U. Eight such systems are started at the states (O,9), (2,5), (0,5), (1,9), (1,5),
Respectively. How much variety is in the t’s ? How much
In the u’s ?
Ex. 2: (Continued.) Find the states at the next step. How much variety has t now?
Predict an upper limit to u’s variety. How much has u now?
Ex. 3: In another system, T has two variables, t1 and t2, and U has two, u1 and
u2. The whole has states ( t1, t2, u1, u2), and transformation t1' = t1t2, t2' = t1,
u1' = u1 + t2u2, u2' = t1u2, so that T dominates U. Three replicas are started
From the initial states (0,0,0,1), (0,0,1,1) and (1,0,0,1). What is T’s variety ?
What is U’s ?
Ex. 4: (Continued.) Find the three states one step later. What is U’s variety now ?
Transmission at second step. We have just seen that, at the
First step, U may gain in variety by an amount up to that in T; what
Will happen at the second step? T may still have some variety: will
This too pass to U, increasing its variety still further ?
Take a simple example. Suppose that every member of the
Whole set of replicates was at one of the six states (Ti,Uk), (Ti, Ul),
(Ti, Um), (Tj,Uk), (Tj,Ul), (Tj,Um), so that the T’s were all at either
T' or T and the U’s were all at Ul, Ul or Um. Now the system as a
Whole is absolute; so all those at, say (Ti, Uk), while they may
Change from state to state, will all change similarly, visiting the
Various states together. The same argument holds for those at each
Of the other five states. It follows that the set’s variety in state can-
Not exceed six, however many replicates there may be in the set,
Or however many states there may be in T and U, or for however
Long the changes may continue. From this it follows that the U’s
Can never show more variety than six U-states. Thus, once U has
Increased in variety by the amount in T, all further increase must
Cease. If U receives the whole amount in one step (as above) then
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T RA N SMISSI O N O F VA R IE TY
U receives no further increase at the second step, even though T
Still has some variety.
It will be noticed how important in the argument are the pair-
Ings between the states of T and the states of U, i.e. which value
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Of T and which of U occur in the same machine. Evidently merely
Knowing the quantities of variety in T and in U (over the set of
Replicates) is not sufficient for the prediction of how they will
Change.
Transmission through a charmer. We can now consider how
Variety, or information, is transmitted through a small intermedi-
Ate transducer— a “channel”— where “small” refers to its number
Of possible states. Suppose that two large transducers Q and S are
Connected by a small transducer R, so that Q dominates R, and R
Dominates S.
Q → R → S
As usual, let there be a great number of replicates of the whole tri-
Ple system. Let R’s number of possible states be r. Put log2r equal
to ρ. Assume that, at the initial state, the Q’s have a variety much
Larger than r states, and that the R’s and S’s, for simplicity,have
None. (Had they some variety, S.8/11 shows that the new variety,
Gained from Q, would merely add, logarithmically, to what they
Possess already.)
Application of S. 8/11 to R and S shows that, at the first step,
S’s variety will not increase at all. So if the three initial varieties,
Measured logarithmically, were respectively N, O and 0, then
after the first step they may be as large as N, ρ, and 0, but cannot
Be larger.
At the next step, R cannot gain further in variety (by S.8/12), but
S can gain in variety from R (as is easily verified by considering
An actual example such as Ex. 2). So after the second step the vari-
eties may be as large as N, ρ and ρ. Similarly, after the third step
they may be as large as N, ρ and 2 ρ; and so on. S’s variety can thus
increase with time as fast as the terms of the series, O, ρ, 2 ρ, 3 ρ,
But not faster. The rule is now obvious: a transducer that can-
Not take more than r states cannot transmit Variety at more than
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