Query Complexity with Unknowns

TL;DR

This paper introduces the u-query model for Boolean functions, where queries can return 'unknown', and explores how this affects query complexity, establishing relationships with standard models and demonstrating exponential differences for some functions.

Contribution

The paper formalizes the u-query model using Kleene's logic, relates it to standard query complexities, and shows polynomial relations among different complexities within this new framework.

Findings

Existence of functions exponentially harder in u-query model

Conditions for u-query complexity to match standard query complexity

Polynomial relations among deterministic, randomized, and quantum complexities in u-query model

Abstract

We initiate the study of a new model of query complexity of Boolean functions where, in addition to 0 and 1, the oracle can answer queries with ``unknown''. The query algorithm is expected to output the function value if it can be conclusively determined by the partial information gathered, and it must output ``unknown'' if not. We formalize this model by using Kleene's strong logic of indeterminacy on three variables to capture unknowns. We call this model the `u-query model'. We relate the query complexity of functions in the new u-query model with their analogs in the standard query model. We show an explicit function that is exponentially harder in the u-query model than in the usual query model. We give sufficient conditions for a function to have u-query complexity asymptotically the same as its query complexity. Using u-query analogs of the combinatorial measures of sensitivity, block sensitivity, and certificate complexity, we show that deterministic, randomized, and quantum u-query complexities of all total Boolean functions are polynomially related to each other, just as in the usual query models.