I’ve written a new paper under the title “Compression-based investigation of the dynamical properties of cellular automata and other systems”.

A pdf version of the paper from a Mathematica notebook is available online at ArXiv

Abstract:
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A method for studying the qualitative dynamical properties of abstract computing machines based on the approximation of their program-size complexity using a general lossless compression algorithm is presented. It is shown that the compression-based approach classifies cellular automata (CA) into clusters according to their heuristic behavior, with these clusters showing a correspondence with Wolfram’s main classes of CA behavior. A Gray code-based numbering scheme for initial conditions and a compression based method to estimate a characteristic exponent to detect phase transitions and measure the resiliency or sensitivity of a system to its initial conditions is also proposed, constituting a compression-based framework for investigating the dynamical properties of cellular automata and other systems.
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Stephen Wolfram proposed a classification of cellular automaton rules into four types, according to the results of evolving the system from a “disordered” (namely random) initial configuration:

1. Evolution leads to a homogeneous state.
2. Evolution leads to a set of separated simple stable or periodic structures.
3. Evolution leads to a chaotic pattern.
4. Evolution leads to complex localized structures, sometimes long-lived.

An interesting question about Wolfram’s classification concerns its dependence on the initial condition–chiefly because the classification was originally meant to be constructed by visual inspection over the evolution of a CA, and as we know, the evolution of a CA depends on its initial condition. This has been a major critic (Eppstein) of Wolfram’s classification, because somehow the classification should be based on the evolution from an unordered (random) configuration and no on initially ordered configuration.

Nevertheless, the classification is based on the potential of a CA to evolve into any of the possible behaviors from at least one initial configuration (the question is of course not finitely answerable since there is an infinite number of possible initial configurations). But that in the end one ends up analyzing only a finite number of particular cases, including an order and a disordered initial configuration. Wolfram’s classification might therefore be seen as being dependent on the initial condition of a CA.

It is not a surprise that one can for example construct a CA belonging to more than one of Wolfram’s four classes when starting from different initial configurations. Think of rule 110 for example. Rule 110 can in principle be made to look as if it belonged to any class because, given its universality, it is capable of simulating any other CA. Rule 110 belongs to class 4 because it is capable of universal computation— one can set up an initial configuration to ‘program’ rule 110 to carry out any computation (it is the very basic concept of a programmable computer).

For every CA rule there is a definite (but in general undecidable ) answer to the question whether or not it is capable of universal computation (or in reachability terms, whether a CA will develop into a certain configuration). The question only makes sense if the evolution of a CA depends on its initial configuration. No rule can be universal that fixes the initial configuration once and for all (there would be no way to input an instruction and carry out an arbitrary computation).

On the other hand, some rules, such as Rule 0, don’t produce one or another configuration relative to variant initial configurations. No matter how you change the initial condition, there is no way to make it compute something other than what it actually computes for every other initial configuration.

In light of all this, David Eppstein’s critique of Wolfram’s classification is vacuous because obvious from my point of view. His main argument is that there are CAs that can be made to look as if they belonged to all classes by modifying their initial conditions. Which is obviously true!

A CA belongs to a certain class until, given another initial configuration, it is made to behave as if it belonged to another, more powerful one (assuming some kind of hierarchy, at least between classes 1 and 2 and classes 3 and 4).

My paper on compression-based investigations shows that Wolfram’s heuristic classification can actually be quantified by a measure which is clearly dependent on the initial conditions while also being capable of detecting sensitivity to initial configurations and hence of replacing the visual inspection.

This represents a formal approach to Wolfram’s classification process, and a method to determine to what class a CA belongs which is compatible with what Stephen Wolfram himself has proposed in his NKS book.

Notice that the paper is a pdf file generated from a Mathematica notebook. Hence some images (e.g. cells indicating an initial configuration) are not optimal. A version in LaTeX is being prepared and will replace this version.

One of the greatest scientific achievements of the last century was the understanding of life in terms of information. We know today that the information for synthesizing the molecules that allow organisms to survive and replicate is encoded in the DNA. In the cell, DNA is copied to messenger RNA, and triplet codons in the messenger RNA are decoded in the process of translation to synthesize polymers of the natural 20 amino acids.

Humans have been intrigued by the origin and mechanisms underlying complexity in nature coming from information contained in repositories such as the DNA. Darwin’s theory of evolution suggests that this complexity could evolve by natural selection acting successively on numerous small, heritable modifications.

Darwin’s theory represents a great leap forward in our understanding of the fundamental processes behind life. However, evolution may not be the main or sole driving force behind the complexity of living organisms [If you wish to know more about the theory of evolution by means of natural selection, three respectable British institutions have set up special websites in celebration of Darwin's 200th. anniversary: the University of Cambridge (with the original scanned text and even an audio version in mp3 format), the Open University and the BBC]. 

Based on my own research interests it is my strong belief that though by no means wrong, Darwin’s theory of evolution belongs within a larger theory of computation, according to which life has managed to speed up its rate of change by channeling information faster and somehow efficiently, and in so doing has benefited from an exchange of information with the outside by a process that while seemingly random, is in fact the consequence of interaction with other algorithmic processes.

