Anima Ex Machina

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, there is a tendency to assume that evolution os the sole factor in designing nature while it may not actually be the main 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].

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.

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 information and computation, according to which life has managed to speed up its rate of change by channeling information efficiently between generations, together with a rich exchange of information with the outside by a process that while seemingly random, is in fact the consequence of interaction with other algorithmic processes.

Think a bit further about it. Evolution seems deeply connected to biology on Earth, but as part of a larger computation theory it might be applied anywhere in the universe just as the laws of physics do. Evolution may be formulated and explained as a problem of information transmission and channeling, pure communication between 2 points in time. If you want to efficiently gather and transmit information it may turn out that biological evolution may be not the cause but the consequence.

The theory of algorithmic information (or simply AIT) on the other hand does not require a random initial configuration (unfortunately perhaps, nor any divine intervention) 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.

To celebrate Gregory Chaitin’s 60th birthday Stephen Wolfram decided to design a medal for him.

In the mid 1960s, while still a teenager, Chaitin created algorithmic information theory (AIT), which combines, among other elements, Shannon’s information theory and Turing’s theory of computability. In the three decades since, he has been the principal architect of AIT. Among his contributions are the definition of a random sequence via algorithmic incompressibility, and his information-theoretic approach to Gödel’s incompleteness theorem. His work on Hilbert’s 10th problem has shown that in a sense there is randomness even in elementary arithmetic.

The idea was to somehow replicate the Gottfried Leibniz medallion, an image of which appears at the bottom of Greg’s home page.

Gregory Chaitin has spent his career working on foundational questions in mathematics and computation, and in some ways he has been a modernizer of Leibnizian ideas. Leibniz may have been the first computer scientist and information theorist. Early in his life he discovered the binary number system and binary arithmetic.

On January 2nd, 1697, Leibniz wrote a letter to Rudolf August, Duke of Braunschweig-Wolfenbüttel, in which he detailed the design of a commemorative coin or medallion which he suggested could be minted in silver. The design he described posited an analogy between “the creation of all from nothing through the omnipotence of God” and the fact that “all numbers [could] be created from zeros and ones”.

So the medal does not commemorate Leibniz’s discovery of binary arithmetic. Rather, his description suggests a medal in which binary arithmetic glorifies God–and the duke. (He proposed that the obverse of the coin bear the Duke’s “face or monogram”).

More on the history of Leibniz’ binary language, the letter and the medallion can be found here (pp. 31-36):

["The binary medallion apparently was never struck*. Numerous writers have based a contrary assumption, in the last analysis, upon having seen some version of its design. The Duke was already 70 years old when he received the medallion proposal in 1697. "(p. 35)

"After a thorough search of the catalogs of applicable coin collections, including all known special Brunswickian collections, Dr. W. Jesse of the Stadtisches Museum Braunschweig reported in his letter of November 2, 1965 that in his opinion, the proposed medallion had never been struck. (p. 51)"

"What actually survives are illustrations in later printings of the letter. Two Versions of Leibniz's Design of the Binary Medallion. They are facsimiles of the ones appearing on the respective title pages of Johann Bernard Wiedeburg's Dissertatio mathematica de praestantia arithmeticae binaria prae decimali (Jena: Krebs, 1718) and Rudolf August Nolte's Leibniz Mathematischer Beweis der Erschaffung und Ordnung der Welt in einem Medallion. Langenheim, 1734. (See pp. 34, 36, 56 for images of the proposed coin, including the obverse side)."]

During the Summer a group of people from Wolfram Research (WRI) led by Stephen Wolfram worked together on the design for Chaitin’s 60th birthday medallion. Stephen and I were keen to incorporate representations of the most definitive elements of Chaitin’s influential career as founder of AIT. It was pretty obvious that Chaitin’s medallion had to include the letter Omega representing his Omega number (Chaitin’s Omega gives the halting probability of a universal Turing machine). We also wanted to show the digits recently calculated by Cristian Calude, since even though the omega number is non-computable, Calude managed to calculate an initial segment by using the binary version of Chaitin’s formula and following Chaitin’s construction with register machine programs (Of course the digits are dependent on the universal Turing machine chosen). The halting and non-halting results for the register machine programs in question were represented by arrows and lines below the letter Omega. Here is the link to Calude’s paper in which he computed the first digits of Chaitin’s Omega number. It includes a section that we used in determining the placement of the arrows in our design:

Cristian S. Calude, Michael J. Dinneen, and Chi-Kou Shu. “Computing a Glimpse of Randomness,” Experimental Mathematics, Vol. 11 (2002), No. 3.

