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cleartomark
%%EndFont 
%%BeginFont: CMMI10
%!PS-AdobeFont-1.1: CMMI10 1.100
%%CreationDate: 1996 Jul 23 07:53:57
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.100) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMMI10) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle -14.04 def
/isFixedPitch false def
end readonly def
/FontName /CMMI10 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 0 /.notdef put
readonly def
/FontBBox{-32 -250 1048 750}readonly def
/UniqueID 5087385 def
currentdict end
currentfile eexec
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cleartomark
%%EndFont 
%%BeginFont: CMR10
%!PS-AdobeFont-1.1: CMR10 1.00B
%%CreationDate: 1992 Feb 19 19:54:52
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.00B) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMR10) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle 0 def
/isFixedPitch false def
end readonly def
/FontName /CMR10 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 0 /.notdef put
readonly def
/FontBBox{-251 -250 1009 969}readonly def
/UniqueID 5000793 def
currentdict end
currentfile eexec
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cleartomark
%%EndFont 
%%BeginFont: CMSY8
%!PS-AdobeFont-1.1: CMSY8 1.0
%%CreationDate: 1991 Aug 15 07:22:10
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.0) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMSY8) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle -14.035 def
/isFixedPitch false def
end readonly def
/FontName /CMSY8 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 0 /.notdef put
readonly def
/FontBBox{-30 -955 1185 779}readonly def
/UniqueID 5000818 def
currentdict end
currentfile eexec
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%%EndFont 
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% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
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currentdict end
currentfile eexec
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cleartomark
%%EndFont 
%%BeginFont: CMMI12
%!PS-AdobeFont-1.1: CMMI12 1.100
%%CreationDate: 1996 Jul 27 08:57:55
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.100) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMMI12) readonly def
/FamilyName (Computer Modern) readonly def
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end readonly def
/FontName /CMMI12 def
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/FontMatrix [0.001 0 0 0.001 0 0] readonly def
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/FontBBox{-30 -250 1026 750}readonly def
/UniqueID 5087386 def
currentdict end
currentfile eexec
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cleartomark
%%EndFont 
%%BeginFont: XYDASH10
%!PS-AdobeFont-1.1: XYDASH10 001.104
%%CreationDate: 1997 Jul 20 21:19:18
%%RevisionDate: 1997 Aug 28 05:34:12
%%RevisionDate: 1997 Sep 18 10:23:31
%
% XYDASH10: line segments for Xy-pic at 10 point
%
% Original Metafont design Copyright (C) 1991-1997 Kristoffer H. Rose.
% PostScript adaptation Copyright (C) 1994-1997 Ross Moore.
% Hinting and ATM compatibility Copyright (C) 1997 Y&Y, Inc.
%
% This file is part of the Xy-pic macro package.
% Xy-pic Copyright (c) 1991-1997 Kristoffer H. Rose <krisrose@brics.dk>
%
% The Xy-pic macro package is free software; you can redistribute it
% and/or modify it under the terms of the GNU General Public License
% as published by the Free Software Foundation; either version 2
% of the License, or (at your option) any later version.
%
% The Xy-pic macro package is distributed in the hope that it will
% be useful, but WITHOUT ANY WARRANTY; without even the implied
% warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.
% See the GNU General Public License for more details.
%
% You should have received a copy of the GNU General Public License
% along with this macro package; if not, write to the
% Free Software Foundation, Inc., 675 Mass Ave, Cambridge, MA 02139, USA.
11 dict begin
/FontInfo 9 dict dup begin
/version (001.104) readonly def
/Notice (Copyright (C) 1996, 1997 Ross Moore and Y&Y, Inc.) readonly def
/FullName (XYDASH10) readonly def
/FamilyName (XYDASH) readonly def
/Weight (Medium) readonly def
/ItalicAngle 0 def
/isFixedPitch false def
/UnderlinePosition -300 def
/UnderlineThickness 150 def
end readonly def
/FontName /XYDASH10 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 78 /d78 put
dup 81 /d81 put
dup 86 /d86 put
dup 104 /d104 put
dup 109 /d109 put
dup 111 /d111 put
dup 113 /d113 put
readonly def
/FontBBox{-40 -520 503 520}readonly def
/UniqueID 5092844 def
currentdict end
currentfile eexec
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cleartomark
%%EndFont 
%%BeginFont: XYBTIP10
%!PS-AdobeFont-1.1: XYBTIP10 001.104
%%CreationDate: 1997 Jul 20 21:19:18
%%RevisionDate: 1997 Sep 14 19:58:47
%
% XYBTIP10: lower arrow tips for Xy-pic at 10 point "technical style".
%
% Original Metafont design Copyright (C) 1991-1997 Kristoffer H. Rose.
% PostScript adaptation Copyright (C) 1994-1997 Ross Moore.
% Hinting and ATM compatibility Copyright (C) 1997 Y&Y, Inc.
%
% This file is part of the Xy-pic macro package.
% Xy-pic Copyright (c) 1991-1997 Kristoffer H. Rose <krisrose@brics.dk>
%
% The Xy-pic macro package is free software; you can redistribute it
% and/or modify it under the terms of the GNU General Public License
% as published by the Free Software Foundation; either version 2
% of the License, or (at your option) any later version.
%
% The Xy-pic macro package is distributed in the hope that it will
% be useful, but WITHOUT ANY WARRANTY; without even the implied
% warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.
% See the GNU General Public License for more details.
%
% You should have received a copy of the GNU General Public License
% along with this macro package; if not, write to the
% Free Software Foundation, Inc., 675 Mass Ave, Cambridge, MA 02139, USA.
11 dict begin
/FontInfo 9 dict dup begin
/version (001.104) readonly def
/Notice (Copyright (C) 1996, 1997 Ross Moore and Y&Y, Inc.) readonly def
/FullName (XYBTIP10) readonly def
/FamilyName (XYBTIP) readonly def
/Weight (Medium) readonly def
/ItalicAngle 0 def
/isFixedPitch false def
/UnderlinePosition -276 def
/UnderlineThickness 138 def
end readonly def
/FontName /XYBTIP10 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 38 /d38 put
dup 39 /d39 put
dup 40 /d40 put
dup 42 /d42 put
dup 47 /d47 put
dup 52 /d52 put
dup 54 /d54 put
dup 55 /d55 put
dup 56 /d56 put
readonly def
/FontBBox{-542 -542 542 542}readonly def
/UniqueID 5092839 def
currentdict end
currentfile eexec
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cleartomark
%%EndFont 
%%BeginFont: XYATIP10
%!PS-AdobeFont-1.1: XYATIP10 001.104
%%CreationDate: 1997 Jul 20 21:19:17
%%RevisionDate: 1997 Sep 14 19:58:47
%
% XYATIP10: upper arrow tips for Xy-pic at 10 point "technical style".
%
% Original Metafont design Copyright (C) 1991-1997 Kristoffer H. Rose.
% PostScript adaptation Copyright (C) 1994-1997 Ross Moore.
% Hinting and ATM compatibility Copyright (C) 1997 Y&Y, Inc.
%
% This file is part of the Xy-pic macro package.
% Xy-pic Copyright (c) 1991-1997 Kristoffer H. Rose <krisrose@brics.dk>
%
% The Xy-pic macro package is free software; you can redistribute it
% and/or modify it under the terms of the GNU General Public License
% as published by the Free Software Foundation; either version 2
% of the License, or (at your option) any later version.