Nature seems to use a specific toolkit of body features rather than totally random shapes. Like units of Lego, Nature assembles its forms from a limited set of elements. For example, despite the variety of living forms on the Earth, they do all seem to have a front-to-back line down the center of the body, and extremities (if any) on the sides, from flies who have a head at one end and a tail at the other, to worms, snakes and humans. Despite the randomness that may undermine any shared regularity among all animals in combinatoric terms, on a certain level, from a certain perspective, we are all similar in shape and features. Why didn’t evolution attempt other, completely different forms? And if it did, why were so few of them successful? Given the improbability of  several other shapes having been put into circulation without any of them winning out save the ones we all know, we could conclude that evolution never did attempt such a path, instead keeping to a small pool of tried and tested basic units whose survival has never been in jeopardy. There are some symmetries and general features that many animals share (more than can be explained by inheritance) that are not so easily explained in purely evolutionist terms. A remarkable example is the resemblance of all animals in their embryonic phase.

Two teams of biologists (Walter Jakob Gehring and colleagues at the University of Basel, Switzerland, and Matthew Scott and Amy Weiner working with Thomas Kaufman at Indiana University, Bloomington) seem to have independently discovered toolkits that Nature appears to use that they have called homeobox containing genes.

This discovery indicates that organisms use a set of very simple rules passed along to them (thus reducing the amount of randomness involved) to build a wide variety of forms from just a few basic possible body parts. To oversimplify somewhat, one can for instance imagine being able to copy/paste a code segment (the homeobox) and cause a leg to grow in the place where an antenna would normally be in an ant.

This begins to sound much more like the footprint of computation rather than a special feature characterizing life, since it turns out that a few simple rules are responsible for the assembly of complex parts. Moreoever, this is consonant with what in Wolfram’s scheme of things life’s guiding force is said to be, viz. computation. And with what Chaitin has proposed as an algorithmic approach to life and evolution, as well as with my own research, which is an attempt to discover Nature’s basic hidden algorithmic nature.  All the operations involved in the replication process of organisms– replacing, copying, appending, joining, splitting–would seem to suggest the algorithmic nature of the process itself. A computational process.

The theory of algorithmic information (or simply AIT) on the other hand does not require a random initial configuration (nor any god, unfortunately!) to have a program, when run, produce complicated output. This is in keeping with Wolfram’s finding that all over the computational universe there are simple programs with simple inputs generating complex output, what in NKS terms is called ‘intrinsic randomness’, yet is purely deterministic. Nor does AIT require the introduction of randomness during the computation itself. In other words, it seems that randomness plays no necessary role in producing complex organisms. Evolution seems to underlie change, its pace and direction, but it does not seem to constitute the driving force behind life.

Evolution seems to be taking advantage of the algorithmic properties of living systems to fabricate new forms of life. To facilitate understanding of these body patterns the University of Utah has set up an illustrative website. Incidentally, this genetic toolkit based on the homeobox concept is surprisingly well captured in the Spore video game.

In a recent article Greg Chaitin has proposed (Speculations on biology, information and complexity) that some of the properties of DNA and the accumulation of information in DNA may be better explained from a software perspective, as a computer program in constant development. When writing software, subroutines are used here and there all the time, and one usually creates an extra module or patch rather than rewrite a subroutine from scratch. This may correspond to what we see in DNA as redundant sections and ‘unused’ sections.

In Chaitin’s opinion, DNA is essentially a programming language for building an organism and then running that organism. One may therefore be able to characterize the complexity of an organism by measuring the program-size complexity of its DNA. This seems to work well for the length of DNA, since the longest known sequence of DNA belongs to what is certainly the most sophisticated organism on this planet, i.e. homo sapiens.
Chaitin proposes the following analogy:

program -> COMPUTER -> output
DNA -> DEVELOPMENT/PREGNANCY -> organism

However, we encounter problems when attempting to view the process of animal replication in the same algorithmic terms. If, as the sophistication of homo sapiens would suggest, human DNA is the most complex repository of information, and given that DNA represents the shortest encoding capable of reproducing the organism itself, we would expect the replication runtime of human DNA to be of the same order relative to other animals’ replication times. But this is not the case. A gestation period table is available here. So what are we to make of the fact that the right complexity measure for living beings (the logical depth of an object as the actual measure of the organizational complexity of a living organism) does not produce the expected gestation times? One would expect the human gestation period to be the longest, but it is not.

Charles Bennett defined the logical depth of an object as the time required by a universal computer to produce the object from its shortest description, i.e. the decompression time taken by the DNA from the fertilized egg of an animal (seen as a universal computer) to produce another organism of the same type. There seems to be more at stake, however, when trying to apply the concept to Chaitin’s replication analogy– issues ranging from when to determine the end of the replication (the gestation period?), to better times to give birth, to gestation times inherited from ancestral species, to the average size of organisms (elephants and giraffes seem to have the longest periods). Some hypotheses on period differences can be found here for example.

If living organisms can be characterized in algorithmic terms as we think they can, we should be able to introduce all these variables and still get the expected values for the complexity measurement of an organism– seen as a computer program–reproducing another organism from its shortest encoding (the DNA being an approximation of it). A complete model encompassing the theory of evolution has yet to emerge. It seems to be on the horizon of AIT, as another application to biology, one that provides a mathematical explanation of life.

