Borges Wrote:The universe (which others call the Library) is composed of an indefinite and perhaps infinite number of hexagonal galleries, with vast air shafts between, surrounded by very low railings. From any of the hexagons one can see, interminably, the upper and lower floors. The distribution of the galleries is invariable. Twenty shelves, five long shelves per side, cover all the sides except two.
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each shelf contains thirty-five books of uniform format; each book is of four hundred and ten pages; each page, of forty lines, each line, of some eighty letters which are black in color. There are also letters on the spine of each book; these letters do not indicate or prefigure what the pages will say.
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The orthographical symbols are twenty-five in number. This finding made it possible, three hundred years ago, to formulate a general theory of the Library and solve satisfactorily the problem which no conjecture had deciphered: the formless and chaotic nature of almost all the books. One which my father saw in a hexagon on circuit fifteen ninety-four was made up of the letters MCV, perversely repeated from the first line to the last. Another (very much consulted in this area) is a mere labyrinth of letters, but the next-to-last page says Oh time thy pyramids. This much is already known: for every sensible line of straightforward statement, there are leagues of senseless cacophonies, verbal jumbles and incoherences.
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For a long time it was believed that these impenetrable books corresponded to past or remote languages …, but four hundred and ten pages of inalterable MCV's cannot correspond to any language, no matter how dialectical or rudimentary it may be. Some insinuated that each letter could influence the following one and that the value of MCV in the third line of page 71 was not the one the same series may have in another position on another page, but this vague thesis did not prevail. Others thought of cryptographs
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Five hundred years ago, the chief of an upper hexagon came upon a book as confusing as the others, but which had nearly two pages of homogeneous lines. He showed his find to a wandering decoder who told him the lines were written in Portuguese; others said they were Yiddish. Within a century, the language was established: a Samoyedic Lithuanian dialect of Guarani, with classical Arabian inflections. The content was also deciphered: some notions of combinative analysis, illustrated with examples of variations with unlimited repetition. These examples made it possible for a librarian of genius to discover the fundamental law of the Library. This thinker observed that all the books, no matter how diverse they might be, are made up of the same elements: the space, the period, the comma, the twenty-two letters of the alphabet. He also alleged a fact which travelers have confirmed: In the vast Library there are no two identical books. From these two incontrovertible premises he deduced that the Library is total and that its shelves register all the possible combinations of the twenty-odd orthographical symbols (a number which, though extremely vast, is not infinite): Everything: the minutely detailed history of the future, the archangels' autobiographies, the faithful catalogues of the Library, thousands and thousands of false catalogues, the demonstration of the fallacy of those catalogues, the demonstration of the fallacy of the true catalogue, the Gnostic gospel of Basilides, the commentary on that gospel, the commentary on the commentary on that gospel, the true story of your death, the translation of every book in all languages, the interpolations of every book in all books.
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At that time it was also hoped that a clarification of humanity's basic mysteries - the origin of the Library and of time - might be found. It is verisimilar that these grave mysteries could be explained in words: if the language of philosophers is not sufficient, the multiform Library will have produced the unprecedented language required, with its vocabularies and grammars. For four centuries now men have exhausted the hexagons ... There are official searchers, inquisitors. I have seen them in the performance of their function: they always arrive extremely tired from their journeys; they speak of a broken stairway which almost killed them; they talk with the librarian of galleries and stairs; sometimes they pick up the nearest volume and leaf through it, looking for infamous words. Obviously, no one expects to discover anything.
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We also know of another superstition of that time: that of the Man of the Book. On some shelf in some hexagon (men reasoned) there must exist a book which is the formula and perfect compendium of all the rest: some librarian has gone through it and he is analogous to a god. In the language of this zone vestiges of this remote functionary's cult still persist. Many wandered in search of Him. For a century they have exhausted in vain the most varied areas. How could one locate the venerated and secret hexagon which housed Him? Someone proposed a regressive method: To locate book A, consult first book B which indicates A's position; to locate book B, consult first a book C, and so on to infinity ... In adventures such as these, I have squandered and wasted my years. It does not seem unlikely to me that there is a total book on some shelf of the universe; I pray to the unknown gods that a man - just one, even though it were thousands of years ago! - may have examined and read it. If honor and wisdom and happiness are not for me, let them be for others. Let heaven exist, though my place be in hell.
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It is useless to observe that the best volume of the many hexagons under my administration is entitled The Combed Thunderclap and another The Plaster Cramp and another Axaxaxas mlö. These phrases, at first glance incoherent, can no doubt be justified in a cryptographical or allegorical manner; such a justification is verbal and, ex hypothesi, already figures in the Library. I cannot combine some characters dhcmrlchtdj which the divine Library has not foreseen and which in one of its secret tongues do not contain a terrible meaning.

