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% Capital Letters
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%Abbreviations
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\newcommand{\cycle}{\mbox{{\rm\sc CYCLE}}}
\newcommand{\datarule}{{\,:\!\!-\,}}			% Datalog command (March 2010)
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\newcommand{\fom}{\mbox{{\rm FOM}}}
\newcommand{\foT}[1]{(\mbox{{\rm FO}}+ \Theta)[#1]}
\newcommand{\fovar}[2]{\mbox{{{\rm FO-VAR}}}[#1,#2]}
\newcommand{\fobinvar}[2]{\mbox{{{\rm FO-BINVAR}}}[#1,#2]}
\newcommand{\fobit}[2]{\mbox{{{\rm FO-VBIT}}}[#1,#2]}
\newcommand{\form}{\mbox{{\rm form}}}
\newcommand{\fosat}{\mbox{{\rm\sc FO-SAT}}}
\newcommand{\fsat}{\mbox{{\rm\sc Finite-FO-SAT}}}
\newcommand{\founsat}{\mbox{{\rm\sc FO-UNSAT}}}
\newcommand{\fovalid}{\mbox{{\rm\sc FO-VALID}}}
\newcommand{\fvalid}{\mbox{{\rm\sc Finite-FO-VALID}}}
\newcommand{\fovarb}[2]{\mbox{{{\rm FO-VAR}}}(\leq,\bit)[#1,#2]}
\newcommand{\fovars}[2]{\mbox{{{\rm FO-VAR}}}(\leq)[#1,#2]}
\newcommand{\fovarw}[2]{\mbox{{{\rm FO-VAR}}}\wo[#1,#2]}
\newcommand{\fosize}{\fo\mbox{-SIZE}}
\newcommand{\fp}{\mbox{{\rm FP}}}
\newcommand{\Fr}{\mbox{Fra\"{\i}ss\'e }}
\newcommand{\free}{\mbox{{\rm free}}}
%\newcommand{\gcd}{\mbox{{\rm\sc GCD}}}
\newcommand{\geo}{\mbox{{\rm\sc GEOGRAPHY}}}
\newcommand{\graphiso}{\mbox{{\rm\sc GRAPH-ISO}}}
\newcommand{\half}{\frac{1}{2}}
\newcommand{\halt}{\mbox{{\rm HALT}}}
\newcommand{\hc}{\mbox{{\rm\sc HC}}}
\newcommand{\head}{\mbox{{\rm head}}}
\newcommand{\hsat}{\mbox{{\rm\sc Horn-SAT}}}
\newcommand{\unsat}{\mbox{{\rm\sc UNSAT}}}
\newcommand{\valid}{\mbox{{\rm\sc VALID}}}
\newcommand{\hyperexp}{\mbox{{\rm hyperExp}}}
\newcommand{\id}{\mbox{{\rm ID}}}
\newcommand{\iden}{\mbox{{\rm id}}}
\newcommand{\ifp}{\mbox{{\rm IFP}}}
\newcommand{\im}{\mbox{$\prod$}}
\newcommand{\ind}{\mbox{{\rm IND}}}
\newcommand{\indvar}[2]{\mbox{{{\rm IND-VAR}}}[#1,#2]}
\newcommand{\init}{\mbox{{\bf init}}}
\newcommand{\inj}{\mbox{{\rm Inj}}}
\newcommand{\inv}{\mbox{{\rm inv}}}
\newcommand{\ip}{\mbox{{\rm IP}}}
\newcommand{\ins}{\mbox{{\rm ins}}}
\newcommand{\insb}{\mbox{{\bf ins}}}
\newcommand{\isomap}{\stackrel{\iso}{\rightarrow}}
\newcommand{\length}{\mbox{{\rm length}}}
\newcommand{\lca}{\mbox{{\rm lca}}}   % least common ancestor relation
\newcommand{\lcaf}{\!\uparrow\!}           % least common ancestor function symbol
\newcommand{\lfp}{\mbox{{\rm LFP}}}
\newcommand{\lt}{\mbox{{\bf let}}}
\newcommand{\logr}{\leq_{\mbox{{\scriptsize log}}}}
\newcommand{\loopy}{\mbox{{\rm loopy}}}
\newcommand{\Accept}{\mbox{{\bf accept }}}
\newcommand{\Else}{\mbox{{ \bf else }}}
\newcommand{\For}{\mbox{{\bf for }}}
\newcommand{\Halt}{\mbox{{\bf halt }}}
\newcommand{\If}{\mbox{{\bf if }}}
\newcommand{\Or}{\mbox{{\bf OR }}}
\newcommand{\Output}{\mbox{{\bf output }}}
\newcommand{\Reject}{\mbox{{\bf reject }}}
\newcommand{\Return}{\mbox{{\bf return}}}
