Let f : {0, 1} n → {0, 1}. Let μ be a product probability measure on {0, 1} n . For ϵ ≥ 0, we define Dϵ ( f ), the ϵ-approximate decision tree complexity of f , to be the minimum depth of a decision tree T with μ ( T ( x ) ≠ f ( x )) ≤ ϵ . For j = 0 or 1 and for δ ≥ 0, we define Cj,δ ( f ), the δ-approximate j -certificate complexity of f , to be the minimum certificate complexity of a set A ⊆ Ω with μ ( AΔf −1 ( j )) ≤ ϵ . Note that if μ ( x ) > 0 for all x then D 0 ( f ) = D ( f ) and Cj ,0 ( f ) = Cj ( f ) are the ordinary decision tree and j -certificate complexities of f , respectively. We extend the well-known result, D ( f ) ≤ C 1 ( f ) C 0 ( f ) [Blum and Impagliazzo 1987; Hartmanis and Hemachandra 1991; Tardos 1989], proving that for all ϵ > 0 there exists a δ > 0 and a constant K = K ( ϵ , δ ) > 0 such that for all n , μ , f , Dϵ ( f ) ≤ K C 1, δ ( f ) C 0, δ ( f ). We also give a partial answer to a related question on query complexity raised by Tardos [1989]. We prove generalizations of these results to general product probability spaces.
Let f : {0, 1} n → {0, 1}. Let μ be a product probability measure on {0, 1} n . For ϵ ≥ 0, we define Dϵ ( f ), the ϵ-approximate decision tree complexity of f , to be the minimum depth of a decision tree T with μ ( T ( x ) ≠ f ( x )) ≤ ϵ . For j = 0 or 1 and for δ ≥ 0, we define Cj,δ ( f ), the δ-approximate j -certificate complexity of f , to be the minimum certificate complexity of a set A ⊆ Ω with μ ( AΔf −1 ( j )) ≤ ϵ . Note that if μ ( x ) > 0 for all x then D 0 ( f ) = D ( f ) and Cj ,0 ( f ) = Cj ( f ) are the ordinary decision tree and j -certificate complexities of f , respectively. We extend the well-known result, D ( f ) ≤ C 1 ( f ) C 0 ( f ) [Blum and Impagliazzo 1987; Hartmanis and Hemachandra 1991; Tardos 1989], proving that for all ϵ > 0 there exists a δ > 0 and a constant K = K ( ϵ , δ ) > 0 such that for all n , μ , f , Dϵ ( f ) ≤ K C 1, δ ( f ) C 0, δ ( f ). We also give a partial answer to a related question on query complexity raised by Tardos [1989]. We prove generalizations of these results to general product probability spaces.
The "direct product problem" is a fundamental question in complexity theory which seeks to understand how the difficulty of computing a function on each of k independent inputs scales with k. We prove the following direct product theorem (DPT) for query complexity: if every T-query algorithm has success probability at most 1 - ε in computing the Boolean function f on input distribution μ, then for α ≤ 1, every αεTk-query algorithm has success probability at most (2αε(1 - ε))kin computing the fc-fold direct product f⊗kcorrectly on k independent inputs from μ. In light of examples due to Shaltiel, this statement gives an essentially optimal tradeoff between the query bound and the error probability. Using this DPT, we show that for an absolute constant α >; 0, the worst-case success probability of any αR2(f)k-query randomized algorithm for f⊗kfalls exponentially with k. The best previous statement of this type, due to Klauck, Spalek, and de Wolf, required a query bound of O(bs(f)k). Our proof technique involves defining and analyzing a collec tion of martingales associated with an algorithm attempting to solve f⊗k. Our method is quite general and yields a new XOR lemma and threshold DPT for the query model, as well as DPTs for the query complexity of learning tasks, search problems, and tasks involving interaction with dynamic entities. We also give a version of our DPT in which decision tree size is the resource of interest.