%
% The Xy-pic macro package is distributed in the hope that it will
% be useful, but WITHOUT ANY WARRANTY; without even the implied
% warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.
% See the GNU General Public License for more details.
%
% You should have received a copy of the GNU General Public License
% along with this macro package; if not, write to the
% Free Software Foundation, Inc., 675 Mass Ave, Cambridge, MA 02139, USA.
11 dict begin
/FontInfo 9 dict dup begin
/version (001.104) readonly def
/Notice (Copyright (C) 1996, 1997 Ross Moore and Y&Y, Inc.) readonly def
/FullName (XYATIP10) readonly def
/FamilyName (XYATIP) readonly def
/Weight (Medium) readonly def
/ItalicAngle 0 def
/isFixedPitch false def
/UnderlinePosition -276 def
/UnderlineThickness 138 def
end readonly def
/FontName /XYATIP10 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 38 /d38 put
dup 39 /d39 put
dup 40 /d40 put
dup 42 /d42 put
dup 47 /d47 put
dup 52 /d52 put
dup 54 /d54 put
dup 55 /d55 put
dup 56 /d56 put
readonly def
/FontBBox{-542 -542 542 542}readonly def
/UniqueID 5092838 def
currentdict end
currentfile eexec
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cleartomark
%%EndFont 
%%BeginFont: CMR12
%!PS-AdobeFont-1.1: CMR12 1.0
%%CreationDate: 1991 Aug 20 16:38:05
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.0) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMR12) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle 0 def
/isFixedPitch false def
end readonly def
/FontName /CMR12 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 0 /.notdef put
readonly def
/FontBBox{-34 -251 988 750}readonly def
/UniqueID 5000794 def
currentdict end
currentfile eexec
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%%EndFont 
%%BeginFont: CMSY7
%!PS-AdobeFont-1.1: CMSY7 1.0
%%CreationDate: 1991 Aug 15 07:21:52
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.0) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMSY7) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle -14.035 def
/isFixedPitch false def
end readonly def
/FontName /CMSY7 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 0 /.notdef put
readonly def
/FontBBox{-15 -951 1252 782}readonly def
/UniqueID 5000817 def
currentdict end
currentfile eexec
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%%EndFont 
%%BeginFont: CMR8
%!PS-AdobeFont-1.1: CMR8 1.0
%%CreationDate: 1991 Aug 20 16:39:40
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.0) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMR8) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle 0 def
/isFixedPitch false def
end readonly def
/FontName /CMR8 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 0 /.notdef put
readonly def
/FontBBox{-36 -250 1070 750}readonly def
/UniqueID 5000791 def
currentdict end
currentfile eexec
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Fn(/)p 2307 3142 V 2531 3183 a Ft(A)m(C)2672 3141 y Fs(1)1322
3502 y Ft(NLOG)1970 3209 y Fo(7)p Fn(7)1933 3228 y Fm(o)1898
3246 y(o)1863 3264 y(o)1828 3282 y(o)1792 3300 y(o)1757
3318 y(o)1722 3336 y(o)1687 3354 y(o)1651 3372 y(o)1616
3390 y(o)1581 3408 y(o)858 3697 y Ft(Figure)f(1:)1285
3577 y
SDict begin H.S end
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SDict begin H.R end
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a
SDict begin [ /View [/XYZ H.V] /Dest (figure.1) cvn H.B /DEST pdfmark
end
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3661 y Fs(1)2525 3697 y Ft(and)e(A)m(C)2854 3661 y Fs(1)257
4000 y
SDict begin H.S end
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SDict begin 14.5 H.A end
 257 4000 a 257 4000 a
SDict begin [ /View [/XYZ H.V] /Dest (subsection.2.1) cvn H.B /DEST
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 257
4000 a 153 x Fl(2.1)131 b(NC)755 4110 y Fk(1)257 4345
y Ft(A)m(t)40 b(the)f(lo)m(w)m(er)g(end,)j(w)m(e)e(b)s(egin)e(with)h
(the)g(class)g(NC)2260 4309 y Fs(1)2300 4345 y Ft(.)63
b(NC)2533 4309 y Fs(1)2612 4345 y Ft(has)40 b(man)m(y)e(equiv)-5
b(alen)m(t)257 4465 y(c)m(haracterizations.)43 b(It)33
b(equals)g(the)g(class)f(of)g(languages)g(accepted)i(b)m(y)403
4686 y Fj(\017)48 b Ft(alternating)37 b(T)-8 b(uring)39
b(mac)m(hines)g(in)f(logarithmic)d(time)j(ALOGTIME)h(\(under)501
4806 y(appropriate)32 b(uniformit)m(y)e(conditions\).)43
b(\([)2114 4806 y
SDict begin H.S end
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SDict begin H.R end
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4742 a 2211 4806 a
SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /Subtype /Link
/Dest (cite.ruzzo81) cvn H.B /ANN pdfmark end
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b Ft(p)s(oly)32 b(size)h(programs)e(o)m(v)m(er)j(\034nite)e(monoids)f
(BP-M.)501 5147 y(A)41 b(program)f(o)m(v)m(er)i(a)f(monoid)e
Fi(M)53 b Fp(=)42 b(\()p Fi(S;)17 b Fj(\016)p Fp(\))41
b Ft(is)g(a)f(list)g(of)h(instructions)f(of)h(the)501
5268 y(form)33 b Fj(h)p Fi(i;)17 b(a;)g(b)p Fj(i)34 b
Ft(where)h Fi(i)c Fj(2)g Fp([)p Fi(n)p Fp(])p Ft(,)k
Fi(a;)17 b(b)31 b Fj(2)g Fi(S)6 b Ft(.)48 b(The)35 b(instruction)e
Fj(h)p Fi(i;)17 b(a;)g(b)p Fj(i)p Ft(,)35 b(on)f(input)501
5388 y Fi(x)28 b Fj(2)g(f)p Fp(0)p Fi(;)17 b Fp(1)p Fj(g)920
5352 y Fh(n)966 5388 y Ft(,)26 b(ev)-5 b(aluates)23 b(to)g
Fi(a)h Ft(if)e Fi(x)1750 5403 y Fh(i)1806 5388 y Fp(=)28
b(1)p Ft(,)d(to)e Fi(b)h Ft(otherwise.)41 b(The)24 b(en)m(tire)g
