allbibs.tex
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commands.tex
\usepackage{amsfonts}
\usepackage{amsmath} 
\usepackage{amssymb}
\usepackage{enumerate}

% Decision Problems 
%\newcommand{\SAT}{\mathrm{SAT}}
%\newcommand{\MAXSAT}{\mathrm{MAXSAT}}
%\newcommand{\MAXTSAT}{\mathrm{MAX3SAT}}
\newcommand{\MAXSAT}{\textsc{MaxSat}}
\newcommand{\MAXTSAT}{\textsc{Max3Sat}}
\newcommand{\SAT}{{\rm SAT}}
\newcommand{\MAXCLIQUE}{\textsc{MaxClique}}
\newcommand{\CLIQUE}{\textsc{Clique}}

% Proof Commands
\newcommand{\PROOF}{\begin{proof}}%{{\noindent\bf{\em Proof:\/}~~}}
\newcommand{\PROOFSKETCH}{{\noindent\bf{\em Proof sketch:\/}~~}}
\newcommand{\QED}{\end{proof}}
\newcommand{\QEDbox}{\nolinebreak{\hfill $\square$}\vspace*{0.3cm}}

% Miscellaneous
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\newcommand{\diam}{\mathrm{diam}}
%\newcommand{\doublespace}{\renewcommand{\baselinestretch}{1.5}\normalsize}
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\newcommand{\RANDb}{\RAND^\vb}

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\newcommand{\CeqP}{{{\rm C}_=\P}}
\newcommand{\MODkP}{{{\rm MOD}_k\P}}
\newcommand{\BP}{\mathrm{BP}\cdot}

% Blackboard Bold Sets
\newcommand{\N}{\mathbb{N}}
\newcommand{\Z}{\mathbb{Z}}
\newcommand{\Q}{\mathbb{Q}}
\newcommand{\R}{\mathbb{R}}

% Measure and Dimension
\newcommand{\p}{{\mathrm{p}}}
\newcommand{\ptwo}{{\p_{\thinspace\negthinspace_2}}}
\newcommand{\pthree}{{\p_{\thinspace\negthinspace_3}}}
\newcommand{\pspace}{{\mathrm{pspace}}}
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\newcommand{\poly}{{\mathrm{poly}}}
\newcommand{\rec}{{\mathrm{rec}}}
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\newcommand{\FS}{{\mathrm{FS}}}
\newcommand{\comp}{{\mathrm{comp}}}
\newcommand{\constr}{{\mathrm{constr}}}
\newcommand{\mup}{\mu_{\mathrm{p}}}
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\newcommand{\mupthree}{\mu_\pthree}
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\newcommand{\dimDeltapthree}{\dim_\Deltapthree}
\newcommand{\Deltaptwo}{{\Delta^\p_2}}
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\newcommand{\DeltaPthree}{{\Delta^\P_3}}

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\newcommand{\ldim}{{\underline{\dim}}}
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\newcommand{\udimmb}{\udim_{\mathrm{MB}}}

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\newcommand{\Dimfs}{{\Dim_\FS}}
\newcommand{\Dimp}{{\Dim_\p}}
\newcommand{\Dimcomp}{{\Dim_\comp}}
\newcommand{\Dimpspace}{{\Dim_\pspace}}

\newcommand{\str}{{\mathrm{str}}}
\newcommand{\regSS}{S^\infty}
\newcommand{\strSS}{S^\infty_{\mathrm{str}}}

\newcommand{\G}{{\mathcal{G}}}
\newcommand{\Ghat}{{\widehat{\G}}}
\newcommand{\hatG}{\Ghat}
\newcommand{\Gstr}{\G^{\mathrm{str}}}
\newcommand{\hatGstr}{\hatG^{\mathrm{str}}}

\newcommand{\D}{{\mathcal{D}}}
\newcommand{\Gconstr}{\mathcal{G}_\mathrm{constr}}
\newcommand{\hatGconstr}{\mathcal{\hatG}_\mathrm{constr}}
\newcommand{\Gstrconstr}{\mathcal{G}_\mathrm{constr}^\mathrm{str}}
\newcommand{\hatGstrconstr}{\mathcal{\hatG}_\mathrm{constr}^\mathrm{str}}

\newcommand{\C}{\mathbf{C}}
\newcommand{\bd}{{\mathbf{d}}}
\newcommand{\FREQ}{{\mathrm{FREQ}}}
\newcommand{\freq}{{\mathrm{freq}}}

