Stephen Wolfram Research Articles

Machine Learning with Quantum Matter: An Example Using Lead Zirconate Titanate

Stephen Wolfram (2002) proposed the concept of computational equivalence, which implies that almost any dynamical system can be considered as a computation, including programmable matter and nonlinear materials such as, so called, quantum matter. Memristors are often used in building and evaluating hardware neural networks. Ukil (2011) demonstrated a theoretical relationship between piezoelectrical materials and memristors. We review that work as a necessary background prior to our work on exploring a piezoelectric material for neural network computation. Our method consisted of using a cubic block of unpoled lead zirconate titanate (PZT) ceramic, to which we have attached wires for programming the PZT as a programmable substrate. We then, by means of pulse trains, constructed on-the-fly internal patterns of regions of aligned polarization and unaligned, or disordered regions. These dynamic patterns come about through constructive and destructive interference and may be exploited as a type of reservoir network. Using MNIST data we demonstrate a learning machine.

Physicists Criticize Stephen Wolfram's 'Theory of Everything'

Physicists Criticize Stephen Wolfram's 'Theory of Everything'

Some Relativistic and Gravitational Properties of the Wolfram Model

The Wolfram Model, which is a slight generalization of the model first introduced by Stephen Wolfram in A New Kind of Science (NKS), is a discrete spacetime formalism in which space is represented by a hypergraph whose dynamics are determined by abstract replacement operations on set systems, and in which the conformal structure of spacetime is represented by a causal graph. The purpose of this article is to present rigorous mathematical derivations of many key properties of such models in the continuum limit, as first discussed in NKS, including the fact that large classes of them obey discrete forms of both special and general relativity. First, we prove that causal invariance (namely, the requirement that all causal graphs be isomorphic, irrespective of the choice of hypergraph updating order) is equivalent to a discrete version of general covariance, with changes to the updating order corresponding to discrete gauge transformations. This fact then allows one to deduce a discrete analog of Lorentz covariance, and the resultant physical consequences of discrete Lorentz transformations. We also introduce discrete notions of Riemann and Ricci curvature for hypergraphs, and prove that the correction factor for the volume of a discrete spacetime cone in a causal graph corresponding to curved spacetime of fixed dimensionality is proportional to a timelike projection of the discrete spacetime Ricci tensor, subsequently using this fact (along with the assumption that the updating rules preserve the dimensionality of the causal graph in limiting cases) to prove that the most general set of constraints on the discrete spacetime Ricci tensor corresponds to a discrete form of the Einstein field equations.

The 2017 Richard D. Jenks Memorial Award

The 2017 Richard D. Jenks Memorial Prize for Excellence in Software Engineering Applied to Computer Algebra has been awarded to Stephen Wolfram for Wolfram—Alpha and Mathematica.

Commentary: Physicists without borders

Commentary: Physicists without borders

Stephen Wolfram, Idea Makers: Personal Perspectives on the Lives & Ideas of Some Notable People

Stephen Wolfram, Idea Makers: Personal Perspectives on the Lives & Ideas of Some Notable People

Creating Order From Chaos

> It’s always seemed like a big mystery how nature, seemingly so effortlessly, manages to produce so much that seems to us so complex. Well, I think we found its secret. It’s just sampling what’s out there in the computational universe. > > –Stephen Wolfram, 2010 Pathogenesis of nearly all arrhythmia relies on a fundamental premise—that there exists within the cellular milieu regions of unexpected or unwanted heterogeneous conduction that may allow for initiation and propagation of wave fronts along a nonstandard pattern. The simplest paradigm for this may be atrioventricular reentrant tachycardias wherein 2 paths exist, both capable of propagating signals in either the antegrade or retrograde direction, but with fundamentally different conductive properties that ultimately allow for development of reentry with an appropriately timed stimulus. This appropriately timed stimulus, or trigger, has for the past 3 decades been the focus of electrophysiologists in the treatment of more complex arrhythmias that exist outside the ability of our modern day understanding and technology to map and ablate. Where a single circuit exists, mapping and ablation is straightforward, and we must prove it exists and that it participates in the tachycardia, identify its location, and then determine the best means by which to ablate it. However, in the case of atrial fibrillation, understanding, identifying, and determining an ideal ablative solution for the substrate is eminently more difficult. This is because the propagation and continuation of atrial fibrillation is fundamentally dependent on the continuous activity of many sites simultaneously. Thus, we have often referred to atrial fibrillation almost as chaos, throwing up arms at the seemingly inexplicable and indescribable complexity and focused instead on the elimination of triggers, whether by pulmonary vein isolation or targeting other sites, rather than the myocardial substrate responsible for arrhythmia maintenance as we would with any other simpler arrhythmia, such …

