In this paper, we study the query complexity of Boolean functions in the presence of uncertainty, motivated by parallel computation with an unlimited number of processors where inputs are allowed to be unknown. We allow each query to produce three results: zero, one, or unknown. The output could also be: zero, one, or unknown, with the constraint that we should output ''unknown'' only when we cannot determine the answer from the revealed input bits. Such an extension of a Boolean function is called its hazard-free extension. - We prove an analogue of Huang's celebrated sensitivity theorem [Annals of Mathematics, 2019] in our model of query complexity with uncertainty. - We show that the deterministic query complexity of the hazard-free extension of a Boolean function is at most quadratic in its randomized query complexity and quartic in its quantum query complexity, improving upon the best-known bounds in the Boolean world. - We exhibit an exponential gap between the smallest depth (size) of decision trees computing a Boolean function, and those computing its hazard-free extension. - We present general methods to convert decision trees for Boolean functions to those for their hazard-free counterparts, and show optimality of this construction. We also parameterize this result by the maximum number of unknown values in the input. - We show lower bounds on size complexity of decision trees for hazard-free extensions of Boolean functions in terms of the number of prime implicants and prime implicates of the underlying Boolean function.
In this thesis we study various models of query complexity. A query algorithm computes a function under the restriction that the input can be accessed only by making probes to the the bits of the input. The query complexity of a function f is the minimum number of probes made by any query algorithm that computes f. In this thesis, we consider three different models of query complexity, (1) deterministic decision tree complexity (query complexity when the underlying algorithm is deterministic), (2) approximate decision tree complexity aka. property testing (query complexity when the underlying algorithm is probabilistic and only expected to ”approximately ” compute f) and quantum query complexity (query complexity when the underlying algorithm is allowed to make quantum queries). The main results in this thesis are: • We study the relation between deterministic decision tree complexity and other combinatorial measures of complexity measures like sensitivity and block sensitivity. We prove that for minterm-transitive functions the sensitivity is quadratically related to block sensitivity which is polynomially
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.
We study the query complexity of Boolean functions
f:{0,1}n→{0,1} in the noisy query model introduced by Feige, Raghavan, Peleg and Upfal [SICOMP 1994]. In this model, an algorithm can adaptively query the bits of an input vector, but each query result is independently flipped with constant probability
p∈(0,1/2); repeated queries are allowed. The noisy query complexity
Np(f) of a function
f is defined as the minimum expected number of queries needed to compute
f(x) with error probability at most
1/3, for the worst case input
x. We prove a general lower bound on
Np(f) based on degree statistics of certain subgraphs of the Boolean hypercube. This is the first general lower bound beyond those implied by the simple observation that
Np(f) is lower bounded by the randomized query complexity. We show that this recovers (up to a constant factor) most previously known lower bounds on the noisy query complexity of Boolean functions, providing a unified framework for understanding these results and simplifying the proofs in several cases. Furthermore, this resolves in the affirmative an open problem of Gu, Li and Xu [COLT 2025] that
Np(f)=Ω(I(f)logI(f)), where
I(f) denotes the total influence of
f. We also apply our general lower bound to obtain tight bounds on the noisy query complexity for several new functions.
Based on the recent breakthrough of Huang (2019), we show that for any total Boolean function
f, the deterministic query complexity,
D(f), is at most quartic in the quantum query complexity,
Q(f):
D(f)=O(Q(f)4). This matches the known separation (up to log factors) due to Ambainis, Balodis, Belovs, Lee, Santha, and Smotrovs (2017). We also use the result to resolve the quantum analogue of the Aanderaa-Karp-Rosenberg conjecture. We show that if
f is a nontrivial monotone graph property of an
n-vertex graph specified by its adjacency matrix, then
Q(f)=Ω(n), which is also optimal.
Given a classical query algorithm as a decision tree, when does there exist a quantum query algorithm with a speed-up over the classical one? We provide a general construction based on the structure of the underlying decision tree, and prove that this can give us an up-to-quadratic quantum speed-up. In particular, we obtain a bounded-error quantum query algorithm of cost O(s) to compute a Boolean function (more generally, a relation) that can be computed by a classical (even randomized) decision tree of size s .Lin and Lin [ToC'16] and Beigi and Taghavi [Quantum'20] showed results of a similar flavor, and gave upper bounds in terms of a quantity which we call the "guessing complexity" of a decision tree. We identify that the guessing complexity of a decision tree equals its rank, a notion introduced by Ehrenfeucht and Haussler [Inf. Comp.'89] in the context of learning theory. This answers a question posed by Lin and Lin, who asked whether the guessing complexity of a decision tree is related to any complexity-theoretic measure. We also show a polynomial separation between rank and randomized rank for the complete binary AND-OR tree.Beigi and Taghavi constructed span programs and dual adversary solutions for Boolean functions given classical decision trees computing them and an assignment of non-negative weights to its edges. We explore the effect of changing these weights on the resulting span program complexity and objective value of the dual adversary bound, and capture the best possible weighting scheme by an optimization program. We exhibit a solution to this program and argue its optimality from first principles. We also exhibit decision trees for which our bounds are asymptotically stronger than those of Lin and Lin, and Beigi and Taghavi. This answers a question of Beigi and Taghavi, who asked whether different weighting schemes could yield better upper bounds.
We define and study the complexity of robust polynomials for Boolean functions and the related fault-tolerant quantum decision trees, where input bits are perturbed by noise. We compare several different possible definitions. Our main results are * For every n-bit Boolean function f there is an n-variate polynomial p of degree O(n) that robustly approximates it, in the sense that p(x) remains close to f(x) if we slightly vary each of the n inputs of the polynomial. * There is an O(n)-query quantum algorithm that robustly recovers n noisy input bits. Hence every n-bit function can be quantum computed with O(n) queries in the presence of noise. This contrasts with the classical model of Feige et al., where functions such as parity need Theta(n*log n) queries. We give several extensions and applications of these results.