\newcommand{\Then}{\mbox{{ \bf then }}}
\newcommand{\While}{\mbox{{\bf while }}}
\newcommand{\Do}{\mbox{{ \bf do }}}
\newcommand{\To}{\mbox{{ \bf to }}}
\newcommand{\Item}{\mbox{{\rm Item}}}
\newcommand{\iter}{\mbox{{\rm ITER}}}
\newcommand{\itera}[1]{\mbox{{\rm ITER}}[\mbox{{\rm arity }}#1]}
\newcommand{\iterd}[2]{\mbox{{\rm ITER}}[#1,\mbox{{\rm arity }}#2]}
\newcommand{\iterp}{\mbox{{\rm ITER}}}
\newcommand{\ket}[1]{|#1\! >}
\newcommand{\ksat}{\mbox{{\rm\sc 2-SAT}}}
\newcommand{\lgc}{\sel_{\hbox{{\scriptsize GC}}}}
\newcommand{\lh}{\mbox{{\rm LH}}}
\newcommand{\logspace}{\mbox{{\rm L}}}
\newcommand{\lollipop}{\multimap}
\newcommand{\ltl}{\mbox{{\rm LTL}}}
\newcommand{\Last}{\mbox{{\it Last}}}
\newcommand{\maj}{\mbox{{\rm\sc MAJ}}}
\newcommand{\Math}{\mbox{{\rm\sc MATH}}}
\newcommand{\maparg}[1]{\stackrel{#1}{\rightarrow}}
\newcommand{\mclique}{\mbox{{\rm\sc MAX-CLIQUE}}}
\newcommand{\maxkf}{\mbox{{\rm\sc MAX-$k$-FSAT}}}
\newcommand{\maxthree}{\mbox{{\rm\sc MAX-$3$-SAT}}}
\newcommand{\mcvp}{\mbox{{\rm\sc MCVP}}}
\newcommand{\mdist}{\mbox{{\rm min-dist}}}
\newcommand{\mn}{{\it min}}  %  and that's final
\newcommand{\modm}[1]{\hspace*{.5em}({\rm mod }\, #1)}
\newcommand{\monadic}{\mbox{{\rm monadic}}}
\newcommand{\mult}{\mbox{{\rm MULT}}}
\newcommand{\mx}{\mbox{{\it max}}}  %  and that's final
\newcommand{\MX}{\mbox{{\sc MAX}}}  % for max degree of a fixed point
\newcommand{\mychoose}[2]{\left({#1 \atop #2}\right)} 
\newcommand{\mymod}{\mbox{{\rm mod }}}
\newcommand{\pctmod}{\,\%\,}
\newcommand{\fmod}[1]{\mbox{{\rm MOD$_{\mbox{fin}}$}}[#1]}
\newcommand{\Mod}[1]{\mbox{MOD}[#1]}
\newcommand{\mvd}{\ensuremath{\rightarrow\!\!\!\!\rightarrow}}  
\newcommand{\nc}{\mbox{{\rm NC}}}
\newcommand{\ncs}[1]{\mbox{{\rm NC$^{#1}$}}}
\newcommand{\acs}[1]{\mbox{{\rm AC$^{#1}$}}}
\newcommand{\sacs}[1]{\mbox{{\rm sAC$^{#1}$}}}
\newcommand{\ncmod}[3]{#1\not\equiv #2\,({\rm mod}\, #3)}
\newcommand{\ncomp}{\mbox{{\rm \#comp}}}
\newcommand{\ndist}{\mbox{{\rm\sc NDIST}}}
\newcommand{\negskip}{\vspace{-1.0em}}
\newcommand{\New}{\mbox{{\rm New}}}
\newcommand{\Next}{\mbox{{\rm\sc NEXT}}}
\newcommand{\nil}{\mbox{{\bf nil}}}
\newcommand{\nl}{\mbox{{\rm NL}}}
\newcommand{\nmax}{\mbox{{\rm Not-Max}}}
\newcommand{\nmsf}{\mbox{{\rm Not-MSF}}}
\newcommand{\nopath}{\mbox{{\rm NOPATH}}}
\newcommand{\np}{\mbox{{\rm NP}}}
\newcommand{\npc}{\mbox{{\rm NPC}}}
\newcommand{\nramtime}{\mbox{{\rm NRAM-TIME}}}
\newcommand{\nspace}{\mbox{{\rm NSPACE}}}
\newcommand{\ntime}{\mbox{{\rm NTIME}}}
\newcommand{\numb}{\mbox{{\rm NUMBER}}}
\newcommand{\nx}{\mbox{{\rm NEXT}}}
\newcommand{\oneLO}{\mbox{{\rm 1LO}}}
\newcommand{\twoLO}{\mbox{{\rm 2LO}}}
\newcommand{\occur}{\mbox{{\rm Occur}}}
\newcommand{\op}{\mbox{{\rm op}}}
\newcommand{\opt}{\mbox{{\rm OPT}}}
\newcommand{\ord}{\mbox{Ord}}
\newcommand{\ov}{\overline}
\newcommand{\p}{\mbox{{\rm P}}}
\newcommand{\parity}{\mbox{{\rm\sc PARITY}}}