In summary:

• So far, what we know is that the DNA is the place where the information for replicating an animal is to be found. What’s being proposed above is that the information content in the DNA can be actually effectively approximated by means of its program-size complexity and logical depth to define a measure of the complexity of an organism. If one can quantify these values one could, for example, actually quantify an evolutionary step in mathematical terms. This would represent a first step toward encompassing Darwin’s theory of evolution within an algorithmic mathematical theory of life. Evolution is not usually seen as part of a computational theory, but as a special feature of life. The above suggests otherwise.
• Randomness has hitherto been thought to play a major role in the evolution of species, as it is mutation that drives the evolutionary process. But I suggest that this is not the case. Rather I suggest that what appears to be random is actually part of a deterministic computation, which means that randomness plays no significant part in the process, while computation does.
• Finally, evolution has hitherto been thought of as a process that advances by very small steps, rather than one that is capable of quickly building over blocks of code, as it might be actually the case. This new understanding favors the computational view I am putting forward here as playing a main role in the process of life, because it is based on what in software technology is the practice of a subroutine orientation programming paradigm: code reuse.

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The Shortest Universal Turing Machine Implementation Contest

                          ANNOUNCEMENT

                          23 Dec – 2008

  http://www.mathrix.org/experimentalAIT/TuringMachine.html

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Contest Overview

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In the spirit of the busy beaver competition though related to program-size complexity, we are pleased to announce the “Shortest Universal Turing Machine Implementation Contest”.

The contest is open-ended and open to anyone. To enter, a competitor must submit a universal machine implementation written in the language specified in the contest web site (C++) with smaller size values than the latest  record published on the web page.

In order to take part in this competition it is necessary to submit the source code, to be compiled using the compiler program and version specified in the contest web site. It is important that you provide documentation of your code, either in an attached file or as commented text in the source code file.

Each submitter must agree to be bound and abide by the rules. Submissions remain the sole property of the submitter(s), but should be released under the GNU General Public License (GPL)  so we may be permitted to make them available on  this web site for downloading and executing.

 

Rules

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http://www.mathrix.org/experimentalAIT/TuringMachine.html (General Rules section)

 

Team composition

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Players may enter alone or as teams of any size. Anyone is eligible to enter.

 

Subscribe to the Newsletter

=============

We have a mailing list that we will use to keep participants informed of news about the contest. You can subscribe to this mailing list at any time:

Subscribe at http://www.mathrix.org/mailinglist/?p=subscribe

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Organizers

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Hector Zenil (IHPST and LIFL, Paris 1 University and Lille 1 University)
Jean-Paul Delahaye (LIFL, Lille 1 University)

Having quit his studies in physics, Theo Jansen became an artist. In this video he demonstrates his amazing life-like kinetic sculptures, built from plastic tubes and bottles. His Beach Creatures or Strandbeest are built to move and even survive on their own:

I’ve been in touch with Theo Jansen recently. For further details about his creations he referred me to his book (available at his web shop ) entitled The Great Pretender. Even more details are provided in Boris Ingram’s thesis on leg designs based on 12-bar linkages, in which he describes Jansen’s walker algorithm. Jansen’s designs are computer-generated using an evolutionary algorithm, and the animals, which are wind powered, are made out of PVC piping.

The valves essentially act like logic gates, allowing water to pass or not depending on the state of the other gates.

Jansen’s creations do not require engines, sensors or any other type of advanced technology in order to walk and react to the environment. As for Boris Ingram’s work, it would be greatly enriched if it were to incorporate a wider range of possible structures and algorithms.

More online references:

• Theo Jansen’s webpage.
• Theo Jansen’s Strandbeests, artfutura.
• Wild Things are on the Beach, Wired Magazine.
• Theo Jansen’s interview.

Originally uploaded by hzenilc.

Models and Simulations 2
11 – 13 October 2007
Tilburg University, The Netherlands

I attended this conference one month ago. Among several interesting talks, one in particular caught my attention. It was given by Michael Seevinck from the Institute for History and Foundations of Science at Utrecht, The Netherlands. His talk was about the foundations of Quantum Mechanics, and there were many NKS related topics that it brought  to mind. He talked about reconstructing Quantum Mechanics (QM) from scratch by exploring several restricted models in order to solve the so-called measurement problem, to deal with the nonlocality of quantum correlations, and with its alleged non-classicality, there being  no consensus on  the meaning of Quantum Mechanics  (Niels Bohr said once: “If you think you have understood quantum mechanics, then you have not understood quantum mechanics.”—More quotes of this sort on QM here).  The restrictons chosen in order to reconstruct the theory must be physical principles and not  theoretical assumptions. In other words, one approaches the problem contrariwise than is traditional, taking the least possible restrictions and exploring the theories that can be built thereon. The speaker characterized  this approach  as the “study [of]  a system from the outside” in order to ”reconstruct the model”. It is basically a pure NKS approach: “Start from a general class of possible models and try to constrain it using some physical principles so as to arrive at the model in question (in this case QM).”