Wetware: A Computer in Every Living Cell. Dennis Bray. (2009, New Haven, London: Yale University Press.) $28 (hardback), 267 pages.

Wetware: A Computer in Every Living Cell. Dennis Bray. (2009, New Haven, London: Yale University Press.) $28 (hardback), 267 pages. Make no mistake, this book falls into the popular science category: It is accessible to the general reader without a scientific background, but it also makes a scientifically interesting argument. At one level, Wetware by Dennis Bray is an easy introduction to systems biology—the relatively new science born from the union of molecular biology with information science. At another, it proposes a model of the cell as a computer, not of the von Neumann kind, but a rather elaborate neural network type of computation system. This idea is expanded to include a few aspects of multicellular information processing, especially with living neurons, but perhaps these are just to support the main argument, as they receive scant attention. The main focus for Bray is the molecular signaling network, especially illustrated by the generation and control of directed movement in bacterial cells. Using no more than first-year university-level science, we are introduced to molecular sensors, messengers, switches, transistors, amplifiers, logic gates, and transducers: a full toolbox of biochemical information-processing components. These labels are just analogies, of course, but identifying functional equivalents to computer hardware components provides a model of molecular networks as “wired-up” wetware. It is the constant adjustment of concentration and state of thousands of computationally functional proteins in the cell that amounts to its living. In the sense that this chemistry of life is information processing, we could conclude that living is computing, thoughBray leaves that to the reader to conclude. Those unfamiliar with cellular biochemistry will no doubt be fascinated to learn howproteins can change shape and therefore subtly alter their chemical behavior in response to specific signals, enabling semantic communications betweenmolecules as the state of one influences that of others in turn. Vast and bewilderingly complex networks of molecular interactions operate and control the inner workings and the behavior of the cell, appearing to give it a life of its own. If you ever ponder on the deep meaning of life, it could be that this will give a significant mechanistic insight. The probabilistic nature of chemical interactions, molecule by molecule, scales up to give even bacteria individual identities, as network connections subtly and randomly differ from one cell to another. This is a recurring theme and one obviously close to Brayʼs heart: Real biology is considerably (almost unimaginably) more complicated than any model that could be useful. Indeed he gets close to accusing some modelers of self-deception (he calls it “reality distortion”), singling Stephen Wolfram out for near-derision; he is far kinder to Stuart Kauffman. Bray is a hugely respected molecular biologist, so what he says about molecular mechanisms is solid and well written, and he seems to judge the tradeoff between technical detail and engagement well. This is good news for the information scientist with little background in biology, because this book really does open the mind to the amazing idea that living is computing. On the other hand, Brayʼs handling of information science seemed more clumsy; for example, the explanation of artificial neural networks was all too familiar, and the description of mechanical tortoises was quaint and amusing, rather than illuminating. Biologists may find some of the explanations in their own field similarly plodding, but plausibly marvel at the tortoises—to be fair, this is the unavoidable problem of multidisciplinary approaches. Wetware introduces the reader to genetic, biochemical, and neural networks in real living organisms, explaining the foundations of their inner workings. It is certainly not a textbook, not just because of the frequent expressions of the authorʼs personal opinions, but more because he is so selective in subject material. Bray uses the book to express a big idea—that of living as computing—so it is more a well-illustrated argument than a tour of cellular biochemistry or systems biology. I think it is all the better for that.

