Degrees of freedom, conflict and tolerance

[MartinT]

[Martin Taylor 2007.12.09.12.33]

This is mostly intended as a test message to try out a new mailing method and address, but I'll include some technical material, too, in case it actually does get distributed (only once, I hope).

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I'm wondering if part of the problem with degrees of freedom in the recent thread "Conflict and tolerance (was Maximization)" is the distinction between perceiving and acting. Let's consider the peceptual side first.

Imagine that there is a trivial game board that consists of four squares in a 2 x 2 array, on each of which a player may place a token that could be red, white, or blue. A player has a stack of tokens of each colour. Let's consider some of the different ways we could think of degrees of freedom for what the player could possibly perceive about this board in the context of the game (meaning that we ignore things such as where the board is placed, its colour and texture, lighting conditions, and so forth). Then we will deal with controlling those perceptions.

One way of looking at the situation is to concentrate on the board. It has four squares, each of which can be unoccupied or be occupied by exactly one token, which could be red, white, or blue. Each square represents one degree of freedom for the board, and each of those four degrees of freedom has a range of four possible values, 0, r, w, and b.

A person looking at the board might see an arrangement {0, 0, 0, 0} in which all four squares are unoccupied, {0, 0, r, r} in which the top row is unoccupied while the bottom row has red tokens on each square, or any of 4^4 different patterns. (If the range for each of the four degrees of freedom had been a real number from, say, 0 to 1, the number of possible patterns an observer might see would have been infinite -- in fact aleph 1).

Now let's consider one person's action possibilities. In the context of the game, there is just one possible action sequence, consisting of two independently controlled actions: (1) to pick up a token from the person's pile or to refrain from picking up a token and (2) to put the selected token (if there is one) on the board. That's one degree of freedom for the choice of token, and one for the choice of square. The token degree of freedom has a range of {0, remaining tokens} and the square degree of freedom has a range of {unoccupied squares}. So at the start of the game the move has a range {no token picked up, red on top-left, white on top-left, .... , blue on bottom right}.

One move (action sequence) has only one degree of freedom. Placing the token on the board cannot happen unless a token has been picked up. Having picked up a token, the person cannot pick up another until that one has been placed. The single degree of freedom for a "move" does affect two different degrees of freedom at lower levels, but as a move, it has only one degree of freedom, with a range that is the Cartesian product of the available tokens and the unoccupied board squares (plus the null option in each case).

In one move, the person can influence exactly one of the degrees of freedom of the board, by restricting the range of that degree of freedom (occupancy of a square) to just one value. Once that value has been decided, the board has one fewer degree of freedom for further change. An observer who has seen that the top-left square has a red token no longer needs to observe that square, because its value is fixed.

Notice that the one degree of freedom for the action sequence affects only one degree of freedom of the Board. This would remain true even if the permitted actions included moving a token from one square to another on the board (as in a chess game). Such a move would seem to affect two of the board's degrees of freedom, but it doesn't, any more than moving one end of a see-saw from down to up while moving the other end from up to down influences two degrees of freedom for the see-saw. In both cases, the changes are dependent on each other, and only one degree of freedom is affected. In the case of the game board, after one token has been placed, three degrees of freedom remain; after that token is moved to a different square, three degrees of freedom still remain.

Now we have in place most of the elements of the control loop. The person has a reference perception for the desired state of the board, which has four degrees of freedom, meaning that there four scalar perceptual signal values to be controlled, one for each square of the board. The person has available only one degree of freedom for action -- a "move". If the person were to attempt to control all the four perceptual degrees of freedom at once (in the same move), the four control systems would be in conflict, and at least three of them would fail to control. Most probably, none of them would be able to control, until some external agency gave one of them prority over the others.

The four control systems can, however, control their four perceptions by time multiplexing. At the first move, three of them use as their action "refrain" while one uses the action sequence "pick up and place", after which, that control system experiences no error and refrains from further action. The other three still experience error, but cannot all succeed in acting on the next move. Any one of them can succeed if the other two "refrain", because it uses one degree of freedom, which is all that it requires. But if the other two do not "refrain", none of the control systems will be successful.

By using the "refrain" option appropriately, the four degrees of freedom of the board can all be set to the desired values in just four moves, each using one degree of freedom, and all of the control systems will have reached a state of zero error.

There are several things to notice in this example:
   1. If the person were allowed to use two hands for each move, two degrees of freedom for action sequence would be available, and two of the four control systems would be able to control in each move. "Four into two won't go" is still true, but only two systems would have to "refrain" on move 1, and both could succeed on move 2.
   2. The degree-of-freedom bottleneck is in the action output, not in the environment. No two control systems are attempting to set the same perceptual variable, and yet there is conflict.
   3. Degrees of freedom is a concept that applies over time as well as over space. The "four-move" sequence that allows the four control systems to be effective has four degrees of freedom, each having a range over the token choice and the choice of square. The same is true for two-handed moves; two degrees of freedom are available for action in each of two moves, making four in all.
   4. The "refrain" pseudo-action is what permits the four control systems all to be effective over a time-period of at least four moves. "Refrain" is a metaphor for "tolerance". It says "Yes there is error, but we won't do anything about it just now". It allows more control systems to control than there are instantaneously available degrees of freedom.
   5. Since none of the control systems can perceive anything about the state of any of the others, there is no reason for any one of them to choose "refrain". In the trivial example, it takes some external agency or an unmentioned internal algorithm, perhaps along the lines of the Ethernet "back-off" algorithm", to induce the choice of "refrain" as opposed to continuing a bull-headed attempt to get through the bottleneck.

Point 5 is, I think, rather important to consider, at least from the Theorist's or Analyst's viewpoint.

I'll continue this later, if this message makes it properly through my changing mailer options. But I'd like comments on it if it does get to the CSGnet distribution, and preferably not only from Bill P. The basic comment I would like is an answer to: "Does this make sense to you?".

Martin

[_Erling_Jorgensen]

[From Erling Jorgensen (2007.12.11 1545 EST)]

Martin Taylor 2007.12.09.12.33

Hi Martin,

In answer to your "basic comment" request, it doesn't (yet) make
sense. It may be an artifact of your rules for this game.

In one move, the person can influence exactly one of the degrees of
freedom of the board, by restricting the range of that degree of
freedom (occupancy of a square) to just one value. Once that value
has been decided, the board has one fewer degree of freedom for
further change.

I thought "unoccupied" was one of the choices. And it can be
chosen by doing nothing. This corresponds with the situation with
a control system, where lack of error does not mean lack of action.
It just means lack of change of action.

To my muddled mind, this suggests that the setting of one's
reference standard is the key "degree of freedom." I'm not sure
if that mixes up your distinction between a degree of freedom vs.
the particular _value_ of a given degree.

Now we have in place most of the elements of the control loop. The
person has a reference perception for the desired state of the board,
which has four degrees of freedom, meaning that there four scalar
perceptual signal values to be controlled, one for each square of the
board. The person has available only one degree of freedom for action
-- a "move". If the person were to attempt to control all the four
perceptual degrees of freedom at once (in the same move), the four
control systems would be in conflict, and at least three of them
would fail to control.

Again, isn't the degree of freedom problem (the bottle-neck) affected
by how many variables need to _change_ from their current state?
If I want all spaces unoccupied, zero moves will get me there, with
no experience of a temporal bottleneck -- unless we redefine "move"
to include checking the conditional gate on controling a program
perception.

Sorry if this confuses where you're trying to get with this discussion.

All the best,
Erling

[MartinT]

[Martin Taylor 2007.12.11.16.57]

[From Erling Jorgensen (2007.12.11 1545 EST)]

Martin Taylor 2007.12.09.12.33

Hi Martin,

In answer to your "basic comment" request, it doesn't (yet) make
sense. It may be an artifact of your rules for this game.

Thanks for letting me know.

In one move, the person can influence exactly one of the degrees of
freedom of the board, by restricting the range of that degree of
freedom (occupancy of a square) to just one value. Once that value
has been decided, the board has one fewer degree of freedom for
further change.

