Related papers: On Function Description
In resolving instances of a computational problem, if multiple instances of interest share a feature in common, it may be fruitful to compile this feature into a format that allows for more efficient resolution, even if the compilation is…
Kolmogorov complexity of a finite binary word reflects both algorithmic structure and the empirical distribution of symbols appearing in the word. Words with symbol frequencies far from one half have smaller combinatorial richness and…
What is the best way to define algorithmic fairness? While many definitions of fairness have been proposed in the computer science literature, there is no clear agreement over a particular definition. In this work, we investigate ordinary…
In this paper, we study classes of structures and individual structures for which programs implementing functions defined everywhere are equivalent to finite tree-programs. The programs under consideration may have cycles and at most…
Algorithmic fractal dimensions quantify the algorithmic information density of individual points and may be defined in terms of Kolmogorov complexity. This work uses these dimensions to bound the classical Hausdorff and packing dimensions…
It is well understood that classification algorithms, for example, for deciding on loan applications, cannot be evaluated for fairness without taking context into account. We examine what can be learned from a fairness oracle equipped with…
In addition to the equations, physicists use the following additional difficult-to-formalize property: that the initial conditions and the value of the parameters must not be abnormal. We will describe a natural formalization of this…
Given only information in the form of similarity triplets "Object A is more similar to object B than to object C" about a data set, we propose two ways of defining a kernel function on the data set. While previous approaches construct a…
Recently Quantum Computation has generated a lot of interest due to the discovery of a quantum algorithm which can factor large numbers in polynomial time. The usefulness of a quantum com puter is limited by the effect of errors. Simulation…
Many theorems about Kolmogorov complexity rely on existence of combinatorial objects with specific properties. Usually the probabilistic method gives such objects with better parameters than explicit constructions do. But the probabilistic…
The question of obtaining well-defined criteria for multiple criteria decision making problems is well-known. One of the approaches dealing with this question is the concept of nonessential objective function. A certain objective function…
One approach to confronting computational hardness is to try to understand the contribution of various parameters to the running time of algorithms and the complexity of computational tasks. Almost no computational tasks in real life are…
This article first presents two examples of algorithms that extracts information on scheme out of its defining equations. We also give a review on the notion of Castelnuovo-Mumford regularity, its main properties (in particular its relation…
Using appropriate notation systems for proofs, cut-reduction can often be rendered feasible on these notations, and explicit bounds can be given. Developing a suitable notation system for Bounded Arithmetic, and applying these bounds, all…
The aim of this paper is to undertake an experimental investigation of the trade-offs between program-size and time computational complexity. The investigation includes an exhaustive exploration and systematic study of the functions…
We introduce an asymmetric distance in the space of learning tasks, and a framework to compute their complexity. These concepts are foundational for the practice of transfer learning, whereby a parametric model is pre-trained for a task,…
Elementary function calls are a common feature in numerical programs. While their implementions in library functions are highly optimized, their computation is nonetheless very expensive compared to plain arithmetic. Full accuracy is,…
One of the most fundamental problems in science is to define {\it quantitatively} the complexity of organized matters, i.e., {\it organized complexity}. Although many measures have been proposed toward this aim in previous decades, there is…
In this paper, we define a Mathematical model of program structure. Mathematical model of program structure defined here provides unified mathematical treatment of program structure, which reveals that a program is a large and finite set of…
One of the most ubiquitous problems in optimization is that of finding all the elements of a finite set at which a function $f$ attains its minimum (or maximum). When the codomain of $f$ is equipped with a total order, it is easy to…