We define and study a measure of information complexity for randomized decision trees. We prove three main results about this complexity measure: Information equals amortized size complexity. We show that the information complexity of randomized decision tree is equal to the logarithm of the amortized worst-case randomized tree size complexity of computing a function f. That is, when computing f on n inputs, the logarithm of the randomized tree size is exactly equal to the amount of information needed to compute the function. Information allows for tree size compression. We show that even when computing f on a single input, the information complexity can be used to compress the size of a tree, if we allow a small loss in success probability. With the recent characterization of Chattopadhyay, Dahiya, Mande, Radhakrishnan, and Sanyal (2023), this result shows that the depth of AND-OR trees can also be compressed in terms of information complexity. Direct Product Theorems. We show that the success-conditioned variant of information complexity satisfies a perfect direct product theorem. This result gives an information complexity analogue of the direct product theorem for success-conditioned randomized query complexity by Ben-David and Blais (2025).
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
Joint computation is the ubiquitous scenario in which a computer is presented with not one, but many computational tasks to perform. A fundamental question arises: when can we cleverly combine computations, to perform them with greater efficiency or reliability than by tackling them separately? This thesis investigates the power and, especially, the limits of efficient joint computation, in several computational models: query algorithms, circuits, and Turing machines. We significantly improve and extend past results on limits to efficient joint computation for multiple independent tasks; identify barriers to progress towards better circuit lower bounds for multiple-output operators; and begin an original line of inquiry into the complexity of joint computation. In more detail, we make contributions in the following areas:
Improved product theorems for randomized query complexity: The direct product problem seeks to understand how the difficulty of computing a function on each of k independent inputs scales with k. We prove the following product theorem (DPT) for query complexity: if every T-query algorithm has success probability at most 1 – e in computing the Boolean function f on input distribution μ, then for α ≤ 1, every αeTk-query algorithm has success probability at most (2αe(1 – e))k in computing the k-fold product f ⊗k correctly on k independent inputs from μ. In light of examples due to Shaltiel, this statement gives an essentially optimal tradeoff between the query bound and the error probability. Using this DPT, we show that for an absolute constant α > 0, the worst-case success probability of any αR 2(f)k-query randomized algorithm for f⊗k falls exponentially with k. The best previous statement of this type, due to Klauck, Spalek, and de Wolf, required a query bound of O( bs(f)k).
Our proof technique involves defining and analyzing a collection of martingales associated with an algorithm attempting to solve f ⊗k. Our method is quite general and yields a new XOR lemma and threshold DPT for the query model, as well as DPTs for the query complexity of learning tasks, search problems, and tasks involving interaction with dynamic entities. We also give a version of our DPT in which decision tree size is the resource of interest.
Joint complexity in the Decision Tree Model: We study the diversity of possible behaviors of the joint computational complexity of a collection f1, …, fk of Boolean functions over a shared input. We focus on the deterministic decision tree model, with depth as the complexity measure; in this model, we prove a result to the effect that the obvious constraints on joint computational complexity are essentially the only ones.
The proof uses an intriguing new type of cryptographic data structure called a mystery bin, which we construct using a polynomial separation between deterministic and unambiguous query complexity shown by Savický. We also pose a conjecture in the communication model which, if proved, would extend our result to that model. (Copies available exclusively from MIT Libraries, Rm. 14-0551, Cambridge, MA 02139-4307. Ph. 617-253-5668; Fax 617-253-1690.) (Abstract shortened by UMI.)
Certificate complexity (C(f)) is a fundamental measure of complexity of Boolean functions f which counts the number of bits of an input that need to be known in order for the value of the function to be determined. A certificate can be viewed as a partial assignment, or a boolean subcube where the function is constant. Certificate complexity is well understood for deterministic query (or decision tree) complexity (D) and other query models such as bounded-error randomized and quantum complexity (R,Q), but not as well for quantum zero-error (Q0) and exact query complexity (QE), where there is no agreed-upon certificate 'object' (even for Q). Instead, we study an operational notion of certification and apply it to various query-based models, with a focus on zero-error and exact quantum query complexity, but also on polynomial degree measures. We give new characterizations of C,RC (randomized certificate complexity) and QC (quantum certificate complexity), in terms of various measures such as classical and quantum sabotage complexity, unambiguous certificate complexity, and variants of polynomial degree. Certification complexity also gives rise to new lower bounds on QE and Q0, the quantum analogues of D and R0, complexity measures for which few lower bound techniques are known which are not already lower bounds for two-sided error quantum query complexity. We exhibit a total Boolean function for which our certification complexity measure gives a tight lower bound for Q0, but rational degree and Q are asymptotically smaller.
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.