Abstract
This work continues the development of hardness magnification. The latter proposes a new strategy for showing strong complexity lower bounds by reducing them to a refined analysis of weaker models, where combinatorial techniques might be successful.
We consider gap versions of the meta-computational problems MKtP and MCSP, where one needs to distinguish instances (strings or truth-tables) of complexity ≤ s1 (N) from instances of complexity ≥ s2(N), and N = 2n denotes the input length. In MCSP, complexity is measured by circuit size, while in MKtP one considers Levin's notion of time-bounded Kolmogorov complexity. (In our results, the parameters s1 (N) and s2 (N) are asymptotically quite close, and the problems almost coincide with their standard formulations without a gap.) We establish that for Gap-MKtP[s1, s2] and Gap-MCSP[s1, s2], a marginal improvement over the state-of-the-art in unconditional lower bounds in a variety of computational models would imply explicit super-polynomial lower bounds.
Theorem. There exists a universal constant c ≥ 1 for which the following hold. If there exists ε > 0 such that for every small enough β > 0
(1) Gap-MCSP[2βn / cn, 2βn] ∉ Circuit[N1+ε], then NP ⊈ Circuit[poly].
(2) Gap-MKtP[2βn, 2βn + cn] ∉ TC0[N1+ε], then EXP ⊈ TC0[poly].
(3) Gap-MKtP[2βn, 2βn + cn] ∉ B2-Formula[N2+ε], then EXP ⊈ Formula[poly].
(4) Gap-MKtP[2βn, 2βn + cn] ∉ U2-Formula[N3+ε], then EXP ⊈ Formula[poly].
(5) Gap-MKtP[2βn, 2βn + cn] ∉ BP[N2+ε], then EXP ⊈ BP[poly].
(6) Gap-MKtP[2βn, 2βn + cn] ∉ (AC0[6])[N1+ε], then EXP ⊈ AC0[6].
These results are complemented by lower bounds for Gap-MCSP and Gap-MKtP against different models. For instance, the lower bound assumed in (1) holds for U2-formulas of near-quadratic size, and lower bounds similar to (3)-(5) hold for various regimes of parameters.
We also identify a natural computational model under which the hardness magnification threshold for Gap-MKtP lies below existing lower bounds: U2-formulas that can compute parity functions at the leaves (instead of just literals). As a consequence, if one managed to adapt the existing lower bound techniques against such formulas to work with Gap-MKtP, then EXP ⊈ NC1 would follow via hardness magnification.
