Slashdot Mirror

Fishbat writes "In a cutting message to the Foundations of Mathematics mailing list, Stanford's Vaughan Pratt has pointed out an elementary mistake in the recently announced proof that Wolfram's (2,3) machine is universal." Update: 10/30 04:18 GMT by KD : Ed Pegg Jr. from Wolfram Research points to this response to Dr. Pratt's note, which has been submitted to the FoM mailing list but has not yet appeared there due to moderation.

Rik702 writes "Wolframscience.com have announced that an undergraduate from Birmingham, UK has proved Wolfram's 2,3 Turing Machine is universal." You can read a pdf of the proof as well as some related coverage.

Rik702 writes "Wolframscience.com have announced that an undergraduate from Birmingham, UK has proved Wolfram's 2,3 Turing Machine is universal." You can read a pdf of the proof as well as some related coverage.

An anonymous reader writes "Stephen Wolfram, creator of Mathematica and author of A New Kind of Science, is offering a prize of $25K to anyone who can prove or disprove his conjecture that a particular 2-state, 3-color Turing machine is universal. If true, it would be the simplest universal TM, and possibly the simplest universal computational system. The announcement comes on the 5-year anniversary of the publication of NKS, where among other things Wolfram introduced the current reigning TM champion — 'rule 110,' with 2 states and 5 colors."

An anonymous reader writes "Stephen Wolfram, creator of Mathematica and author of A New Kind of Science, is offering a prize of $25K to anyone who can prove or disprove his conjecture that a particular 2-state, 3-color Turing machine is universal. If true, it would be the simplest universal TM, and possibly the simplest universal computational system. The announcement comes on the 5-year anniversary of the publication of NKS, where among other things Wolfram introduced the current reigning TM champion — 'rule 110,' with 2 states and 5 colors."

waynegoode writes "Stephen Wolfram's Wolfram Research has produced an new application: WolframTones-- 'An Experiment in A New Kind of Music'. It combines the principles in Stephen's book, 'A New Kind of Science' and Mathematica to 'instantly create unique music' in many different styles. They describe it as pretty neat as well as being scientifically interesting, and useful. After listening to some compositions and creating a few random ones myself, I must agree that it is. And anyone who has listen to the radio the last few years could certainly use some unique music."

gotscheme writes "When Stephen Wolfram of Mathematica fame self-published A New Kind of Science in 2002, he raised the suspicions of many in scientific communities that he was taking advantage of a lot of other people's work for his sole financial gain and that he was going against the open nature of academia by using restrictive copyright. Yesterday, Wolfram and company released the entire contents of NKS for free on the Web (short registration required). Perhaps Wolfram is giving back to the scientific community; perhaps it is simply clever marketing for a framework that is beginning to gain momentum. For any matter, the entire encyclopedic volume is online, and this appears to be a positive step for scientific writing."

Stephen Wolfram, subject of much discussion here, once known solely as the creator of Mathematica, now also known as the author of A New Kind of Science (/. review here), gave a lecture at Boston University this past spring on that book's subject matter. The audio of the lecture was broadcast this evening on the program World of Ideas on WBUR-FM out of Boston. If you don't live in the Boston area, if you missed the program, or if like me you were listening in your car while driving and found that two activites incompatible, the hour-long recording is also available for download in RealMedia format.

An Anonymous Coward writes "The New York Times has had a couple of articles about Stephen Wolfram in the last couple of weeks. Is he self-aggrandizing or brilliant? Or both? And is God a software engineer?" I thought our reader-contributed review of ANKOS was quite good.

cybrpnk2 writes: "The story is one of epic proportions: Boy genius gets PhD from Cal Tech at age 20, is the youngest recipient ever of the MacArthur Foundation Genius Grant, writes the Mathematica simulation software used by millions of people, makes millions of dollars in the process, becomes enticed by the seductive lure of the Game of Life, and goes into a decade of seclusion to discover the secrets of the universe. You can catch up on the resulting speculation and hype here. The years of anticipation and publication delays came to an end Tuesday, May 14, 2002 with Stephan Wolfram's release of his opus, A New Kind of Science." Read on for cybrpnk2's review of Wolfram's much-heralded work. A New Kind Of Science author Stephen Wolfram pages 1197 (plus 62 page index) publisher Wolfram Media, Inc. rating 10 reviewer cybrpnk2 ISBN 1-57955-008-8 summary A long awaited treatise that cellular automations, not mathematics, holds the key to understanding reality

