 y Fu(and)f(an)m(y)h(nonsingular)e Ft(k)23 b Fr(\002)f Ft(k)35 b Fu(matrix)30 b(is)i(an)f(automorphism.) 257 1512 y(Otherwise,)45 b(if)40 b Ft(g)902 1527 y Fp(i)971 1512 y Fu(has)i(order)g(more)e(than)i Fs(2)f Fu(then)h(pic)m(k)g(a)f(p) s(ositiv)m(e)g(in)m(teger)g Ft(a)i(>)f Fs(1)257 1632 y Fu(suc)m(h)d(that)f Fs(gcd\()p Ft(a;)17 b(n)p Fs(\))36 b(=)g(1)h Fu(b)m(y)i(randomly)d(pic)m(king)h Ft(a)f Fr(2)h Fs([)p Ft(n)26 b Fr(\000)g Fs(1])p Fu(.)58 b(Then,)40 b(with)d(high)257 1752 y(probabilit)m(y)f Ft(g)808 1767 y Fp(i)872 1752 y Fr(6)p Fs(=)g Ft(g)1035 1716 y Fp(a)1031 1777 y(i)1114 1752 y Fu(and)i Ft(g)1356 1767 y Fp(i)1421 1752 y Fu(and)g Ft(g)1667 1716 y Fp(a)1663 1777 y(i)1745 1752 y Fu(ha)m(v)m(e)h(the)f(same)g(order.)59 b(No)m(w,)39 b Ft(g)3008 1767 y Fp(i)3073 1752 y Fr(7!)d Ft(g)3260 1716 y Fp(a)3256 1777 y(i)3338 1752 y Fu(and)257 1873 y Ft(g)304 1888 y Fp(j)368 1873 y Fr(7!)28 b Ft(g)543 1888 y Fp(j)579 1873 y Ft(;)17 b(j)33 b Fr(6)p Fs(=)28 b Ft(i)k Fu(de\034nes)j(a)d(non)m(trivial)e(automorphism.)404 1996 y(Ho)m(w)m(ev)m(er,)40 b(if)c(an)g(ab)s(elian)f(group)i Ft(G)g Fu(is)f(giv)m(en)h(b)m(y)h(a)f(generating)f(set)h(\(not)g (neces-)257 2116 y(sarily)32 b(indep)s(enden)m(t\))h(then)g(the)g (complexit)m(y)f(of)g(the)h(problem)e(is)i(op)s(en.)p Black 257 2333 a Fj(Problem)k(15.)p Black 43 w Fn(F)-7 b(or)35 b(ab)-5 b(elian)34 b(gr)-5 b(oups,)36 b(is)g(the)g(pr)-5 b(oblem)35 b(of)h(\034nding)e(a)i(nontrivial)f(au-)257 2453 y(tomorphism)f(har)-5 b(der)34 b(than)h(inte)-5 b(ger)34 b(factoring?)44 b(Is)34 b(it)i(har)-5 b(der)34 b(than)g(discr)-5 b(ete)35 b(lo)-5 b(g?)404 2670 y Fu(Another)32 b(observ)-5 b(ation)31 b(is)g(that)h(the)g(problem)e(of)i(group)f (isomorphism)e(testing)j(is)257 2790 y(harder)e(than)g(the)g(decision)f (v)m(ersion)h(of)f(discrete)h(log:)40 b(giv)m(en)30 b Ft(a;)17 b(b)28 b Fr(2)g Fk(Z)2882 2754 y Fl(\003)2882 2815 y Fp(n)2926 2790 y Fu(,)j(the)e(problem)257 2910 y(is)37 b(to)f(c)m(hec)m(k)j(if)d Ft(a)h Fu(is)f(in)g(the)h(cyclic)g (group)f(generated)i(b)m(y)f Ft(b)h Fu(\(i.e.)e Ft(a)f Fr(2)g(h)p Ft(b)p Fr(i)p Fu(\).)56 b(Clearly)-8 b(,)257 3031 y Ft(a)28 b Fr(2)g(h)p Ft(b)p Fr(i)h Fu(i\033)f Fr(h)p Ft(a;)17 b(b)p Fr(i)28 b Fu(is)h(isomorphic)d(to)j Fr(h)p Ft(b)p Fr(i)p Fu(.)42 b(More)29 b(generally)-8 b(,)28 b(the)i(mem)m(b)s(ership)e(testing)257 3151 y(problem)k(for)g (groups)g(reduces)j(to)d(group)g(isomorphism.)404 3274 y(F)-8 b(or)32 b(b)s(oth)h(ring)f(and)h(group)g(isomorphism)d(the)k (relativ)m(e)e(complexities)f(of)i(searc)m(h)257 3395 y(and)48 b(decision)f(remains)g(op)s(en.)89 b(In)48 b(the)g(case)g(of)f (graph)h(isomorphism,)h(searc)m(h)g(is)257 3515 y(p)s(olynomial-time)28 b(reducible)34 b(to)f(decision.)46 b(The)35 