The first 64 bits of Chaitin’s Omega from the paper are:
000000100000010000011000100001101000111111…
0010111011101000010000
However, we decided to use the 40 digits from the standard binary formula version (Chaitin’s original formulation), also calculated by Calude in the same seminal paper:
0001000000010000101001110111000011111010

The upper background of the medallion is a binary circular array conceived by Michael Schreiber and generated with the following code in Mathematica:
Manipulate[Graphics[
{Black, Disk[{0, 0}, p + 2], Table[
Table[{GrayLevel[Mod[a, 2]],
Disk[{0, 0}, q + 1, {2 Pi (a - 1)/(2^q), 2 Pi a/(2^q)}]}, {a, 1, 2^(q),
1}],
{q, p, 1, -1}], White, Disk[]}],
{{p, 3, “bits”}, 1, 8, 1}]

Like Leibniz, we wanted an inscription in timeless Latin, so we began looking for a text to inscribe on Greg’s medallion, one that was related to his seminal work.

One year previously, when I met Chaitin at his office in IBM’s Thomas J. Watson Research Center in Yorktown Heights, New York, he invited me to his home and kindly gave me some of his published books (I already had a couple of them but he completed my collection). In return I sent him a very rare limited edition of a book by Jorge Luis Borges and Alfonso Reyes entitled “La máquina de pensar” (“The thinking machine”). Needless to say I kept a copy for myself! As everybody knows, Borges is a famous Argentinian writer. Reyes is a Mexican writer whom Borges credits as an important influence. Indeed their styles show a degree of similarity. In any case, it turned out that like me, Chaitin liked Borges a lot, but he had never heard of Reyes, whom I happen to like as much as Borges. He told me he had enjoyed the book very much, so some of the first inscriptions proposed for the medal were quotes from Borges. But soon we decided that one of the Leibniz quotations appearing on Chaitin’s webpage would be more appropriate:

*Dieu a choisi celuy qui est… le plus simple en hypotheses et le plus riche en phenomenes.
[God has chosen that which is the most simple in hypotheses and the most rich in phenomena.]
*Mais quand une regle est fort composée, ce qui luy est conforme, passe pour irrégulier.
[But when a rule is extremely complex, that which conforms to it passes for random.]

Greg has suggested that these quotes from Leibniz, among others, are early anticipations of his AIT.

But after further discussions with Stephen, we agreed on two of Chaitin’s own most often quoted statements encapsulating his most seminal contributions: “Everything can be summarized in one thing, but that thing cannot be reached” (In other words: All computable facts can be summarized in Chaitin’s Omega number, but that number is not itself computable); and “Mathematical facts are true for no reason” (or by accident).

Stephen decided to consult a world expert—a friend of his from high school named Armand d’Angour who is now a Classics professor at Oxford. In 2004 he was commissioned by the International Olympic Committee to compose a Pindaric Ode to Athens which was recited at the Olympic Games. The first thing he pointed out was that Leibniz’s inscription (‘omnibus ex nihilo ducendis sufficit unum’) was a hexameter. D’Angour quickly came up with a pentameter as well for Greg, in his words a “perfect classical one-liner” of the kind that kings in antiquity used to reward poets for. Thus we had a full elegiac couplet, the first line of which read as follows:

Everything can be summarized in one thing, but the thing itself cannot be reached
OMNE UNO IMPLICITUR QUOD NON ATTINGITUR IPSUM.
D”Angour suggested that we replace the “o” in “uno” with an Omega letter (‘Everything can be summarised in one Omega, which itself cannot be attained’).
He added that Latin verse aficionados would enjoy the way the first three words ran into each other, thus demonstrating what the phrase connoted.

The second line which at first read:
Mathematical facts are true by chance
MATHEMATICAE PRINCIPIA FORTUITO VERA

was later turned into the pentametric
FORTUITA EVENIUNT VERA MATHEMATICAE.
The truths of mathematics turn out to be fortuitous.

And beneath this the medal read:
Celebrating the work(s) of Gregory Chaitin MMVII:
AD LAUDEM GC MMVII (where the Leibniz version has IMAGO CREATIONIS INVEN GGL).

D’Angour claims that if he were Greg Chaitin, he would be happy to have all this inscribed on his tombstone. If he were Maecenas, he would consider rewarding the poet with a Sabine Farm.

The Latin inscription on Leibniz’s medallion can be rendered thus: “To make all things from nothing unity suffices” (i.e. You can represent every number using just the digit 1). The inscription on Chaitin’s medallion says: “Everything can be summarized in one [Omega], which cannot itself be attained/ The truths of mathematics turn out to be fortuitous”.

