Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Succinct Hitting Sets and Barriers to Proving Algebraic Circuits Lower
  Bounds   We formalize a framework of algebraically natural lower bounds for algebraic
circuits. Just as with the natural proofs notion of Razborov and Rudich for
boolean circuit lower bounds, our notion of algebraically natural lower bounds
captures nearly all lower bound techniques known. However, unlike the boolean
setting, there has been no concrete evidence demonstrating that this is a
barrier to obtaining super-polynomial lower bounds for general algebraic
circuits, as there is little understanding whether algebraic circuits are
expressive enough to support "cryptography" secure against algebraic circuits.
  Following a similar result of Williams in the boolean setting, we show that
the existence of an algebraic natural proofs barrier is equivalent to the
existence of succinct derandomization of the polynomial identity testing
problem. That is, whether the coefficient vectors of polylog(N)-degree
polylog(N)-size circuits is a hitting set for the class of poly(N)-degree
poly(N)-size circuits. Further, we give an explicit universal construction
showing that if such a succinct hitting set exists, then our universal
construction suffices.
  Further, we assess the existing literature constructing hitting sets for
restricted classes of algebraic circuits and observe that none of them are
succinct as given. Yet, we show how to modify some of these constructions to
obtain succinct hitting sets. This constitutes the first evidence supporting
the existence of an algebraic natural proofs barrier.
  Our framework is similar to the Geometric Complexity Theory (GCT) program of
Mulmuley and Sohoni, except that here we emphasize constructiveness of the
proofs while the GCT program emphasizes symmetry. Nevertheless, our succinct
hitting sets have relevance to the GCT program as they imply lower bounds for
the complexity of the defining equations of polynomials computed by small
circuits.

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The categories of this paper are: math
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The categories of this paper are: chem-ph
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The categories of this paper are: math-ph
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The categories of this paper are: q-fin
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The categories of this paper are: nucl-th
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The categories of this paper are: mtrl-th
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The categories of this paper are: econ
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The categories of this paper are: stat
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The categories of this paper are: cmp-lg
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The categories of this paper are: nucl-ex
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The categories of this paper are: solv-int
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The categories of this paper are: nlin
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The categories of this paper are: cond-mat
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The categories of this paper are: eess
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The categories of this paper are: plasm-ph
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The categories of this paper are: patt-sol