I thought "unoccupied" was one of the choices. And it can be
chosen by doing nothing. This corresponds with the situation with
a control system, where lack of error does not mean lack of action. It just means lack of change of action.

Quite true. An oversight in my statement of the rules. In my mind I had the idea that the player wanted to have each square occupied, for two reasons:

   (1) to make the point, the rules must not allow a placed token to be removed, because if they do, the available degrees of freedom are not reduced by the placement of a token. If the player cannot correct an error when the reference value is "unoccupied" that's a difficulty of a different kind, which I didn't want to address yet (though it is one that needs to be addressed after the points in this message are understood).
   (2) to make the point, the player must need action to eliminate error, which means that all four of the relevant control systems must initially experience error.

To my muddled mind, this suggests that the setting of one's
reference standard is the key "degree of freedom." I'm not sure
if that mixes up your distinction between a degree of freedom vs.
the particular _value_ of a given degree.

No, it doesn't. The player has four reference values, one for the colour of the token on each square. They all can be chosen independently from the range {r, w, b}, and thus there are four degrees of freedom among the reference values.

>Now we have in place most of the elements of the control loop. The

person has a reference perception for the desired state of the board,
which has four degrees of freedom, meaning that there four scalar
perceptual signal values to be controlled, one for each square of the
board. The person has available only one degree of freedom for action
-- a "move". If the person were to attempt to control all the four
perceptual degrees of freedom at once (in the same move), the four
control systems would be in conflict, and at least three of them
would fail to control.

Again, isn't the degree of freedom problem (the bottle-neck) affected
by how many variables need to _change_ from their current state?

Yes, in the particular case. In the general case, you consider which ones _might_ need to change.

If I want all spaces unoccupied, zero moves will get me there, with
no experience of a temporal bottleneck -- unless we redefine "move"
to include checking the conditional gate on controling a program
perception.

I'm not sure what you mean by this redefinition, but perhaps my previous responses, above, answer the point. I was assuming that to correct all the errors would require four moves, but mis-stated the rules that would require it. At any move, a control system is still allowed to "refrain", but only at the cost of sustaining error for at least that move.

Sorry if this confuses where you're trying to get with this discussion.

No it doesn't confuse at all. You have pointed out inadequacies in my presentation. That's what might be confusing. I hope I have done a little toward making it less confusing.

Also, I'm glad to see that my message got through, apparently only once, to judge from what came back to me!

Martin

[MartinT]

[Martin Taylor 2007.12.11.17.48]

[Martin Taylor 2007.12.11.16.57]

[From Erling Jorgensen (2007.12.11 1545 EST)]

Martin Taylor 2007.12.09.12.33

Hi Martin,

In answer to your "basic comment" request, it doesn't (yet) make
sense. It may be an artifact of your rules for this game.

I realized I made a slight mis-statement in answering one of Erling's questions.

Again, isn't the degree of freedom problem (the bottle-neck) affected
by how many variables need to _change_ from their current state?

Yes, in the particular case. In the general case, you consider which ones _might_ need to change.

What I should have said is that the bottleneck isn't at all affected by how many variables need to change from their current state. Whether the bottleneck matters is what is affected. If there's a dorrway wide enough for three people to go through at the same time, there's no problem if only two people want to pass, but there's a big bottleneck there if a crowd of people is trying escape from a fire in the nightclub. The bottleneck was there all along, but it wan't a problem when people wanted to go in or out in twos and threes.

The bottleneck is a characteristic of the set of environmental feedback paths of the control systems. There always is one, somewhere. Whether it affects the ability of any control system to bring its perception to the reference depends on what all the control systems using the bottleneck are doing at any given moment.

Martin

[Bill_Powers1]

I’m wondering if part of the
problem with degrees of freedom in the recent thread “Conflict and
tolerance (was Maximization)” is the distinction between perceiving
and acting. Let’s consider the peceptual side first.

