This dissertation presents several results at the intersection ofcomplexity theory and algorithm design. Complexity theory aims tolower-bound the amount of computational resources (such as time andspace) required to solve interesting problems. Algorithm design aimsto upper-bound the amount of computational resources required to solveinteresting problems. These pursuits appear opposed. However, somealgorithm design and complexity lower bound problems are inextricablyconnected.This dissertation explores several such connections. From "natural"proofs of circuit-size lower bounds, we create learningalgorithms. From the exact hardness of problems in polynomial time, wecreate algorithms of estimating the acceptance probability ofcircuits. Finally, from algorithms for testing the identity ofairthmetic circuits over finite fields, we create arithmeticcircuit-complexity lower bounds
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