(15-01-2026, 09:50 AM)MarcoP Wrote: You are not allowed to view links. Register or Login to view.Another interesting detail is: "[a book] made up of the letters MCV, perversely repeated from the first line to the last" - the most dramatic example of a hopelessly meaningless text that has character conditional entropy equal to zero.

(15-01-2026, 10:11 AM)Jorge_Stolfi Wrote: You are not allowed to view links. Register or Login to view.

(15-01-2026, 09:50 AM)MarcoP Wrote: You are not allowed to view links. Register or Login to view.Another interesting detail is: "[a book] made up of the letters MCV, perversely repeated from the first line to the last" - the most dramatic example of a hopelessly meaningless text that has character conditional entropy equal to zero.

To be pedantic, the average character entropy would be about 20/N bits, where N is the length of the book.  The 20 bits is a guess at the size of a program that would print out that text, on the Librar'y's Official Turing Machine.

All the best, --stolfi

(15-01-2026, 01:27 PM)Mauro Wrote: You are not allowed to view links. Register or Login to view.

(15-01-2026, 10:11 AM)Jorge_Stolfi Wrote: You are not allowed to view links. Register or Login to view.

(15-01-2026, 09:50 AM)MarcoP Wrote: You are not allowed to view links. Register or Login to view.Another interesting detail is: "[a book] made up of the letters MCV, perversely repeated from the first line to the last" - the most dramatic example of a hopelessly meaningless text that has character conditional entropy equal to zero.

To be pedantic, the average character entropy would be about 20/N bits, where N is the length of the book.  The 20 bits is a guess at the size of a program that would print out that text, on the Librar'y's Official Turing Machine.

All the best, --stolfi

Should'nt Shannon 1st-order entropy be -3*(1/3*ln(1/3)) =~ 1.099, independent from the length of the book?

(15-01-2026, 08:24 PM)Jorge_Stolfi Wrote: You are not allowed to view links. Register or Login to view.

(15-01-2026, 01:27 PM)Mauro Wrote: You are not allowed to view links. Register or Login to view.

(15-01-2026, 10:11 AM)Jorge_Stolfi Wrote: You are not allowed to view links. Register or Login to view.

(15-01-2026, 09:50 AM)MarcoP Wrote: You are not allowed to view links. Register or Login to view.Another interesting detail is: "[a book] made up of the letters MCV, perversely repeated from the first line to the last" - the most dramatic example of a hopelessly meaningless text that has character conditional entropy equal to zero.

To be pedantic, the average character entropy would be about 20/N bits, where N is the length of the book.  The 20 bits is a guess at the size of a program that would print out that text, on the Librar'y's Official Turing Machine.

All the best, --stolfi

Should'nt Shannon 1st-order entropy be -3*(1/3*ln(1/3)) =~ 1.099, independent from the length of the book?

The Shannon entropy of a stream of characters is how many bits you need, on average, to specify the next character.   IIRC, in Borges story the alphabet was only the 26 capital Latin letters.  It seems reasonable to assume that each of these letters can occur at each position in any book with the same probability, independently of the rest of the text in that book.  Then the Shannon entropy per character (for any order k), will be log_2(26) = ln(26)/ln(2) = ~4.700 bits.