guest post by Leonard Adleman

About 50 years ago Kolomogorov assigned to each finite binary string a non-negative integer that measured that string’s ‘descriptive complexity’. Informally, is the length (in binary) of the shortest (Turing machine) program that with the empty string as input, outputs and halts. A related measure of descriptive complexity is , where denotes the length of . A simple string like:

can be produced by a very short program; hence is near 0. But if is a ‘random string’ (e.g. obtained by flipping coins), then, with high probability, it cannot be produced by a program significantly shorter than itself; hence will be near 1.

If I asked you to produce (using a computer) a thousand strings of length one million with near 1, it would be easy to do; just flip a lot of coins. If I asked you to produce a thousand strings with near 0, that would also be easy. For example, you could start with a short random string and repeat it a lot. Actually, if I chose my favorite and wanted a thousand strings of length one million with near , then a mix of the preceding approaches can be used to produce them. So strings with a desired are not rare.

Now let’s consider ‘deep strings’*. I will be informal here, but the underlying theory can be found in my Time space and randomness.

For all binary strings , we assign a value that measures the ‘depth’ of . is obtained by considering both the size and the number of steps used by each program that with the empty string as input, outputs and halts**. has the following properties:

If there is no short program to produce , then is small.

, then is small. If there is a short program to produce and it uses few steps, then is small.

and it uses few steps, then is small. If there is a short program to produce , but all short programs to produce use lots of steps, then is large. Roughly speaking, the more steps small programs use to produce , the larger will be.

Informally, we call a string with large ‘deep’ and one with small ‘shallow’. A few examples may help.

Consider a string obtained by flipping coins. With high probability there is no short program to produce , hence is shallow.

Now consider above. Since there is a short program to produce and that program uses few steps, is shallow.

Now treat as a number in binary (i.e. ) and consider the prime factorization. The fundamental theorem tells us it exists and will be about one million bits long. But, unless is somehow special (e.g. a prime times a very smooth number), its prime factorization may be very very deep. A short program can generate the prime factorization (just generate one million 1s with a short program and then give it to a short factoring program). But if it turns out that factoring can’t be done in polynomial time, then perhaps all short programs that generate the prime factorization use a huge number of steps. So the prime factorization would have a very large . Conceivably, since steps on a computer use time and energy, the prime factorization can never be realized. It is not a long string (only one million bits) but it may exist only in theory and not in reality.

If I asked you to produce even one string of length one million with near that of the prime factorization of , could you do it? I would not know how and I suspect that as a practical matter it cannot be done. So strings with such a are rare.

Here is a string that does exist in our universe and that (I suspect) is quite deep:









In fact, this string is the prime factorization of . We should not expect it to be as deep as the prime factorization of , but we should still expect it to have considerable depth. There is even some experimental evidence that supports this view. How did I get this prime factorization? I got it from: Factorization of a 1061-bit number by the Special Number Field Sieve, by Greg Childers. So it was easy to get. Well, easy for me anyway; not so easy for Childers. He reports that the factorization took about `3 CPU-centuries’ using the Special Number Field Sieve. Other than by factoring , how could you ever come to write this pair of primes? I venture to say that since the Special Number Field Sieve is the fastest known algorithm for factoring numbers of this kind, no method available today could have written these primes (for the first time) using fewer steps (and hence less time/energy).

The situation might be compared to that of atomic physics. I am not a physicist, but I suppose it is possible to theorize about an atomic nucleus with a million protons. But what if I want to create one? It appears that producing transuranic elements takes huge amounts of time/energy and the greater the number of protons, the more time/energy it takes. It is even conceivable (to me at least) that there is not enough time/energy available (at least on earth) to actually produce one. Like the prime factorization of , it may exist in theory but not in reality. On the other hand, physicists from Russia and America, using lots of time/energy, have created an atomic nucleus with 118 protons called Ununoctium. Ununoctium is analogous to Childers’ prime factorization; both exist in reality; both were very costly to create.

But my focus here is neither ontology nor physics; it is money. Recently Bitcoins and Bitcoin-like things have captured the world’s attention. I suspect that this kind of currency will revolutionize the nature of economies, and consequently the nature of governance. Personally, I am agnostic on whether this is a good thing or a bad thing. But, in any case, I am wondering if ‘depth’ might form part of a theoretical basis for understanding new currencies. I must confess to knowing few details about Bitcoins; consequently, I will start from first principles and consider some of the properties of currency:

Mass : A currency may have mass or may be massless. U.S. dollars and gold have mass. Bitcoins are massless. The efficiencies of the Internet market require a massless currency. Imagine running eBay or Amazon using gold. U.S. dollars have mass, so we do not actually use them for transactions on the Internet, rather we use credit card numbers to create massless IOUs that promise to deliver (but, in fact, seldom actually deliver) U.S. dollars in the future. These IOUs are essentially a massless currency.