\newcommand{\Path}{\mbox{{\rm\sc PATH}}}
\newcommand{\pay}{\mbox{$\mapsto$}}  % arrow in todo list
\newcommand{\pcp}{\mbox{{\rm PCP}}}
\newcommand\pF{{\cal PF}}        % the set of partial, recursive functions
\newcommand{\ph}{\mbox{{\rm PH}}}
\newcommand{\plus}{\mbox{{\rm PLUS}}}
\newcommand{\poly}{/\mbox{{\rm poly}}}
\newcommand{\polylog}{\mbox{{\rm polylog}}}
\newcommand{\power}{\wp}
\newcommand{\primrec}{\mbox{{\bf Primitive Recursive}}}
\newcommand{\primrecF}{\mbox{{\bf PrimRecFcns}}}
\newcommand{\primrecP}{\mbox{{\bf PrimRecPreds}}}
\newcommand{\recF}{\mbox{{\rm Partial
RecursiveFcns}}}
\newcommand{\grecF}{\mbox{{\rm G\"odelRecFcns}}}
\newcommand{\pr}{\leq_{\mbox{{\scriptsize p}}}}
\newcommand{\prob}{\mbox{{\rm Prob}}}
\newcommand{\prog}{\mbox{{\rm\sc PROG}}}
\newcommand{\pspace}{\mbox{{\rm PSPACE}}}
\newcommand{\psubset}{{\raisebox{.03in}{\mbox{{\tiny \begin{tabular}{c}
 $\subset$\\[-.5ex]
 $\ne$
 \end{tabular}}}}}}
\newcommand{\pstc}{\mbox{{\rm pos STC}}}
\newcommand{\ptc}{\mbox{{\rm pos TC}}}
\newcommand{\qb}{\mbox{{\rm QB}}}
\newcommand{\uqb}{\mbox{{\rm UQB}}}  % universal quantifier block
\newcommand{\pce}{+\!:=}
\newcommand{\spce}{\;+\!:=\;}
\newcommand{\mce}{-\!:=}
\newcommand{\smce}{\;-\!:=\;}
\newcommand{\pe}{+\!=}
\newcommand{\spe}{\;+\!=\;}
\newcommand{\me}{-\!=}
\newcommand{\sme}{\;-\!=\;}
\newcommand{\qe}{\quad =\quad}
\newcommand{\qE}{\quad \equiv\quad}
\newcommand{\qqe}{\qquad =\qquad}
\newcommand{\qqE}{\qquad \equiv\qquad}
%\newcommand{\qfr}{\leq_{\hbox{{\scriptsize qf}}}}
\newcommand{\qsep}{\, . \,}
\newcommand{\qLra}{\quad\Leftrightarrow\quad}
\newcommand{\qlra}{\quad\leftrightarrow\quad}
\newcommand{\qqLra}{\qquad\Leftrightarrow\qquad}
\newcommand{\qqlra}{\qquad\leftrightarrow\qquad}
\newcommand{\qr}{\mbox{qr}}
\newcommand{\qra}{\quad \rightarrow\quad}
\newcommand{\qR}{\quad \Rightarrow\quad}
\newcommand{\qRa}{\quad \Rightarrow\quad}
\newcommand{\qsat}{\mbox{{\rm\sc QSAT}}}
\newcommand{\query}{\mbox{{\bf query}}}
\newcommand{\ramtime}{\mbox{{\rm RAM-TIME}}}
\newcommand{\rank}{\mbox{rank} }
\newcommand{\rng}{\mbox{{\rm rng}}}
\newcommand{\Rdual}{\raisebox{0.2ex}{$\stackrel{\sim}{{\cal R}}$}}
\newcommand{\reach}{\mbox{{\rm\sc REACH}}}
\newcommand{\RE}{\mbox{{\rm r.e.}}}
\newcommand{\reacha}{\mbox{{\rm\sc REACH$_a$}}}
\newcommand{\reachd}{\mbox{{\rm\sc REACH$_d$}}}
\newcommand{\reachdl}{\mbox{{\rm\sc REACH$_{d\ell}$}}}
\newcommand{\reachl}{\mbox{{\rm\sc REACH$_\ell$}}}
\newcommand{\reachu}{\mbox{{\rm\sc REACH$_u$}}}
\newcommand{\recursive}{\mbox{{\rm Recursive}}}
\newcommand{\regular}{\mbox{{\rm Regular}}}
\newcommand{\Res}{\mbox{{\rm Res}}}
\newcommand{\rst}[2]{#1|_{#2}}
\newcommand{\rr}{\leq_{\mbox{{\scriptsize r}}}}
\newcommand{\sac}{\mbox{{\rm sAC}}}
\newcommand{\sat}{\mbox{{\rm\sc SAT}}}
\newcommand{\Sat}{\mbox{{\rm\sc 3-SAT}}}
\newcommand{\sbar}{\; |\;}
\newcommand{\sdot}{\; .\;}
\newcommand{\se}{\; =\;}
\newcommand{\sE}{\; \equiv\;}
