Query Complexity with Unknowns

Nikhil S. Mande; Karteek Sreenivasaiah

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arXiv · 2412.06395

Query Complexity with Unknowns

Nikhil S. Mande·Karteek Sreenivasaiah

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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.

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Nikhil S. Mande, Karteek Sreenivasaiah. 2024-12-09. Query Complexity with Unknowns. https://arxiv.org/abs/2412.06395

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