% Prediction
\newcommand{\pred}{\mathrm{pred}}
\newcommand{\Pred}{\mathrm{Pred}}
\newcommand{\Predp}{\mathrm{Pred}_\p}
\newcommand{\predp}{\mathrm{pred}_\p}
\newcommand{\unpred}{\mathrm{unpred}}
\newcommand{\Unpred}{\mathrm{Unpred}}
\newcommand{\unpredlog}{\mathrm{unpred}^\mathrm{log}}
\newcommand{\Unpredlog}{\mathrm{Unpred}^\mathrm{log}}
\newcommand{\unpredlogp}{\mathrm{unpred}^\mathrm{log}_\mathrm{p}}
\newcommand{\unpredabs}{\mathrm{unpred}^\mathrm{abs}}
\newcommand{\unpredabsp}{\mathrm{unpred}^\mathrm{abs}_\mathrm{p}}
\newcommand{\predcomp}{\pred_\comp}
\newcommand{\predfs}{\pred_\mathrm{FS}}
\newcommand{\Ik}{{\cal I}_k}
\newcommand{\Ikbar}{\overline{\Ik}}
\newcommand{\Hk}{{\cal H}_k}

\newcommand{\calL}{{\mathcal{L}}}
\newcommand{\logL}{\calL^{\log}}
\newcommand{\strlogL}{\calL^{\log}_{\str}}
\newcommand{\abs}{\mathrm{abs}}
\newcommand{\absL}{\calL^{\abs}}

\newcommand{\loss}{{\mathit{loss}}}
\newcommand{\losslog}{\loss^{\log}}
\newcommand{\lossabs}{\loss^\abs}


% Complexity Classes
\newcommand{\co}[1]{\mathrm{co}#1}
\newcommand{\strings}{\{0,1\}^*}
\newcommand{\ALL}{\mathrm{ALL}}
\newcommand{\RAND}{\mathrm{RAND}}
\newcommand{\RANDweak}{\mathrm{RAND}_\mathrm{weak}}
\newcommand{\DIM}{\mathrm{DIM}}
\newcommand{\DIMstr}{\DIM_\str}
\newcommand{\coDIM}{\mathrm{coDIM}}
\newcommand{\REC}{\mathrm{REC}}
\newcommand{\DEC}{\mathrm{DEC}}
\newcommand{\CE}{\mathrm{CE}}
\newcommand{\coCE}{\mathrm{coCE}}
\newcommand{\RE}{\mathrm{RE}}
\newcommand{\coRE}{\mathrm{coRE}}
\newcommand{\SPARSE}{\mathrm{SPARSE}}
\newcommand{\DENSE}{\mathrm{DENSE}}
\newcommand{\DENSEio}{\mathrm{DENSE}_\mathrm{i.o.}}
\newcommand{\DTIME}{\mathrm{DTIME}}
\newcommand{\NTIME}{\mathrm{NTIME}}
\newcommand{\DTIMEF}{\mathrm{DTIMEF}}
\newcommand{\NP}{{\ensuremath{\mathrm{NP}}}} 
\newcommand{\coNP}{\co{\NP}}
\newcommand{\AM}{{\rm AM}}
\newcommand{\MA}{{\rm MA}}
\newcommand{\DeltaPTwo}{\Delta^{\mathrm{P}}_2}
\newcommand{\DP}{\mathrm{DP}}
\newcommand{\SUBEXP}{\mathrm{SUBEXP}}
\newcommand{\SIZE}{{\rm SIZE}}
\newcommand{\SIZEio}{\SIZE_{\rm i.o.}}
\newcommand{\NSIZE}{{\rm NSIZE}}
\newcommand{\NSIZEio}{\NSIZE_{\rm i.o.}}
\newcommand{\BPP}{{\rm BPP}}
\newcommand{\PF}{{\rm PF}}
\newcommand{\ZPP}{{\rm ZPP}}
\newcommand{\RP}{{\rm RP}}
\newcommand{\coRP}{\co{\RP}}
\newcommand{\APP}{{\rm APP}}
\newcommand{\AP}{{\rm AP}}
\newcommand{\E}{{\rm E}}
\newcommand{\NE}{{\rm NE}}
\newcommand{\EE}{{\rm EE}}
\newcommand{\ESPACE}{{\rm ESPACE}}
\newcommand{\PSPACE}{{\rm PSPACE}}
\renewcommand{\P}{\ensuremath{{\mathrm P}}}
\newcommand{\QP}{{\rm QP}}
\newcommand{\NQP}{{\rm NQP}}
\newcommand{\Etwo}{{\rm E_2}}
\newcommand{\Ethree}{{\rm E_3}}
\newcommand{\EtwoSPACE}{{\rm E_2SPACE}}
\newcommand{\EXP}{{\rm EXP}}
\newcommand{\NEXP}{{\rm NEXP}}
\newcommand{\DSPACE}{{\rm DSPACE}}
\newcommand{\DSPACEF}{{\rm DSPACEF}}
\newcommand{\NEE}{{\rm NEE}}
\newcommand{\Poly}{{\rm P}}
\newcommand{\Pc}{{\rm P}}
\newcommand{\PPoly}{{\rm {P/Poly}}}
\newcommand{\Psel}{{\rm {P-sel}}}
\newcommand{\WS}{{\rm WS}}
\newcommand{\Ei}{\E_i}
\newcommand{\EiSPACE}{\mathrm{E}_i\mathrm{SPACE}}
\newcommand{\EoneSPACE}{\mathrm{E}_1\mathrm{SPACE}}
\newcommand{\linear}{\mathrm{linear}}
\newcommand{\polynomial}{\mathrm{polynomial}}
\newcommand{\EXPSPACE}{\mathrm{EXPSPACE}}
\newcommand{\PP}{{\rm PP}}
\newcommand{\SPP}{{\rm SPP}}
\newcommand{\GapP}{{\rm GapP}}
\newcommand{\PH}{{\rm PH}}
\newcommand{\Ppoly}{{\rm P/poly}}
\newcommand{\EH}{{\rm EH}}

