Abstract
Williams’s celebrated circuit lower bound technique works by showing that the existence of certain small enough nonuniform circuits implies that nondeterministic exponential time can be speeded up in such a way that it implies a contradiction with the nondeterministic time hierarchy.
We apply Williams’s technique by speeding up instead (i) deterministic exponential-time computations and (ii) nondeterministic exponential-time computations that use only a limited number of nondeterministic bits. From (i), we obtain that \(\mathrm{EXP}\subseteq \text {ACC}^0\) has a consequence that might seem unlikely, while (ii) yields an exponential \(\text {ACC}^0\) size-depth tradeoff for \(\mathrm{E}^\mathrm{NP[2^{n^{c\delta }}]}\), which is the class of exponential-time computation with access to an NP oracle where the number of oracle queries is bounded.
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Notes
- 1.
Williams actually takes \(L \in \mathrm{NTIME}(2^n)\). As in [1], we instead use \(\mathrm{NTIME}(2^n/n^{10})\) so that the circuit C in SUCCINCT-3SAT has exactly n inputs instead of \(n+c\log n\) inputs.
- 2.
If nothing else is said then \(\text {ACC}^0\) stands for the class of languages that have \(\text {ACC}^0\) circuits of polynomial size.
- 3.
Alternatively, \(C_x'\) could have been constructed to be \(\text {ACC}^0\) right away using the recent result by Jahanjou, Miles, and Viola [6], but we don’t know if the required witness circuit can be shown to exist also in this case.
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Spakowski, H. (2016). On Limited Nondeterminism and ACC Circuit Lower Bounds. In: Dediu, AH., Janoušek, J., Martín-Vide, C., Truthe, B. (eds) Language and Automata Theory and Applications. LATA 2016. Lecture Notes in Computer Science(), vol 9618. Springer, Cham. https://doi.org/10.1007/978-3-319-30000-9_25
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