One can then proceed to ask such questions as how one might identify QM uniquely, what it is that makes QM quantum, what set of axioms in the model is to be used, and which of them are necessary and sufficient? The question of meaning, previously asked of the formalism, is removed, and bears, if at all, only on the selection and justification of  first principles. Seevinck came up with the following interesting statement: “The partially ordered set of all questions in QM is isomorphic to the partially ordered set of all closed subspaces of a separable Hilbert space” (one of Mackey’s axioms in his axiomatisation of 1957). He added: “They (the principles)have solely an epistemic status. The personal motives for adopting certain first principles should be bracketed. One should be ontologically agnostic. The principles should be free of ontological commitment.” And further: “…axioms are neutral towards philosophical positions: they can be adopted by a realist, instrumentalist, or subjectivist.” He cited Clifton, Bub and Halverson who provided the following quantum information constraints used to derive quantum theory:

1. No superluminal information transfer via measurement.

2. No broadcasting

3. No secure bit commitment

Seevinck’s methodology in further detail is: Start with a general reconstruction model with a very weak formalism. Gradually see what (quantum) features are consequences of what added physical principles, and also see which features are connected and which features are a consequence of adding which principle. One thereby learns which principle is responsible for which element in the (quantum) theoretical structure.

One can generate further foundational questions over the whole space of restricted models, e.g.  how many of them:

- forbid superluminal signalling?

- allow nonlocality, and to what extent?

- solve NP-complete problems in polynomial time?

An important question which arises concerns whether intrinsic randomness would be of a different nature in different models or whether all of them would yield to deterministic randomness.

His talk slides are available online. Highly recommended.

Among other interesting people I met was Rafaela Hillebrand, of  the Institute for The Human Future at Oxford University. The Institute’s director, Nick Bostrom, has proposed an interesting theory concerning the likelihood that our reality is actually  a computer simulation. I have myself approached the  question in my work on experimental algorithmic complexity, in particular in my work on  the testability and the skepticism content of the simulation hypothesis. I will post on that subject later. The subject of thought experiments–in which I have an interest– was one that came up frequently.

Researchers at Berkeley working to unlock the potential of nanoscience:

High Definition Nanotechnology video from KQEDAmazing how nature produces its own nanodevices, such as motors like the flagella that allow spermatozoa to swim. Imagine how many structures can be found by exploring the universe of possible simple nanostructures! We also know that given a few elements, computing devices are capable of universal computation (see my previous post on the smallest universal Turing machine). So one could potentially provide  nanomachines with coded instructions to  perform just about any task–of course within the constraints of their mechanical capabilities.Further references available online from molecular to nano-computing:

- Tseng and Ellenbogen, Toward Nanocomputers, Science 9 November 2001.
- The world’s smallest computer made entirely of biological molecules, News Medica, 2004.
- Beckett and Jennings, Towards Nanocomputer Architecture
- DNA Computer Works in Human Cells, Scientific American 2007.

My paper On the Kolmogorov-Chaitin complexity for short sequences, coauthored with my PhD thesis advisor Jean-Paul Delahaye has been published as a book chapter in:RANDOMNESS AND COMPLEXITY, FROM LEIBNIZ TO CHAITIN, edited by Cristian S. Calude (University of Auckland, New Zealand) and published by World Scientific.

An extended draft version of this paper can be found in arXiv here and the webpage we have set up for our research on what we call Experimental Algorithmic Theory can be accessed here. The results of our ongoing experiments will be frequently published on this site.The book is a collection of papers contributed by eminent authors from around the world in honor of Gregory Chaitin’s birthday. It is a unique volume including technical contributions, philosophical papers and essays.

I presented our paper at the NKS Science Conference 2007 held at the University of Vermont, Burlington, U.S. The conference blog has an entry describing my participation.

From left to right: Hector Zenil, Stephen Wolfram, Paul Davies, Ugo Pagallo, Gregory Chaitin, Cristian Calude, Karl Svozil, Gordana Dodig-Crnkovic and John Casti.

In the “Period 1 Cycle English Dictionary” published by “No way, Inc.” (it’s been said to be the most accurate dictionary ever) one can read:

dog (Noun) : a dog is a dog.

The lazy creators of this dictionary appear to have forgotten what is broadly accepted by common sense. A definition would not be a definition if it were cyclical and self-referential, simply because one would, theoretically, fall into an infinite loop trying to define an unknown word that has as part of its definition the word itself.

Let’s leave aside the fictional Period 1 Cycle English Dictionary, which I just made up, and consider how things work in a regular dictionary. Assume you want to know the definition of a word. Looking it up, you’ll encounter a set of ordered words in the language in question(if the dictionary is not a bilingual one). Let’s prove that in the end, any definition of a word w in a closed finite dictionary is cyclic with period p_w:

Proof sketch:

We may hypothesize that the dictionary (D) is closed under the operation “the definition of” (i.e. all words in the definition of a word have a definition in the dictionary). One can also safely assume that all words in D have a non-empty definition (given the definition of “dictionary” itself!) and that the dictionary is finite. Then if w is a word in D, the orbit of the successive definitions d_1,d_2,…d_n, … for w (listed as sets of words themselves) will end up eventually crossing the path d_w={d_1,d_2,…d_n, …} because d_w cannot be an infinite sequence without repeating elements in D– the finite nature of D would necessitate  coming back to the original word itself after a certain period p depending on the word, making every word w in the dictionary cyclic with period p_w. 

