Query Complexity with Unknowns
Nikhil S. Mande, Karteek Sreenivasaiah
TL;DR
The authors propose the $u$-query model, a three-valued extension of Boolean query complexity based on Kleene's logic of indeterminacy, with hazard-free extension $f_\mathsf{u}\u001b$ and input domain $\{0,1,\mathsf{u}\}^n\u001b$. They establish the relationships between $D_\mathsf{u}, R_\mathsf{u}, Q_\mathsf{u}$ and the standard complexities, introducing $u$-analogs of sensitivity, block sensitivity, and certificate complexity, and show that these measures yield polynomial relationships among the $u$-query complexities for all total Boolean functions. A key result is an explicit exponential separation for the classic Indexing function in the $u$-model, namely $\mathsf{D}_\mathsf{u}(\mathsf{IND}_n) = \Theta(2^n), \mathsf{Q}_\mathsf{u}(\mathsf{IND}_n) = \Theta(2^{n/2})$, while a monotone variant $\mathsf{mIND}$ exhibits logarithmic $u$-query complexity, illustrating that monotone functions behave similarly to the standard model under $u$-queries. The paper proves that $\mathsf{D}_\mathsf{u}(f) = O(\mathsf{C}_\mathsf{u}(f)\mathsf{bs}_\mathsf{u}(f))$ and $\mathsf{D}_\mathsf{u}(f) = O(\mathsf{R}_\mathsf{u}(f)^3) = O(\mathsf{Q}_\mathsf{u}(f)^6)$, mirroring classical bounds in the $u$-model, with the proofs relying on $u$-variations of block sensitivity and certificates. These results open avenues to characterize $u$-query complexity via standard complexity measures and to extend the framework to other logics.
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.