First things first - have I read this book? Hell, no, and if anybody else says THEY have in the next year, they're lying thru their teeth. This book is so dense that if Wolfram had added a single additional page, the whole thing would have imploded into a black hole. That's got to be the only reason he quit writing and finally went to press.

I've been waiting for years for ANKOS to come out. I ordered my copy Tuesday when it was released, got it on Thursday and I've been skimming it like mad since. To give you some idea of how engrossing this book is, I was reading it Friday morning at 4 AM in the bathroom of a Motel 6, curled up in a bedspread on the tile floor to keep from disturbing my wife and stepdaughter during a trip to my stepson's graduation. I've got four college degrees, one in math and two from MIT, and bottom line - this sucker's gonna take a while to digest. However, it's theoretically straightforward enough that anybody with a high enough level of obsession and a few years to stay glued to it can follow it in its entirety. In ANKOS, Wolfram certainly comes across as arrogantly cocky but in the final analysis is he a crank or a revolutionary genius? Who knows, but it's going to be a new nerd pastime for the next decade to argue that point.

ANKOS is 1250+ pages divided into 850 pages of breezy exposition followed by 350 pages of fine-print notes. The exposition is composed of 12 chapters and the notes have about a paragraph per page of topic- and name-dropping technobabble to let you know where to go next for more details on whichever of Wolfram's tangents strike your fancy. Topping the whole thing off is a 60+ page index with thousands of entries in even smaller typeface than the notes.

Despite its length, ANKOS is not a rigorous mathematical proof of anything as much as it is a superficial survey of a vast new intellectual landscape. And what a landscape Wolfram has laid before us. It's all about cellular automations, which have traditionally been relegated to the realm of mathematical recreations. Start with a black square in the center grid square (cell) on the top line of a sheet of graph paper. Think up a few rules about whether a square gets colored black or white on the next line down depending on the colors of its neighbors. Apply these rules to the squares on the next line of the sheet of graph paper. Repeat. Watch what happens. Sounds simple. It isn't.

The first short chapter outlines Wolfram's central thesis: That three hundred years of mathematics based on the equals sign have failed to provide true insight into various complex systems in nature, and that algorithms based on the DO loop can succeed in this endeavor where mathematics has failed. The reason, claims Wolfram, is that deceptively simple algorithms can produce heretofore undreamed of levels of complexity. He claims that while frontier intellectual efforts such as chaos theory, fractals, AI, cybernetics and so forth have hinted at this concept for years, his decade of isolation studying cellular automata has taken the idea of simple algorithms or rules embodying universal complexity to the level of a new paradigm.

The second chapter outlines what Wolfram calls his crucial experiment: the systematic analysis of the 256 simplest rule sets for the most basic cellular automatons. He discovers this "universe" of rules is sufficient to produce his four so-called "classes" of complex systems: order, self-similar nested patterns, structures and most importantly, true randomness. The first two lead to somewhat familiar checkerboard-type patterns and leaf-type fractals; the last two, unforeseen unique shapes and unpredictable sequences. Wolfram stresses that the ability of simple iterative algorithms to produce complex and unique non-fractal shapes as well as truly random sequences of output is in fact a revolutionary new discovery with subtle and profound implications.

The third chapter expands his initial 256-rule-set universe of simple algorithms with many others Wolfram has researched for years in the dead of night while others slept. Rule sets involving multiple colors beyond black-and-white, rule sets that update only one grid square instead of a whole row, rule sets that embody full-blown Turing machines, rule sets that substitute entire sets of patterned blocks into single grid cells, that tag end point grid squares with new patterns, that implement "registers" and "symbols" - Wolfram has examined them all in excruciating detail. And no matter how complex the rule set is he explores, it ends up generating still more and more unexpected complex behavior with many notable features as the rule sets are implemented. This ever-escalating spiral of complexity leads Wolfram to believe that cellular automatons are a viable alternative to mathematics in modeling - in fact, embodying - the inherent complexity of the natural world.

