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- W2100965091 abstract "Complexity theory attempts to classify problems into classes according to their resource requirements such as time and space, and to understand the relative power of these resources. This dissertation is motivated, in particular, by the study of the limitations of Boolean and arithmetic circuits as models of computation. The goal of this area is to show that there are explicit problems that cannot be solved by small circuits. To this end we make progress along several avenues for proving such circuit lower bounds. Polynomial Identity Testing is the fundamental problem of testing whether a given multivariate polynomial is identically zero. There is a natural efficient randomized algorithm. Showing that there is an efficient deterministic identity test, in a sufficiently general setting, implies long elusive circuit lower bounds. In the first part of this dissertation, we develop new deterministic identity tests for bounded-read multilinear arithmetic formulas, an interesting class of polynomials that encompasses and significantly generalizes several prior works. Locality is a property of logical formulas that expresses that the formula cannot distinguish between two inputs that appear the same up to some distance parameter. Once established for a given set of logical formulas, the property can often be used to quickly argue separations from that set. In this dissertation we tightly characterize the locality of a logical system corresponding to AC0, the class of languages solvable by families of constant-depth polynomial-size Boolean circuits. In doing so we give a meta-theorem for proving that certain graph properties cannot be computed in AC0. Kernelization is the process of transforming an equivalent instance of one problem into an instance of another problem where the size of the latter instance depends only on a single parameter. We look at a set system that is motivated by the study of kernelization, and give a tight bound on the size of that set system." @default.
- W2100965091 created "2016-06-24" @default.
- W2100965091 creator A5079027478 @default.
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- W2100965091 date "2012-01-01" @default.
- W2100965091 modified "2023-09-22" @default.
- W2100965091 title "Advancing algebraic and logical approaches to circuit lower bounds" @default.
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