Earthquake prediction, Wolfram Alpha, neuroimaging

PurposeThe purpose of this paper is to review proposals for earthquake prediction and study using accelerometers incorporated in laptop computers. Also reviewed is a lecture by Stephen Wolfram, creator of the Mathematica software package, introducing a new venture called Wolfram Alpha that combines computation with access to real‐world data. Developments in neural imaging techniques and some findings are discussed, including a remarkable manifestation of neural plasticity.Design/methodology/approachThe aim is to review developments on the internet, especially those of general cybernetic interest.FindingsUse of accelerometers, incorporated in laptop computers for a different purpose, appears to offer valuable opportunities for earthquake prediction and study. Exactly what will emerge from the Wolfram‐Alpha initiative remains to be seen but it holds great promise. New neuroimaging techniques have shown their worth in producing new findings, including one that reveals interesting neural plasticity.Practical implicationsEarthquake prediction and study, especially by a means that can be implemented quickly and cheaply, is of obvious value in quake‐prone areas such as California. Wolfram Alpha may come to be as familiar a tool as search engines are today. A primary aim of cybernetics is unravelling the working of the central nervous system, and both ethical and practical considerations favour non‐invasive means of investigation such as new neuroimaging techniques that have already proved their worth.Originality/valueIt is hoped this is a valuable periodic review.

Wolfphram|Alpha: A Brief Introduction

On May 18, 2009, British computer scientist Stephen Wolfram officially launched a new search product called Wolfram|Alpha (WA). This launch was preceded by months of speculation and hype online about exactly what WA would be and how it would compare to Google and other search engines. This article will explore the basic features of WA, show some example queries and results, and discuss the usefulness and limitations of this new tool.

Stephen Wolfram reveals his radical new web search engine

Stephen Wolfram reveals his radical new web search engine

A Conversation with Stephen Wolfram

A Conversation with Stephen Wolfram

A Conversation with Stephen Wolfram

A Conversation with Stephen Wolfram

Complexity and Universality of Iterated Finite Automata

The iterated finite automaton (IFA) was invented by Stephen Wolfram for studying the conventional finite state automaton (FSA) by means of A New Kind of Science methodology. An IFA is a composition of an FSA and a tape with limited cells. The complexity of behaviors generated by various FSAs operating on different tapes can be visualized by two-dimensional patterns. Through enumerating all possible two-state and three-color IFAs, this paper shows that there are a variety of complex behaviors in these simple computational systems. These patterns can be divided into eight classes such as regular patterns, noisy structures, complex behaviors, and so forth. Also they show the similarity between IFAs and elementary cellular automata. Furthermore, any cellular automaton can be emulated by an IFA and vice versa. That means IFAs support universal computation.

Beyond continuous mathematics and traditional scientific analysis: Understanding and mining Wolfram's A New Kind of Science

In A New Kind of Science, Stephen Wolfram recommends abandoning traditional scientific analysis and the continuous mathematical description that it affords in favor of the study of simple rules. He focuses on a machine known as a cellular automaton as the prototype generator of complex phenomena such as those we see in nature. The simplest cellular automaton consists of a row of cells, each existing in one of two states. The states of the cells are updated from moment to moment by simple rules. Wolfram shows that these machines and their many variations can generate a host of outcomes ranging from very simple to extremely complex. He argues that among these outcomes representations of all the phenomena in the universe will be found, including presumably the behavior of organisms. The output of cellular automata can be mapped to behavior by considering, for example, one of the states of a cell to represent the emission of a behavior. For some cellular automaton rules, these mappings generate cumulative records and inter-response time distributions that are similar to those produced by live organisms. In addition, at least one cellular automaton generates the Herrnstein hyperbola as an emergent outcome. These results suggest that Wolfram's program and its mainstream version, which is known as complexity theory, is worth pursuing as a possible means of understanding and accounting for the behavior of organisms.

Discrete Sequence Rule Models as a Social Science Methodology: An Exploratory Analysis of Foreign Policy Rule Enactment within Palestinian–Israeli Event Data