Once we had finalized the design, we wondered about the obverse of the medallion. We realized that this was the chance to finally cast Leibniz’ medallion after almost three hundred years! So I went about reconstructing it, noting every single detail. I wrote some Mathematica code incorporating all these details which could be used for an electronic design and finally struck it. Here is the Mathematica notebook. Stephen Wolfram presented the medallion to Chaitin during the NKS Science Conference on the 15th. of July, 2007 at the University of Vermont, Burlington, U.S. The original solid silver medallion was delivered to him on November the 2nd of the same year. Nine more copies were made of Merlin gold, one of which belongs to me (pictures below). The others were given to Chaitin’s relatives, and to Armand D’Angour, Cristian Calude, Jeremy Davis and Stephen Wolfram. Two were retained by WRI’s design department for the archive.

In an exchange of emails, Seth Lloyd and I discussed the topic I wrote about some posts ago. Here is some of it.

According to Lloyd, there is a perfectly good definition of a quantum Turing machine (basically, a Turing machine with qubits and extra instructions to put those qubits in superposition, as above). A universal quantum computer is a physical system that can be programmed (i.e., whose state can be prepared) to simulate any quantum Turing machine. The laws of physics support universal quantum computation in a straightforward way, which is why my colleagues and I can build quantum computers. So the universe is at least as powerful as a universal quantum computer. Conversely, he says, a number of years ago he proved that quantum computers could simulate any quantum system precisely, including one such as the universe that abides by the standard model. Accordingly, the universe is no more computationally powerful than a quantum computer.

The chain of reasoning, to jump to the quantum computer universe view, seems to be 1 and 2 implies 3 where 1, 2 premises and the conclusion 3 are:

1 the universe is completely describable by quantum mechanics
2 standard quantum computing completely captures quantum mechanics
3 therefore the universe is a quantum computer.

Seth Lloyd claims to have proved the connection between 1 and 2, which probably puts the standard (or some standard) theory of quantum mechanics and the standard quantum computing model in an isomorphic relation with each other.

Lloyd’s thesis adds to the conception of the Universe as a Turing computer an important and remarkable claim (albeit one that depends on the conception of the quantum computer), viz. that the Universe is not only Turing computable, but because it is constituted by quantum particles which behave according to quantum mechanics, it is a quantum computer.

In the end, the rigid definition of qubit together with the versatility of possible interpretations of quantum mechanics allows, makes difficult to establish the boundaries of the claim that the universe is a quantum computer. If one does assume that it is a standard quantum computer in the sense of the definition of a qubit then a description of the universe in these terms assumes that quantum particles encode only a finite amount of information as it does the qubit, and that the qubit can be used for a full description of the world.

Quantum computation may have, however, another property that may make it more powerful than Turing machines as Cristian Calude et al. have suggested. That is the production of indeterministic randomness for free. Nevertheless, no interpretation of quantum mechanics rules out the possibility of deterministic randomness even at the quantum level. Some colleagues, however, have some interesting results establishing that hidden variables theories may require many more resources in memory to keep up with known quantum phenomena. In other words hidden variable theories are more expensive to assume, and memory needed to simulate what happens in the quantum world grows as bad as it could be for certain deterministic machines. But still, that does not rule out other possibilities, not even the hidden variables theories, even if not efficient in traditional terms.

This is important because this means one does not actually need ‘true’ randomness, the kind of randomness assumed in quantum mechanics. So one does not really need quantum mechanics to explain the complexity of the world or to underly reality to explain it, one does require, however, computation, at least in this informational worldview. Unlike Lloyd and Deutsch, it is information that we think may explain some quantum phenomena and not quantum mechanics what explains computation (neither the structures in the world and how it seems to algorithmically unfold), so we put computation at the lowest level underlying physical reality.

Lloyd’s thesis adds to the conception of the Universe as a Turing computer an important and remarkable claim (albeit one that depends on the conception of the quantum computer), viz.  that the Universe is not only Turing computable, but because it is constituted by quantum particles which behave according to quantum mechanics, it is a quantum computer computing its future state from its current one. The better we understand and master such theories, the better prepared we would be to hack the universe in order to perform the kind of computations–quantum computations–we would like to perform.

I would agree with Rudy Rucker too as to why Seth Lloyd assigns such an important role to quantum mechanics in this story. Rudy Rucker basically says that being a subscriber to quantum mechanics, Lloyd doesn’t give enough consideration to the possibility of deterministic computations. Lloyd writes, “Without the laws of quantum mechanics, the universe would still be featureless and bare.” However, though I am one among many (including Stephen Wolfram) who agree  that it is unlikely that the universe is a cellular automaton, simply because cellular automata are unable to reproduce quantum behavior from empirical data (but note that Petri and Wolfram himself attempt explanations of quantum processes based on nets), there’s  absolutely no need to rush headlong into quantum mechanics. If you look at computer simulations of physical systems, they don’t use quantum mechanics as a randomizer, and they seem to be able to produce enough variations to feed a computational universe. Non-deterministic randomness is not neccesary; pseudorandomness or unpredictable computation seem to be enough.