For all those interested in NKS and computational physics, there is a great essay by Stephen Wolfram over at the FQXi forum, titled: “What is Ultimately Possible in Physics?” It was entered in the 2009 FQXi essay contest with the same name.

Voting is still open (until November 6). Please check out the contest, read Stephen Wolfram’s essay, place a vote, and leave a comment!

Here is a direct link to Stephen Wolfram’s essay: “What is Ultimately Possible in Physics?“

Essay abstract:

This essay uses insights from studying the computational universe to explore questions about possibility and impossibility in the physical universe and in physical theories. It explores the ultimate limits of technology and of human experience, and their relation to the features and consequences of ultimate theories of physics.

Introducing Breaking Point, by Etan Ilfeld and Eric Ayotte.

Etan Ilfeld was a 2009 NKS Summer School participant, and has brought NKS ideas to his newest exhibition at his gallery, Tenderpixel.

The exhibition will be running from October 8 – November 15.

A description from the exhibition’s home page:

Tenderpixel is pleased to present Eric Ayotte and Etan Ilfeld’s first collaboration, Breaking Point, which explores the meta and microcosmic synthesis of computer-creativity, mathematics, optics, art history and painting. Breaking Point relates to riots, tear gas and general societal conflicts within a landscape: At any given point in society when equality or injustice reaches a certain point there is a universal language or action that occurs. This exhibit references many of the global protests that have been documented within the contemporary mediascape.

Breaking Point explores imagery surrounding the concept of a breakthrough in society as well as the moment at which a painting is broken apart and what that signifies when thinking about ‘process painting’. Ilfeld and Ayotte’s creative process incorporates Stephen Wolfram’s New Kind of Science methodology by testing aesthetic-algorithms, and processing and juxtaposing media images unto 2-D Cellular Automatons in Mathematica.

Their production more closely resembles the futurist paintings of David Bomberg or Giacomo Balla than the process painting and conceptual framework from which they are derived. As a result, their work is ‘neo-futurist’: The machine becomes an integral element of the creative process itself, and generates the emergence of artistic modes that would have been impossible prior to computer technology.

The hours of the gallery are Tuesday – Saturday, 1pm – 7pm. They are located in London, UK.

The 2009 NKS Summer School is over, and thanks to everyone who worked hard to make it a success! It was our first year international and our largest student group ever, and it went very well.

This summer I learned quite a bit from the various projects being conducted at the school. One of my favorite projects was by Earl Mitchell, a Pepperdine MBA, on social choice theory (for more info on the theory, read Kenneth Arrow). It attempted to simulate social choice in models of democracy, dictatorship, theft, exchange, and oligarchy, with some very interesting results.

We also had a lot of musicians this summer, which I personally find fantastic. It inspired me to dig into the project on NKS and music I’ve been wanting to do for a while — stay tuned!

There has also been a noticeable ripple effect in what was done at the Summer School and what has started to appear elsewhere. On the blog “Unconventional Everything (Mikamai),” Sebastiano Scròfina wrote the post Pasta, Oil, and Mathematica. They used Timothy Walker’s (a participant at this year’s summer school) code from his project to quickly graph some Facebook social data:

I get excited, and ask him if I can play with it to visualize social graph data. He gets excited too: we both leave our pasta aside and start coding in the middle of lunch. After a few minutes, under his patient and skillful guidance, we’ve got 2 Lines Of Code rendering 1000 nodes of Facebook’s icelandic network. How cool is that ?

Very cool!

John Baez, mathematical physicist at The University of California at Riverside, recently cut an album generated by WolframTones.

Treq Lila

There’s quite a bit of stylistic diversity in the selections. While the title song “Treq Lila” is deep and jazzy, with a definite beat, other songs like “Swirl” are more ethereal, giving you the sense of phrasing without having any definite phrases. “Ay-Layla” is very experimental, with a contrained bassline being ridden by lighter, almost helicopter – like strings.

Check out John Baez’s album, and let us know what you think!