\newcommand{\Sin}{\;\in\;}
\newcommand{\lsyn}{\lbrack\!\lbrack}
\newcommand{\rsyn}{\rbrack\!\rbrack}
\newcommand{\semp}[1]{\lsyn #1 \rsyn}
\newcommand{\sIn}{\;\in\;}
\newcommand{\qIn}{\quad\in\quad}
\newcommand{\sLra}{\;\Leftrightarrow\;}
\newcommand{\sla}{\;\leftarrow\;}
\newcommand{\slra}{\;\leftrightarrow\;}
\newcommand{\sra}{\;\rightarrow\;}
\newcommand{\sepimplies}{-\!\!\star\;}
\newcommand{\Set}{\mbox{{\bf set}}}
\newcommand{\size}{\mbox{{\rm SIZE}}}
\newcommand{\smb}[1]{\;\mbox{{\rm #1}}\;}
\newcommand{\so}{\mbox{{\rm SO}}}
\newcommand{\soa}{\mbox{{\rm SO}$\forall$}}
\newcommand{\soe}{\mbox{{\rm SO}$\exists$}}
\newcommand{\soh}{\mbox{{\rm SO-Horn}}}
\newcommand{\soeh}{\mbox{\soe{\rm -Horn}}}
\newcommand{\sok}{\mbox{{\rm SO-Krom}}}
\newcommand{\som}{\mbox{{\rm SOM}}}
\newcommand{\Space}{\mbox{{\rm SPACE}}}
\newcommand{\ess}{\mbox{{\red\bf S}}}
\newcommand{\dee}{\mbox{{\green\bf D}}}
\newcommand{\spec}{\mbox{{\rm spec}}}
\newcommand{\sq}{\mbox{{\rm sq}}}
\newcommand{\sRa}{\; \Rightarrow\;}
\newcommand{\sss}{\;\subseteq\;}
\newcommand{\qsubseteq}{\quad\subseteq\quad}
\newcommand{\Sol}{\mbox{{\rm Sol}}}
\newcommand{\St}{\mbox{{\rm \,s.t.\,}}}
\newcommand{\straight}{\mbox{{\rm\sc STRAIGHT}}}
\newcommand{\strs}{\mbox{{\rm Stars}}}
\newcommand{\struct}[1]{\mbox{{\rm STRUC}}[#1]}
\newcommand{\strucn}[2]{\mbox{{\rm STRUC}}_{#1}[#2]}
\newcommand{\fstruc}[1]{\mbox{{\rm STRUC$_{\mbox{fin}}$}}[#1]}
\newcommand{\struc}[1]{\mbox{{\rm STRUC}}[#1]}
\newcommand{\sx}[1]{_{\mbox{{\scriptsize #1}}}}
\newcommand{\suc}{\mbox{{\rm Suc}}}
\newcommand{\succinct}{\mbox{{\rm\sc SUCCINCT }}}
\newcommand{\sub}[2]{#1/#2}         % substitute a/x
\newcommand{\superset}{\supset}
\newcommand{\superseteq}{\supseteq}
% temporal logic operators
\newcommand{\tA}{\mbox{{\bf A}}}
\newcommand{\tB}{\mbox{{\bf B}}}
\newcommand{\tE}{\mbox{{\bf E}}}
\newcommand{\tF}{\mbox{{\bf F}}}
\newcommand{\tG}{\mbox{{\bf G}}}
\newcommand{\tR}{\mbox{{\bf R}}}
\newcommand{\tU}{\mbox{{\bf U}}}
\newcommand{\tW}{\mbox{{\bf W}}}
\newcommand{\tX}{\mbox{{\bf X}}}
\newcommand{\ta}{\mbox{{\scriptsize\bf A}}}
\newcommand{\tb}{\mbox{{\scriptsize\bf B}}}
\newcommand{\te}{\mbox{{\scriptsize\bf E}}}
\newcommand{\tu}{\mbox{{\scriptsize\bf U}}}
\newcommand{\tx}{\mbox{{\scriptsize\bf X}}}
\newcommand{\tail}{\mbox{{\rm tail}}}
\newcommand{\taut}{\mbox{{\rm\sc TAUT}}}
\newcommand{\tc}{\mbox{{\rm TC}}}
\newcommand{\term}{\mbox{{\rm term}}}
\newcommand{\Th}{\mbox{{\rm Theory}}}
\newcommand{\thc}{\mbox{{\rm ThC}}}
\newcommand{\thOrd}{\mbox{$^{\rm th}$}}
\newcommand{\Times}{\mbox{{\rm TIMES}}}
\newcommand{\tr}{\mbox{{\bf Tr}}}  % trace
\newcommand{\trianglelt}{\lhd}
\newcommand{\trianglele}{\unlhd}
\newcommand{\trianglegt}{\rhd}
\newcommand{\trianglege}{\unrhd}
\newcommand{\tridown}{\bigtriangledown}
\newcommand{\triup}{\bigtriangleup}
%\newcommand{\true}{\mbox{{\bf true}}}
\newcommand{\tsp}{\mbox{{\rm\sc TSP}}}
