Symmetry of Information (SoI) [ZL70] is a fundamental theorem of classical Kolmogorov complexity whose status for time-bounded computation is closely related to central open problems in complexity theory and cryptography. In a recent development, [HMO26] showed that SoI fails for $rKt$ complexity, namely in the setting of randomized computation. By contrast, many basic properties of Kolmogorov complexity are known to remain valid in the presence of nondeterminism [HLO26], suggesting that SoI may provide a fruitful structural principle for exploring the relative power of randomness and nondeterminism in computation. Our results reveal that the validity of SoI for nondeterministic computation, namely for $nKt$ complexity, is a subtle question connected to several frontiers. We show that symmetry of information holds on average for randomized extensions of $nKt$, while its worst case validity would imply breakthrough complexity separations. Although the standard cryptographic argument for asymmetry does not extend to the nondeterministic setting, we provide evidence that information remains asymmetric in nondeterministic computations. Among other results, we show that SoI would imply a collapse of exponential time classes to low levels of the polynomial hierarchy. Then, building on the work of [Hir22b, GK22, KK26, HMO26], we develop a general framework connecting symmetry of information, meta-complexity, and explicit constructions in the nondeterministic setting. More precisely, we prove that suitable formulations of these three notions are quantitatively equivalent. We apply different aspects of this correspondence to make progress on complexity lower bounds and on the failure of SoI for nondeterministic computation. These consequences are unconditional and provide some of the strongest known lower bounds in meta-complexity.

翻译：暂无翻译