From Almost Optimal Algorithms to Logics for Complexity Classes via Listings and a Halting Problem
Journal of the ACM (JACM)
A parameterized halting problem
The Multivariate Algorithmic Revolution and Beyond
Hard instances of algorithms and proof systems
CiE'12 Proceedings of the 8th Turing Centenary conference on Computability in Europe: how the world computes
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There are standard logics DTC, TC, and LFP capturing the complexity classes L, NL, and P on ordered structures, respectively. In [4] we have shown that LFP_{inv}, the ``order-invariant least fixed-point logic LFP,'' captures P (on all finite structures) if and only if there is a listing of the P subsets of the set TAUT of propositional tautologies. We are able to extend the result to listings of the L-subsets (NL-subsets) of TAUT and the logic DTC_{inv} (TC_{inv}). As a byproduct we get that LFP_{inv} captures P if DTC_{inv} captures L. Furthermore, we show that the existence of a listing of the L-subsets of TAUT is equivalent to the existence of an almost space optimal algorithm for TAUT. To obtain this result we have to derive a space version of a theorem of Levin on optimal inverters.