Existing formal models of political behavior have followed the lead of the natural sciences and generally focused on methods that use continuous-variable mathematics. In 2002, Stephen Wolfram produced an extended critique of that approach in the natural sciences, and suggested that a great deal of natural behavior can be accounted for using rules that produce discrete patterns. This paper reports some initial findings designed to apply this pattern-based method to political event data. We believe that discrete sequence rule (DSR) models can provide a new social science methodology that is capable of preserving the agential basis of social interaction, tracking multiple agents as they enact rules through behavior directed at one another, and capturing the evolution of such interaction over time. The core of this project is a new, publicly accessible Web-based tool designed for the visualization and analysis of event data patterns (http://www.nkss.org). Using event data on the Israel–Palestine conflict generated by the TABARI automated coding program of the Kansas Event Data System (KEDS) for the period 1979–2004, we perform an initial exploration of this methodology. Specifically, we identify patterned behavior for which specific rule use can be imputed, and then examine several agent-based rules, plus four “meta-rules,” to parse Israeli–Palestinian interaction over time. Face validity of the analysis is apparent, and we also find the qualitative historical record can be augmented through observation of rule enactment in the event stream. Several descriptive empirical applications are demonstrated, including moving totals and increasingly complex sequences of rule enactment that go beyond the simple variations on tit-for-tat responses. While this paper represents an exploratory analysis of the method, the results are promising enough to warrant further investigation beyond its use in thick description as demonstrated here, to ultimately include hypothesis generation and falsification.

How do simple programs behave

AbstractWhereas once the world view of the ancient world was shifted by Pythagorean mathematics, scientific knowledge and human perception has been contested in the 21st century by developments in computation. In his ground‐breaking book Stephen Wolfram, the British physicist and creator of the Mathematica program, asserted that the complexity of the universe could be clearly understood in terms of simple programs. Here in an exclusive extract, Wolfram describes how one of the most straightforward programs, cellular automata, despite adhering to simple rules, yields some surprisingly complex results. Copyright © 2006 John Wiley & Sons, Ltd.

Cellular automata and ecology

A fascinating and potentially very fertile new approach for solving various scientific problems in mathematics, physics, cosmology, chemistry, biology, psychology and economics, to mention only the most important ones, is that pioneered by Stephan Wolfram. In the following, I give a brief outline of his “NKS” (new kind of science”) with emphasis on cellular automata; some of his conclusions concerning evolutionary biology; a comparison with other approaches; and an application of the method to some problems of ecology.The principles of NKS (“new kind of science”) developed by Stephen Wolfram are discussed, with emphasis on cellular automata. Wolfram's conclusions concerning optimisation and the evolution of complexity in biological systems are outlined and compared with those from some other recent approaches. NKS is applied to some central problems of ecology, i.e. the possibility of establishing general ecological laws, the existence of vacant niches, the importance of interspecific competition and the causes of latitudinal gradients in species diversity.

Universality in Elementary Cellular Automata

The purpose of this paper is to prove a conjecture made by Stephen Wolfram in 1985, that an elementary one dimensional cellular automaton known as "Rule 110" is capable of universal computation. I developed this proof of his conjecture while assisting Stephen Wolfram on research for A New Kind of Science [1].

Book Review: A NEW KIND OF SCIENCE BY STEPHEN WOLFRAM

This is a very unusual book. In 846 pages of text and 349 pages of notes, author Stephen Wolfram claims no less than that we have all been doing science with an outmoded set of analytical tools and that he has invented a new approach that can lead to rapid progress in fields ranging from mathematics to biology. This new way of doing science is, in short, to run computer simulations of complex systems. Wolfram’s claim is based on two major subclaims: First, that essentially all phenomena of interest to scientists can be described and studied as computations; and second, in what Wolfram calls the Principle of Computational Equivalence, that above a certain level of simplicity, essentially all computational systems embody a similar level of complexity. In an idea that seems close in spirit to Turing’s treatment of the halting problem, he further suggests that the only way to predict the behavior of such systems is to run them on a computer and see what happens. Embedded in this innocent-sounding proposal is a very deep criticism of the way theoretical science, particularly physics and mathematics, has developed up to the present day, namely, that scientists have always selected which theories to pursue based on a kind of aesthetic principle which favors those theories that most easily yield predictions via standard mathematical analysis. The reason for this idea of beauty in theories is that before the advent of modern computers it was not practical to develop predictions from theories by working out the consequences of simple rules at very large numbers of points in space and time; rather, short cuts, for example, the analytical solution of equations that might be more complex, had to be used to reduce the amount of work involved in projecting the outcome of a theory to distant times and places. Wolfram’s main thesis, and, indeed, I think the main contribution of this book, is to point out that we now have the computational means to look at theories based on simpler rules that may not admit of shortcut routes to their predictions.