Numerical relations in logics are known to
characterize, via the finite models of their sentences,
polynomial advice nonuniform complexity classes. These are known
to coincide with reduction classes of tally sets. Our
contributions here are: 1/ a refinement of that characterization
that individualizes the reduction class of each tally
set, and 2/ characterizing logarithmic advice classes via
numerical constants, both in the (rather easy) case of C/log
and in the more complex case of Full-C/log; this proof requires
to extend to classes below P the technical characterizations
known for the class Full-P/log.
CitationAtserias, A., Balcazar, J. L. "Refining logical characterizations of advice complexity classes". 1997.
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