b(reduction)e(uses)i(graph) f(gadgets)g(to)257 3635 y(guide)f(a)f(pre\034x)i(searc)m(h.)45 b(It)33 b(is)f(not)h(clear)f(ho)m(w)i(to)e(build)f(similar)f(gadgets)j (for)f(groups)257 3756 y(and)h(rings.)p Black 257 3972 a Fj(Problem)k(16.)p Black 42 w Fn(Is)f(se)-5 b(ar)g(ch)35 b(p)-5 b(olynomial-time)34 b(r)-5 b(e)g(ducible)35 b(to)h(de)-5 b(cision)35 b(for)g(gr)-5 b(oup)36 b(iso-)257 4093 y(morphism)e(and)g (ring)g(isomorphism?)257 4441 y Fv(7)156 b(Derandomization)257 4666 y Fu(Babai)43 b(classi\034ed)h(in)g([7)o(])h(the)f(graph)g (non-isomorphism)d(problem)i(in)g(AM,)i(a)f(ran-)257 4786 y(domized)j(v)m(ersion)i(of)e(NP)i(that)e(can)i(b)s(e)f(describ)s (ed)g(in)f(terms)h(of)g(Arth)m(ur)g(Merlin)257 4907 y(proto)s(cols.)73 b(Sev)m(eral)43 b(authors)f(\(e.g.)h([3,)g(22)o(,)g(30)o(]\))g(ha)m(v)m (e)h(studied)e(derandomization)257 5027 y(of)g(AM)h(to)e(NP)i(under)g (suitable)e(hardness)j(assumptions,)g(th)m(us)f(sho)m(wing)g(that)f(GI) 257 5147 y(b)s(elongs)30 b(to)f Fs(NP)18 b Fr(\\)f Fs(coNP)q Fu(.)42 b(This)31 b(derandomization)c(w)m(orks)32 b(for)d(the)i(en)m (tire)f(class)g(AM.)257 5268 y(It)41 b(is)f(natural)f(to)i(ask)g(if)e (the)i(AM)g(proto)s(col)e(for)h(graph)h(non-isomorphism)c(can)k(b)s(e) 257 5388 y(unconditionally)30 b(derandomized.)p Black Black eop %%Page: 16 16 16 15 bop Black Black Black 257 573 a Fj(Problem)47 b(17.)p Black 47 w Fn(Can)d(the)g Fs(AM)g Fn(pr)-5 b(oto)g(c)g(ol)44 b(for)g(gr)-5 b(aph)43 b(non-isomorphism)e(b)-5 b(e)44 b(der)-5 b(an-)257 693 y(domize)g(d)34 b(unc)-5 b(onditional)5 b(ly?)45 b(Or)35 b(under)g(we)-5 b(aker)34 b(har)-5 b(dness)34 b(assumptions)g(that)i(those)257 814 y(use)-5 b(d)35 b(in)g([3)o(,)g(22,)f(30]?)404 978 y Fu(W)-8 b(e)25 b(considered)g(in)f ([6])h(the)g(question)g(of)g(whether)h(the)f(group)f(isomorphism)f (prob-)257 1098 y(lem)j(\(for)h(the)h(case)g(of)f(groups)g(giv)m(en)h (b)m(y)g(m)m(ultiplication)23 b(tables\))j(lies)h(in)f(NP)i Fr(\\)g Fu(coNP)-8 b(.)257 1218 y(This)27 b(migh)m(t)e(b)s(e)h(easier)g (to)g(sho)m(w)h(than)g(for)e(the)i(case)g(of)f(GI)g(since)h(group)f (isomorphism)257 1339 y(in)38 b(the)h Fn(table)h(r)-5 b(epr)g(esentation)44 b Fu(app)s(ears)39 b(to)f(b)s(e)h(an)f(easier)g (problem.)59 b(F)-8 b(ollo)m(wing)36 b(the)257 1459 y(same)d(approac)m (h)g(as)g(it)f(has)i(b)s(een)f(done)h(for)e(the)h(case)h(of)e(GI)h(w)m (e)h(sho)m(w)m(ed)h(that)d(group)257 1580 y(non-isomorphism)j(has)k(an) e(Arth)m(ur-Merlin)g(proto)s(col)f(with)i(the)g(prop)s(ert)m(y)h(that)e (on)257 1700 y(input)h(groups)g(of)g(size)g Ft(n)p Fu(,)i(Arth)m(ur)f (uses)g Ft(O)s Fs(\(log)2059 1657 y Fo(6)2115 1700 y Ft(n)p Fs(\))g Fu(random)e(bits)h(and)g(Merlin)f(uses)257 1820 y(only)24 b Ft(O)s Fs(\(log)704 1778 y Fo(2)760 1820 y Ft(n)p Fs(\))h Fu(nondeterministic)d(bits.)41 