(program,)1852 5637 y(2)p eop end
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257 266 a
SDict begin H.R end
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SDict begin [ /View [/XYZ H.V] /Dest (page.3) cvn H.B /DEST pdfmark
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b(input)e Fi(x)j Fj(2)f(f)p Fp(0)p Fi(;)17 b Fp(1)p Fj(g)1358
537 y Fh(n)1404 573 y Ft(,)43 b(th)m(us)e(constructs)h(a)e(w)m(ord)h
Fi(w)j Fj(2)d Fi(S)2796 537 y Fg(\003)2835 573 y Ft(.)67
b(The)42 b(program)501 693 y(accepts)34 b Fi(x)f Ft(if)e
Fi(w)k Ft(ev)-5 b(aluates)33 b(to)f(a)g(designated)h(v)-5
b(alue)32 b(in)g Fi(M)10 b Ft(.)501 854 y(A)36 b(simple)d
(divide-and-conquer)i(approac)m(h)h(establishes)g(that)f(programs)f(o)m
(v)m(er)501 974 y(\034nite)c(monoids)f(can)h(b)s(e)g(ev)-5
b(aluated)30 b(in)f(NC)2125 938 y Fs(1)2165 974 y Ft(.)43
b(Barrington)29 b([)2757 974 y
SDict begin H.S end
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y
SDict begin H.R end
 2806 910 a 2806 974 a
SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /Subtype /Link
/Dest (cite.bar) cvn H.B /ANN pdfmark end
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1095 y(con)m(v)m(erse)43 b(b)m(y)e(sho)m(wing)f(that)g(o)m(v)m(er)h(an)
m(y)f(non-solv)-5 b(able)39 b(group,)j(w)m(ords)f(can)f(b)s(e)501
1215 y(constructed)c(to)f(co)s(de)g(logical)c(AND)k(and)g(negation.)49
b(In)35 b(particular,)e(he)i(used)501 1335 y(the)30 b(p)s(erm)m
(utation)f(group)g Fi(S)1555 1350 y Fs(5)1595 1335 y
Ft(;)h(th)m(us)h(programs)e(o)m(v)m(er)h(this)g(group)f(are)h(complete)
501 1456 y(for)i(NC)793 1419 y Fs(1)833 1456 y Ft(.)403
1656 y Fj(\017)48 b Ft(b)s(ounded-width)33 b(p)s(oly)f(size)g(branc)m
(hing)h(programs)e(BWBP)-8 b(.)501 1777 y(These)44 b(are)d(constan)m
(t-width)h(la)m(y)m(ered)h(graphs)f(\(one)g(for)f(eac)m(h)i(input)e
(length\))501 1897 y(with)47 b(designated)h(start)g(and)g(\034nish)f(v)
m(ertices)i Fi(s;)17 b(t)p Ft(,)51 b(and)d(edges)h(lab)s(elled)c(b)m(y)
501 2017 y(literals)34 b(or)h(constan)m(ts.)54 b(An)36
b(input)g(is)f(accepted)i(if)e(the)h(corresp)s(onding)f(graph)501
2138 y(has)e(an)g Fi(st)f Ft(path)h(where)h(all)c(edges)k(are)e(lab)s
(elled)f(b)m(y)i(1)f(or)g(true)h(literals.)501 2298 y(It)47
b(is)g(folklore)e(\(and)i(easy)h(to)f(see\))h(that)f(programs)f(o)m(v)m
(er)i(monoids)e(can)h(b)s(e)501 2419 y(describ)s(ed)34
b(in)d(this)h(form,)g(and)g(vice)h(v)m(ersa.)501 2579
y(Suc)m(h)c(programs)d(are)i(also)e(equiv)-5 b(alen)m(t)27
b(to)h(sk)m(ew)h(circuits,)f(where)g(OR)g(gates)f(are)501
2700 y(unrestricted)36 b(but)g(AND)g(gates)g(can)g(ha)m(v)m(e)g(at)g
(most)e(one)i(input)f(that)g(is)g(not)g(a)501 2820 y(literal)30
b(or)i(a)g(constan)m(t.)403 3021 y Fj(\017)48 b Ft(b)s(ounded-width)33
b(p)s(oly)f(size)g(circuits)g(SC)2043 2984 y Fs(0)2083
3021 y Ft(.)501 3141 y(SC)25 b(is)f(the)h(class)g(of)f(p)s(olynomial)c
(size)25 b(p)s(oly)f(logarithmic)c(width)25 b(circuits)e(\(width)501
3261 y Fi(O)s Fp(\(log)742 3218 y Fh(i)787 3261 y Fi(n)p
Fp(\))38 b Ft(for)f(SCi\).)59 b(\(Again,)38 b(wlog)f(the)h(circuits)f
(can)h(b)s(e)g(assumed)g(to)g(ha)m(v)m(e)501 3382 y(negations)c(only)g
(at)g(the)h(lea)m(v)m(es.\))50 b(In)35 b(sim)m(ulating)c(a)j(circuit)f
(b)m(y)j(a)e(T)-8 b(uring)34 b(ma-)501 3502 y(c)m(hine,)26
b(width)e(roughly)f(translates)h(to)f(space)i(and)f(size)g(to)g(time;)h
(th)m(us)g(SC)f(corre-)501 3623 y(sp)s(onds)j(in)e(the)h(uniform)d
(setting)j(to)f(a)g(sim)m(ultaneous)g(time-space)f(b)s(ound.)42
b(\(SC)501 3743 y(stands)36 b(for)d(Stev)m(e's)k(Classes,)f(named)e
(after)g(Stephen)i(Co)s(ok)e(who)h(pro)m(v)m(ed)h(the)501
3863 y(\034rst)27 b(non-trivial)22 b(result)k(ab)s(out)g(p)s(olynomial)
c(time)i(log-squared)i(space)h(PLoSS,)501 3984 y(i.e.)37
b(SC2,)i(in)d([)1047 3984 y
SDict begin H.S end
 1047 3984 a Ft(11)1145 3919
y
SDict begin H.R end
 1145 3919 a 1145 3984 a
SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /Subtype /Link
/Dest (cite.Cook79) cvn H.B /ANN pdfmark end
 1145 3984 a Ft(].)57 b(See)38
b(for)e(instance)i([)1999 3984 y
SDict begin H.S end
 1999 3984 a Ft(22)2096
3919 y
SDict begin H.R end
 2096 3919 a 2096 3984 a
SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /Subtype /Link
/Dest (cite.Johnson90) cvn H.B /ANN pdfmark end
 2096 3984 a Ft(]\).)57
b(As)38 b(describ)s(ed)g(ab)s(o)m(v)m(e,)h(Barring-)501
4104 y(ton's)44 b(result)e(places)h(NC)1473 4068 y Fs(1)1556
4104 y Ft(inside)f(sk)m(ew)j(SC)2211 4068 y Fs(0)2251
4104 y Ft(.)75 b(But)43 b(ev)m(en)h(non-sk)m(ew)h(SC)3347
4068 y Fs(0)3430 4104 y Ft(is)501 4224 y(easily)32 b(seen)j(to)d(b)s(e)
h(inside)f(NC)1655 4188 y Fs(1)1695 4224 y Ft(,)h(since)h(only)e(a)h
(constan)m(t)g(amoun)m(t)g(of)f(memory)501 4345 y(is)h(needed)i(to)e
(ev)-5 b(aluate)33 b(the)h(gates)g(of)f(the)h(circuit)e(la)m(y)m(er)i
(b)m(y)g(la)m(y)m(er.)46 b(Th)m(us)35 b(SC)3455 4309
y Fs(0)501 4465 y Ft(equals)e(NC)940 4429 y Fs(1)980
4465 y Ft(.)403 4666 y Fj(\017)48 b Ft(p)s(oly)32 b(size)h(form)m(ulae)
e(F,)h(ev)m(en)i(when)g(restricted)f(to)f(log-width)e(form)m(ula)h(L)
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(eac)m(h)g(gate)f(has)g(fanout)f(at)h(most)f(1.)48 b(NC)3281
4750 y Fs(1)3354 4786 y Ft(cir-)501 4907 y(cuits)25 b(can)h(b)s(e)f
(con)m(v)m(erted)i(to)d(form)m(ulae)g(b)m(y)i(duplication;)f(the)h(blo)
m(w-up)e(in)g(size)i(is)501 5027 y(still)j(within)i(a)g(p)s(olynomial.)
