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Departing from previous approaches, we exploit the technique of shrinkage under random restrictions to obtain a variety of new results for this model. Among them, we show\n \n Average-case Lower Bounds<\/jats:italic> For every $$k = k(n)$$<\/jats:tex-math>\n \n k<\/mml:mi>\n =<\/mml:mo>\n k<\/mml:mi>\n (<\/mml:mo>\n n<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> with $$k \\geqslant \\log n$$<\/jats:tex-math>\n \n k<\/mml:mi>\n \u2a7e<\/mml:mo>\n log<\/mml:mo>\n n<\/mml:mi>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>, there exists a polynomial-time computable function $$f_k$$<\/jats:tex-math>\n \n f<\/mml:mi>\n k<\/mml:mi>\n <\/mml:msub>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> on n<\/jats:italic> bits such that, for every comparator circuit C<\/jats:italic> with at most $$n^{1.5}\/O\\!\\left( k\\cdot \\sqrt{\\log n}\\right) $$<\/jats:tex-math>\n \n \n n<\/mml:mi>\n \n 1.5<\/mml:mn>\n <\/mml:mrow>\n <\/mml:msup>\n \/<\/mml:mo>\n O<\/mml:mi>\n \n \n k<\/mml:mi>\n \u00b7<\/mml:mo>\n \n \n log<\/mml:mo>\n n<\/mml:mi>\n <\/mml:mrow>\n <\/mml:msqrt>\n <\/mml:mfenced>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> gates, we have $$\\begin{aligned} \\mathop {{{\\,\\mathrm{\\textbf{Pr}}\\,}}}\\limits _{x\\in \\left\\{ 0,1\\right\\} ^n}\\left[ C(x)=f_k(x)\\right] \\leqslant \\frac{1}{2} + \\frac{1}{2^{\\Omega (k)}}. \\end{aligned}$$<\/jats:tex-math>\n \n \n \n \n \n \n \n \n Pr<\/mml:mi>\n \n <\/mml:mrow>\n \n x<\/mml:mi>\n \u2208<\/mml:mo>\n \n \n 0<\/mml:mn>\n ,<\/mml:mo>\n 1<\/mml:mn>\n <\/mml:mfenced>\n n<\/mml:mi>\n <\/mml:msup>\n <\/mml:mrow>\n <\/mml:munder>\n \n C<\/mml:mi>\n \n (<\/mml:mo>\n x<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n =<\/mml:mo>\n \n f<\/mml:mi>\n k<\/mml:mi>\n <\/mml:msub>\n \n (<\/mml:mo>\n x<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:mfenced>\n \u2a7d<\/mml:mo>\n \n 1<\/mml:mn>\n 2<\/mml:mn>\n <\/mml:mfrac>\n +<\/mml:mo>\n \n 1<\/mml:mn>\n \n 2<\/mml:mn>\n \n \u03a9<\/mml:mi>\n (<\/mml:mo>\n k<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:msup>\n <\/mml:mfrac>\n .<\/mml:mo>\n <\/mml:mrow>\n <\/mml:mtd>\n <\/mml:mtr>\n <\/mml:mtable>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:disp-formula> This average-case lower bound matches the worst-case lower bound of G\u00e1l and Robere by letting $$k=O\\!\\left( \\log n\\right) $$<\/jats:tex-math>\n \n k<\/mml:mi>\n =<\/mml:mo>\n O<\/mml:mi>\n \n \n log<\/mml:mo>\n n<\/mml:mi>\n <\/mml:mfenced>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>.<\/jats:p>\n <\/jats:list-item>\n \n $$\\#$$<\/jats:tex-math>\n #<\/mml:mo>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>SAT Algorithms<\/jats:italic> There is an algorithm that counts the number of satisfying assignments of a given comparator circuit with at most $$n^{1.5}\/O\\!\\left( k\\cdot \\sqrt{\\log n}\\right) $$<\/jats:tex-math>\n \n \n n<\/mml:mi>\n \n 1.5<\/mml:mn>\n <\/mml:mrow>\n <\/mml:msup>\n \/<\/mml:mo>\n O<\/mml:mi>\n \n \n k<\/mml:mi>\n \u00b7<\/mml:mo>\n \n \n log<\/mml:mo>\n n<\/mml:mi>\n <\/mml:mrow>\n <\/mml:msqrt>\n <\/mml:mfenced>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> gates, in time $$2^{n-k}\\cdot {{\\,\\textrm{poly}\\,}}(n)$$<\/jats:tex-math>\n \n \n 2<\/mml:mn>\n \n n<\/mml:mi>\n -<\/mml:mo>\n k<\/mml:mi>\n <\/mml:mrow>\n <\/mml:msup>\n \u00b7<\/mml:mo>\n \n \n poly<\/mml:mtext>\n \n <\/mml:mrow>\n \n (<\/mml:mo>\n n<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>, for any $$k\\leqslant n\/4$$<\/jats:tex-math>\n \n k<\/mml:mi>\n \u2a7d<\/mml:mo>\n n<\/mml:mi>\n \/<\/mml:mo>\n 4<\/mml:mn>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>. The running time is non-trivial (i.e., $$2^n\/n^{\\omega (1)}$$<\/jats:tex-math>\n \n \n 2<\/mml:mn>\n n<\/mml:mi>\n <\/mml:msup>\n \/<\/mml:mo>\n \n n<\/mml:mi>\n \n \u03c9<\/mml:mi>\n (<\/mml:mo>\n 1<\/mml:mn>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:msup>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>) when $$k=\\omega (\\log n)$$<\/jats:tex-math>\n \n k<\/mml:mi>\n =<\/mml:mo>\n \u03c9<\/mml:mi>\n (<\/mml:mo>\n log<\/mml:mo>\n n<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>.<\/jats:p>\n <\/jats:list-item>\n \n Pseudorandom Generators and <\/jats:italic>$$\\textsf {MCSP} $$<\/jats:tex-math>\n MCSP<\/mml:mi>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>L<\/jats:italic>ower Bounds There is a pseudorandom generator of seed length $$s^{2\/3+o(1)}$$<\/jats:tex-math>\n \n s<\/mml:mi>\n \n 2<\/mml:mn>\n \/<\/mml:mo>\n 3<\/mml:mn>\n +<\/mml:mo>\n o<\/mml:mi>\n (<\/mml:mo>\n 1<\/mml:mn>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:msup>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> that fools comparator circuits with s<\/jats:italic> gates. 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