% Calligraphy
\newcommand{\calA}{{\cal A}}
\newcommand{\calB}{{\cal B}}
\newcommand{\calC}{{\cal C}}
\newcommand{\calD}{{\cal D}}
\newcommand{\F}{{\mathcal{F}}}
\newcommand{\calH}{{\cal H}}
\newcommand{\calP}{{\cal P}}

% Kolmogorov Complexity
\newcommand{\calKS}{{\cal KS}}
\newcommand{\calKT}{{\cal KT}}
\newcommand{\calK}{{\cal K}}
\newcommand{\calCD}{{\cal CD}}
\newcommand{\calCND}{{\cal CND}}
\newcommand{\K}{{\mathrm{K}}}
\newcommand{\KS}{{\mathrm{KS}} }
\newcommand{\KT}{{\mathrm{KT}} }
\newcommand{\CD}{{\mathrm{CD}} }
\newcommand{\CND}{{\mathrm{CND}} }

% Reductions
\newcommand{\leqpm}{\leq^\p_\mathrm{m}}
\newcommand{\leqmp}{\leqpm}
\newcommand{\leqpT}{\leq^\mathrm{p}_\mathrm{T}}
\newcommand{\leqpr}{\leq^\mathrm{p}_r}
\newcommand{\leqonettp}{\leq_{1-\mathrm{tt}}^{\mathrm{P}}}
\newcommand{\murec}[1]{{\mu({#1}\mid \REC)}}
\newcommand{\muex}[1]{{\mu({#1}\mid \Etwo)}}
\newcommand{\redu}[2]{{\leq^{\rm {{#1}}}_{{\rm{#2}}}}}
\newcommand{\predu}[1]{{\redu{P}{#1}}}
\newcommand{\pmred}{{\predu{m}}}
\newcommand{\ptred}{{\predu{T}}}

%pair reductions

\newcommand{\leqppm}{\leq^{\p\p}_\mathrm{m}}
\newcommand{\leqppum}{\leq^{\p\p}_\mathrm{um}}

\newcommand{\leqppsm}{\leq^{\p\p}_\mathrm{sm}}
\newcommand{\leqppT}{\leq^{\mathrm{pp}}_\mathrm{T}}
\newcommand{\leqppuT}{\leq^{\mathrm{pp}}_\mathrm{uT}}



% Reduction Closures
\newcommand{\Pm}{\mathrm{P}_{\mathrm{m}}}
\newcommand{\PT}{\mathrm{P}_{\mathrm{T}}}
\newcommand{\Ptt}{\mathrm{P}_{\mathrm{tt}}}
\newcommand{\Pctt}{\mathrm{P}_{\mathrm{ctt}}}
\newcommand{\Pdtt}{\mathrm{P}_{\mathrm{dtt}}}
\newcommand{\Ponett}{\mathrm{P}_{1-\mathrm{tt}}}
\newcommand{\Pmi}{\Pm^{-1}}
\renewcommand{\Pr}{\mathrm{P}_r}
\newcommand{\Pri}{\Pr^{-1}}
% Degrees
\newcommand{\degpm}{\mathrm{deg}^{\mathrm{p}}_{\mathrm{m}}}
\newcommand{\degpr}{\mathrm{deg}^\mathrm{p}_r}
\newcommand{\equivpm}{\equiv^\p_\mathrm{m}}
% Hard and Complete sets
\newcommand{\m}{\mathrm{m}}
\newcommand{\Cpm}{\calC^\p_\m}
\newcommand{\Hpm}{\calH^\p_\m}
\newcommand{\Cpr}{\calC^\p_r}
\newcommand{\Hpr}{\calH^\p_r}

