Skip to main content
\(\newcommand{\set}[1]{\{1,2,\dotsc,#1\,\}} \newcommand{\ints}{\mathbb{Z}} \newcommand{\posints}{\mathbb{N}} \newcommand{\rats}{\mathbb{Q}} \newcommand{\reals}{\mathbb{R}} \newcommand{\complexes}{\mathbb{C}} \newcommand{\twospace}{\mathbb{R}^2} \newcommand{\threepace}{\mathbb{R}^3} \newcommand{\dspace}{\mathbb{R}^d} \newcommand{\nni}{\mathbb{N}_0} \newcommand{\nonnegints}{\mathbb{N}_0} \newcommand{\dom}{\operatorname{dom}} \newcommand{\ran}{\operatorname{ran}} \newcommand{\prob}{\operatorname{prob}} \newcommand{\Prob}{\operatorname{Prob}} \newcommand{\height}{\operatorname{height}} \newcommand{\width}{\operatorname{width}} \newcommand{\length}{\operatorname{length}} \newcommand{\crit}{\operatorname{crit}} \newcommand{\inc}{\operatorname{inc}} \newcommand{\HP}{\mathbf{H_P}} \newcommand{\HCP}{\mathbf{H^c_P}} \newcommand{\GP}{\mathbf{G_P}} \newcommand{\GQ}{\mathbf{G_Q}} \newcommand{\AG}{\mathbf{A_G}} \newcommand{\GCP}{\mathbf{G^c_P}} \newcommand{\PXP}{\mathbf{P}=(X,P)} \newcommand{\QYQ}{\mathbf{Q}=(Y,Q)} \newcommand{\GVE}{\mathbf{G}=(V,E)} \newcommand{\HWF}{\mathbf{H}=(W,F)} \newcommand{\bfC}{\mathbf{C}} \newcommand{\bfG}{\mathbf{G}} \newcommand{\bfH}{\mathbf{H}} \newcommand{\bfF}{\mathbf{F}} \newcommand{\bfI}{\mathbf{I}} \newcommand{\bfK}{\mathbf{K}} \newcommand{\bfP}{\mathbf{P}} \newcommand{\bfQ}{\mathbf{Q}} \newcommand{\bfR}{\mathbf{R}} \newcommand{\bfS}{\mathbf{S}} \newcommand{\bfT}{\mathbf{T}} \newcommand{\bfNP}{\mathbf{NP}} \newcommand{\bftwo}{\mathbf{2}} \newcommand{\cgA}{\mathcal{A}} \newcommand{\cgB}{\mathcal{B}} \newcommand{\cgC}{\mathcal{C}} \newcommand{\cgD}{\mathcal{D}} \newcommand{\cgE}{\mathcal{E}} \newcommand{\cgF}{\mathcal{F}} \newcommand{\cgG}{\mathcal{G}} \newcommand{\cgM}{\mathcal{M}} \newcommand{\cgN}{\mathcal{N}} \newcommand{\cgP}{\mathcal{P}} \newcommand{\cgR}{\mathcal{R}} \newcommand{\cgS}{\mathcal{S}} \newcommand{\bfn}{\mathbf{n}} \newcommand{\bfm}{\mathbf{m}} \newcommand{\bfk}{\mathbf{k}} \newcommand{\bfs}{\mathbf{s}} \newcommand{\bijection}{\xrightarrow[\text{onto}]{\text{$1$--$1$}}} \newcommand{\injection}{\xrightarrow[]{\text{$1$--$1$}}} \newcommand{\surjection}{\xrightarrow[\text{onto}]{}} \newcommand{\nin}{\not\in} \newcommand{\prufer}{\mbox{prüfer}} \DeclareMathOperator{\fix}{fix} \DeclareMathOperator{\stab}{stab} \DeclareMathOperator{\var}{var} \newcommand{\inv}{^{-1}} \newcommand{\lt}{<} \newcommand{\gt}{>} \newcommand{\amp}{&} \)

Section11.7Discussion

Zori started the conversation with “Who in their right mind would trust their lives to an algorithm that used random methods?” Xing quickly responded “Everyone. At least everyone should. We routinely deal with probabilistic concepts, like getting run over by a bus when crossing the street or having a piano fall on our head. The general public is much more comfortable with notions of probability, even though they may never know the formal definition of a probability space. I for one am completely comfortable taking an airline flight if I can be assured that the probability of a disaster is less than \(10^{-20}\text{.}\)”

Dave wasn't biting on this topic. Instead he offered “You have to be struck by the statements that it appears difficult to construct objects which you can prove exist in abundance. I wonder why this is so.” Alice said “We all find your brain to be a totally random thing, sometimes making sense but often not.” There was laughter or at least some snickering. But after a bit, Carlos said “There's something fundamental here. Maybe one could prove that there are easily stated theorems which only have long proofs.” Bob blurted “That doens't make any sense.” Zori saw an opportunity where a client would, at considerable expense, commission her to solve a problem (at least better than the competition) that was readily understood but somehow difficult in the end. She knew about the class \(\cgN\cgP\) but maybe there were even bigger challenges (and bigger paychecks) out there.