Reducing computational complexity with array predicates

Reducing the Computational Complexity of Two-Dimensional LSTMs

Long Short-Term Memory Recurrent Neural Networks (LSTMs) are good at modeling temporal variations in speech recognition tasks, and have become an integral component of many state-of-the-art ASR systems. More recently, LSTMs have been extended to model variations in the speech signal in two dimensions, namely time and frequency [1, 2]. However, one of the problems with two-dimensional LSTMs, suc...

Counting Predicates of Conjunctive Complexity One

Complexity and Hierarchy in Truth Predicates∗

Since seminal work of Tarski (e.g. Tarski, 1935), hierarchies have been much discussed in the literature on truth and paradox. Especially in recent years, this discussion has been decidedly negative. Tarski’s hierarchy of languages is sometimes described as the “orthodox” response to the Liar paradox (e.g. Kripke, 1975), but it is an orthodoxy many authors have gone to great lengths to avoid. F...

The Complexity of Somewhat Approximation Resistant Predicates

A boolean predicate f : {0, 1} → {0, 1} is said to be somewhat approximation resistant if for some constant τ > |f −1(1)| 2k , given a τ -satisfiable instance of the MAX k-CSP(f) problem, it is NP-hard to find an assignment that strictly beats the naive algorithm that outputs a uniformly random assignment. Let τ(f) denote the supremum over all τ for which this holds. It is known that a predicat...

The Complexity of Computing Hard Core Predicates

We prove that a general family of hard core predicates requires circuits of depth (l-o(l)) ~ or super-polynomial size to be realized. This lower bound is essentially tight, l~or constant depth circuits, an exponential lower bound on the size is obtained. Assuming the existence of one-way functions, we explicitly construct a one-way function f(x) such that for any circuit c from a family of circ...