Abstract
Existing definitions of the relativizations of NC 1, L and NL do not preserve the inclusions NC 1 ⊆ L, NL ⊆ AC 1. We start by giving the first definitions that preserve them. Here for L and NL we define their relativizations using Wilson’s stack oracle model, but limit the height of the stack to a constant (instead of log(n)). We show that the collapse of any two classes in {AC 0(m), TC 0, NC 1, L, NL} implies the collapse of their relativizations. Next we develop theories that characterize the relativizations of subclasses of P by modifying theories previously defined by the second two authors. A function is provably total in a theory iff it is in the corresponding relativized class. Finally we exhibit an oracle α that makes AC k(α) a proper hierarchy. This strengthens and clarifies the separations of the relativized theories in [Takeuti, 1995]. The idea is that a circuit whose nested depth of oracle gates is bounded by k cannot compute correctly the (k + 1) compositions of every oracle function.
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Aehlig, K., Cook, S., Nguyen, P. (2007). Relativizing Small Complexity Classes and Their Theories. In: Duparc, J., Henzinger, T.A. (eds) Computer Science Logic. CSL 2007. Lecture Notes in Computer Science, vol 4646. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-74915-8_29
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DOI: https://doi.org/10.1007/978-3-540-74915-8_29
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