Imagine that there is a trivial game board that consists of four squares
in a 2 x 2 array, on each of which a player may place a token that could
be red, white, or blue. A player has a stack of tokens of each colour.
Let’s consider some of the different ways we could think of degrees of
freedom for what the player could possibly perceive about this board in
the context of the game (meaning that we ignore things such as where the
board is placed, its colour and texture, lighting conditions, and so
forth). Then we will deal with controlling those perceptions.

One way of looking at the situation is to concentrate on the board. It
has four squares, each of which can be unoccupied or be occupied by
exactly one token, which could be red, white, or blue. Each square
represents one degree of freedom for the board, and each of those four
degrees of freedom has a range of four possible values, 0, r, w, and
b
.

A person looking at the board might see an arrangement {0, 0, 0, 0} in
which all four squares are unoccupied, {0, 0, r, r} in which the top row
is unoccupied while the bottom row has red tokens on each square, or any
of 4^4 different patterns. (If the range for each of the four degrees of
freedom had been a real number from, say, 0 to 1, the number of possible
patterns an observer might see would have been infinite – in fact aleph
1).

Now let’s consider one person’s action possibilities. In the context of
the game, there is just one possible action sequence, consisting of two
independently controlled actions: (1) to pick up a token from the
person’s pile or to refrain from picking up a token and (2) to put the
selected token (if there is one) on the board. That’s one degree of
freedom for the choice of token, and one for the choice of square. The
token degree of freedom has a range of {0, remaining tokens} and the
square degree of freedom has a range of {unoccupied squares}. So at the
start of the game the move has a range {no token picked up, red on
top-left, white on top-left, … , blue on bottom right}.

One move (action sequence) has only one degree of freedom. Placing the
token on the board cannot happen unless a token has been picked up.
Having picked up a token, the person cannot pick up another until that
one has been placed. The single degree of freedom for a “move”
does affect two different degrees of freedom at lower levels, but as a
move, it has only one degree of freedom, with a range that is the
Cartesian product of the available tokens and the unoccupied board
squares (plus the null option in each case).

In one move, the person can influence exactly one of the degrees of
freedom of the board, by restricting the range of that degree of freedom
(occupancy of a square) to just one value. Once that value has been
decided, the board has one fewer degree of freedom for further change. An
observer who has seen that the top-left square has a red token no longer
needs to observe that square, because its value is fixed.

Notice that the one degree of freedom for the action sequence affects
only one degree of freedom of the Board. This would remain true even if
the permitted actions included moving a token from one square to another
on the board (as in a chess game). Such a move would seem to affect two
of the board’s degrees of freedom, but it doesn’t, any more than moving
one end of a see-saw from down to up while moving the other end from up
to down influences two degrees of freedom for the see-saw. In both cases,
the changes are dependent on each other, and only one degree of freedom
is affected. In the case of the game board, after one token has been
placed, three degrees of freedom remain; after that token is moved to a
different square, three degrees of freedom still remain.

Now we have in place most of the elements of the control loop. The person
has a reference perception for the desired state of the board, which has
four degrees of freedom, meaning that there four scalar perceptual signal
values to be controlled, one for each square of the board. The person has
available only one degree of freedom for action – a “move”. If
the person were to attempt to control all the four perceptual degrees of
freedom at once (in the same move), the four control systems would be in
conflict, and at least three of them would fail to control. Most
probably, none of them would be able to control, until some external
agency gave one of them prority over the others.

The four control systems can, however, control their four perceptions by
time multiplexing. At the first move, three of them use as their action
“refrain” while one uses the action sequence “pick up and
place”, after which, that control system experiences no error and
refrains from further action. The other three still experience error, but
cannot all succeed in acting on the next move. Any one of them can
succeed if the other two “refrain”, because it uses one degree
of freedom, which is all that it requires. But if the other two do not
“refrain”, none of the control systems will be
successful.

By using the “refrain” option appropriately, the four degrees
of freedom of the board can all be set to the desired values in just four
moves, each using one degree of freedom, and all of the control systems
will have reached a state of zero error.

There are several things to notice in this example:

1. If the person were allowed to use two hands for each move, two
   degrees of freedom for action sequence would be available, and two of the
   four control systems would be able to control in each move. “Four
   into two won’t go” is still true, but only two systems would have to
   “refrain” on move 1, and both could succeed on move 2.