\newcommand{\tw}{\mbox{{\rm tw}}}
\newcommand{\two}{\mbox{{\rm\sc TWO}}}
\newcommand{\ultra}{\mbox{{{\rm ULTRA}}}}
\newcommand{\un}[1]{\underline{#1}}
\newcommand{\unary}{\mbox{{\rm unary}}}
\newcommand{\union}{\mbox{{\rm union}}}
\newcommand{\undir}{\mbox{{\rm undir}}}
\newcommand{\yesC}{\mbox{{\rm YesComp}}}
\newcommand{\val}{\mbox{{\rm val}}}
\newcommand{\Value}{\mbox{{\rm Value}}}
\newcommand{\var}{\mbox{{\rm var}}}
\newcommand{\vargr}{\mbox{{\it var}}}
\newcommand{\vc}{\mbox{{\it vc}}}
\newcommand{\vbit}{\mbox{{\rm VBIT}}}
\newcommand{\varsz}[2]{\mbox{{{\rm VAR-SZ}}}[#1,#2]}
\newcommand{\varszw}[2]{\mbox{{{\rm VAR-SZ}}}\wo[#1,#2]}
\newcommand{\wo}{\mbox{{\rm \small (wo$\leq$)}}}
\newcommand{\wob}{\mbox{{\rm \small (wo BIT)}}}
\newcommand{\wtc}{\mbox{{\rm W$_5$TC}}} 
\newcommand{\zpa}{\hspace*{.25in}}
\newcommand{\zpb}{\hspace*{.5in}}
\newcommand{\zpc}{\hspace*{.75in}}
\newcommand{\zpd}{\hspace*{1.0in}}
\newcommand{\zpl}{\mbox{{\rm ZPL}}}
\newcommand\pf{\smallskip\noindent{\bf Proof:}}
%\newcommand{\oldqed}{\penalty 1000 \hfill\penalty 1000\rule{5pt}{8pt}\par\medskip}
\newcommand{\qed}{\penalty 1000 \hfill\penalty 1000$\Box$\par\medskip}
\newenvironment{proof}{\pf }{\qed }
\newenvironment{slide}{\newpage
}{}
\newenvironment{framework}{\medskip\noindent\refstepcounter{theorem}{\bf
Framework \thetheorem\ }}{\qed }
\newenvironment{sproof}{\pf }{ }
\newenvironment{sql}{\medskip\noindent\refstepcounter{theorem}{\bf
SQL Query \thetheorem\ }}{ }
\newenvironment{datalog}{\medskip\noindent\refstepcounter{theorem}{\bf
Datalog Query \thetheorem\ }}{ }
\newenvironment{definition}{\medskip\noindent\refstepcounter{theorem}{\bf
Definition \thetheorem\ }}{\qed }
\newenvironment{remark}{\medskip\noindent\refstepcounter{theorem}{{\color{dg}\bf
Remark \thetheorem\ }}}{{\color{dg}\qed}}
\newenvironment{problem}{\medskip\noindent\refstepcounter{theorem}{\bf
Open Problem \thetheorem\ }}{ }
\newenvironment{sdefinition}{\medskip\noindent\refstepcounter{theorem}{\bf
Definition \thetheorem\ }}{ }
\newenvironment{invariant}{\medskip\noindent\refstepcounter{theorem}{\bf
Invariant \thetheorem\ }}{\qed }
\newenvironment{sinvariant}{\medskip\noindent\refstepcounter{theorem}{\bf
Invariant \thetheorem\ }}{ }
\newenvironment{sexample}{\medskip\noindent\refstepcounter{theorem}{\bf
Example \thetheorem\ }}{ }
\newenvironment{example}{\medskip\noindent\refstepcounter{theorem}{\bf
Example \thetheorem\ }}{\qed }
\newenvironment{exercise}{\medskip\noindent\refstepcounter{theorem}{\bf
Exercise \thetheorem\ }}{\qed }
\newenvironment{sexercise}{\medskip\noindent\refstepcounter{theorem}{\bf
Exercise \thetheorem\ }}{ }
\newcounter{myfig}
\newcommand{\startfig}{\begin{figure}\refstepcounter{theorem}
                      \setcounter{myfig}{\value{theorem}} }
\newcommand{\eJJ}{\end{figure}}
\newenvironment{myfigure}{\startfig}{\eJJ}     
\newcommand{\pdfFig}[3]{\refstepcounter{theorem}\label{#1}
