Transcomputational problem

In computational complexity theory, a transcomputational problem is a problem that requires processing of more than 10⁹³ bits of information. Any number greater than 10⁹³ is called a transcomputational number. The number 10⁹³, called Bremermann's limit, is, according to Hans-Joachim Bremermann, the total number of bits processed by a hypothetical computer the size of the Earth within a time period equal to the estimated age of the Earth. The term transcomputational was coined by Bremermann.

Examples

Testing integrated circuits

Exhaustively testing all combinations of an integrated circuit with 309 boolean inputs and 1 output requires testing of a total of 2³⁰⁹ combinations of inputs. Since the number 2³⁰⁹ is a transcomputational number, the problem of testing such a system of integrated circuits is a transcomputational problem. This means that there is no way one can verify the correctness of the circuit for all combinations of inputs through brute force alone.

Pattern recognition

Consider a q×''q array of the chessboard type, each square of which can have one of k'' colors. Altogether there are kⁿ color patterns, where n = q². The problem of determining the best classification of the patterns, according to some chosen criterion, may be solved by a search through all possible color patterns. For two colors, such a search becomes transcomputational when the array is 18×18 or larger. For a 10×10 array, the problem becomes transcomputational when there are 9 or more colors.
This has some relevance in the physiological studies of the retina. The retina contains about a million light-sensitive cells. Even if there were only two possible states for each cell the processing of the retina as a whole requires processing of more than 10^300,000 bits of information. This is far beyond Bremermann's limit.

General systems problems

A system of n variables, each of which can take k different states, can have
kⁿ possible system states. To analyze such a system, a minimum of kⁿ bits of information are to be processed. The problem becomes transcomputational when kⁿ > 10⁹³. This happens for the following values of k and n:

k	2	3	4	5	6	7	8	9	10
n	308	194	154	133	119	110	102	97	93

Implications

The existence of real-world transcomputational problems implies the limitations of computers as data processing tools. This point is best summarized in Bremermann's own words: