Throughout Part I. If the probabilities are all O or I then the two are
Identical. If the probabilities are all very near to O or I, we get a
Machine that is almost determinate in its behaviour but that occa-
Sionally does the unusual thing. As the probabilities deviate fur-
Ther and further from O and 1, so does the behaviour at each step
Become less and less determinate, and more and more like that of
One of the insects considered in S.9/4.
It should be noticed that the definition, while allowing some
Indeterminacy, is still absolutely strict in certain respects. If the
machine, when at state x, goes on 90% of occasions to y and on
10% of occasions to z, then those percentages must be constant
(in the sense that the relative frequencies must tend to those per-
Centages as the sequence is made longer; and the limits must be
Unchanging as sequence follows sequence). What this means in
Practice is that the conditions that determine the percentages must
Remain constant.
The exercises that follow will enable the reader to gain some
Familiarity with the idea.
Ex. 1: A metronome-pendulum oscillates steadily between its two extreme
states, R and L, but when at the right (R) it has a 1% chance of sticking there
At that step. What is its matrix of transition probabilities ?
Ex. 2: A determinate machine α has the transformation
*Ex. 4: (Continued.) What general rule, using matrix multiplication, allows the
Answer to be written down algebraically? (Hint: Ex. 9/6/8.)
*Ex. 5: Couple the Markovian machine (with states a, 6, c and input-states α,
β)
α :
↓
a
b
c
a
b
c
0.2 0.3 0.3
. 0.7 0.2
0.8 . 0.5
β :
↓
a
b
c
a
b
c
0.3 0.9 0.5
0.6 0.1 0.5
0.1 ..
to the Markovian machine (with states e, f and input-states δ, ε, θ)
↓
e
f
↓
e
f
↓
e
f
δ :
e
f
0.7 0.5
0.3 0.5
ε :
e
f
0.2 0.7
0.8 0.3
θ :
e
f
0.5 0.4
0.5 0.6
By the transformations
↓ ε δ θ
a
b
c
↓ β α
e
f
↓ B D D D
A Markovian machine β has the matrix of transition probabilities
|
|
|
ABCD ↓
A0000
B 0.9 000
C00 0.2 0
D 0.1 1.0 0.8 1.0
How do their behaviours differ? (Hint: Draw α’s graph and draw β’s graph
After letting the probabilities go to 1 or 0.)
Ex. 3: A Markovian machine with input has a parameter that can take three val-
Ues— p, q, r— and has two states, a and b, with matrices
A
B
C D
What is the Markovian machine (without input) that results ? (Hint: Try
Changing the probabilities to O and 1, so as to make the systems determi-
Nate, and follow S.4/8; then make the probabilities fractional and follow
The same basic method.)
*Ex. 6: (Continued.) Must the new matrix still be Markovian?
*Ex. 7: If M is a Markovian machine which dominates a determinate machine N,
Show that N’s output becomes a Markov chain only after M has arrived at
Statistical equilibrium (in the sense of S.9/6).
↓
a
b
(p)
a
1/2
1/2
b
1
0
↓
a
b
(q)
a
b
1/4 3/4
3/4 1/4
↓
a
b
(r)
a
b
1/3 3/4
2/3 1/4
It is started at state b, and goes one step with the input at q, then one step with
It at r, then one step with it at p. What are the probabilities that it will now be
At a or b?
Whether a given real machine appears Markovian or deter-
Minate will sometimes depend on how much of the machine is
Observable (S.3/11); and sometimes a real machine may be such
That an apparently small change of the range of observation may
Be sufficient to change the appearances from that of one class to
The other.
Thus, suppose a digital computing machine has attached to it a
Long tape carrying random numbers, which are used in some proc-
Ess it is working through. To an observer who cannot inspect the
Tape, the machine’s output is indeterminate, but to an observer
|
|
|
Who has a copy of the tape it is determinate. Thus the question “Is
This machine really determinate?” is meaningless and inappropri-
Ate unless the observer’s range of observation is given exactly. In
Other words, sometimes the distinction between Markovian and
Determinate can be made only after the system has been defined
Accurately. (We thus have yet another example of how inadequate
Is the defining of “the system” by identifying it with a real object.
227
226
A N I N T R O D UC T I O N T O C Y B E R NE T I C S
Дата добавления: 2019-11-16; просмотров: 199; Мы поможем в написании вашей работы! |
Мы поможем в написании ваших работ!