computational complexity theory In theoretical computer science and mathematics, computational complexity theory focuses on classifying computational problems according to their resource usage, and explores the relationships between these classifications. A computational problem ...

, a transcomputational problem is a

problem Problem solving is the process of achieving a goal by overcoming obstacles, a frequent part of most activities. Problems in need of solutions range from simple personal tasks (e.g. how to turn on an appliance) to complex issues in business an ...

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 Earth is the third planet from the Sun and the only astronomical object known to Planetary habitability, harbor life. This is enabled by Earth being an ocean world, the only one in the Solar System sustaining liquid surface water. Almost all ...

within a time period equal to the estimated age of the Earth.Bremermann, H.J. (1962
''Optimization through evolution and recombination''
In: Self-Organizing systems 1962, edited M.C. Yovitts et al., Spartan Books, Washington, D.C. pp. 93–106. The term ''transcomputational'' was coined by Bremermann.

Examples

Testing integrated circuits

Exhaustively testing all combinations of an

integrated circuit An integrated circuit (IC), also known as a microchip or simply chip, is a set of electronic circuits, consisting of various electronic components (such as transistors, resistors, and capacitors) and their interconnections. These components a ...

with 309 boolean inputs and 1

output Output may refer to: * The information produced by a computer, see Input/output * An output state of a system, see state (computer science) * Output (economics), the amount of goods and services produced ** Gross output in economics, the valu ...

requires testing of a total of 2³⁰⁹ combinations of inputs. Since the number 2³⁰⁹ is a transcomputational number (that is, a number greater than 10⁹³), the problem of testing such a system of

s 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. While the source uses 308 as the number of inputs, this number is based on an error: 2³⁰⁸ < 10⁹³.

Pattern recognition

Consider a ''q''×''q'' array of the chessboard type, each square of which can have one of ''k''

color Color (or colour in English in the Commonwealth of Nations, Commonwealth English; American and British English spelling differences#-our, -or, see spelling differences) is the visual perception based on the electromagnetic spectrum. Though co ...

s. Altogether there are ''k''^''n''

pattern A pattern is a regularity in the world, in human-made design, or in abstract ideas. As such, the elements of a pattern repeat in a predictable manner. A geometric pattern is a kind of pattern formed of geometric shapes and typically repeated l ...

s, where ''n'' = ''q''². 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 (; or retinas) is the innermost, photosensitivity, light-sensitive layer of tissue (biology), tissue of the eye of most vertebrates and some Mollusca, molluscs. The optics of the eye create a focus (optics), focused two-dimensional ...

. The retina contains about a million light-sensitive cells. Even if there were only two possible states for each cell (say, an active state and an inactive state) the processing of the

as a whole requires processing of more than 10^300,000 bits of information. This is far beyond Bremermann's limit.

General systems problems

system A system is a group of interacting or interrelated elements that act according to a set of rules to form a unified whole. A system, surrounded and influenced by its open system (systems theory), environment, is described by its boundaries, str ...

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

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: :"The experiences of various groups who work on problem solving, theorem proving and

pattern recognition Pattern recognition is the task of assigning a class to an observation based on patterns extracted from data. While similar, pattern recognition (PR) is not to be confused with pattern machines (PM) which may possess PR capabilities but their p ...

all seem to point in the same direction: These problems are tough. There does not seem to be a royal road or a simple method which at one stroke will solve all our problems. My discussion of ultimate limitations on the speed and amount of data processing may be summarized like this: Problems involving vast numbers of possibilities will not be solved by sheer data processing quantity. We must look for quality, for refinements, for tricks, for every ingenuity that we can think of. Computers faster than those of today will be a great help. We will need them. However, when we are concerned with problems in principle, present day computers are about as fast as they ever will be. :We may expect that the technology of data processing will proceed step by step – just as ordinary technology has done. There is an unlimited challenge for ingenuity applied to specific problems. There is also an unending need for general notions and theories to organize the myriad details."

References