2. The degree-of-freedom bottleneck is in the action output, not
   in the environment. No two control systems are attempting to set the same
   perceptual variable, and yet there is conflict.

3. Degrees of freedom is a concept that applies over time as well
   as over space. The “four-move” sequence that allows the four
   control systems to be effective has four degrees of freedom, each having
   a range over the token choice and the choice of square. The same is true
   for two-handed moves; two degrees of freedom are available for action in
   each of two moves, making four in all.

4. The “refrain” pseudo-action is what permits the four
   control systems all to be effective over a time-period of at least four
   moves. “Refrain” is a metaphor for “tolerance”. It
   says “Yes there is error, but we won’t do anything about it just
   now”. It allows more control systems to control than there are
   instantaneously available degrees of freedom.

5. Since none of the control systems can perceive anything about
   the state of any of the others, there is no reason for any one of them to
   choose “refrain”. In the trivial example, it takes some
   external agency or an unmentioned internal algorithm, perhaps along the
   lines of the Ethernet “back-off” algorithm", to induce the
   choice of “refrain” as opposed to continuing a bull-headed
   attempt to get through the bottleneck.

Point 5 is, I think, rather important to consider, at least from the
Theorist’s or Analyst’s viewpoint.

I’ll continue this later, if this message makes it properly through my
changing mailer options. But I’d like comments on it if it does get to
the CSGnet distribution, and preferably not only from Bill P. The basic
comment I would like is an answer to: “Does this make sense to
you?”.

Martin

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AM
[From Bill Powers (2007.12.11.1438 MST)]

Martin Taylor 2007.12.09.12.33 –

My analysis differs from yours.

The board consists of four squares (nothing about your rules says they
can’t just be arranged in a straight line). Without going any farther we
can say that these squares form a constant array [0,1,2,3]-- there is
nothing variable about them. The board itself has no degrees of
freedom.

The relationship of any token to the board is the same as the
relationship of a variable to the coordinate system in which it’s
measured. Any given token can have the board position of 0,1,2, or 3.
That is one degree of freedom for a token’s position on the
board.

Likewise, in any one position on tbe board, a token having that position
can range through the color values r,w,b. So we can construct an
independent dimension at right angles to the dimension of position. We
now have two degrees of freedom for a token. If N is one of the four
positions and C is one of the three colors, then (N,C) identifies the
state of a token in both dimensions.

Each token placed on the board can therefore have any one of 12 locations
in this two-dimensional space.

So far there are no constraints: we could stack up any number of tokens
in any way we liked, locating each token with two numbers. Any number of
controlling agents, each with a control system for each dimension, could
place tokens of any desired color in any desired position.

However, there is a constraint in the form of another rule that says only
one token can occupy a board position at a time. One color can be chosen
for each position, and once a position is chosen it becomes unavailable
as a location for any other token of any color. This means that only one
controlling agent controlling in both dimensions can place a token of any
possible color in any possible position on the board. Adding even one
more agent with two similar control systems means loss of range over
which reference conditions can freely be varied. Four such agents is the
maximum possible, and the number of available reference conditions for
each one would shrink to one. There would be no ability on the part of
any agent to vary the controlled variable, since only one reference
condition can be achieved by each one.

This brings up a point that arose some time ago. Unconflicted control
doesn’t just mean avoiding collisions – it means being organized so each
controlled variable can be varied freely in a dimension orthogonal to all
others. This means there is complete freedom of choice of reference
signals (within the range of physically achievable values), because all
possible values of each controlled variable are available independently
of any other variable. But we can have almost-complete freedom as
well.

In your game-board example, the discrete nature of the variables makes
“interactions” into either-or situations. We can’t have one
square partly occupied mostly by one token and a little bit by another.
We can’t have errors that are non-zero but small. In a system with
continuous variables, it can be possible for n systems to control a range
of reference conditions of n variables because exact solutions of the
system of equations are not necessary. In a discrete system, variables
can have only the exact values specified, never any value that is
“almost” right. So either there is conflict or there is no
conflict; degrees of conflict don’t exist as they do in continuous
systems.