\begin{center}
\includegraphics[width=#3\textwidth]{#1.pdf}
\end{center}
\vspace*{-.3in}
\noindent{\bf Figure \thetheorem}: {\small #2}
    \par\smallskip}
\newcommand{\epsFig}[3]{\refstepcounter{theorem}\label{#1}
\begin{center}
\includegraphics[width=#3\textwidth]{#1.eps}
\end{center}
\vspace*{-.3in}
\noindent{\bf Figure \thetheorem}: {\small #2}
    \par\smallskip}
\newcounter{myeqn}
\newcommand{\myequation}[2]{\par\refstepcounter{theorem}
   \setcounter{myeqn}{\value{theorem}}\label{#1} 
\noindent\makebox[\textwidth]{\hfill${\displaystyle #2 }$\hfill
            {\mbox{\rm (\ref{#1})}}}
    \par\smallskip}
% same for two-column, conferences
\newcounter{dmyeqn}
\newcommand{\dmyequation}[2]{\par\refstepcounter{theorem}
   \setcounter{dmyeqn}{\value{theorem}}\label{#1} 
   \smallskip\noindent\makebox[3.25in]{\hfill${\displaystyle #2 }$\hfill
   {\rm (\ref{#1})}} 
    \par\smallskip}
\newcommand{\regequation}[2]{\begin{equation}\label{#1}
                                    #2
                             \end{equation}}     
\newcommand{\lline}[1]{\par\medskip
    \noindent\hspace*{-.4in}\makebox[\textwidth]{#1\hfill}
    \par\medskip\noindent}
\newcommand{\labeleqn}[2]{\par\medskip
    \noindent\makebox[\textwidth]{\hfill${\displaystyle #2 }$\hfill $#1$}
    \par\medskip\noindent}
\newcommand{\oneLine}[1]{\par\smallskip
    \noindent\makebox[\textwidth]{#1}
    \par\smallskip\noindent}
\newcommand{\seqn}[1]{\labeleqn{\Box}{#1}}
\newcommand{\narrowlabeleqn}[2]{\par\medskip
    \noindent\makebox[6.8in]{\hfill${\displaystyle #2 }$\hfill $#1$}
    \par\medskip\noindent}
% MYfig:  same as myfig but it is not a figure so it can't move where it shouldn't
\newcounter{MYfig}
\newcommand{\Startfig}{\par\smallskip\refstepcounter{theorem}
                      \setcounter{MYfig}{\value{theorem}} }
\newcommand{\Ejj}{\par\smallskip}
\newenvironment{MYfigure}{\Startfig}{\Ejj}

\newcommand{\snieqn}{\begin{equation}\refstepcounter{theorem}}
\newcommand{\enieqn}{\end{equation}}
\newenvironment{nieqn}{\snieqn}{\enieqn}
\newcommand{\sniarray}{\begin{eqnarray}\refstepcounter{theorem}}
\newcommand{\eniarray}{\end{eqnarray}}
\newenvironment{niarray}{\sniarray}{\eniarray}
\renewcommand{\theequation}{\thetheorem}
\newcommand{\nilabel}[1]{\\ \refstepcounter{theorem}\label{#1}}  % place at end of
						       % each additional
						       % labeled line in
						       % an niarray
\newcommand{\nn}{\nonumber \\}


% right justified equation for two-column, conferences
\newcommand{\drteqn}[1]{\noindent\makebox[3.25in]{\hfill${\displaystyle #1 }$}}
%left justified equation for two-column, conferences
\newcommand{\dlteqn}[1]{\noindent\makebox[3.25in]{${\displaystyle #1 }$\hfill}}
\hyphenation{Abite-boul Bool-ean it-er-at-ed poly-no-mial poly-no-mial-time
   quan-ti-fier quan-ti-fier-free De-li-lah Sam-son}
%the following makes atheorem numbered numerically:
\newtheorem{atheorem}{Theorem}
