Factor corresponds, of course, to a parameter, giving as many par-
Ticular codings (or transformations) U1, U2, ... as there are values
To the factor.
Decoding” means applying such a transformation to the trans-
Form Ci as will restore the original message Mi :
C1 C2 C3 …
M1 M2 M3 …
Such a transformation V is said to be the inverse of U; it may then
Be written as U-1. In general, only one-one transformations have
Single-valued inverses.
If the original message Mi is to be recoverable from the coded
Form Ci, whatever value i may have, then both U and U-1 must be
One-one; for if both Mi and Mj were to be transformed to one form
Ck, then the receiver of Ck could not tell which of the M’s had
Been sent originally, and Ck cannot be decoded with certainty.
Next suppose that a set of messages, having variety v, is sent
Coded by a one-one transformation U. The variety in the set of
Coded forms will also be v. Variety is not altered after coding by
A one-one transformation.
It follows that if messages of variety v are to pass through sev-
Eral codes in succession, and are to be uniquely restorable to their
Original forms, then the process must be one that preserves the
Variety in the set at every stage.
V: ↓
Ex. 1: Is the transformation x' = log10 x, applied to positive numbers, a one-one
Coding? What is “decoding” it usually called?
Ex. 2: Is the transformation x' = sin x, applied to the positive numbers, a one-one
Coding?
Ex. 3: What transformation results from the application of, first, a one-one trans-
Formation and then its inverse ?
Ex. 4: What transformation is the inverse of n' = n + 7?
Ex. 5: What transformation is the inverse of x' = 2x + y, y' = x + y?
Ex. 6: If the coded form consists of three English letters, e.g. JNB, what is the
Variety of the possible coded forms (measured logarithmically) ?
Ex. 7: (Continued.) How many distinct messages can be sent through such a
Code, used once?
Ex. 8. Eight horses are running in a race, and a telegram will tell Mr. A. which
Came first and which second. What variety is there in the set of possible mes-
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Sages ?
Ex. 9: (Continued.) Could the set be coded into a single letter, printed either as
Capital or
Ex. 10: The concentrations “high” or “low” of sex-hormone in the blood of a cer-
Tain animal determines whether it will, or will not, go through a ritual of
Courtship. If the sex-hormone is very complicated chemically and the ritual
Very complicated ethnologically, and if the variable “behaviour” is regarded
As a coded form of the variable “concentration”, how much variety is there
In the set of messages ?
Coding by machine. Next we can consider what happens when
A message becomes coded by being passed through a machine.
That such questions are of importance in the study of the brain
Needs no elaboration. Among their other applications are those
Pertaining to “instrumentation”— the science of getting informa-
Tion from some more or less inaccessible variable or place, such
As the interior of a furnace or of a working heart, to the observer.
The transmission of such information almost always involves
Some intermediate stage of coding, and this must be selected suit-
Ably. Until recently, each such instrument was designed simply on
The principles peculiar to the particular branch of science; today,
However, it is known, after the pioneer work of Shannon and
Wiener, that certain general laws hold over all such instruments.
What they are will be described below.
A “machine” was defined in S.3/4 as any set of states whose
Changes in time corresponded to a closed single-valued transfor-
Mation. This definition applies to the machine that is totally iso-
Lated i.e. in constant conditions; it is identical with the absolute
System defined in Design.... In S.4/ I the machine with input was
Defined as a system that has a closed single-valued transformation
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For each one of the possible states of a set of parameters. This is
Identical with the “transducer” of Shannon, which is defined as a
System whose next state is determined by its present state and the
Present values of its parameters. (He also assumes that it can have
A finite internal memory, but we shall ignore this for the moment,
Returning to it in S.918.)
Assume then that we have before us a transducer M that can be
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