39 b(Con)m(v)m(ersely)-8 b(,)34 b(an)m(y)e(form)m(ula)d(can)j(b)s(e)g
(restructured)501 5147 y(in)m(to)27 b(an)h(equiv)-5 b(alen)m(t)27
b(one)h(with)f(p)s(olynomial)d(blo)m(w-up)j(in)g(size)h(and)g
(logarithmic)501 5268 y(depth;)i(a)d(non-uniform)f(w)m(a)m(y)i(to)f(do)
h(this)f(w)m(as)h(\034rst)g(describ)s(ed)g(in)f([)2990
5268 y
SDict begin H.S end
 2990 5268 a Ft(7)3039 5204 y
SDict begin H.R end
 3039 5204 a 3039
5268 a
SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /Subtype /Link
/Dest (cite.Brent) cvn H.B /ANN pdfmark end
 3039 5268 a Ft(,)3093 5268 y
SDict begin H.S end
 3093 5268 a Ft(35)3190
5204 y
SDict begin H.R end
 3190 5204 a 3190 5268 a
SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /Subtype /Link
/Dest (cite.Spira) cvn H.B /ANN pdfmark end
 3190 5268 a Ft(],)i(while)501
5388 y(it)f(can)g(b)s(e)h(done)g(uniformly)d(as)j(in)e([)1819
5388 y
SDict begin H.S end
 1819 5388 a Ft(28)1917 5324 y
SDict begin H.R end
 1917 5324 a 1917
5388 a
SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /Subtype /Link
/Dest (cite.BIS) cvn H.B /ANN pdfmark end
 1917 5388 a Ft(].)42 b(F)-8 b(urther,)29 b(an)m(y)g(log)e(depth)
i(form)m(ula)e(can)1852 5637 y(3)p eop end
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257 266 a
SDict begin H.R end
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SDict begin [ /View [/XYZ H.V] /Dest (page.4) cvn H.B /DEST pdfmark
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b(restructured)i(to)d(log)f(width)i(\(at)f(the)i(exp)s(ense)g(of)f
(depth,)i(of)d(course\),)k(as)501 693 y(observ)m(ed)35
b(in)d([)1044 693 y
SDict begin H.S end
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SDict begin H.R end
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SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /Subtype /Link
/Dest (cite.IZ) cvn H.B /ANN pdfmark end
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b(=)f(F)g(=)h(NC)2145 657 y Fs(1)2185 693 y Ft(.)403
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1063 y(ti\034er)52 b(or)h(a)f(monoidal)e(quan)m(ti\034er)j
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1199 y Fh(G)802 1184 y Ft(].)43 b([)926 1184 y
SDict begin H.S end
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a Ft(28)1024 1120 y
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/Dest (cite.BIS) cvn H.B /ANN pdfmark end
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a Ft(].)257 1396 y
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1396 a
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Fk(1)257 1734 y Ft(A)m(t)41 b(the)g(higher)f(end,)j(w)m(e)f(ha)m(v)m(e)
f(A)m(C)1633 1697 y Fs(1)1674 1734 y Ft(.)67 b(Less)41
b(is)f(kno)m(wn)i(ab)s(out)e(A)m(C)2839 1697 y Fs(1)2879
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b(of)f(languages)g(accepted)i(b)m(y)403 2104 y Fj(\017)48
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(alternations)i(b)s(et)m(w)m(een)j(univ)m(ersal)e(and)h(existen)m(tial)
e(states)501 2345 y(on)27 b(an)m(y)g(computation)f(path)g(ASP)-8
b(,AL)g(T\(log,log\))25 b(\(under)j(appropriate)e(unifor-)501
2465 y(mit)m(y)32 b(conditions\).)403 2715 y Fj(\017)48
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(orking)g(in)f(logarithmic)d(time)501 2835 y(with)32
b(p)s(olynomially)d(man)m(y)j(pro)s(cessors.)404 3085
y(No)m(w)h(consider)g(the)g(in)m(termediate)e(classes.)257
3272 y
SDict begin H.S end
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SDict begin [ /View [/XYZ H.V] /Dest (subsection.2.3) cvn H.B /DEST
pdfmark end
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3272 a 159 x Fl(2.3)131 b(DLOG)257 3635 y Ft(DLOG)27
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h(augmen)m(ted)h(with)f(deterministic)257 3996 y(transitiv)m(e)40
b(closure)g(F)m(O[DTC],)h([)1532 3996 y
SDict begin H.S end
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3932 y
SDict begin H.R end
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/Dest (cite.imm87) cvn H.B /ANN pdfmark end
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(ws)g(from)g([)2621 3996 y
SDict begin H.S end
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y
SDict begin H.R end
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/Dest (cite.reingold) cvn H.B /ANN pdfmark end
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SDict begin H.S end
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257 4328 a
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pdfmark end
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(nondeterministic)e(logspace)i(ma-)257 4786 y(c)m(hines.)84
b(The)46 b(inductiv)m(e)g(coun)m(ting)f(tec)m(hnique)i(of)e(Immerman)e
(and)j(Szelep)s(cs\351n)m(yi)257 4907 y([)284 4907 y
SDict begin H.S end

284 4907 a Ft(19)382 4842 y
SDict begin H.R end
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SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /Subtype /Link
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4907 a Ft(,)444 4907 y
SDict begin H.S end
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SDict begin H.R end
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4842 a 542 4907 a
SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /Subtype /Link
/Dest (cite.Szelep) cvn H.B /ANN pdfmark end
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(closed)h(under)g(complemen)m(tation.)50 b(An)35 b(equiv)-5
b(alen)m(t)257 5027 y(form)m(ulations)24 b(of)h(NLOG)h(is)g(the)g
(class)g(of)g(languages)f(accepted)j(b)m(y)e(uniform)f(p)s(oly)g(size)
257 5147 y(sk)m(ew)42 b(circuits)e(\(or)g(branc)m(hing)g(programs\);)j
(see)e([)2170 5147 y
SDict begin H.S end
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SDict begin H.R end
 2268
5083 a 2268 5147 a
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4842 y
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/Dest (cite.Cook79) cvn H.B /ANN pdfmark end