If all the definitions in a dictionary are then cyclic and enclosed in the dictionary itself, where does the analytic knowledge that a dictionary seems to provide come from? When one consults a dictionary in a given language, one already knows some words in that language, but assuming one doesn’t, would a dictionary be completely useless, as the issue of ultimately cyclic self-referential definitions seems to suggest?

One could conceive of a dictionary as a whole as a syntactical knowledge container with no actual knowledge in it– to someone who does not bring to it any knowledge of the relevant language. One may wonder how something so perfectly self-referential could actually be of use, as dictionaries seem to be. Is it indeed because you always know some, or indeed many, words of a language already when you consult a dictionary? For since every word is defined by other words in the same language, looking up the meaning of a given word would lead you to other words and yet others and eventually back to the first word, the word whose meaning you set out to learn. This would be the case even if  the dictionary were bilingual, and the meaning of the word you wished to check was given in a second language. Thus all dictionaries are perfectly circular, closed, self-referential sources.

However, the analytical knowledge in a dictionary does not come from the definitions as words, but from the word net underneath, where the words are connected in some, perhaps unique fashion(modulo exact synonyms) to each other. That’s what quite successful projects such as WordNet and functions like WordData in Mathematica are about. The power of being able to analyze the language as a net in a computable way is priceless for the progress of computational linguistics and linguistics in general.

 

 

 

For example, if “chair” is connected to “table”, “office”, “dining room”, etc. it should be easy to map it to its equivalent in any other language. Unless the target language doesn’t have the concept “chair” as referring to a seat placed at a table in a dining room (which was perhaps the case in some Asian countries before colonialism), together with an explicit cognitive representation of it.

Of course problems arise when considering the mappings between words having the same written form but different senses. The word “chair,” for example, is a seat, but may also mean the officer who presides at a meeting. Also posing a problem would be cases of words being mapped to or from words belonging to different parts of speech, such as ”personne” in French, which  maps onto two completely different words in Spanish: ”persona” and “nadie”,  a noun and an adjective respectively, with completely different connections and different supporting nets. So even when it seems that most relations might be surjective, the general case is certainly not bijective, and that applies to homonyms too, which often creates ambiguities in translation. However, the supporting network  would be able to uncover this fact and solve a possible ambiguity based on  context by extending the word network to encompass the ambiguity. In other words, if a subnet cannot be uniquely mapped, extending it should eventually solve the ambiguity. What one would need  is a corpus big enough to build such a network once and for all and then simply make comparisons at the network level. This could work even for completely new or unknown languages, either dead, living or artificial, assuming that they share a part of our actual reality and hence some part of our mental representations  (In a sense this is what Champollion did when he deciphered the Rosetta stone– he discovered a partial mapping of a subnetwork of words from an unknown language – Egyptian – to a subnetwork of a known one – Greek). In the final analysis, each language has a single unique network (changing slightly through time but remaining well connected and strong enough to make  it unique and recognizable while being isomorphic with that of any other language).  All languages could be identified by their fingerprints -their word net. This kind of analysis would identify the lack of a word net structure in hoax languages, such as perhaps the Voynich manuscript.

Having  established that, what about mining the world of all possible meanings, the world of all possible translations, and the world of all possible ideas? We wouldn’t have the problem of distinguishing between a coherent idea and a non-coherent one since the network would provide some minimal coherence. Thus the net-into-the -net approach would give us a way of translating from word to word and from phrase to phrase and from idea to idea as faithfully as possible in most cases, since in the end all of us as human beings share a single reality, though we perhaps approach it from different points of view.

Again, the analytical knowledge in a dictionary comes from the net connecting the words, so even if someone does not know English at all, I would say that he would be able, albeit with considerable difficulty, to learn English just by deducing the net connecting objects, in other words, by mapping his own mental  representations of objects onto words in the  English dictionary. In the process he could encounter some ambiguities, but the further he goes, the more of these he would be able to resolve. On the other hand,  speakers of those languages in which “chair” does not exist, both in the language itself and as a real object in the culture,   would be able to deduce what  a chair is by tracking its  relations with the objects they know and for which they do have mental representations and the phonemes to externalize them. So the problem of translation, which began with the mapping of word onto word and then phrase onto phrase  with statistical tools,  becomes, with this approach, a matter of  mapping net to net.  Indeed this seems to be the approach adopted  by Meaningful Machines.  

These ideas could be carried to the limit by taking the sum total of human languages and enquiring into the mapping between such a network and our cognitive representations. Such a move would provide grounds  for rebutting the Chinese room argument, since in the end it does not matter whether someone inside the room has no knowledge at all of a language; he would be able to map what he is mechanically translating onto his own mental representations, generating what, according to the argument, could not be generated: understanding. Because Searle’s  idea was, as I recall, to build up a case against A.I. in terms of the meaningless of the Turing test and true AI in general.

One may actually use a dictionary without knowing a single word in it! That is because there is a mapping between the word net in the dictionary and one’s own language, or even better, a mapping (not necessarily injective or surjective) between the word net of the dictionary and your cognitive personal word net.

Oversimplifying, translation might be reduced to the search for the homomorphism between algebraic groups, with each group G={W,d} being a language dictionary, with W the set of words in that language and d the closed operation “definition of”. One can then see each definition as an oversimplification of a directed, ordered graph g={w,s}, with w the set of vertex words involved in the definition and s the ordered (since definitions may be not commutative) set of edges for an ideally formal- enough language dictionary.