% Stuff from CPED
\newcommand{\CH}{\mathcal{H}}
\newcommand{\CHdec}{\mathcal{H}_\DEC}
\newcommand{\CHce}{\mathcal{H}_\CE}
\newcommand{\Pione}{\Pi^0_1}
\newcommand{\Pitwo}{\Pi^0_2}
\newcommand{\BorelPi}[2]{{\bf \Pi}^ #1 _#2}
\newcommand{\BorelSigma}[2]{{\bf \Sigma}^ #1_#2}
\newcommand{\Sigmaone}{\Sigma^0_1}
\newcommand{\Sigmatwo}{\Sigma^0_2}
\newcommand{\PioneC}{\Pione}% \upharpoonright \C}
\newcommand{\PitwoC}{\Pitwo}% \upharpoonright \C}
\newcommand{\SigmaoneC}{\Sigmaone}% \upharpoonright \C}
\newcommand{\SigmatwoC}{\Sigmatwo}% \upharpoonright \C}
\newcommand{\io}{{\mathrm{i.o.}}}
\newcommand{\aev}{{\mathrm{a.e.}}}
\newcommand{\ioA}{A^\io}
\newcommand{\ioB}{B^\io}
\newcommand{\ioC}{C^\io}
\newcommand{\aeA}{A^\aev}
\newcommand{\aeB}{B^\aev}


% Names
\newcommand{\MartinLof}{Martin-L\"{o}f}

\newcommand{\nth}{n^{\mathrm{th}}}


\newcommand{\supp}{\mathrm{supp}}
\newcommand{\eH}{\mathrm{H}}

numbering.tex
\usepackage{amsthm}

\newtheorem{theorem}{Theorem}[section]
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem*{claim}{Claim}

\newenvironment{theorem_cite}[1]
{\begin{theorem}  {\rm (#1)}}
{\end{theorem}}
\newenvironment{lemma_cite}[1]
{\begin{lemma}  {\rm (#1)}}
{\end{lemma}}
\newenvironment{corollary_cite}[1]
{\begin{corollary}  {\rm (#1)}}
{\end{corollary}}
\newenvironment{proposition_cite}[1]
{\begin{proposition}  {\rm (#1)}}
{\end{proposition}}

\newenvironment{theorem_named}[1]
{\begin{theorem} {\rm (#1)}}
{\end{theorem}}
\newenvironment{lemma_named}[1]
{\begin{lemma} {\rm (#1)}}
{\end{lemma}}

\newenvironment{proof_claim}
{\noindent{\em Proof of Claim.}}
{\hfill $\square$ {\em Claim.}}

\newenvironment{Proof}
{\begin{proof}}
{\end{proof}}

\newenvironment{Proofnamed}[1]
{\begin{proof}[Proof #1.]}
{\end{proof}}

\newenvironment{pf}
{\begin{proof}[Pf.]}
{\end{proof}}

\newtheorem{observation}[theorem]{Observation}
\newtheorem{observations}[theorem]{Observations}

\newtheorem{openproblem}[theorem]{Open Problem}
\newtheorem{openquestion}[theorem]{Open Question}
\newtheorem{question}[theorem]{Question}

\newtheorem{fact}[theorem]{Fact}
\newtheorem{facts}[theorem]{Facts}

\newtheorem{problem}[theorem]{Problem}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\theoremstyle{definition}

\newtheorem*{definition}{Definition}
\newtheorem*{defn}{Definition}
\newtheorem*{definitions}{Definitions}

\newtheorem{example}[theorem]{Example}
\newtheorem{examples}[theorem]{Examples}
\newtheorem*{example*}{Example}
\newtheorem*{examples*}{Examples}

\newenvironment{example_cite}[1]
{\begin{example}  {\rm (#1)}}
{\end{example}}

\newtheorem{construction}[theorem]{Construction}
\newtheorem{extension}[theorem]{Extension}

\newtheorem*{remarks}{Remarks}
\newtheorem*{remark}{Remark}

\newtheorem*{notation}{Notation}
\newtheorem*{terminology}{Terminology}

\newtheorem*{ack}{Acknowledgment}
\newtheorem*{acks}{Acknowledgments}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\theoremstyle{remark}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\numberwithin{equation}{section}
\numberwithin{figure}{section}

dc.bbl
\begin{thebibliography}{10}

\bibitem{All01}
E.~Allender.
\newblock When worlds collide: Derandomization, lower bounds, and kolmogorov
  complexity.
\newblock In {\em Proceedings of the 21st annual Conference on Foundations of
  Software Technology and Theoretical Computer Science}, volume 2245 of {\em
  Lecture Notes in Computer Science}, pages 1--15. Springer-Verlag, 2001.