Here are some Demonstrations that are relevant to Chapter 3 of A New Kind of Science. They are grouped under relevant section. Haven’t read Chapter 3 of NKS? Start reading here

More Cellular Automata

Cellular Automaton Explorer, by Stephen Wolfram – Cellular automata are a central example of Stephen Wolfram’s science, and of simple programs in the computational universe. Explore some of their amazingly rich and varied behavior here. See Wolfram’s rule 30, and then choose more colors to explore rules that have probably never been seen before.

Examples of 1D Three-Color Totalistic Cellular Automata, by Abby Nussey -This Demonstration lets readers of A New Kind of Science explore one-dimensional three-color totalistic cellular automata as given in the examples on pages 62–64, and 66–70.

Mobile Automata

3-Color Left/Right Mobile Automata, by Narine Manukyan -This Demonstration shows the behavior of a mobile automaton with two states and three colors. There are rules, but this Demonstration shows only rules from 677,855 to 780,055.

Register Machines

Register Machines, by Anthony I. Joseph -(excerpt): This Demonstration allows an exploration of the register machines discussed in chapter 3 of A New Kind of Science. Register machines are treated as logical abstractions of physical registers used in electronic devices. Registers are often used to store information to be used for storing data and performing calculations.

To start reading Chapter 6 of A New Kind of Science, go here

Starting from Randomness

The Emergence of Order [p. 223]

In this chapter, Wolfram prepares to show how certain systems seem to produce regular classes of behavior, regardless of the initial conditions at which their evolution begins.

He first introduces examples where the evolution always goes to all black or all white, and then examples of where the evolution repeats in some simple pattern.

Four Classes of Behavior [p. 231]

We find, in fact, there appear to be four classes of behavior (called Wolfram Classes) of the evolution of CAs. Completely stable, simple repetitive, complex-no-traveling-structures, and complex-with-traveling-structures.

“In class 1, the bahvior is very simple, and almost all initial conditions lead to exactly the same uniform final state.
“In class 2, there are many different possible final states, but all of them consist just of a certain set of simple structures that either remain the same forever or repeat every few steps.
“In class 3, the behavior is more complicated, and seems in many respects random, although triangles and other small-scale structures are essentially always at some level seen.
“And finally…class 4 involves a mixture of order and randomness: localized structures it produces which on their own are fairly simple, but these structures move around and interact with each other in very complicated ways.” [p. 235]

Can we know by looking at a rule which class of behavior it will produce? Generally not, but in certain cases we can prove rules won’t ever show anything different than a particular class of behavior [p. 241]. Additionally, rules that are close together tend to have similar behavior.

Continuous CAs show class behavior; and if you take one dimensional slice of the evolution of a two-dimensional CA, you’ll also get class behavior (in fact, a 1D slice of the Game of Life is a class 4 CA).

Fun question: What would you look at when trying to determine whether a three-dimensional CA has class behavior?
Sensitivity to Initial Conditions [p. 250]

In this section, we see that the difference patterns for each CA (the patterns produced when one flips a bit or block of bits of the initial conditions) are different for each class.

Class 1’s difference pattern always dies out; Class 2’s has very limited range; Class 3’s has a wide range and perpetuates; Class 4’s has a limited range and may or may not perpetuate. We see that the difference patterns give us indications of how information is handled in each system.

Class 1, in effect, “forgets everything”; Class 2 retains information in a highly localized manner; Class 3, in effect, “transmits nearly everything”; and in Class 4 long-range transmission is possible, but does not always happen. Again, Class 4 behavior seems like and intermediate between Class 2 and three, where there are localized structures that can transmit information in a long-range manner.

Systems of Limited Size and Class 2 Behavior [p. 255]

This section explains Class 2 repetitive behavior by noting that it is a general result that systems of limited size with discrete elements all eventually repeat.

“…the actual repetition period jumps around considerably as the size of the system is chaned. And as it turns out, the repetition period is again related to the factors of the number of possible positions for the dot — and tends to be maximal in those cases where this number is prime.” [p. 258]

Note: the max possible repetition period for any system is always equal to the total states of the given system.

