# Between the deterministic and non-deterministic query complexity

## TL;DR

This paper explores the spectrum of query complexities between deterministic and non-deterministic models, introducing intermediate measures and providing bounds for various problems to understand their query requirements.

## Contribution

It introduces a new hierarchy of query complexities $D_k(P,n)$ between deterministic and non-deterministic cases, with bounds for different problems.

## Key findings

- Established bounds for $D_k(P,n)$ across multiple problems.
- Defined a hierarchy of query complexities interpolating between known extremes.
- Provided insights into how input modifications affect query complexity.

## Abstract

We consider problems that can be solved by asking certain queries. The deterministic query complexity $D(P,n)$ of a problem $P$ is the smallest number of queries needed to ask in order to find the solution with an input of size $n$ (in the worst case), while the non-deterministic query complexity $D_0(P,n)$ 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, $D(P,n)$ 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, $D_k(P,n)$ 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 $D_k(P,n)$ for various problems $P$.