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b(sub)s(class)g(of)f(NC)h(con)m(taining)f(LogDCFL)g(is)h(kno)m(wn)h(to)
e(b)s(e)h(inside)257 5027 y(SC,)37 b(though)g(a)f(p)s(ossibly)g
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5147 y
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5147 a
SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /Subtype /Link
/Dest (cite.nisan-sc) cvn H.B /ANN pdfmark end
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b(,)35 b(w)m(e)h(do)g(not)f(y)m(et)h(kno)m(w)g(ho)m(w)g(to)f(com)m
(bine)g(these)i(t)m(w)m(o)f(constructions)g(to)e(place)257
5388 y(randomized)e(p)s(oly)g(time)f(AuxPD)m(A)j(inside)e(SC.)1852
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b(A)44 b(formal)g(language)f(view)257 760 y Ft(F)-8 b(or)39
b(man)m(y)h(reasons,)j(A)m(C)1231 724 y Fs(1)1311 760
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1597 y(DCFL)3109 1830 y Fo(\()p Fn(\()3071 1814 y Fm(Q)3034
1798 y(Q)2998 1782 y(Q)2962 1766 y(Q)2925 1750 y(Q)2889
1734 y(Q)2853 1718 y(Q)2817 1702 y(Q)2780 1686 y(Q)2744
1670 y(Q)2708 1653 y(Q)2672 1637 y(Q)2635 1621 y(Q)282
1914 y Ft(Regular)838 1890 y Fo(/)p Fn(/)p 639 1891 200
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1608 y Fo(4)p Fn(4)2302 1620 y Fm(h)2263 1631 y(h)2225
1642 y(h)2187 1653 y(h)2149 1664 y(h)2110 1675 y(h)2072
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1731 y(h)1881 1742 y(h)1843 1753 y(h)1804 1764 y(h)1766
1775 y(h)1728 1786 y(h)1689 1797 y(h)1651 1808 y(h)1613
1819 y(h)1575 1830 y(h)2297 2158 y Fo(*)p Fn(*)2257 2146
y Fm(V)2219 2135 y(V)2181 2124 y(V)2143 2113 y(V)2105
2102 y(V)2067 2091 y(V)2029 2080 y(V)1991 2069 y(V)1954
2058 y(V)1916 2048 y(V)1878 2037 y(V)1840 2026 y(V)1802
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1971 y(V)1612 1960 y(V)1575 1949 y(V)3126 1924 y Ft(CFLs)2126
2251 y(Linear)32 b(CFLs:)44 b(Lin)3108 1949 y Fo(6)p
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2014 y(m)2925 2030 y(m)2889 2046 y(m)2853 2062 y(m)2817
2078 y(m)2780 2094 y(m)2744 2110 y(m)2708 2126 y(m)2672
2142 y(m)2635 2158 y(m)1124 2436 y Ft(Figure)32 b(2:)1551
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SDict begin H.S end
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SDict begin H.R end
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SDict begin [ /View [/XYZ H.V] /Dest (figure.2) cvn H.B /DEST pdfmark
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2678 y(All)k(the)i(con)m(tainmen)m(ts)f(are)h(prop)s(er,)h(and)e(DCFL)h
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y(exactly)38 b(the)g(complexit)m(y)f(classes)i(of)e(Figure)2017
2919 y
SDict begin H.S end
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2919 a
SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /Subtype /Link
/Dest (figure.1) cvn H.B /ANN pdfmark end
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2883 y Fs(1)2897 2919 y Ft(is)f(not)g(co)m(v)m(ered)257
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3160 y(of)36 b(CSLs)h(giv)m(es)f(all)e(of)i(PSP)-8 b(A)m(CE.)38
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257 3280 y(exactly)d(A)m(C)733 3244 y Fs(1)774 3280 y
Ft(.)257 3446 y
SDict begin H.S end
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SDict begin 14.5 H.A end
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3446 a
SDict begin [ /View [/XYZ H.V] /Dest (subsection.2.8) cvn H.B /DEST
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3764 y Ft(Here)34 b(is)e(a)g(partial)e(list)h(of)h(problems)g(complete)
g(for)g(eac)m(h)h(of)f(these)i(classes:)257 3974 y Ff(LogCFL)50
b Ft(Blo)s(c)m(kChoice\(Dyc)m(k-2\),)25 b(the)d(hardest)h(CFL:)f(Giv)m
(en)f(a)h(sequence)i(of)d(blo)s(c)m(ks,)501 4095 y(eac)m(h)41
b(con)m(taining)d(a)h(list)f(of)h(strings,)i(can)f(w)m(e)g(pic)m(k)g
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b(blo)s(c)m(k)e(so)h(that)g(their)f(concatenation,)h(in)f(that)g
(order,)i(is)e(in)g(Dyc)m(k-2)i(\(the)501 4335 y(language)g(of)g
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(paren)m(theses\)?)501 4456 y(Non-zero)d(T)-8 b(ame)29
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y
SDict begin H.S end
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SDict begin H.R end
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a
SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /Subtype /Link
/Dest (cite.dhm) cvn H.B /ANN pdfmark end
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(satisfying)501 4576 y(a)33 b(certain)f(\020tameness\021)40
b(prop)s(ert)m(y)-8 b(,)33 b(determine)f(whether)i(it)e(is)g(non-zero.)
501 4697 y(Semi-extended)40 b(regular)e(expression)i(mem)m(b)s(ership)e
([)2555 4697 y
SDict begin H.S end
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SDict begin H.R end
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a 2653 4697 a
SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /Subtype /Link
/Dest (cite.Petersen02a) cvn H.B /ANN pdfmark end
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4817 y(sion)32 b Fi(r)i Ft(o)m(v)m(er)f(some)f(alphab)s(et)f
Fp(\006)h Ft(where)h Fi(r)i Ft(is)c(lik)m(e)g(a)h(regular)e(expression)
k(but)e(is)501 4937 y(also)38 b(allo)m(w)m(ed)g(to)g(use)i
Fj(\\)p Ft(,)h(and)e(giv)m(en)g(a)f(string)g Fi(x)h Fj(2)f
Fp(\006)2585 4901 y Fg(\003)2625 4937 y Ft(,)j(determine)d(whether)501
5058 y Fi(x)28 b Fj(2)g Fi(L)p Fp(\()p Fi(r)s Fp(\))p
Ft(.)257 5268 y Ff(NLOG)49 b Ft(Reac)m(habilit)m(y)31
b(in)h(a)g(directed)h(acyclic)f(graph.)501 5388 y(2-CNF-SA)-8
b(T.)1852 5637 y(7)p eop end
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b(expression)i(mem)m(b)s(ership)f([)1900 573 y
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a Ft(21)1997 509 y
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b(Giv)m(en)29 b(a)g(regular)f(expression)i Fi(r)i Ft(o)m(v)m(er)501