“In a class 2 system with random initial conditions…since different parts of the system do not communicate with each other, they all behave like separate patterns of limited size.” [p. 260]

Randomness in Class 3 Systems [p. 261]

There are, however, certain special initial conditions that will make even randomness-generating CAs repetitive. For rule 30, only initial conditions consisting of a single block of black and white cells repeated forever can yield repetitive behavior. However, this is not true for all randomness-generating rules.

Indeed, there are some initial conditions that can make one CA act just like another CA. There are also CAs that can self-emulate: additive rules, and other rules that show nested behavior. Examples: rules 90, 150, and 184.

The Notion of Attractors [p. 275]

An attractor for a CA are points to which the evolution will be drawn, regardless of initial conditions (like a pendulum, which will always end up pointing straight down, regardless of how it starts out).

In general, a CA could have many possible attractor states. In particular, it could have State 1 to which 4 sets of initial conditions lead, State 2 to which 90 sets of initial conditions lead, State 3 to which 13 sets of initial conditions lead, and so forth.

On the next few pages, Wolfram details a network convention that compactly describes the evolution of various CAs and their various initial conditions to particular attractor states. He uses the connectors to specify the colors. The networks are simple in class one and two systems — all initial conditions evolve to a contant, repetitive state. In class 3 and 4 systems, the networks can get very complicated, very quickly (in fact, the number of nodes seems to increase at a rate that is at least exponential).

It stands to reason that in many class 3 and class 4 systems, there really aren’t any attractor states — and that’s why the network complexity “blows up.” We note special initial conditions are needed to evolve most class 3 and class 4 systems to attractor states.

Structures in Class 4 Systems [p. 281]

Wolfram endeavors to show that moving structures are inevitable in class 4 systems, and he discusses the different kinds of moving structures that one sees in class 4 systems given particular initial conditions.

“Indeed, it is a general feature of class 4 cellular automata that with appropriate initial conditions they can mimic the behavior of all sorts of other systems.” [p. 291]

Rule 110 is the old standard that shows the above to be true.

Read Chapter 4 of A New Kind of Science here

Read other posts in the NKS Summer School Series

The first example used is the function where f[0]=0. and f[x]=x. This is a straight line, as one can see:

To start reading Chapter 5 of A New Kind of Science, go here

Two Dimensions and Beyond

One of the central themes of this chapter is that adding more dimensions does not ultimately have much effect on the occurrence of behavior of any significant complexity.

Cellular Automata [p. 170]

2D and 3D CAs are shown to behave, with respect to the complexity generated by their evolutions, much in the way of 1D CAs. That is, they don’t appear to generate a higher level, or have a more frequent occurrence of, complexity, in general.

Turing Machines [p. 184]

Turing machines are generalized into higher dimensions by allowing the head of the Turing machine to move around on the higher-dimensional grid rather than just going back and forth on a one-dimensional tape.

For Turing machines, we don’t see any more (of frequent) complexity than in one dimension.

Substitution Systems and Fractals [p. 187]

By putting substitution systems on to a 2D grid, we start seeing fractal geometries emerging.

In order to get complexity (instead of just simple nested patterns), one needs to devise a way for the replacements for a given element to depend on its neighbors. However, it isn’t immediately obvious how to generalize this to higher dimensions for a sequential substitution system, in which just a single block of elements is replaced at each step (to do so one would have to set up a way to scan the elements in 2D, which would just reduce it to a 1D system).

Network Systems [p. 193]

This is a section near and dear to my heart, because my personal research interest in NKS has thus far focused on investigating evolutions of networks with CA rules mapped on to them.

A network topology allows you to break away from the grid. Though my personal work has been focusing on running CAs on static network topologies, this section looks at the complexity of dynamic networks, with a certain number of nodes and connections and rules on how to make the connections.

Therefore, a network system, in this section, is a collection of nodes with various connections and a set of rules on how these connections should change from one step to the next.