Best,

Bill P.

[MartinT]

[From Bill Powers (2007.12.11.1438 MST)]

Martin Taylor 2007.12.09.12.33 --

I'm wondering if part of the problem with degrees of freedom in the recent thread "Conflict and tolerance (was Maximization)" is the distinction between perceiving and acting. Let's consider the peceptual side first.

Imagine that there is a trivial game board that consists of four squares in a 2 x 2 array, on each of which a player may place a token that could be red, white, or blue. A player has a stack of tokens of each colour. Let's consider some of the different ways we could think of degrees of freedom for what the player could possibly perceive about this board in the context of the game (meaning that we ignore things such as where the board is placed, its colour and texture, lighting conditions, and so forth). Then we will deal with controlling those perceptions.

One way of looking at the situation is to concentrate on the board. It has four squares, each of which can be unoccupied or be occupied by exactly one token, which could be red, white, or blue. Each square represents one degree of freedom for the board, and each of those four degrees of freedom has a range of four possible values, 0, r, w, and b

My analysis differs from yours.

I'll comment on your analysis, which does not seem to make a clear distinction between perceiving and acting. Would you mind commenting

they can't just be arranged in a straight line). Without going any farther we can say that these squares form a constant array [0,1,2,3]-- there is nothing variable about them. The board itself has no degrees of freedom.

True. I made the board a square because I wanted to introduce the concept of higher-order patterns after the basic argument was understood.

True also that the board itself has no degrees of freedom. If I used that language it was shorthand for saying that there are just so many "degrees of freedom for the perception of the occupancy of the board squares by tokens of different colours". It's shorter to say "board degrees of freedom".

The relationship of any token to the board is the same as the relationship of a variable to the coordinate system in which it's measured. Any given token can have the board position of 0,1,2, or 3. That is one degree of freedom for a token's position on the board.

Yes, if you want to consider the perception of the set of tokens, you can do that. It's not inconsistent with looking at the occupancies of the squares. I didn't specify how many tokens the player starts with. Let's say there are N, so until we know something about the constraints, any one of them could be in-hand, or on square 0, 1, 2, or 3. That would be N degrees of freedom, each with a range of 5 possible values.

Likewise, in any one position on tbe board, a token having that position can range through the color values r,w,b.

(or the square might be unoccupied)

So we can construct an independent dimension at right angles to the dimension of position. We now have two degrees of freedom for a token.

For the perception of where a token is on the board and what colour it is, yes. Tokens can't change their colours. I don't know if this is a niggle or a point.

If the situation were one in which token colours could be changed by some action of a control system, as could their locations, then the colour and location would be two degrees of freedom. I noted that in my analysis, since one control system selects the colour of token to pick up, and the action of that control system is followed by the action of another that controls where it should be placed. Both are lower-level systems that support a control system that controls a colour-location complex perception. This upper level perception is a one df scalar, as is any other controlled perception in classic HPCT.

Is that your point?

If N is one of the four positions and C is one of the three colors, then (N,C) identifies the state of a token in both dimensions.

Each token placed on the board can therefore have any one of 12 locations in this two-dimensional space.

That's true, you seem to be taking some implication for granted that I don't see. Tokens have no degrees of freedom any more than does the board. It's the description of the board occupancy, or from the token point of view, the token's colour and its placement, that has degrees of freedom. The end result is the same, but getting at it from the token viewpoint seems to me to be rather more complicated than getting at it the way I did.

I think what you have done is substitute a 12-square (4 x 3) board and replace the coloured tokens with identical pebbles. Why it becomes more complicated when you do so is that the system doesn't have 12 degrees of freedom, because there are pattern constraints that limit where pebbles can be placed on the board -- only one per column. I didn't want to get into pattern constraints at all until a later message in the set, but since you brought it up ....

So far there are no constraints: we could stack up any number of tokens in any way we liked, locating each token with two numbers. Any number of controlling agents, each with a control system for each dimension, could place tokens of any desired color in any desired position.

