Dimentionality. In rectangular-grid based CA, there is a dimentionality. However, the dimentionality concept may become unclear or merge with when the CA is based on arbitrary period tilings (1 or higher dimention), or when the topology of neighborhood is arbitrarily defined.
Number of States. Each cell can have n states, n ≥ 2.
Discrete vs Continuous. In continuous↗ , the cell has infinite states as a real number from 0 to 1.
Topology of neighborhood. In simple rectangular-grid based 2D CA, the neighborhood is commonly a cross (4), or all (8), or X (4). But what cells are considered as neighbors can be arbitrarily defined. The concept of neighborhood may mix when the grid is based on arbitrary periodic tilings. (for example of periodic tilings, see Tilings Gallery, Go Variations on Tilings)
Whole topology of the grid. e.g. torus, or any surfaces or manifold. (in conjunction with dimensions greater than 2)
Finitness of the gridspace. i.e. bounded plane or unbounded.
Totalistic CA. In simple 1D CA, the Position of neighbood states is part of the rule. In “Game of Life”, the position doesn't matter, only their totaly number. So, the latter is called Totalistic CA. “Totalistic” CA may be better called neighbor-locality-insensitive CA.
Continuous spatial automaton↗. i.e. the grid is not discrete, but continuous. In a sense, the cells are points in a plane or higher dimentional space.
Some words Wolfram likes to use in his writings: fundamental, quite, rather, “it turns out”, “kind”. Usually used in a manner to qualify his sentence so that it is still true in a broader, general context, but often just makes his statements hazy.
Here's a typical example from p170: “... adding more dimensions does not ultimately seem to have much effect on the occurrence of behavior of any significant complexity.”.
I've read KNOS chapters 1 to 6, about 3 times in different years, during 2002 to 2007, and scanned or skim'd other parts of the book. (Wolfram himself sent me a copy in about 2005.) I'm currently making another pass to write this notes.
Chapters 1 to 6 contains the bulk of technical info of the book that are mathematical. (and subsections in chapter 12: “implications for mathematics and its foundations”. Also, much of the General Notes section (p849 to contains significant amount of math, perhaps not new but can serve as a book of its own as one person's high-level survey of history of math and some science.)
The exploration of simple systems is interesting in at least as a subject of creational mathematis. In particular, systematic approach is noteworthy. However, after a while, one forgot what systems the book actually explored. Re-reading the book is problematic because it is too bloated. So, part of the purpose of this note is to summarize the significant info in the book. Also, it is my habit to write notes when learning.
The following are from: “Critical Review of “A New Kind of Science””, 12 July 2002, by David Drysdale
Part of the problem is Wolfram's insistance on using his own terminology for concepts and ideas that have perfectly good names in regular mathematics and science. Examples: he always referring to fractals as "nested" (and never makes clear whether the term includes less structured fractals or not), he doesn't refer to well-known pictures by their common name in the main text (such as the Sierpinski gasket or the Koch curve), he calls refers to lossy compression as "irreversible", he insists on using Mathematica notation rather than standard mathematical notation (there may indeed be a million Mathematica users, but there are considerably more who understand normal notation and don't have access to this expensive tool).
There are good justification on the way Wolfram uses his own terms, at least some of them. Usually, Wolfram terms, in my opinion, are better technical terminologies, judged from the point of view on the quality of scientific terminologies. For example, “Sierpinski gasket” or the “Koch curve” conveys nothing about what they are. (one class of scientific jargons that are bad quality is those after a person's name.) Most scientific terms arose and got adopted are due to circumstancial happenings (much like natural languages such as English), not due to design or thought. Thus, a significant portion of scientific terminologies, are rather misleading or blank label, and in general contribute to confusion of the field, in particular in education and for students. (“student” here are anyone learning the field, including expert in other fields) See Math Terminology and Naming of Things.
The term “lossy compression” vs “irreversible compression” are somewhat a tie with regards to the term's quality. They just emphasize 2 different aspects.
Wolfram's use of Mathematica notation instead of traditional math notation, in my opinion, is a good move. First of all, there are selfish motive since afterall his company sells Mathematica. But, traditional math notation, as a evolutionary product like languages and terminologies, has a lot ambiguities. Mathematica's notation, is in a sense a actual formal system since it runs as a computer program. In practice, when math notation is used in the book, it doesn't matter whether traditional math notation is used or Mathematica. If the notation is about math, there's little difference other than a few Captilization or parenthesis (e.g. Sin[x] vs sin(x)). If the notation is a computing code, one has to chose a language anyway. See The Problems of Traditional Math Notation.
It is true that the book is full of aggrandizing bloat. I think, the texts in the book can be made to be 3/4 of its size without losing any info, including how Wolfram thinks of himself and his discovery, and maitaining a readibility at the generally intelligent. The bloat and repeatition really makes the book hard to read and fuzzy. It is difficult to discern what is exactly claimed.
The justification for his writing style, given on p849., are not that convincing. It basically just further aggrandize himself more.
