Related papers: On the parity complexity measures of Boolean funct…
The tree-width of a multivariate polynomial is the tree-width of the hypergraph with hyperedges corresponding to its terms. Multivariate polynomials of bounded tree-width have been studied by Makowsky and Meer as a new sparsity condition…
We show that the deterministic decision tree complexity of a (partial) function or relation $f$ lifts to the deterministic parity decision tree (PDT) size complexity of the composed function/relation $f \circ g$ as long as the gadget $g$…
Sensitivity conjecture is a longstanding and fundamental open problem in the area of complexity measures of Boolean functions and decision tree complexity. The conjecture postulates that the maximum sensitivity of a Boolean function is…
Computational models typically assume that operations are applied in a fixed sequential order. In recent years several works have looked at relaxing this assumption, considering computations without any fixed causal structure and showing…
The goal of the paper is to relate complexity measures associated with the evaluation of Boolean functions (certificate complexity, decision tree complexity) and learning dimensions used to characterize exact learning (teaching dimension,…
In this paper, we consider classes of decision tables with many-valued decisions closed under operations of removal of columns, changing of decisions, permutation of columns, and duplication of columns. We study relationships among three…
We study the relationship between various one-way communication complexity measures of a composed function with the analogous decision tree complexity of the outer function. We consider two gadgets: the AND function on 2 inputs, and the…
We introduce a restriction of the classical 2-party deterministic communication protocol where Alice and Bob are restricted to using only comparison functions. We show that the complexity of a function in the model is, up to a constant…
In the decision tree computation model for Boolean functions, the depth corresponds to query complexity, and size corresponds to storage space. The depth measure is the most well-studied one, and is known to be polynomially related to…
Sensitivity, block sensitivity and certificate complexity are basic complexity measures of Boolean functions. The famous sensitivity conjecture claims that sensitivity is polynomially related to block sensitivity. However, it has been…
We compute the nonlinearity of Boolean functions with Groebner basis techniques, providing two algorithms: one over the binary field and the other over the rationals. We also estimate their complexity. Then we show how to improve our…
A divide-and-conquer cryptanalysis can often be mounted against some keystream generators composed of several (nonlinear) independent devices combined by a Boolean function. In particular, any parity-check relation derived from the periods…
The tree share structure proposed by Dockins et al. is an elegant model for tracking disjoint ownership in concurrent separation logic, but decision procedures for tree shares are hard to implement due to a lack of a systematic theoretical…
In a recent result, Knop, Lovett, McGuire and Yuan (STOC 2021) proved the log-rank conjecture for communication complexity, up to log n factor, for any Boolean function composed with AND function as the inner gadget. One of the main tools…
The decision tree recursively partitions the input space into regions and derives axis-aligned decision boundaries from data. Despite its simplicity and interpretability, decision trees lack parameterized representation, which makes it…
An open problem in communication complexity proposed by several authors is to prove that for every Boolean function f, the task of computing f(x AND y) has polynomially related classical and quantum bounded-error complexities. We solve a…
We completely characterise the complexity in the decision tree model of computing composite relations of the form h = g(f^1,...,f^n), where each relation f^i is boolean-valued. Immediate corollaries include a direct sum theorem for decision…
In this paper we study the separation between the deterministic (classical) query complexity ($D$) and the exact quantum query complexity ($Q_E$) of several Boolean function classes using the parity decision tree method. We first define the…
We define decision trees for monotone functions on a simplicial complex. We define homology decidability of monotone functions, and show that various monotone functions related to semimatroids are homology decidable. Homology decidability…
We generalize and extend the ideas in a recent paper of Chiarelli, Hatami and Saks to prove new bounds on the number of relevant variables for boolean functions in terms of a variety of complexity measures. Our approach unifies and refines…