: A currency may have mass or may be massless. U.S. dollars and gold have mass. Bitcoins are massless. The efficiencies of the Internet market require a massless currency. Imagine running eBay or Amazon using gold. U.S. dollars have mass, so we do not actually use them for transactions on the Internet, rather we use credit card numbers to create massless IOUs that promise to deliver (but, in fact, seldom actually deliver) U.S. dollars in the future. These IOUs are essentially a massless currency. Production : Who can produce or destroy the currency? This has been an important political, legal and economic issue for millennia. In the west, coins began to appear at about the time Solon ruled Athens. Solon realized that he could manipulate the value of coins. And so he did. Solon’s lesson has not been lost on governments ever since. In its simplest form: if you owe dollars, you print dollars, and voila! no more debt. Creditors don’t like debtors to do that, so they want to be paid in a currency that no one can produce more of. Gold comes close. You can produce more gold but you have to spend a lot of time/energy in a mine to do so. Production also includes counterfeiting. Counterfeiting comes in at least two important forms: de novo-counterfeiting (build a press) and duplication-counterfeiting (get a xerox machine). For now, all massless currencies are in digital form and are vulnerable to duplication-counterfeiting since computers make it cheap and easy to create perfect copies of digital notes (a unit of a currency will be called a ‘note’). As a result, massless currencies will typically be associated with systems that mitigate this threat. Perhaps the elaborate ledgers implemented by the creators of Bitcoin are an attempt to deal with the threat of duplication-counterfeiting.

: Who can produce or destroy the currency? This has been an important political, legal and economic issue for millennia. In the west, coins began to appear at about the time Solon ruled Athens. Solon realized that he could manipulate the value of coins. And so he did. Solon’s lesson has not been lost on governments ever since. In its simplest form: if you owe dollars, you print dollars, and voila! no more debt. Creditors don’t like debtors to do that, so they want to be paid in a currency that no one can produce more of. Gold comes close. You can produce more gold but you have to spend a lot of time/energy in a mine to do so. Production also includes counterfeiting. Counterfeiting comes in at least two important forms: de novo-counterfeiting (build a press) and duplication-counterfeiting (get a xerox machine). For now, all massless currencies are in digital form and are vulnerable to duplication-counterfeiting since computers make it cheap and easy to create perfect copies of digital notes (a unit of a currency will be called a ‘note’). As a result, massless currencies will typically be associated with systems that mitigate this threat. Perhaps the elaborate ledgers implemented by the creators of Bitcoin are an attempt to deal with the threat of duplication-counterfeiting. Abundance: As is often said, money can be used to ‘lubricate the economy’. To accomplish this a currency has to be sufficiently abundant. For example, Ununoctium, might be attractive to creditors because, compared to gold, it is far more costly in time/energy to produce more. However, it is not desirable for lubricating the economy because it is so costly to produce that less than 100 atoms have ever been created.

The transition from mass to masslessness will lead to currencies with uses and properties we do not normally associate with money. For example, using the techniques of secret-sharing, it becomes possible to create digital currencies where a single note can be jointly owned by 1000 individuals; any 501 of whom could cooperate to spend it, while any 500 of whom would be unable to do so.

What is the perfect currency? This is probably the wrong question, rather we should ask what properties may currency have, in theory and in reality. Let’s consider massless currencies.

Is there a massless currency similar to the U.S. dollar? I think yes. For example, the Government could simply publish a set of numbers and declare the numbers to be currency. Put another way, make the U.S. dollar smaller and smaller to decrease mass until asymptotically all that is left is the serial number. With regard to abundance, like the U.S. dollar, the Government is free to determining the number of notes available. With regard to production, as with the U.S. dollar, the Government can print more by simply publishing new numbers to be added to the set (or destroy some by declaring that some numbers have been removed from the set). With regard to counterfeiting, the U.S. dollar has some advantages. The mass of the U.S. dollar turns counterfeiting from a digital problem to a physical one. This provides the Government with the ability to change the U.S. dollar physically to defeat technology that might arise to produce faithful (either de novo or duplication) counterfeits.