In chapter four, he begins this process, by linking cellular automatons to the natural world concept of numbers. Automatons that multiply and divide, that calculate prime numbers and generate universal constants like pi, that calculate square roots and even more complex numerical functions like partial differential equations - Wolfram details them all. Who needs conscious human minds like those of Pythagoras or Newton to laboriously work out over thousands of years the details of things like trigonometry or calculus? Set up dominos in just the right way, flip the first one and stand back - nature can do such calculations automatically, efficiently and mindlessly.

Chapter five broadens the natural scope of cellular automations from one-dimensional numbers to multi-dimensional entities. Simple X-Y Cartesian coordinates are left behind as Wolfram defines "networks" and "constraints" as the canvas on which updated cellular automatons flourish - always generating the ever-higher levels of complexity. More Turing machines and fractals such as snowflakes and biological cells forming organs spontaneously spring forth. So far we've seen some really neat sleight-of-hand that Martin Gardner or Michael Barnsley might have written. But we're only on page 200 of 850 with seven chapters to go, and Wolfram is just now getting warmed up.

Chapter six is where Wolfram begins to lay the foundation for what he believes is so special about his insights and discoveries. Instead of using rigid and fixed initial conditions as the starting points for the cellular automations he has described, he now explores what happens using random and unknown initial conditions in each of his previously defined four "classes" of systems. He finds that while previously explored checkerboard (Class 1) and fractal (Class 2) systems yield few surprises, his newly-discovered unique (Class 3) and random (Class 4) cellular automaton systems generate still higher levels of complexity and begin to exhibit behavior that can simulate any of the four classes - a telltale hint of universality. Furthermore, their behavior starts to be influenced by "attractors" that guide them to "structure" and self-organization.

With the scent of universality and self-organization in the air, Wolfram begins in chapter seven to compare and contrast his cellular automations to various real-world topics of interest. Billiards, taffy-making, Brownian motion, casino games, the three-body problem, pachinko machines - randomness is obviously a factor in all of these. Yet, Wolfram notes, while randomness is embedded in the initiation and influences the outcomes of each of these processes, none of them actually generate true randomness in the course of running the process itself. The cellular automations he has catalogued, particularly his beloved Rule 30, do. The realization that cellular automations can uniquely serve as an initiator or generator of true randomness is a crucial insight, leading to the difference between continuity and discreteness and ultimately to the origins of simple behaviors. How, you ask? Hey, Wolfram takes most of the chapter to lay it out in a manner that I'm still trying to follow: no way can I summarize it in a sentence or two.

By chapter eight, Wolfram believes he has laid out sufficient rationale for why you, me and everybody else should think cellular automations are indeed the mirror we should be looking in to find true reflections of the world around us. Forget the Navier-Stokes equations - if you want to understand fluid flow, you have to think of it as a cellular automation process. Ditto for crystal growth. Ditto for fracture mechanics. Ditto for Wall Street. Most definitely ditto for biological systems like leaf growth, seashell growth and pigmentation patterns. This is very convincing stuff - tables of Mathematica-generated cellular automation shapes side by side with the photos of corresponding leaves or seashells or pigment patterns found in nature. Yes, you've seen this before in all of the fractals textbooks. The difference between fractals and cellular automations: fractals are a way to mathematically catalog the points that make up the object while cellular automations are a way to actually physically create the object via a growth process. It's a somewhat subtle difference - and a key Wolfram point.

