Hardness magnification near state-of-the-art lower bounds

This work continues the development of hardness magnification. The latter proposes a 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[s1s2] and Gap-MCSP[s1s2], 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 c1 for which the following hold. If there exists 0 such that for every small enough 0

1. Gap-MCSP[2ncn2n] Circuit[N1+], then NP is not contained in Circuit[poly].
2. Gap-MKtP[2n2n+cn] TC0[N1+], then EXP is not contained in TC0[poly].
3. Gap-MKtP[2n2n+cn]B2-Formula[N2+], then EXP is not contained in Formula[poly].
4. Gap-MKtP[2n2n+cn]U2-Formula[N3+], then EXP is not contained in Formula[poly].
5. Gap-MKtP[2n2n+cn] BP[N2+], then EXP is not contained in BP[poly].
6. Gap-MKtP[2n2n+cn] (AC0[6])[N1+], then EXP is not contained in 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.

Going beyond the standard boolean devices, we 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 is not contained in NC1 would follow via hardness magnification.