Bad practices of cyclic definitions in a dictionary should rather be expressed as the practice of period 1 cycles, i.e. words that have in their  immediate definition the word itself.

This post is  related to a previous post titled “Meaning against A.I.“

Ist das Universum ein Computer?

http://www.dtmb.de/Aktuelles/Aktionen/Informatikjahr-Zuse/

Germany, November 2006, Informatik Jahr
Deutschen Technikmuseum Berlin
From Konrad Zuse’s Invention of the Computer to his “Calculating Space” to Quantum Computing.

Lesson One: For someone with a hammer in his hand the world seems to be a  nail. Joseph Weizenbaun.

Lesson Two: Knowing the input and the transition function of a Turing machine we know everything about it. Marvin Minsky.

Dr. Zuse’s futuristic drawing

- The first talk entitled  ”What Can We Calculate With?” by Prof. Dr. Bernd Mahr from the Technische Universitat of Berlin was a very good introduction to the standard theory of computation based on Turing’s model and classical mathematical logic. His remarks on the time when computing arose from math because the Greeks discovered they were unable to compute the square root of 2 were interesting. He pointed out some evident but not always explicit facts: Calculation has a subject (the individual who calculates), an object (what is calculated),  a medium (how it is calculated), and a symbolic representation (the language -binary, for instance). His use of the Leibniz medallion for explaining   starting points, ending points and transitions in a calculation was elementary but interesting (transitions: intermediate calculations). Further explanations of reversibility and non-linearity using transition nets were also illuminating. The context of a calculation (or computation) and the strong relation between the computation itself and its context is  such that it is sometimes difficult to distinguish them. Since any process can be seen as an object in itself, the context can become the calculation and the context of the context too. In some way, as we know, the concept of calculation is a constraint of a part of a calculation, and then it is defined in terms of an input, an ouput and a transition. He pointed out too that behind many of these complicated systems there is a Turing machine. It  is no longer visible from the top, but it is there.

Dr. Konrad Zuse

- Dr. Horst Zuse’s  talk titled “Konrad Zuse’s Calculating Space (Der Rechnende Raum)”:
Dr. Konrad Zuse’s son, Dr. Horst Zuse made some interesting remarks about his father’s book “Calcualting Space”,  in which  Dr. Zuse proposes studying  the universe as a digital system, specifically a cellular automaton. Dr. Horst Zuse is a professor at the Technische Universitat of Berlin and his personal webpage can be found at: www.cs-tu-berlin.de/~zuse and www.zuse.info

Dr. Konrad Zuse’s son, Dr. Horst Zuse

Dr. Zuse’s father’s main  question was: “Is Nature Digital, Analog or Hybrid?” It seems that he tended to answer “Digital,” proposing a No-Yes value language (binary). His thoughts were published in the Nova Acta Leopoldina. He evidently did acknowledge that there could be problems attempting to reconcile an explanation of the Universe in terms of Cellular Automata with Quantum Mechanics and General Relativity.

According to Konrad Zuse, laws of physics could be explained in terms of laws of switches (in the computational sense). Physical laws are computing approximations captured by the formulas in our models. He saw that differential equations could be solved by digital (hence discrete) systems.

Dr. Carl Adam Petri at the Berlin conference

- Dr. Carl Adam Petri (yes, the creator of the Petri nets!) on “Rechnender Netzraum” or “Computing Net Universe”:
According to Dr. Petri, at the Planck length quantum computing can be described by digital systems using combinatorial models (net models, kennings, combinat), and therefore the universe can be studied using discrete nets which are even capable of explaining quantum and relativistic fundamentals like Bell’s theorem and Heisenberg’s uncertainty principle. That would mean that discrete systems (for instance those proposed by Stephen Wolfram) would suffice to explain even quantum and relativistic phenomena.

According to Petri, measurement is equivalent to counting. For instance in S.I. one second is 9192631770 Cs periods. In fact Norbert Wiener proposed some axiomatics of measurement.

The correspondence of Petri nets with the Heisenberg uncertainty principle arises from the limitations of our observational capacities when carrying out measurements. When two different types of observations are performed, -for example momentum p and position q- we can only see p or q in a chain of succesive events related by a causality net. His nets as well as his explanations of such phenomena are very neat and elegant. The relevant slides  on causality and linear logic may be found  at:

http://www.informatik.uni-hamburg.de/TGI/mitarbeiter/profs/petri/slides/

He also distributed a CD with his slide presentation at the conference.

For Petri, the Universe is a Petri Net.

Dr. Seth Lloyd’s presentation at the conference in Berlin, Germany

- Dr. Seth Lloyd’s talk entitled “The Universe is a Quantum Computer”:
According to Dr. Seth Lloyd, professor at MIT, because quantum mechanics is the most fundamental and foundational theory of the universe,  assuming it would lead us to conclude that  the whole universe is a quantum computer computing itself. The input consists of basically random processes (which he characterizes using the metaphor of  programmer monkeys) and the outcome is all that we see around us. Because an elementary particle interacts with others and changes its state, he argues that each particle can be seen as information, as a qubit (for quantum binary digit), which unlike a bit  can be in 0,1 or both states at the same time according to the quantum property known as  entaglement or superposition. When a particle interacts with other particles they change their states according to a quantum logical gate.