Yes, but in this paragraph you begin to merge degrees of freedom for perception with degrees of freedom of the external environment, while disregarding the concept of degrees of freedom for action, which is the location of the bottleneck in this example.

It's important to keep these three separate. It's particularly important to keep distinct the concept of degrees of freedom for perception and the degrees of freedom involved in manipulating the values of those perceptual degrees of freedom. The key to my analysis is that the person can use only one hand, and by conflating the degrees of freedom in the three different conceptual areas, you miss that.

only one token can occupy a board position at a time. One color can be chosen for each position, and once a position is chosen it becomes unavailable as a location for any other token of any color.

Yes. Whereas beforehand there were four degrees of freedom for "colour at location", with a wide range of possible values for the number of tokens of any colour at each location, adding the constraint that only one colour can occupy any one square means that there are only three degrees of freedom left after one of them is fixed. On the pebble board, without the constraints you could have put any number of pebbles on any of the 12 squares, but with the constrain, once a column is occupied, that degree of freedom is lost.

I used the rule of one token per square because I was thinking toward future discussions. The array of values on the squares was intended as an analogue of the values of samples in a waveform; you can't have more than one value for any one sample. To expand the analogy for a moment: on the "pebble-board" imagine an infinitely long column with locations not in labelled boxes, but identified by real numbers. On the column you can still put only one pebble, its location being given by the real number corresponding to the "box" on which it sits. The column represents one degree of freedom, the boxes or real numbers representing its range, and the waveform sample being represented by the position of the pebble on the column.

This means that only one controlling agent controlling in both dimensions can place a token of any possible color in any possible position on the board. Adding even one more agent with two similar control systems means loss of range over which reference conditions can freely be varied.

This is where I lose you.

Four such agents is the maximum possible, and the number of available reference conditions for each one would shrink to one.

I suppose not understanding the previous sentence is the reason this one makes no sense. So far as I can see, all four control systems can have any reference values at all, without constraint. They just can't all satisfy their references by any legal actions in the game.

There would be no ability on the part of any agent to vary the controlled variable, since only one reference condition can be achieved by each one.

No, I really don't follow. My last paragraph comment wouldn't lead to this. Four "agents" can control the four degrees of freedom perfectly well. Five couldn't.

Maybe you could restate this paragraph, which I cut to show where I get lost.

In your game-board example, the discrete nature of the variables makes "interactions" into either-or situations.

That was a deliberate choice, for the very reason that it makes the situation clear. There's no question of nearly parallel perceptual vectors, fractional degrees of freedom, exponentially increasing forces, or anything like that. The didactic idea was like that of Shannon: deal with the discrete case where the concepts are clear (or I hope they become so if they are not), and then generalize to the continuous case when the basic ideas are well understood.

Actually, I didn't talk about interactions at all, since in the usual sense there are none. There's only one player, and the four control units don't disturb each other's perceptions at all, since each perceives and influences only the occupancy of one square on the board, each one perceiving a different square. What they do is act through the same one hand, and if controller 3 wants to pick up a red token while controller 2 wants to pick up a blue, they can't both do it at the same time. That's why I quantized the time dimension into discrete moves as well as specifying a discrete board with quantized values.

In a discrete system, variables can have only the exact values specified, never any value that is "almost" right. So either there is conflict or there is no conflict; degrees of conflict don't exist as they do in continuous systems.

Precisely

And I'm glad to see you use the term "degrees of conflict".

I'd appreciate a clarification of the paragraph where I started to fail to follow, and I'd also appreciate your comments on my analysis of the degrees of freedom for perception, action, and in the environment for the board-game example (as well as comments on the cow-distribution example I posted a few days ago). I'd like to know you understand what I'm saying, even if it isn't the approach you would have taken, and especially if you disagree with what I say. There are legitimately different approaches to the analysis, but if they are legitimate, they should wind up with the same result.

Thanks.