b(F)-8 b(or)23 b(the)i(case)g(of)f(solv)-5 b(able)23 b(groups)h(w)m(e)i(could)257 1941 y(derandomize)i(this)f(restricted)i (proto)s(col)d(applying)h(t)m(w)m(o)h(di\033eren)m(t)g(metho)s(ds)g (sho)m(wing)257 2061 y(that:)p Black 403 2237 a Fr(\017)p Black 48 w Fu(there)44 b(is)e(a)g(nondeterministic)f(p)s(olynomial)e (time)j(algorithm)d(for)j(the)h(group)501 2357 y(non-isomorphism)22 b(problem)i(restricted)h(to)f(solv)-5 b(able)24 b(groups)h(that)f(is)g (incorrect)501 2478 y(for)32 b(at)g(most)g Fs(2)1057 2442 y Fo(log)1148 2416 y Fh(O)r Fm(\(1\))1291 2442 y 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Black 50 w Fn(Do)49 b(the)h(ab)-5 b(ove)48 b(der)-5 b(andomization)48 b(r)-5 b(esults)50 b(hold)f(for)g(the)h(c)-5 b(ase)49 b(of)257 3731 y(gener)-5 b(al)34 b(gr)-5 b(oups?)404 3895 y Fu(As)42 b(men)m(tioned)f(in)f([6])h(the)h(derandomization)d(do)s(es)j(w)m(ork)g (for)f(general)g(groups)257 4015 y(assuming)32 b(the)h(short)g(presen)m (tation)g(conjecture.)404 4135 y(T)-8 b(urning)31 b(to)h(ring)f (isomorphism)f(in)h(the)i(table)e(represen)m(tation)i(the)g(ab)s(o)m(v) m(e)g(prob-)257 4256 y(lem)h(is)g(easy)i(to)e(resolv)m(e.)50 b(As)35 b(the)g(additiv)m(e)f(group)h(is)f(ab)s(elian,)f(rings)h(ha)m (v)m(e)i(succinct)257 4376 y(represen)m(tations)28 b(of)e(the)h (appropriate)e(t)m(yp)s(e)j(of)e(p)s(olylogarithmic)c(size)k(in)g(the)h (n)m(um)m(b)s(er)257 4497 y(of)g(ring)f(elemen)m(ts.)42 b(Therefore)28 b(Problem)e(18)h(can)g(b)s(e)g(answ)m(ered)i (a\036rmativ)m(ely)d(for)h(the)257 4617 y(case)36 b(of)f(rings)f(with)g (addition)f(and)i(m)m(ultiplication)30 b(tables)35 b(giv)m(en)g 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b(of)e(k-Con)m(tractible) j(Graphs.)e(A)g(Generalization)h(of)447 2604 y(Bounded)d(V)-8 b(alence)43 b(and)g(Bounded)g(Gen)m(us.)76 b Fd(Information)41 b(and)i(Contr)-5 b(ol,)45 b Fz(56\(1/2\):)447 2716 y(1-20,)31 b(1983.)p Black 257 2867 a([30])p Black 50 w(P)-8 b(.B.)36 b(Miltersen)f(and)g(N.)g(Vino)s(dc)m(handran,)j(Derandomizing)f(Arth)m (ur-Merlin)g(games)447 2980 y(using)f(hitting)i(sets,)g(in)e Fc(Pro)s(c.)h(40th)g(IEEE)h(Symp)s(osium)e(on)h(F)-8 b(oundations)38 b(of)e(Com-)447 3093 y(puter)31 b(Science,)g Fz(1999,)g(71\02580.)p Black 257 3243 a([31])p Black 50 w(M.)37 b(Mosca.)61 b Fd(Quantum)40 b(Computer)f(algorithms)p Fz(.)61 b(PhD)38 b(thesis,)h(Oxford)f(Univ)m(ersit)m(y)-8 b(,)447 3356 y(1999.)p Black 257 3506 a([32])p Black 50 w(M.)29 b(R\366tteler)h(and)h(T.)e(Beth,)i(P)m(olynomial-Time)h (Solution)f(to)f(the)g(Hidden)h(Subgroup)447 3619 y(Problem)24 b(for)f(a)g(Class)g(of)f(non-ab)s(elian)j(Groups.)e(ArXiv)f(preprin)m (t)j(quan)m(t-ph/9812070,)447 3732 y(1998.)p Black 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