Therefore, his conclusion is that the universe is not only a computer but a quantum computer. However some questions arise:

1. What does he mean by quantum computing? According to the standard model (by Deutsch) quantum computing is Turing computable (disregarding run time). If Lloyd is assuming the standard model then the universe is indeed a quantum computer, but even more remarkably (since we have scientific and philosophical hypotheses like the Turing thesis) it is Turing computable. However, if he is assuming the more general quantum mechanics model, let’s say the standard model in physics (which basically assumes the possibility of harmless continuity rather than inquiring into it) he is saying that the universe is not a computer (since the term derives  from what we mean by Turing computable and hence covers  digital computers too). So the assumptions made are significant and cannot be glossed over if one wishes  to argue convincingly that  the universe is  a computer in some standard sense. If what we  assume to be computing is something that seems to be deterministic or rule-based, the concept becomes fuzzy and  additional remarks need to be made.

2. What if a quantum particle encodes more information than just a 1 or 0 for the spin or any other quantum property? Let’s say a third value, or even worse, a non-computable number of values. In quantum mechanics for example, the superposition of a particle assumes an infinite and non-countable number of places since it is in all the spectra at the same time. If space/time is a continuum then it is evidently in a non -countable number of positions, which leaves us with  a non-computable model, or at least with something that’s neither a Turing-computable model nor a standard quantum-computable (namely Deutsch) model. And this is not a simple assumption since it requires anotherTuring-type thesis which in the final analysis does not answer the most fundamental  question, i.e. whether the universe is a computer or not and if it is, what kind of computer (in the computational power sense) it is.

I raised these questions with Seth Lloyd and I will be posting his answers soon.

Seth Lloyd at the conference in Berlin

A remarkable idea proposed by Seth Lloyd concerned “hacking the universe”. As Charles Bennett used to say, a computer is a handicapped quantum computer. So if Lloyd is right, a computer is not only a handicapped quantum computer but it is not taking advantage of the full computational power of the universe and it is just patching the universe instead of hacking it, as it would be in its power to do. By contrast, a quantum computer uses some particles that are already computing “something” (nothing less and nothing more than the processes in the universe ) to perform the computation that we want it to perform. It can be said to be  ”hacking the universe” in Lloyd’s terms.

On the other hand, if the notion of programmer monkeys is valid it should be possible to test it experimentally. Under the supervision of M. Jean-Paul Delahaye, computer science professor at the University of Lille I (http://www2.lifl.fr/~delahaye/) we are undertaking this task. We are exploring  Lloyd’s quantum computational universe (or at least a handicapped but representative part, the recursive computational universe), applying some complexity measures (universal distribution, average-case complexity or Levin’s measure) in order to uncover the monkeys behind the Universe, or in other terms, to analyse the average distribution of randomly discrete systems with random inputs.

Is Seth Lloyd falling into the carpenter’s problem of thinking that the universe is a nail and the moon made of wood?  Is it because he is a quantum computer scientist that he thinks the universe is a quantum computer? He argues of course that the charge is unfair, but then we have been told by Dr Petri  that the Universe is in fact a Petri Net which probably  needs neither strong randomness nor quantum mechanics!

Here  is a video online in which he explains much of this:

http://www.edge.org/video/dsl/EF02_Lloyd.html

Dr. Zuse’s futuristic drawing 2

- Jurgen Schmidhuber reprised his algorithmic approach to the theory of everything in his talk entitled   “The program that computes all computable universes”.
Jurgen Schmidhuber’s major contribution probably is his Speed Prior concept, a complexity measure similar to Algorithmic Information Complexity, except that it is based on computation speed and not program length. i.e. the fastest way of describing objects rather than the shortest.
There is more information on his website: http://www.idsia.ch/~juergen/ (where he includes an unfavorable review of  NKS) and in his slide presentation on the Speed Prior at: http://www.idsia.ch/~juergen/speedprior/sld001.htm
Of course Schmidhuber himself has identified a problem with the Prior measure: If every possible future exists, how can we predict anything?

Other interesting talks on philosophical issues: If the Universe is a computer, therefore the human mind should be a computer too.
Is “the Universe is a  computer” a metaphor?
My answer: The metaphor is “The Universe is not a Computer”

Lesson Three: Metaphors can be reversed.

Kovas Boguta’s  talk was titled ”Is the Computer a Universe?” In it he pointed out the richness of mining the computational universe of simple programs.

Because we were together during the gala dinner I had an interesting exchange with Dr. Konrad Zuse’s son, Dr. Horst Zuse (Also at our table were the Technikmuseum director Dr. Dirk Bondel and  my colleague Kovas Boguta from Wolfram Research, among others). He shed some light on his father’s interactions with Alan Turing ( none apparently),  with von Neumann (some interaction regarding the controversy over who first built a digital computer and concerning von Neumann’s architecture, which  our current digital computers do not conform to, the ALU being separated from the memory as it is in Zuse’s conception but not in  von Neumann’s original design).

Zuse’s Z1 first computer “the input device, something equivalent to the keyboard, at the Technikmuseum in Berlin”

- Symbolic Visualizations, University of Texas:http://cvcweb.ices.utexas.edu/ccv/projects/VisualEyes/SymbVis/index.php- Proof nets and zero-knowledge proofs.