\bibitem{ABKMR02}
E.~Allender, H.~Buhrman, M.~Kouck\'y, D.~van Melkebeek, and D.~Ronneburger.
\newblock Power from random strings.
\newblock In {\em Proceedings of the 43rd Annual IEEE Symposium on Foundations
  of Computer Science}, pages 669--678, 2002.

\bibitem{Athreya:ESDAICC}
K.~B. Athreya, J.~M. Hitchcock, J.~H. Lutz, and E.~Mayordomo.
\newblock Effective strong dimension, algorithmic information, and
  computational complexity.
\newblock In {\em Proceedings of the 21st International Symposium on
  Theoretical Aspects of Computer Science}. Springer-Verlag, 2004.
\newblock to appear.

\bibitem{Haus19}
F.~Hausdorff.
\newblock Dimension und {\"a}usseres {M}ass.
\newblock {\em Mathematische Annalen}, 79:157--179, 1919.

\bibitem{Hitchcock:FDLLU}
J.~M. Hitchcock.
\newblock Fractal dimension and logarithmic loss unpredictability.
\newblock {\em Theoretical Computer Science}, 304(1--3):431--441, 2003.

\bibitem{Hitchcock:SSSD}
J.~M. Hitchcock.
\newblock Small spans in scaled dimension.
\newblock Technical Report cs.CC/0304030, Computing Research Repository, 2003.

\bibitem{Hitchcock:SDNC}
J.~M. Hitchcock, J.~H. Lutz, and E.~Mayordomo.
\newblock Scaled dimension and nonuniform complexity.
\newblock {\em Journal of Computer and System Sciences}.
\newblock To appear. Preliminary version appeared in {\em Proceedings of the
  30th International Colloquium on Automata, Languages, and Programming}, pages
  278--290, 2003.

\bibitem{Hitchcock:DERC}
J.~M. Hitchcock and N.~V. Vinodchandran.
\newblock Dimension, entropy rates, and compression.
\newblock In {\em Proceedings of the 19th IEEE Conference on Computational
  Complexity}, 2004.
\newblock To appear.

\bibitem{Kann82}
R.~Kannan.
\newblock Circuit-size lower bounds and non-reducibility to sparse sets.
\newblock {\em Information and Control}, 55:40--56, 1982.

\bibitem{Lutz:CMCC}
J.~H. Lutz.
\newblock Category and measure in complexity classes.
\newblock {\em SIAM Journal on Computing}, 19(6):1100--1131, 1990.

\bibitem{Lutz:AEHNC}
J.~H. Lutz.
\newblock Almost everywhere high nonuniform complexity.
\newblock {\em J.~Comput. Syst. Sci.}, 44(2):220--258, 1992.

\bibitem{Lutz:GCDIS}
J.~H. Lutz.
\newblock Gales and the constructive dimension of individual sequences.
\newblock In {\em Proceedings of the 27th International Colloquium on Automata,
  Languages, and Programming}, pages 902--913, 2000.
\newblock Revised as \cite{Lutz:DISS}.

\bibitem{Lutz:DCC}
J.~H. Lutz.
\newblock Dimension in complexity classes.
\newblock {\em SIAM Journal on Computing}, 32:1236--1259, 2003.
\newblock Preliminary version appeared in {\em Proceedings of the Fifteenth
  Annual IEEE Conference on Computational Complexity}, pages 158--169, 2000.

\bibitem{Lutz:DISS}
J.~H. Lutz.
\newblock The dimensions of individual strings and sequences.
\newblock {\em Information and Computation}, 187:49--79, 2003.
\newblock Preliminary version appeared as \cite{Lutz:GCDIS}.

\bibitem{Sull84}
D.~Sullivan.
\newblock Entropy, {Hausdorff} measures old and new, and limit sets of
  geometrically finite {Kleinian} groups.
\newblock {\em Acta Mathematica}, 153:259--277, 1984.

\bibitem{Tricot82}
C.~Tricot.
\newblock Two definitions of fractional dimension.
\newblock {\em Mathematical Proceedings of the Cambridge Philosophical
  Society}, 91:57--74, 1982.

\end{thebibliography}