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b(determine)f(whether)i Fi(x)28 b Fj(2)g Fi(L)p Fp(\()p
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(b)s(e)f(o)m(v)m(er)501 1010 y Fj(f)p Fi(a;)17 b(b;)g(c;)g(d)p
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b(.)501 1131 y(Giv)m(en)31 b(a)g(string)f Fi(x)1192 1146
y Fs(0)1260 1131 y Fj(2)e Fp(\()p Fi(a)19 b Fp(+)g Fi(c)p
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(blo)s(c)m(ks)i Fi(B)2903 1146 y Fs(1)2942 1131 y Fi(;)17
b(:)g(:)g(:)f(;)h(B)3235 1146 y Fh(k)3308 1131 y Ft(eac)m(h)501
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SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /Subtype /Link
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b([)2774 1930 y
SDict begin H.S end
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y(The)h(Bo)s(olean)d(F)-8 b(orm)m(ula)30 b(V)-8 b(alue)32
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Ft(.)257 3141 y
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SDict begin [ /View [/XYZ H.V] /Dest (section.3) cvn H.B /DEST pdfmark
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b(classes)257 3517 y Ft(By)25 b(v)-5 b(arying)24 b(parameters)g
(appropriately)e(b)s(et)m(w)m(een)27 b(NC)2344 3481 y
Fs(1)2408 3517 y Ft(and)d(LogCFL,)g(w)m(e)h(get)f(some)257
3638 y(lesser-kno)m(wn)34 b(classes)g(in)e(this)g(range:)257
3793 y
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SDict begin [ /View [/XYZ H.V] /Dest (subsection.3.1) cvn H.B /DEST
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a Ft(41)2518 4284 y
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a Ft(])e(that)g(for)f(eac)m(h)h(p)s(olylog)501 4469 y
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b(nothing)f(m)m(uc)m(h)h(more)f(is)g(kno)m(wn.)44 b(F)-8
b(or)26 b(instance,)j(ev)m(en)h(the)e(smallest)e(class)501
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5027 y Fj(\017)48 b Ft(OR)24 b(fanin:)38 b(Within)22
b(p)s(oly)i(size)g(log)e(depth)j(circuits)e(with)h(constan)m(t)h(AND)f
(fanin,)501 5147 y(v)-5 b(arying)38 b(OR)g(fanin)g(from)f(constan)m(t)i
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5111 y Fs(1)3409 5147 y Ft(to)501 5268 y(SA)m(C)695 5232
y Fs(1)766 5268 y Ft(\(i.e.)30 b(LogCFL\).)g(What)g(ab)s(out)g(OR)g
(fanin)f Fi(f)11 b Fp(\()p Fi(n)p Fp(\))30 b Ft(where)h
Fi(f)41 b Ft(is,)31 b(sa)m(y)-8 b(,)32 b(p)s(oly-)501
5388 y(log?)52 b(Again,)36 b(Vina)m(y)f(sho)m(w)m(ed)j(closure)e(under)
g(complemen)m(t,)g([)2879 5388 y
SDict begin H.S end
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5324 y
SDict begin H.R end
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SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /Subtype /Link
/Dest (cite.vinay96) cvn H.B /ANN pdfmark end
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b(just)1852 5637 y(8)p eop end
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b(SA)m(C)817 537 y Fs(1)892 573 y Ft(con)m(tains)g(NLOG)f(\(at)g
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(class)h(describ)s(ed)h(ab)s(o)m(v)m(e.)42 b(But)27 b(do)501
814 y(they)34 b(capture)f(an)m(y)g(natural)e(problems?)403
1057 y Fj(\017)48 b Ft(Circuit-width:)64 b(Constraining)42
b(circuit)h(width)g(alone)g(to)g(p)s(olylog)f(giv)m(es)i(the)501
1177 y(SC)29 b(hierarc)m(h)m(y)-8 b(,)30 b(and)f(v)m(ery)h(little)c(of)
i(the)h(NC)g(hierarc)m(h)m(y)g(is)f(kno)m(wn)i(to)e(lie)f(within)501
1297 y(it.)51 b(Ho)m(w)m(ev)m(er,)39 b(the)c(de\034ning)h(prop)s(ert)m
(y)g(separating)e(LogCFL)h(from)f(P)i(is)f(p)s(oly)501
1418 y(size)e(pro)s(of)e(trees)i(\(also)e(referred)i(to)f(as)g(p)s(oly)
g(degree\).)44 b(One)32 b(could)g(com)m(bine)f(a)501
1538 y(width)h(restriction)e(with)h(a)h(degree)g(restriction)f(to)g
(obtain)g(a)g(sub-hierarc)m(h)m(y)i(of)501 1658 y(SC)26
b(within)f(LogCFL.)g(Lima)m(y)m(e)f(et)i(al)e([)1921
1658 y
SDict begin H.S end
 1921 1658 a Ft(26)2019 1594 y
SDict begin H.R end
 2019 1594 a 2019
1658 a
SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /Subtype /Link
/Dest (cite.LMR07) cvn H.B /ANN pdfmark end
 2019 1658 a Ft(])i(de\034ne)g(what)g(they)g(call)e(small)f(SC)j
(de-)501 1779 y(noted)i(sSC:)g(its)f Fi(i)p Ft(th)g(lev)m(el)g(has)h(p)
s(oly)e(size)i(p)s(oly)e(degree)i Fi(O)s Fp(\(log)2794
1736 y Fh(i)2823 1779 y Fp(\))f Ft(width)g(circuits.)501
1899 y(Again,)47 b(eac)m(h)f(lev)m(el)e(here)h(con)m(tains)g(the)g
(corresp)s(onding)g(width-constrained)501 2020 y(BPs,)36
b(though)f(no)g(relationship)e(with)h(the)h(constrained-OR-fanin)e
(circuits)h(is)501 2140 y(kno)m(wn.)76 b(Though)43 b(these)h(classes)g
(are)f(not)f(y)m(et)i(kno)m(wn)g(to)e(b)s(e)h(closed)g(under)501
2260 y(complemen)m(t,)31 b([)1099 2260 y
SDict begin H.S end
 1099 2260 a
Ft(26)1197 2196 y
SDict begin H.R end
 1197 2196 a 1197 2260 a
SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /Subtype /Link
/Dest (cite.LMR07) cvn H.B /ANN pdfmark end
 1197 2260 a
Ft(])g(sho)m(ws)i(that)e(co-sSC)2031 2224 y Fh(i)2092
2260 y Ft(is)g(in)f(sSC)2463 2224 y Fs(2)p Fh(i)2528
2260 y Ft(.)43 b(A)m(t)32 b(the)g(smallest)e(lev)m(el,)501
2381 y(sSC)663 2345 y Fs(0)742 2381 y Ft(equals)38 b(SC)1167
2345 y Fs(0)1207 2381 y Ft(,)i(but)e(it)f(is)h(not)g(kno)m(wn)h
(whether)g(sSC)2699 2345 y Fs(1)2778 2381 y Ft(is)f(as)g(p)s(o)m(w)m
(erful)g(as)501 2501 y(SC)625 2465 y Fs(1)665 2501 y
Ft(.)257 2681 y
SDict begin H.S end
 257 2681 a 257 2681 a
SDict begin 14.5 H.A end
 257 2681 a 257
2681 a
SDict begin [ /View [/XYZ H.V] /Dest (subsection.3.2) cvn H.B /DEST
pdfmark end
 257 2681 a 158 x Fl(3.2)131 b(Language/Automata-theoretic)42
b(constructs)403 3040 y Fj(\017)48 b Ft(Let)42 b(us)g(tak)m(e)g(a)f
(closer)h(lo)s(ok)e(at)h(Figures)2113 3040 y
SDict begin H.S end
 2113 3040
a Ft(1)2162 2975 y
SDict begin H.R end
 2162 2975 a 2162 3040 a
SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /Subtype /Link