Having established some credibility for his ideas, Wolfram stretches that credibility to the limit in chapter nine, where he applies his cellular automation ideas to fundamental physics. It was practically inevitable he would do this - his first published paper as a teenager was on particle physics, and that's the field he got his PhD in from Cal Tech at age 20 before going on to write the Mathematica software program and make his millions as a young businessman. Despite his solid background in physics, this seems at first blush to be pretty speculative stuff. He shifts his focus on the cellular automations from randomness to reversibility, and describes several rule-sets that both lead to complexity and are reversible. This behavior is an apparent violation of the Second Law of Thermodynamics. From Wolfram's way of thinking, if the universe is indeed some kind of ongoing cellular automation, then it may well be reversible and the Second Law must not be the whole story, so there must be something more we have yet to learn about the nature of the universe itself. He continues extensive speculations on what this may be, and how space, time, gravity, relativity and quantum mechanics must all be manifestations of this underlying Universal Cellular Automation. The rule set for this ultimate automation, which Wolfram believes might ultimately be expressed as only a few lines of code in Mathematica, takes the place of a mathematically-defined unified field theory in Wolfram's world. This is mind-blowing stuff, but ultimately boils down to Wolfram's opinion. I have great difficulty in comprehending space and time and matter and energy as "mere" manifestations of some cellular automation - if so, what is left to be the "system" on which the automation itself is running? I'm reduced to one of Clarke's Laws: The universe is not only stranger than we imagine, it is stranger than we CAN imagine ...

Wolfram shifts from Kubrick-style religion back to mere philosophy in chapter ten, where he explores how cellular automations are perceived by the human mind. Visual image perception, the human perception of complexity and randomness, cryptography, data compression, statistical analysis, and the nature of mathematics as a mental artifact are all explored. The chapter ends on a discussion of language and the mechanics of thinking itself. Wolfram reaches no real concrete conclusions on any of these, except that once again cellular automation is a revolutionary new tool to use in achieving new insights on all of these topics.

Chapter eleven jumps from the human mind to the machine mind by exploring not the nature of consciousness but the nature of computation instead. He goes here into somewhat deeper detail on ideas he has introduced earlier, about how cellular automations can perform mathematical calculations, emulate other computational systems, and act as universal Turing machines. He focuses on the implications of randomness in Class 4 systems and the universality embodied in systems like that of his Rule 110. His arguments lead up to a closing realization, what he does not call but may one day be named Wolfram's Law.

The final chapter, chapter twelve, discusses what all of Wolfram's years of isolation and work have led him to conclude. He calls it the Principle of Computational Equivalence. What follows is an unavoidably oversimplified distillation of Wolfram's thoughts on the PCE. If indeed cellular automations are somehow at the heart of the universe around us, then the human effort to reduce the universe to understandable models and formulas and simulations is ultimately doomed to failure. Because of the nature of cellular automation computation, there is no way to come up with a shortcut method that will deduce the final outcome of a system in advance of it actually running to completion. We can currently compute a rocket trajectory or a lens shape or a skyscraper framework in advance using mathematics merely because these are ridiculously simple human efforts. New technologies based not on mathematics but instead on cellular-automations like wind-tunnel simulators and nanobot devices will be exciting technological advances but will not lead to a fundamentally new understanding of nature. Issues that humans define as undecidability and intractability will always limit the level of understanding we will ultimately achieve, and will always have impacts on philosophical questions such as predestination and free will. To conclude with Wolfram's own final paragraph in the book:

"And indeed in the end the PCE encapsulates both the ultimate power and the ultimate weakness of science. For it implies that all the wonders of the universe can in effect be captured by simple rules, yet it shows that there can be no way to know all the consequences of these rules, except in effect just to watch and see how they unfold."

As noted above, 350+ pages of notes follow this exposition, and trust me, there's no way they can be summarized. To mention one nugget I found amusing as I envisioned Wolfram working towards endless dawns on ANKOS, he thinks sleep has no purpose except to allow removal of built-up brain wastes that cannot be removed while conscious. So much for dreaming.

So what is the bottom line on ANKOS? It is a towering piece of work and an enduring monument to what a focused and disciplined intellect can achieve. It is very thought provoking. It will definitely lead to new work and progress on cellular automation theory and some interesting technological applications we should all look forward to with anticipation. But is it the next Principia, the herald of a new scientific revolution?

Read and decide for yourself. Only time, and a lot of it, will tell.

To read it yourself, you can purchase A New Kind of Science at bn.com. You can read your own book reviews in this space by submitting your reviews after reading the book review guidelines.