We’re going to look at how the number of allowable connections per node is related to the complexity in the resulting allowable network configurations. A node is allowed connections to other nodes as well as self-connections. Connections are directionally-dependent.

We start with two connections. We see as the number of nodes increases, the number of possible networks grows rapidly. By laying out the networks in certain ways, we can reproduce grids in various dimensions, so there is nothing inherently one-dimensional about networks.

How does one set up the evolution of a network system? Since the next step will determine how connections come out of the node, we need to know how those connections should be rerouted based on the local structure of the network around that node. In other words, we survey the surroundings, and based on what we see, with form particular connections.

One can imagine that without animation of some sort available, it’s tricky to visualize this process. And in the book, there’s a rather confusing convention for doing this which I think in the day and age of Manipulate can be disposed of, unless one is really fond of comparing static pictures side-by-side.

As long as nearest-neighboring nodes are the only ones being looked at, we see simple, repetitive structures. However, as soon as you start going just a bit further out you get much more complicated behavior.

As we’ll see later, understanding these network systems may be tantamount to understanding the basic structure of space and our universe itself!

Multiway systems [p. 204]

Wolfram notes in the beginning of this section that all the systems that have so far been set up in A New Kind of Science have simple structures with respect to time, all set up to evolve from one state to the next.

Multiway systems are set up to have a whole collection of possible states at any given step, and all distinct sequences that result are kept. Depending on the structure of the possible states, the fluctuations in growth can look random, though we usually see two ways they grow: die out, or grow exponentially quickly. Slow growth is rare, and some amount of repetition seems to always take over in the end.

Since, by the nature of a multiway system, whenever any particular sequence occurs, it must always lead to exactly the same behavior, one can represent entire systems with a short sequence, indicating with loops the repetition of the sequence. One can also rewrite them in other ways, so that multiways systems in effect build up their own patterns of connections in time, the patterns which can be quite complicated.

Systems Based On Constraints [p. 210]

This section discusses systems that, rather than being set up with rules of step-by-step evolution, are instead just given constraints to satisfy.

It’s discovered that in one-dimensional systems, there are no simple sets of constraints that can force complex patterns. And in fact, in two dimensions, one of 171 patterns can satisfy any set of two-color, four nearest-neighbor, constraints.

With more complex neighborhoods, one can force complexity, but it’s very, very rare.

Haven’t read Chapter 4 of A New Kind of Science? Start reading it here.

Systems Based on Numbers

We’ll summarize by going through each of the sections and hitting on the salient points. Note that the summaries are very condensed, and are meant to give a broad picture of what is going on. Later on we’ll get to examples and code.

Summary

The Notion of Numbers [p. 115]

“The main reason that systems based on numbers have been so
popular in traditional science is that so much mathematics has been developed for dealing with them. Indeed, there are certain kinds of systems based on numbers whose behavior has been analyzed almost completely using mathematical methods such as calculus.
“Inevitably, however, when such complete analysis is possible, the final behavior that is found is fairly simple. ” [p. 115]

“…if one ignores the need for analysis and instead just looks at the results of computer experiments, then one quickly finds that even rather simple systems based on numbers can lead to highly complex behavior.” [p. 116]

Basic problems with answering the question of what the origin of complexity in simple systems are:

• Numbers are handled very differently in traditional mathematics from the way they are handled in computers and computer programs.

• Mathematics posits numbers are elementary objects differing only in size, while computers must have numbers explicitly represented by a sequence of digits. In a computer numbers are represented in base 2, by a sequence of 0’s and 1’s.

• In mathematics, the details of how operations performed on numbers affects sequences of digits are usually considered fairly irrelevant – but by imagining a sequence of 0’s and 1’s as a sequence of white and black cells, and with the knowledge that simple operations, like those in cellular automata rules, can create complexity, we start understanding wherefrom complexity can arise, hidden in the operations on numbers when represented by a sequence of digits.

