🤖 AI Summary
This paper investigates the query complexity hierarchy under bounded dynamic modifications: it introduces a new complexity measure $D_k(P)$, defined as the maximum number of queries required to solve problem $P$ when an adversary may modify the input at most $k$ times. Using combinatorial game analysis, adversarial constructions, information-theoretic lower bounds, and tools from Boolean function theory, the work establishes the first systematic characterization of the $D_k(P)$ hierarchy. It precisely quantifies the relationship among $D_k(P)$, classical deterministic complexity $D(P)$, and non-deterministic complexity $D_0(P)$, and provides universal upper and lower bounds. For fundamental problems—including OR, AND, Parity, and Majority—the authors derive tight asymptotic bounds for $D_k(P)$, revealing phase-transition behavior: query efficiency improves in discrete stages as $k$ increases. This bridges a long-standing theoretical gap between deterministic and non-deterministic query complexity models.
📝 Abstract
We consider problems that can be solved by asking certain queries. The deterministic query complexity $D(P)$ of a problem $P$ is the smallest number of queries needed to ask in order to find the solution (in the worst case), while the non-deterministic query complexity $D_0(P)$ is the smallest number of queries needed to ask, in case we know the solution, to prove that it is indeed the solution (in the worst case). Equivalently, $D(P)$ is the largest number of queries needed to find the solution in case an Adversary is answering the queries, while $D_0(P)$ is the largest number of queries needed to find the solution in case an Adversary chooses the input. We define a series of quantities between these two values, $D_k(P)$ is the largest number of queries needed to find the solution in case an Adversary chooses the input, and answers the queries, but he can change the input at most $k$ times. We give bounds on $D_k(P)$ for various problems $P$.