/Dest (figure.1) cvn H.B /ANN pdfmark end
 2162 3040
a Ft(,)2230 3040 y
SDict begin H.S end
 2230 3040 a Ft(2)2279 2975 y
SDict begin H.R end
 2279
2975 a 2279 3040 a
SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /Subtype /Link
/Dest (figure.2) cvn H.B /ANN pdfmark end
 2279 3040 a Ft(.)70 b(NC)2519 3003
y Fs(1)2601 3040 y Ft(equals)41 b(the)h(closure)g(of)501
3160 y(regular)36 b(languages.)56 b(Y)-8 b(et)38 b(some)f(non-trivial)d
(non-regular)h(CFL)i(families)d(are)501 3280 y(included)39
b(in)g(it.)63 b(These)42 b(include)d(paren)m(thesis)h(languages)f([)
2801 3280 y
SDict begin H.S end
 2801 3280 a Ft(8)2850 3216 y
SDict begin H.R end
 2850 3216 a
2850 3280 a
SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /Subtype /Link
/Dest (cite.Bus87) cvn H.B /ANN pdfmark end
 2850 3280 a Ft(],)i(visibly)e(push-)501 3401
y(do)m(wn)49 b(languages)d(VPLs)j([)1547 3401 y
SDict begin H.S end
 1547
3401 a Ft(3)1596 3336 y
SDict begin H.R end
 1596 3336 a 1596 3401 a
SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /Subtype /Link
/Dest (cite.AM04) cvn H.B /ANN pdfmark end
 1596
3401 a Ft(,)1670 3401 y
SDict begin H.S end
 1670 3401 a Ft(14)1768 3336 y
SDict begin H.R end

1768 3336 a 1768 3401 a
SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /Subtype /Link
/Dest (cite.Dym88) cvn H.B /ANN pdfmark end
 1768 3401 a Ft(],)i(linear)46
b(CFLs)i(with)f(an)g(LL[1])h(condition)501 3521 y([)528
3521 y
SDict begin H.S end
 528 3521 a Ft(17)626 3457 y
SDict begin H.R end
 626 3457 a 626 3521
a
SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /Subtype /Link
/Dest (cite.IJR88) cvn H.B /ANN pdfmark end
 626 3521 a Ft(].)63 b(\(Imp)s(osing)37 b(an)i(LR[1])g(condition)f(is)
g(what)h(corresp)s(onds)i(to)d(determin-)501 3641 y(ism.)62
b(Th)m(us)41 b(CFLs)e(with)g(an)g(LR[1])g(condition)f(equal)g(DCFLs,)k
(linear)c(CFLs)501 3762 y(with)k(an)h(LR[1])f(condition)f(equal)i
(languages)f(accepted)i(b)m(y)f(1-turn)f(DPD)m(A,)501
3882 y(usually)32 b(referred)h(to)g(as)f(deterministic)f(linear)g
(languages.\))501 4064 y(Let)39 b(me)f(highligh)m(t)e(the)j(mem)m(b)s
(ership)f(in)f(NC)2250 4028 y Fs(1)2329 4064 y Ft(of)h(VPLs.)62
b(Firstly)-8 b(,)38 b(what)h(are)501 4184 y(VPLs?)64
b(These)41 b(are)e(languages)f(accepted)j(b)m(y)f(visibly)e(pushdo)m
(wn)j(automata)501 4305 y(VP)-8 b(As.)59 b(So)37 b(what)h(are)f(VP)-8
b(As?)58 b(These)39 b(are)f(PD)m(As)h(with)d(no)i Fi(")f
Ft(mo)m(v)m(es,)i(where)501 4425 y(the)31 b(stac)m(k)h(mo)m(v)m(emen)m
(t)e(\(push)i(/)e(no)g(c)m(hange)h(/)f(p)s(op\))g(is)g(dictated)g
(solely)g(b)m(y)h(the)501 4545 y(input)c(letter)h(b)s(eing)e(read.)42
b(They)30 b(are)d(clearly)g(stronger)h(than)f(NF)-11
b(As)28 b(\(they)h(can)501 4666 y(accept)37 b Fi(a)859
4630 y Fh(n)907 4666 y Fi(b)948 4630 y Fh(n)995 4666
y Ft(:)51 b(push)38 b(on)e Fi(a)p Ft(,)h(p)s(op)g(on)f
Fi(b)p Ft(\),)i(but)e(also)g(w)m(eak)m(er)i(than)e(PD)m(As)i(\(they)501
4786 y(cannot)32 b(accept)h Fi(a)1174 4750 y Fh(n)1222
4786 y Fi(ba)1314 4750 y Fh(n)1361 4786 y Ft(:)44 b(is)31
b Fi(a)h Ft(a)g(push)h(letter)f(or)g(a)g(p)s(op)g(letter?\).)42
b(In)33 b([)3108 4786 y
SDict begin H.S end
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SDict begin H.R end

3157 4722 a 3157 4786 a
SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /Subtype /Link
/Dest (cite.AM04) cvn H.B /ANN pdfmark end
 3157 4786 a Ft(],)f(it)f(w)m(as)501
4907 y(sho)m(wn)i(that)f(VP)-8 b(As)32 b(can)g(b)s(e)g(determinized;)f
(th)m(us)i(VPLs)f(are)g(in)f(DCFLs.)44 b(But)501 5027
y(w)m(ell)d(b)s(efore)g(this)g(w)m(as)h(kno)m(wn,)j(these)e(languages)d
(had)i(b)s(een)g(studied)f(under)501 5147 y(the)i(name)f(input-driv)m
(en)h(languages.)73 b(Dymond)43 b(ga)m(v)m(e)g(a)g(nice)f(construction)
501 5268 y([)528 5268 y
SDict begin H.S end
 528 5268 a Ft(14)626 5204 y
SDict begin H.R end
 626
5204 a 626 5268 a
SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /Subtype /Link
/Dest (cite.Dym88) cvn H.B /ANN pdfmark end
 626 5268 a Ft(])32 b(sho)m(wing)g(that)g(they)h(are)f
(in)g(fact)g(in)f(NC)2210 5232 y Fs(1)2250 5268 y Ft(.)43
b(His)32 b(approac)m(h)g(is)g(generic)g(and)501 5388
y(w)m(orks)i(not)e(just)h(for)f(VP)-8 b(As)34 b(but)f(for)f(an)m(y)h
(PD)m(A)h(satisfying)d(the)i(follo)m(wing:)1852 5637
y(9)p eop end
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SDict begin H.S end
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a 257 266 a
SDict begin H.R end
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SDict begin H.S end
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473 a 257 473 a
SDict begin 14.5 H.A end
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end
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a Ft(1.)49 b(no)32 b Fi(")g Ft(mo)m(v)m(es,)257 614 y
SDict begin H.S end

257 614 a 257 614 a
SDict begin 14.5 H.A end
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SDict begin [ /View [/XYZ H.V] /Dest (Item.2) cvn H.B /DEST pdfmark
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737 a Ft(2.)49 b(an)32 b(accepting)h(run)g(should)f(end)h(with)f(an)h
(empt)m(y)g(stac)m(k,)257 779 y
SDict begin H.S end
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SDict begin 14.5 H.A end
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779 a 257 779 a
SDict begin [ /View [/XYZ H.V] /Dest (Item.3) cvn H.B /DEST pdfmark
end
 257 779 a 591 902 a Ft(3.)49 b(the)29
b(heigh)m(t)f(of)f(the)i(pushdo)m(wn,)i(after)d(pro)s(cessing)g
Fi(i)h Ft(letters)f(of)g(the)h(input,)716 1022 y(should)48
b(b)s(e)g(computable)f(in)h(NC)2003 986 y Fs(1)2043 1022
y Ft(.)90 b(If)48 b(there)h(is)f(more)f(than)i(one)f(run)716
1143 y(\(nondeterministic)27 b(PD)m(A\),)j(and)f(if)f(the)h(heigh)m(t)g
(pro\034les)f(across)i(di\033eren)m(t)716 1263 y(runs)e(are)f
(di\033eren)m(t,)h(then)f(the)h(heigh)m(ts)f(computed)g(should)g(b)s(e)
g(consisten)m(t)716 1384 y(with)k(a)h(single)f(run.)44