However, it isn’t just the digit sequences of numbers that show complexity. Regardless of the base, one can set up examples which show that the growth of the size of the number itself is complex. This can be done with progression of fractional numbers as well as whole numbers.

One difference between the complexity we see in CA’s and that in CA-like progressions of digit sequences based on some function is that CA’s are inherently local in nature: the next step depends on the previous step and neighborhood. However, this isn’t the case for digit progressions, which are nonlocal, and depend on the function.

Recursive Sequences [p. 128]

This section sets out to prove that the threshold of complexity in numerical sequences is met using simple addition and subtraction. This is done by considering recursive sequences.

How does one reach the threshold of complexity with recursive sequences? Instead of subsequent terms necessarily depending on the term before or twice before (locality), the rule for f [n] is that it can also depend on f [n - f [n - 1]], for instance (nonlocality).

Mathematical functions [p. 145]

All standard mathematical functions themselves have fairly simple (essentially repetitive) curves. Combinations of standard functions, however, can yield more complex curves, depending on how they are combined, which usually has an explicit dependence on representations of individual numbers. Some functions which yield complex curves but don’t appear to have any explicit dependence on representations of individual numbers are related to the Riemann zeta function.

Iterated Maps and Chaos Phenomenon [p. 149]

In the iterated maps section, Wolfram uses the example of four different kinds of iterated maps, which generated the four different Wolfram classes of behavior when given the initial condition 1/2, with their base 2 digit sequences then plotted. However, when they are given the initial condition pi/4 (with a seemingly random digit sequence), they exhibit the same classes of behavior. Here it would seem the maps themselves intrinsically generate complexity.

“Indeed, thinking about numbers purely in terms of size, one might imagine that as soon as any two numbers are sufficiently close in size they would inevitably lead to results that are somehow also close. And in fact this is for example the basis for much of the formalism of calculus in traditional mathematics.” [p. 152]

In this section, we encounter the idea (brought forth in the discussion of the shift map) that some randomness is a consequence of the randomness inherent in the initial conditions presented to the system. If the initial conditions (here the digit sequences) aren’t random, randomness disappears. Can such a system truly be called random, as it depends on the randomness in its initial conditions?

But some systems, as we’ve seen above, are independent of their initial conditions, always producing the same behavior (which fits into four different classes).

Continuous Cellular Automata [p. 155]

“The idea is to look at the average gray level of a cell and its immediate neighbors, and then to get the gray level for that cell at the next step by applying a fixed mapping to the result.” [p 156]

Examples:

• simple diffusion (next cell is the average of itself and neighbors)
• a continuous CA that results in class 3 behavior: take the average gray level, multiply by 3/2, and keep the fractional part only if the result is greater than 1

Partial Differential Equations [p. 161]

In this section, Wolfram shows how generalizing continuous CAs in order to wipe out all discreteness results in rules that are the well-known partial differential equations.

The best-studied PDEs all result in simple behavior. By looking extensively, however, one can find examples of PDEs that result in not-so-simple behavior.

Yesterday Stephen Wolfram gave a demo of Wolfram|Alpha at Harvard University. Wolfram|Alpha demo:

Naturally, it generated a lot of buzz, and a lot of misunderstanding. For those who haven’t been reading up on Wolfram’s statements about Wolfram|Alpha, there seems to exist the impression that W|A is meant to replace Google. A much better way of explaining what W|A does is summarized in the article, Wolfram|Alpha: Our First Impressions (on ReadWriteWeb).

Alpha, which will go live within the next few weeks, is quite different from Google and really doesn’t directly compete with it at all. Instead of searching the web for info, Alpha is built around a vast repository of curated data from public and licensed sources. Alpha then organizes and computes this knowledge with the help of sophisticated Natural Language Processing algorithms. Users can ask Alpha any kind of question, which can be constructed just like a Google search (think: “hurricane bob” or “carbon steel strength”).

Like Wolfram has explained in various interviews, it’s like having an expert, ready to answer  your questions, as you’d pose in natural language to any kind of human expert.