Related papers: A mathematical definition of "simplify"
Reductions combine collections of input values with an associative and often commutative operator to produce collections of results. When the same input value contributes to multiple outputs, there is an opportunity to reuse partial…
In the first part of this paper, we present a unified framework for analyzing the algorithmic complexity of any optimization problem, whether it be continuous or discrete in nature. This helps to formalize notions like "input", "size" and…
The concept of effective complexity of an object as the minimal description length of its regularities has been initiated by Gell-Mann and Lloyd. The regularities are modeled by means of ensembles, that is probability distributions on…
Can AI solve all math? What do we actually mean by doing mathematics? How do we communicate mathematics? What is mathematics beyond problem solving? This essay is my attempt to answer these questions.
A simple permutation is one which maps no proper non-singleton interval onto an interval. We consider the enumeration of simple permutations from several aspects. Our results include a straightforward relationship between the ordinary…
Functions that are piecewise defined are a common sight in mathematics while convexity is a property especially desired in optimization. Suppose now a piecewise-defined function is convex on each of its defining components - when can we…
We explore several concepts for analyzing the intuitive notion of computational irreducibility and we propose a robust formal definition, first in the field of cellular automata and then in the general field of any computable function f…
We provide an implicit characterization of polynomial time computation in terms of ordinary differential equations: we characterize the class $\operatorname{PTIME}$ of languages computable in polynomial time in terms of differential…
The classical approach to solvability of a mathematical problem is to define a method which includes certain rules of operation or algorithms. Then using the defined method, one can show that some problems are solvable or not solvable or…
A measure of complexity based on a probabilistic description of physical systems is proposed. This measure incorporates the main features of the intuitive notion of such a magnitude. It can be applied to many physical situations and to…
Symbolic Mathematical tasks such as integration often require multiple well-defined steps and understanding of sub-tasks to reach a solution. To understand Transformers' abilities in such tasks in a fine-grained manner, we deviate from…
In studying the complexity of iterative processes it is usually assumed that the arithmetic operations of addition, multiplication, and division can be performed in certain constant times. This assumption is invalid if the precision…
A new mathematical notation is proposed for the iteration of functions. It facilitates the application of the iteration of functions in mathematical and logical expressions, definitions of sets, and formulations of algorithms. Illustrations…
An algorithm $M$ is described that solves any well-defined problem $p$ as quickly as the fastest algorithm computing a solution to $p$, save for a factor of 5 and low-order additive terms. $M$ optimally distributes resources between the…
It is generally accepted that all models are wrong -- the difficulty is determining which are useful. Here, a useful model is considered as one that is capable of combining data and expert knowledge, through an inversion or calibration…
There is a problem with the foundations of classical mathematics, and potentially even with the foundations of computer science, that mathematicians have by-and-large ignored. This essay is a call for practicing mathematicians who have been…
The method of exhaustion is generalized to a simple formula that can be used to integrate functions under very general conditions, provided that the integral exists. Both a geometric proof (following the usual procedure for the method of…
Mathematical optimization, although often leading to NP-hard models, is now capable of solving even large-scale instances within reasonable time. However, the primary focus is often placed solely on optimality. This implies that while…
We consider representing of natural numbers by arithmetical expressions using ones, addition, multiplication and parentheses. The (integer) complexity of n -- denoted by ||n|| -- is defined as the number of ones in the shortest expressions…
In this paper we give a definition for the Kolmogorov complexity of a pure quantum state. In classical information theory the algorithmic complexity of a string is a measure of the information needed by a universal machine to reproduce the…