Related papers: Invitation to Real Complexity Theory: Algorithmic …
For nearly any challenging scientific problem evaluation of the likelihood is problematic if not impossible. Approximate Bayesian computation (ABC) allows us to employ the whole Bayesian formalism to problems where we can use simulations…
We describe an approximate rational arithmetic with round-off errors (both absolute and relative) controlled by the user. The rounding procedure is based on the continued fraction expansion of real numbers. Results of computer experiments…
We develop a theory of complexity for numerical computations that takes into account the condition of the input data and allows for roundoff in the computations. We follow the lines of the theory developed by Blum, Shub, and Smale for…
Until now, Computer Scientists have concerned themselves with identifying efficient algorithms for solving the general case of some problem -- that is finding one which performs well when the size of the input tends to infinity. In this…
The theory of computational complexity focuses on functions and, hence, studies programs whose interactive behavior is reduced to a simple question/answer pattern. We propose a broader theory whose ultimate goal is expressing and analyzing…
Previously, the author has developed a framework within which to quantify and compare the resources consumed during computational-especially unconventional computational-processes (adding to the familiar resources of run-time and memory…
The Causal Roadmap outlines a systematic approach to asking and answering questions of cause-and-effect: define the quantity of interest, evaluate needed assumptions, conduct statistical estimation, and carefully interpret results. To…
We introduce a lightweight and accessible approach to computation over the real numbers, with the aim of clarifying both the underlying concepts and their relevance in modern research. The material is intended for a broad audience,…
Primitive recursion, mu-recursion, universal object and universe theories, complexity controlled iteration, code evaluation, soundness, decidability, G\"odel incompleteness theorems, inconsistency provability for set theory, constructive…
Depth is a complexity measure for natural systems of the kind studied in statistical physics and is defined in terms of computational complexity. Depth quantifies the length of the shortest parallel computation required to construct a…
Mathematics has many useful properties for developing of complex software systems. One is that it can exactly describe a physical situation of the object or outcome of an action. Mathematics support abstraction and this is an excellent…
In this article, we consider a simple representation for real numbers and propose top-down procedures to approximate various algebraic and transcendental operations with arbitrary precision. Detailed algorithms and proofs are provided to…
Compositionality is a key property for dealing with complexity, which has been studied from many points of view in diverse fields. Particularly, the composition of individual computations (or programs) has been widely studied almost since…
The traditional foundation of science lies on the cornerstones of theory and experiment. Theory is used to explain experiment, which in turn guides the development of theory. Since the advent of computers and the development of…
We revisit the fundamentals of Circuit Complexity and the nature of efficient computation from a fresh perspective. We present a framework for understanding Circuit Complexity through the lens of Information Theory with analogies to results…
Computational complexity is examined using the principle of increasing entropy. To consider computation as a physical process from an initial instance to the final acceptance is motivated because many natural processes have been recognized…
Based on a new coinductive characterization of continuous functions we extract certified programs for exact real number computation from constructive proofs. The extracted programs construct and combine exact real number algorithms with…
#SMT, or model counting for logical theories, is a well-known hard problem that generalizes such tasks as counting the number of satisfying assignments to a Boolean formula and computing the volume of a polytope. In the realm of…
The past century has seen a steady increase in the need of estimating and predicting complex systems and making (possibly critical) decisions with limited information. Although computers have made possible the numerical evaluation of…
This paper covers two topics: first an introduction to Algorithmic Complexity Theory: how it defines probability, some of its characteristic properties and past successful applications. Second, we apply it to problems in A.I. - where it…