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|Title of Book||In Proc. 15th Symposium on Theoretical Aspects of Computer Science|
|Series||Lecture Notes in Computer Science|
|Abstract||In 1979 Valiant developed an algebraic analogue of the theory of NP-completeness for computations of polynomials over a field. We further develop this theory in the spirit of structural complexity and obtain analogues of well-known results by Baker, Gill, and Solovay (1975), Ladner (1975), and Schöning (1982, 1984). We show that if Valiant's hypothesis is true, then there is a $p$-definable family, which is neither $p$-computable nor $VNP$-complete. More generally, we define the posets of $p$-degrees and $c$-degrees of $p$-definable families and prove that any countable poset can embedded in either of them, provided Valiant's hypothesis is true. Moreover, we establish the existence of minimal pairs for $VP$ in $VNP$. Over finite fields, we give a specific example of a family of polynomials which is neither $VNP$-complete nor $p$-computable, provided the polynomial hierarchy does not collapse. We define relativized complexity classes $VP^h$ and $VNP^h$ and construct complete families in these classes. Moreover, we prove that there is a $p$-family $h$ satisfying $VP^h = VNP^h$.|
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