\newtheorem{acorollary}[atheorem]{Corollary}
\newtheorem{alemma}[atheorem]{Lemma}
\newtheorem{aprop}[atheorem]{Proposition}
\newtheorem{afact}[atheorem]{Fact}
\newtheorem{aconjecture}[atheorem]{Conjecture}
\newcommand{\amyequation}[2]{\par\refstepcounter{atheorem}
   \setcounter{myeqn}{\value{atheorem}}\label{#1} 
   \smallskip\noindent\makebox[\textwidth]{\hfill${\displaystyle #2 }$\hfill {\rm (\ref{#1})}} }

\newcounter{myafig}
\newcommand{\startafig}{\begin{figure}\refstepcounter{atheorem}
                      \setcounter{myafig}{\value{atheorem}} }
\newenvironment{myafigure}{\startafig}{\eJJ}     
\newcounter{myaeqn}
\newcommand{\myaequation}[2]{\par\refstepcounter{atheorem}
   \setcounter{myaeqn}{\value{atheorem}}\label{#1} 
   \medskip
    \noindent\makebox[\textwidth]{\hfill${\displaystyle #2 }$\hfill
            {\mbox{\rm (\ref{#1})}}}
    \par\medskip\noindent}
\newcommand{\narrowEquation}[2]{\par\refstepcounter{atheorem}
   \setcounter{myaeqn}{\value{atheorem}}\label{#1} 
   \medskip
    \noindent\makebox[6.5in]{\hfill${\displaystyle #2 }$\hfill
            {\mbox{\rm (\ref{#1})}}}
    \par\medskip\noindent}

\newcommand{\leftPicture}[2]{\par\noindent\begin{minipage}[t]{.48\textwidth} #1
                             \end{minipage} \ 
			     \begin{minipage}[t]{.48\textwidth} #2
                             \end{minipage}}
%\renewcommand{\Box}{\fbox{\rule{0cm}{.04cm}\rule{.04cm}{0cm}}}% theorem_section.tex
%the following make everything numbered like theorems, within sections.
\newtheorem{theorem}{Theorem}[section]
\newtheorem{algorithm}[theorem]{Algorithm}
\newtheorem{claim}[theorem]{Claim}
\newtheorem{conjecture}[theorem]{Conjecture}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{fact}[theorem]{Fact}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{meta-proposition}[theorem]{Meta-Proposition}
\newtheorem{notation}[theorem]{Notation}
\newtheorem{note}[theorem]{Note}
\newtheorem{observation}[theorem]{Observation}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{proviso}[theorem]{Proviso}         
\newtheorem{question}[theorem]{Question}         
%\newtheorem{remark}[theorem]{Remark}         
\setcounter{theorem}{0}

% new 12/17/13:
\newcommand{\colonDash}{\;\mbox{{\rm :-}}\;}
% 250 abbreviations:
\newcommand{\pvar}{\mbox{\bf P$_{\textrm{var}}$}}
\newcommand{\pwff}{\mbox{\bf P$_{\textrm{wff}}$}}
\newcommand{\pfmla}{\mbox{\bf P$_{\textrm{fmla}}$}}
\newcommand{\true}{\mbox{{\bf T}}}
\newcommand{\bT}{\mbox{{\bf T}}}
\newcommand{\false}{\mbox{{\bf F}}}
\newcommand{\bF}{\mbox{{\bf F}}}
\newcommand{\bbf}[1]{{\color{blue}\bf #1}}  %boldface and blue
\newcommand{\gbf}[1]{{\color{dg}\bf #1}}  %boldface and dark green
\newcommand{\gf}[1]{{\color{dg} #1}}  %dark green
\newcommand{\mbf}[1]{{\color{magenta}\bf #1}}  %boldface and magenta
\newcommand{\tbf}[1]{\mbox{{\color{magenta}\fbox{{#1}}}}} % TM box