b(F)-8 b(urthermore,)31 b(if)g(there)i(is)e(an)h(accepting)g(run,)h
(then)716 1504 y(the)41 b(heigh)m(ts)g(computed)g(should)f(b)s(e)h
(consisten)m(t)h(with)e(some)g(accepting)716 1624 y(run.)501
1835 y(F)-8 b(or)38 b(suc)m(h)i(PD)m(A,)g(Dymond)f(transforms)f(the)h
(problem)f(of)g(recognition)f(to)h(an)501 1955 y(instance)31
b(of)g(form)m(ula)d(v)-5 b(alue)31 b(problem,)e(and)i(then)h(in)m(v)m
(ok)m(es)g(Buss's)h(ALogTime)501 2075 y(algorithm)d([)973
2075 y
SDict begin H.S end
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SDict begin H.R end
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a
SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /Subtype /Link
/Dest (cite.Bus87) cvn H.B /ANN pdfmark end
 1021 2075 a Ft(])j(for)f(it.)501 2241 y(VP)-8 b(As)27
b(satisfy)e(these)i(conditions)d(\(with)h(appropriate)f(padding)h(to)g
(satisfy)g(con-)501 2361 y(dition)39 b(\(2\)\).)69 b(But)41
b(m)m(uc)m(h)g(more)g(can)g(b)s(e)g(ac)m(hiev)m(ed)h(via)e(condition)g
(\(3\).)68 b(The)501 2481 y(heigh)m(t)44 b(pro\034les)f(of)g(all)f
(runs)i(in)f(a)g(VP)-8 b(A)44 b(are)g(the)g(same,)i(and)e(can)g(b)s(e)f
(com-)501 2602 y(puted)36 b(in)e(TC0.)52 b(Understanding)35
b(exactly)h(what)f(can)h(b)s(e)f(placed)g(inside)f(NC)3455
2566 y Fs(1)501 2722 y Ft(b)m(y)g(carefully)d(using)h(Dymond's)i(pro)s
(of)d(is)i(a)f(nice)g(question.)501 2887 y(An)40 b(in)m(teresting)g
(prop)s(er)g(generalization)d(of)i(VP)-8 b(As)41 b(are)f(what)g(Caucal)
g(in)m(tro-)501 3008 y(duced)31 b(in)e([)918 3008 y
SDict begin H.S end
 918
3008 a Ft(9)967 2944 y
SDict begin H.R end
 967 2944 a 967 3008 a
SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /Subtype /Link
/Dest (cite.Cau06) cvn H.B /ANN pdfmark end
 967 3008
a Ft(])h(and)g(calls)e(sync)m(hronized)k(PD)m(A.)f(Languages)f
(accepted)h(b)m(y)g(these)501 3128 y(are)36 b(con)m(tained)h(in)e(DPD)m
(A)j(but)e(incomparable)e(with)i(DLin.)54 b(Do)s(es)37
b(their)f(clo-)501 3249 y(sure)30 b(create)f(a)f(new)i(class)f(b)s(et)m
(w)m(een)h(NC)2001 3212 y Fs(1)2070 3249 y Ft(and)f(LogDCFL,)f(or)h(do)
s(es)g(it)e(collapse)501 3369 y(to)32 b(one)h(of)f(these)i(or)e(ev)m
(en)i(to)f(DLOG?)403 3579 y Fj(\017)48 b Ft(The)35 b(fact)g(that)f(the)
g(logspace)g(closure)h(of)e(Lin)h(is)g(NLOG)f(is)h(in)m(teresting.)48
b(The)501 3700 y(mac)m(hine)36 b(mo)s(del)f(for)i(Linear)f(CFLs)h(is)f
(PD)m(A)i(whic)m(h,)g(on)f(eac)m(h)h(run,)g(mak)m(e)f(at)501
3820 y(most)e(1)h(turn)g(on)f(the)h(stac)m(k.)55 b(That)36
b(is,)g(no)f(stac)m(k)i(sym)m(b)s(ols)f(are)f(pushed)i(after)501
3940 y(the)29 b(\034rst)f(p)s(op)g(mo)m(v)m(e.)43 b(Th)m(us)29
b(the)g(mac)m(hine)e(mo)s(del)g(for)g(the)i(logspace)e(closure)i(of)501
4061 y(Lin)e(is)g(AuxPD)m(A\(p)s(oly\))i(making)d(1-turn)g(in)h(stac)m
(k)i(mo)m(v)m(emen)m(t.)42 b(This)28 b(suggests)501 4181
y(a)45 b(\034ne)h(gradation)e(b)s(et)m(w)m(een)j(NLOG)d(and)i(LogCFL)e
(parameterized)h(b)m(y)h(the)501 4302 y(n)m(um)m(b)s(er)33
b(of)e(turns)i(the)f(AuxPD)m(A)i(is)d(allo)m(w)m(ed)g(to)h(mak)m(e.)43
b(A)32 b(similar)d(gradation)501 4422 y(arises)35 b(b)s(et)m(w)m(een)h
(DLOG)e(and)g(LogDCFL)h(b)m(y)g(considering)e(the)i(deterministic)501
4542 y(coun)m(terpart.)257 4708 y
SDict begin H.S end
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a
SDict begin 14.5 H.A end
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SDict begin [ /View [/XYZ H.V] /Dest (subsection.3.3) cvn H.B /DEST
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b(constructs)403 5027 y Fj(\017)k Ft(Unam)m(biguit)m(y:)43
b(Bet)m(w)m(een)34 b(DLOG)f(and)g(NLOG)f(lies,)g(quite)g(naturally)-8
b(,)31 b(unam-)501 5147 y(biguous)38 b(logspace)f(ULOG.)h(Similarly)-8
b(,)35 b(LogUCFL)i(lies)g(b)s(et)m(w)m(een)j(LogDCFL)501
5268 y(and)26 b(LogCFL.)f(In)m(terestingly)-8 b(,)26
b(the)g(corresp)s(ondence)i(b)s(et)m(w)m(een)f(the)f(formal)d(lan-)501
5388 y(guage)41 b(class)g(and)f(the)h(complexit)m(y)f(class)h(is)f(not)
h(kno)m(wn)h(to)e(hold)g(here:)61 b(the)1828 5637 y(10)p
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SDict begin H.S end
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a 257 266 a
SDict begin H.R end
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SDict begin [ /View [/XYZ H.V] /Dest (page.11) cvn H.B /DEST pdfmark
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b(closure)g(of)g(unam)m(biguous)f(CFLs)i(viz.)f(LogUCFL,)g(is)g(con)m
(tained)g(in)501 693 y(unam)m(biguous)f(logspace)f(mac)m(hines)h
(UAuxPD)m(A\(p)s(oly\),)i(but)f(the)f(con)m(v)m(erse)j(is)501
814 y(not)44 b(kno)m(wn,)49 b(and)c(similarly)40 b(for)k(unam)m
(biguous)g(Linear)f(CFLs.)80 b(There)45 b(are)501 934
y(also)34 b(close)h(relationships)e(in)h(the)h(PRAM)h(mo)s(del:)46
b(while)34 b(LogDCFL)h(is)f(c)m(har-)501 1054 y(acterized)28
b(b)m(y)h(log)e(time)g(CR)m(O)m(W)h(PRAMs,)i(LogUCFL)e(is)f(con)m
(tained)h(in)g(b)m(y)h(log)501 1175 y(time)38 b(CREW)i(PRAMs,)h(whic)m
(h)f(corresp)s(ond)g(to)f(a)f(strong)h(form)f(of)h(unam)m(bi-)501
1295 y(guit)m(y)d(in)f(A)m(C)1011 1259 y Fs(1)1088 1295
y Ft(circuits.)53 b(There)37 b(are)f(sev)m(eral)h(subtleties)f(in)f
(the)i(de\034nition)e(of)501 1416 y(unam)m(biguous)i(mac)m
(hines/circuits:)52 b(is)37 b(there)h(at)f(most)g(one)h(accepting)f
(path,)501 1536 y(or)g(at)g(most)g(one)h(path)f(from)f(the)i(initial)c
(to)j(an)m(y)h(con\034guration,)g(or)f(at)g(most)501
1656 y(one)43 b(path)f(b)s(et)m(w)m(een)i(an)m(y)f(pair)e(of)g
(con\034gurations?)72 b(F)-8 b(or)41 b(more)g(ab)s(out)h(these)501
1777 y(n)m(uances,)35 b(see)e([)1079 1777 y
SDict begin H.S end
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a Ft(23)1177 1712 y
SDict begin H.R end
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SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /Subtype /Link
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a Ft(,)1236 1777 y
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