Kurt Godel workshop for studying his legacy and writings. Lille, France, May 19-21, 2006

My thoughts, ideas, references, comments and informal notes:

- The wheel machine, a machine for real computation which I am proposing -as a thought experiment- in a forthcoming paper  on the Church-Turing thesis -Yes, one more paper on the CT thesis!- with comments on Wilfried Sieg’s paper entitled “Church Without Dogma: Axioms for Computability”

- “In case Cantor’s continuum problem should turn out to be undecidable from the accepted axioms of set theory, the question of its truth would loose its meaning, exactly as the question of the truth of Euclid’s fifth postulate in Euclidian geometry did”. Godel replies: “It has meaning anyway, as Euclid’s fifth postulate gave rise to other now accepted mathematical fields.”

- Godel Gibbs Lecture and his dicotomy on absolutely undecidable propositions and the computational power of the human mind (Turing did great work… but he was wrong when he proposed his formal theory as a model of human thought…)

- New contacts and references: Olivier Souan, Rudy Rucker, Karl Svozil

Mark van Atten’s “On Godel’s awareness of Skolem’s lecture”.
Rick Tieszen

- Herbrand on general recursive functions, letter to Godel.

- Leibniz’ influence on Godel’s arithmetization?

- Sources: Godel Editorial Project. Firestone Library, Princeton University. I.A.S. Marcia Tucker, librarian for Godel papers.

- Godel’s concept of finite procedure as the most satisfactory definition of computation. “A machine with a finite number of parts as Turing did” or “finite combinatorial procedure” as a definition of an algorithm, mechanical or computational procedure.

- Computation’s main constraints: boundness and locality (paper from Hernandez-Quiroz and Raymundo Morado).

- Aphorisms and autoreference (Gabriel Sandu and Hinttika)

- Feferman on Turing

- Is Sieg’s paper and the question of “finite machine=effective procedure” a tautology? In fact such an approach seems to be one of the most strict versions of the Turing Thesis, and even though both Church and Turing probably did propose it in such a strict sense, extensive versions of the thesis have traditionaly covered more content, but even when it is strictly stated that there is still space for a thesis, it is neither proved nor provable from my point of view, and most authors would concur, though some clearly would not. I will comment on this more extensively later, since this was one of my Master’s topics and merits a post by itself.

- Putnam’s thought experiment on cutting all sensorial inputs. Solution: It is impossible in practice. However, machines are an example in a sense, and that is why we do not recognize intelligence in them – they are deprived of  sensorial capabilities.

Yes, Godel found an inconsistency in the U.S. constitution. My answer: One? Certainly a bunch. That’s why we need lawyers, who make them even worse.

A novel approach to cooking

The algorithm:

1. Consider the space of all possible recipes.
2. Enumerate the ingredients per food style (Mexican, Chinese, French, Italian, and so on).
3. Take the subsets of the ingredients per food-style set.
4. Begin with a random choice of initial ingredients.
5. Mix them by following a set of simple cooking rules like baking, sprinkling or heating.
6. Season with  salt and pepper to taste.

One could use a robotic device and see what results. One would also have to set up an evaluation scheme to decide which dishes are good and which are bad– either a human tester or a reliable sensor such as a modern “artificial nose” (essentially a microarray-like object with a number of different receptors).

However, since the testing procedure is undecidable in general, in the case of any given recipe, neither a human nor an automatic device can decide whether or not the end result will taste good.

It would also be of interest to find out which “cooking rules” lead to which patterns of smell and taste. For instance,  one would expect  any dish involving sugar, butter, and flour  to taste reasonably good provided the rules do not yield a “burnt offering”.  Of course, to make things really interesting, one should introduce additional ingredients, including ingredients  which aren’t traditionally combined with the basic ones.

For French cuisine, butter would be a primitive, for American food, ketchup, for  Mexican, chili, for Chinese,  rice,  for Italian, tomato. French and Italian food share  cheese as a primitive; cooking “au gratin” would always be a reasonably safe bet.  Since primitives are important, you cannot cook pancakes au gratin, and ketchup would be safer if you wanted your dish to meet with the approval of the American tester.

An ingredient is a primitive in the sense that any dish in a given style can use either more or less of it and still retain  its character as an exemplar of that style.  Primitives can be used at will,  including in emergencies or when the rules happen to yield nasty results.

One might investigate reducible shortcuts, such as preparing something faster or skipping steps. However, most of the recipes are irreducible, making it impossible to take shortcuts without going through the whole preparation.

With some effort one can also prove that there exists a universal algorithm which,  given a recipe and the initial ingredients,  is able to reproduce any other recipe.

Concerning time/space complexity, it remains an open question whether the preparation time can always be reduced to a polynomial.

One can also ask about the algorithmic complexity of a dish, since one can measure its information content–the number of ingredients involved, the number of rules and steps applied, whether the rules were iterated, nested or random- looking.  However, since, as noted above, the procedure is undecidable, the only known way to approach the question of the complexity of a dish is to compress it into a piece of universal tupper-ware and see what happens.

Another open question: Given a compressed ingredient, how many other ingredients can one compress? It is believed that this yields  a classification of  food types, defining a threshold between those in a lower/trivial class and  more sophisticated items.