\newcommand{\mdeg}{\mbox{{\color{magenta}\bf deg}}}  % undirected degree of a vertex
\newcommand{\rbf}[1]{{\color{red}\bf #1}}  %boldface and red
\newcommand{\brbf}[1]{{\color{brown}\bf #1}}  %boldface and brown
\newcommand{\sbf}[1]{{\color{silver}\bf #1}}  %boldface and silver\
\newcommand{\jbf}[1]{{\bf #1}}  %just boldface 
\newcommand{\bfb}[1]{\mbox{ {\bf #1} }}
\newcommand{\rmb}[1]{\mbox{{\rm #1}}} % roman in a box
\newcommand{\hrb}[1]{\mbox{\color{blue} \(#1\)}}

% EPR paper:
\newcommand{\QFR}{\rmb{QF}^R}
\newcommand{\AFR}{\rmb{AF}^R}
\newcommand{\AER}{\rmb{AE}^R}
\newcommand{\QFAR}{\rmb{QF}^{AR}}
\newcommand{\AFAR}{\rmb{AF}^{AR}}
\newcommand{\AEAR}{\rmb{AE}^{AR}}
\newcommand{\qfr}{\rmb{QF}^R}
\newcommand{\afr}{\rmb{AF}^R}
\newcommand{\aer}{\rmb{AE}^R}
\newcommand{\qfar}{\rmb{QF}^{AR}}
\newcommand{\afar}{\rmb{AF}^{AR}}
\newcommand{\aear}{\rmb{AE}^{AR}}

\newcommand{\black}{\color{black}}
\newcommand{\blue}{\color{blue}}
\newcommand{\red}{\color{red}}
\newcommand{\magenta}{\color{magenta}}
\newcommand{\darkgreen}{\color{dg}}
\newcommand{\fromSlide}[2]{#2}
\newcommand{\overlays}[2]{#2}
\newcommand{\ap}{\mbox{{\rm AP}}}
\newcommand{\siff}{\mbox{{\rm $\;$ iff $\;$}}}

\newcommand{\labelAbove}[2]{\draw (#1.north) node[above] {#2};}   % mark node above
\newcommand{\labelBelow}[2]{\draw (#1.south) node[below] {#2};}   % mark node below
\newcommand{\labelLeft}[2]{\draw (#1.west) node[left] {#2};}   % mark node left
\newcommand{\labelRight}[2]{\draw (#1.east) node[right] {#2};}   % mark node right
\newcommand{\slabelAbove}[1]{\draw (#1.north) node[above] {{\bf s}};}   % label s above
\newcommand{\slabelBelow}[1]{\draw (#1.south) node[below] {{\bf s}};}   % label s below
\newcommand{\stlabelAbove}[1]{\draw (#1.north) node[above] {{\bf s t}};}   % label st above
\newcommand{\indexLabelBelow}[2]{\draw (#1.south) node[below] {{\bf #2}};}   % label s below
\newcommand{\tlabelAbove}[1]{\draw (#1.north) node[above] {{\bf t}};}   % label t above
\newcommand{\finalState}[2]{\draw (#1) circle (.2 * #2);}  % final state
\newcommand{\startState}[2]{\draw[line width=2pt] ([shift=(225:.14 * #2)]#1.west) -- (#1.west) --
  ([shift=(135:.14 * #2)]#1.west);}  % start state
\newcommand{\startMyState}[2]{\draw[line width=2pt,color=dg] ([shift=(225:2ex * #2)]#1.west) -- (#1.west) --
  ([shift=(135:2ex * #2)]#1.west);}  % start state
\newcommand{\vState}[1]{\draw (#1.north east) --  (#1.south east)}  % draw vertical state
\newcommand{\hState}[1]{\draw (#1.north east) --  (#1.north west)}  % draw vertical state
\newcommand{\labelGraph}[2]{\draw (#1.west) node[left,xshift=-2ex] {{\Large #2}};}
\newcommand{\labelGraphDist}[3]{\draw (#1.west) node[left,xshift=-#3] {{\Large #2}};}
\newcommand{\emmaAutomataStyle}{
\tikzstyle{every initial by arrow}=[-triangle 90,initial text=,initial distance=0.01,fill=white]
\tikzstyle{every picture}=[->,>=stealth',shorten >=1pt,auto,node distance=2.8cm,semithick]}