Is there a massless currency similar to gold? I think yes. I think this is what Bitcoin-like currencies are all about. They are ‘deep-currencies’, sets of deep strings. With regard to abundance, they are superior to gold. The total number of deep strings in the set can be chosen at initiation by the creator of the currency. The abundance of gold on the other hand has already been set by nature (and, at least as currently used, gold is not sufficiently abundant to lubricate the world economy). With regard to production, as with gold, making notes requires time/energy. With regard to counterfeiting, gold has an advantage. Counterfeit gold is an absurdity, since all gold (by the ounce, not numismatically), no matter how it arises, is the same atomically and is perceived to have the same value. On the other hand, as stated above, massless currencies are vulnerable to duplication-counterfeiting. Interestingly, deep-currencies may be resistant to de novo-counterfeiting, since the creator of the deep-currency is free to choose the depth of the notes, and consequently the cost of producing new notes.

The value of a massless currency in our information based world is clear. Deep-currencies such as Bitcoin offer an attractive approach. But there is an interesting issue that may soon arise. The issue stems from the fact that the production of each note in currencies like Bitcoin requires a large investment of time/energy, and as with gold, this deprives governments of the prerogative to print money. Creditors may like this, but governments will not. What will governments do? Perhaps they will want to create a ‘Dual-currency’. A Dual-currency should be massless. It should come with a secret key. If you do not possess the secret key, then, like gold, it should be very costly to produce a new note, but if you possess the secret key, then, like the U.S. dollar, it should be inexpensive to produce a new note. Is a Dual-currency possible? Here is an example of an approach I call aurum:

Generate a pair of RSA keys: a public key , and a secret key . Publish the public key .

, and a secret key . Publish the public key . Declare that notes are exactly those integers such that and (the least positive residue of) is less than or equal to .

So, choosing a random integer such that , and then computing has about one chance in a billion of producing a note. Hence the expected number of modular exponentiations to produce a note is about one billion. On the other hand, those who possess the secret key can chose an integer such that and calculate to produce a note after just one modular exponentiation. There are many bells and whistles that can be accommodated with aurum, but here is what is ‘really’ going on. The depth of a string is obtained by considering programs running with the empty string as input, but we can consider a more general concept: ‘relative depth’. Given a pair of strings and , the depth of relative to is obtained by considering programs running with as the input. Hence depth as we have been discussing it is the same as depth relative to the empty string. In the example above, we have made `dual-strings’; strings that are deep relative to the public key, but shallow relative to the secret key.

One of the interesting phenomena of theoretical computer science is that you can sometimes turn bad news into good news. If factoring is hard, then we are deprived of the ability to do something we (at least number theorists) would like to do. Bad news. But surprisingly, we acquire public-key cryptography and the ability to preserve our privacy. Good news. Similarly, strings that are very hard to produce seem useless, but Bitcoins have revealed that such strings can provide a new and useful form of currency. Now that we are aware that deep strings can have value, I expect that clever people will find many new uses for them.

After thinking about deep strings for many years, I see them everywhere – I invite you to do the same. I will finish with one of my favorite observations. The watchmaker analogy is well known and is frequently used in arguing the existence of a creator. If you stumble upon a watch, you recognize from its complexity that it did not form by chance, and you conclude that there must have been a watchmaker. A human is more complex than a watch, so there must have been a ‘human-maker’ – a creator. An alternative view is that the complexity you recognize is actually ‘depth’ and the conclusion you should reach is that there must have been a computational process sustained through a great many steps. In the case of humans, the process is 3.6 billion years of evolution and the depth can be read in the genome. The watch is deep as well, but much of its depth is acquired from the human genome relative to which it is not nearly so deep.

*It has been brought to my attention that others have considered similar concepts including Charles Bennett, Murray Gell-Mann, and Luis Antunes, Lance Fortnow, Dieter Van Melkebeek, and N. Variyam Vinodchandran. Indeed, the latter group has even used the term ‘deep string’, hence it is to them we owe the name. **For all : denotes the set of Turing machine programs that with the empty string as input, output and halt.

denotes the set of Turing machine programs that with the empty string as input, output and halt. denotes , where denotes the length of , and denotes the number of steps used by with the empty string as input. It may be convenient for the reader to assume that is approximately ; however, the proper theory of deep strings extends the notion of depth to sets of strings and accommodates the use of randomness in computation. Depth in the context of quantum computation may also be of interest.

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