Nader Bshouty, Lynn Burroughs

We study the proper learnability of axis parallel concept classes

in the PAC learning model and in the exact learning model with

membership and equivalence queries. These classes include union of boxes,

DNF, decision trees and multivariate polynomials.

For the {\it constant} dimensional axis parallel concepts $C$

we ...
more >>>

Detlef Sieling

Decision trees are representations of discrete functions with widespread applications in, e.g., complexity theory and data mining and exploration. In these areas it is important to obtain decision trees of small size. The minimization problem for decision trees is known to be NP-hard. In this paper the problem is shown ... more >>>

Avishay Tal

For Boolean functions $f:\{0,1\}^n \to \{0,1\}$ and $g:\{0,1\}^m \to \{0,1\}$, the function composition of $f$ and $g$ denoted by $f\circ g : \{0,1\}^{nm} \to \{0,1\}$ is the value of $f$ on $n$ inputs, each of them is the calculation of $g$ on a distinct set of $m$ Boolean variables. Motivated ... more >>>

Albert Atserias, Neil Thapen

The ordering principle states that every finite linear order has a least element. We show that, in the relativized setting, the surjective weak pigeonhole principle for polynomial time functions does not prove a Herbrandized version of the ordering principle over $\mathrm{T}^1_2$. This answers an open question raised in [Buss, Ko{\l}odziejczyk ... more >>>

Avishay Tal

The sensitivity of a Boolean function $f:\{0,1\}^n \to \{0,1\}$ is the maximal number of neighbors a point in the Boolean hypercube has with different $f$-value. Roughly speaking, the block sensitivity allows to flip a set of bits (called a block) rather than just one bit, in order to change the ... more >>>

Anurag Anshu, Dmytro Gavinsky, Rahul Jain, Srijita Kundu, Troy Lee, Priyanka Mukhopadhyay, Miklos Santha, Swagato Sanyal

Let the randomized query complexity of a relation for error probability $\epsilon$ be denoted by $\R_\epsilon(\cdot)$. We prove that for any relation $f \subseteq \{0,1\}^n \times \mathcal{R}$ and Boolean function $g:\{0,1\}^m \rightarrow \{0,1\}$, $\R_{1/3}(f\circ g^n) = \Omega(\R_{4/9}(f)\cdot\R_{1/2-1/n^4}(g))$, where $f \circ g^n$ is the relation obtained by composing $f$ and $g$. ... more >>>

Yogesh Dahiya, Meena Mahajan

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 several non-computational complexity measures of functions such as certificate complexity. The size measure ... more >>>

Rahul Chugh, Supartha Poddar, Swagato Sanyal

Relations between the decision tree complexity and various other complexity measures of Boolean functions is a thriving topic of research in computational complexity. While decision tree complexity is long known to be polynomially related with many other measures, the optimal exponents of many of these relations are not known. It ... more >>>

Ond?ej Ježil

For a class of finite graphs, we define a limit object relative to some computationally restricted class of functions. The properties of the limit object then reflect how a computationally restricted viewer "sees" a generic instance from the class. The construction uses Krají?ek's forcing with random variables [7]. We prove ... more >>>

Yogesh Dahiya, Vignesh K, Meena Mahajan, Karteek Sreenivasaiah

We show that polynomial-size constant-rank linear decision trees (LDTs) can be converted to polynomial-size depth-2 threshold circuits LTF$\circ$LTF. An intermediate construct is polynomial-size decision lists that query a conjunction of a constant number of linear threshold functions (LTFs); we show that these are equivalent to polynomial-size exact linear decision lists ... more >>>

Vladimir Podolskii, Dmitrii Sluch

Boolean function $F(x,y)$ for $x,y \in \{0,1\}^n$ is an XOR function if $F(x,y) = f(x\oplus y)$ for some function $f$ on $n$ input bits, where $\oplus$ is a bit-wise XOR. XOR functions are relevant in communication complexity, partially for allowing Fourier analytic technique. For total XOR functions it is known ... more >>>

Bruno Loff, Alexey Milovanov

Let $f$ be a Boolean function given as either a truth table or a circuit. How difficult is it to find the decision tree complexity, also known as deterministic query complexity, of $f$ in both cases? We prove that this problem is $NC$-hard and PSPACE-hard, respectively. The second bound is ... more >>>