Particularly annoying to me is that paragarphs are just 1 to 3 short sentences, the tireless repeatition that simple program generates complex behavior, insistance of his genius. This adds a lot burden to extracting the technical info in the text.
However, i do support certain aspect of the writing style being non-conformal to standard styles as seen in math journals. Here's few elements that i despise in standard style:
In relation to the above ideas, see:
Many reviewers, in particular some mathematicians and scientists, completely dismiss the book as crank. (e.g. A Rare Blend of Monster Raving Egomania and Utter Batshit Insanity (2005-10), by Cosma Shalizi.) Perhaps the falsity of these reviewers are obvious, or that they really just wanted to attack Stephen Wolfram. (it is well known that Wolfram are cocky and have perhaps offended many. But also, fighting and power struggle in the history of math and science is common and well known) However, here i like to list some less commonly cited reasons why the book is considerable. (the commonly cited reasons include the proof of 110 CA↗ by Matthew Cook↗ and Wolfram's shortest axioms for logic. (p808.)
• A Survey of Simple Systems. The book exams thousands of CA or other simple systems and made a report of it. (esp chapters 1 to 6) These explorations, are not trivial. From this aspect, the book can be considered significant at least as a average scientific publication at the research level.
• A Recreational Math Text. The General Notes section (p849 to p1197), of 348 pages in small print, can be considered significant at the level of text book, of a person's survey of the sciences and mathematics in relation to CA.
For example, there are lots of books published aimed at the general public, including science popularizing books, recreational math books, computer programing books, and many received great accolades and awards. (e.g.Douglas R Hofstadter↗'s Gödel, Escher, Bach↗; Martin Gardner↗'s creational math collections; William Poundstone↗'s Labyrinths of Reason. Tristan Needham↗'s Visual Complex Analysis. etc. ) These books are not scientific publication or news breaking discovery, but nevertheless are valuable for at least education. Considering the actual info content of Wolfram's book esp the General Notes, it has value as significant as these books.
• Exemplary Exhibition of Scientific Visualization. Edward Tufte↗ has written books on Scientific Visualization↗ that are widely acclaimed. In my opinion, it is rather trivial to exposit on this matter, as i believe any intelligent person who has put thoughts into creating her illustrations should arrive at the same conclusions or practices. Nevertheless, Edward's book is widely acclaimed in the scientific community. Also, there are awards and seminars in the scientific community for visualization. Wikipedia goes as far to have a article on Information design↗ with citations of journals, organizations, and books published by MIT press. So, these facts tells us that scientific visualization are highly valued by the scientific community. Wolfram's book, in its literally over a thousand illustrations, are examples of the finest quality of scientific visualization. Mathematica is a powerful tool in creating illustrations, but even for a expert Mathematica programer, a significant percentage of illustrations contained in Wolfram's book are not trivial to generate or program. Some types of the illustrations, e.g. involving networks, Ant CA, 3D CA, complex graph of replacement systems with arrows, etc, will take a expert programer days to create, for each. (or months each, to actually create a robust program as seen in many Java applets on the web created by programers) Perhaps most of these are coded by Wolfram's employees, but nevertheless, the book itself is significant at least for its exemplary exhibition of highest quality and quantity of scientific visualization. If Wolfram published the book normally, it would no doubt win awards.
• As Social Movement. Entrepreneurs, even though have contributed nothing to science, but are highly valued by society. In fact, science would make little progress without entrepreneurs. This is a basic understanding for anyone aquainted with sociology. Likewise are great figures in history remembered as educators, reformers. Stephen, due to his wealth, his success with Mathematica, and at least his recognization as a genius for his past contribution to physics, has made his book a relative major impact in society. He also, subsequently started a series of websites, lectures, and school for the type of research he is interested in (i.e. what he calls the New Kind of Science). Unless the study of Cellular Automatica is completely worthless for anything whatsoever in science (which is not true), one must then give credit to Stephen at least as a educator or science popularizer, who made the general public aware of a particular math subject, or made a movement in scientific community, or the programing community, on CA. In short, he made a significant influence. Arguably, popularizing science is only a side effect of his book, where he actually intented to popularize himself. But nevertheless, the end results of his book and websites and investiments into simple programs research, has made non-trivial contribution to science. For example, in 2007 Wolfram offered $25000 US dollars for proving or disproving whether Wolfram's 2-state 3-symbol Turing machine↗ is universal. This award is won by Alex Smith, but the proof is however disputed by a well-known computer scientist Vaughan Pratt↗. If the proof is correct, then this is a significant contribution to science instigated by money (e.g. the Millennium Prize↗ ). If this proof is incorrect, arguably, it contributed to science of more precise understanding of what is meant by universal turing machine.
Collection of books Wolfram owns: http://www.wolframscience.com/reference/books/.
“Notes on “A New Kind of Science”” (2002-2004) by David Drysdale. Detailed notes on each chapter. http://www.lurklurk.org/wolfram/notes.html
Collection of reviews by Edwin Clark: http://shell.cas.usf.edu/~eclark/ANKOS_reviews.html