Martin

···

on mine? >The board consists of four squares (nothing about your rules says On re-reading, I'm wondering if this, rather than later, isn't actually the place where my understand of your writing begins to go awry. I don't think it is, but the possibility is there. >However, there is a constraint in the form of another rule that says

[Bill_Powers1]

True also that the board itself
has no degrees of freedom. If I used that language it was shorthand for
saying that there are just so many “degrees of freedom for the
perception of the occupancy of the board squares by tokens of different
colours”. It’s shorter to say “board degrees of
freedom”.
Martin Taylor (12:15 AM 12/12/2007 -0500) –

Yes, but very misleading when you’re constructing an explanation.
Several times you have said there are degrees of freedom in the
environment where there are no variables to vary. There is a similar
problem with “occupancy,” which has characteristics in common
with the smile of the Cheshire Cat. To be precise, you must explicitly
introduce the relationship between the tokens and the spaces on the
board, as in fact you just did above. And then, as you note later on, you
will end up talking about perceptions of the relationship between tokens
and spaces instead of about the board or the actions. This is not
surprising, since actions are controlled perceptions, too, and game
boards by themselves have no degrees of freedom.

[My board game Trippples, however, is an exception, because the game
board itself was made of movable tiles that the players took turns
placing in a frame before each game).

The relationship of
any token to the board is the same as the relationship of a variable to
the coordinate system in which it’s measured. Any given token can have
the board position of 0,1,2, or 3. That is one degree of freedom for a
token’s position on the board.

Yes, if you want to consider the perception of the set of tokens, you can
do that. It’s not inconsistent with looking at the occupancies of the
squares.

If you look just at the board, you can’t see or measure anything
called “occupancy.” You’re introducing a second element other
than the board “O.P.”, so to speak; alluding to it “off
proscenium” without bringing it into view. Something must occupy the
space or not occupy it, so its location relative to the space can vary.

So I have difficulty with your exposition at this point. You are talking
about a relationship between a space on the board and some unmentioned
other thing that is evidently present in your mind, but is absent from
your description.

I didn’t
specify how many tokens the player starts with. Let’s say there are N, so
until we know something about the constraints, any one of them could be
in-hand, or on square 0, 1, 2, or 3. That would be N degrees of freedom,
each with a range of 5 possible values.

There is a logical contradiction here, that led me to leave out
“absent” as one of the states of a token. The problem is that
these degrees of freedom are not independent, so they are not all
legitimate degrees of freedom.

Suppose we said there are five positions for a token, with -1 meaning off
the board, or “in hand.” Unfortunately, that doesn’t work if
you make a rule saying that only one token can be in a given position. If
that were true there could be no more than 5 tokens: four on the board
and one in hand. The number of tokens, however, was not limited by the
rules.

If a token with one of three colors can be either on the board in one of
four positions, or not on the board, we then have the 12 combinations I
defined, plus what appears to be one more degree of freedom: on the board
or off the board (“in hand”). The state “not on the
board” is not a separate degree of freedom, however, because a token
can’t be off the board and at the same time in one of the four board
locations. True degrees of freedom allow for free choice of a state
without regard to other degrees of freedom. A true third degree of
freedom would be, e.g., shape: round or square. Obviously a token of any
color could be on any square and have either a round or a square shape.
That’s not the case if we replace “round” by “off the
board”.

The real problem here is one that arises often in designing switching
circuits – failure to account for all logical states of each variable.
There is a tacitly assumed physical relationship among states of
occupancy such that if a token occupies one square, it does not occupy
any of the other squares. If it is in hand, it does not occupy any square
at all. So the rules actually imply statements like “0 AND NOT (1 OR
2 OR 3 OR -1)” which says that if a token is in one place, it can’t
also be in another place. This is not true in electrical circuits; a
voltage measured in one place is also present in many other places. These
statements have to be made about each variable in the system if all
ambiguity is to be eliminated. What happens in examples like yours is
that we tend to take some things for granted and forget that they have to
be explicitly stated in analyzing a system, or designing one.

If you substitute dry-markers of three colors for the set of colored
tokens, “off the board” ceases to have a meaning, or to cause
problems.

I believe the only real degrees of freedom are [0,1,2,3] for positions
relative to the board (zero means one of the board positions) and [r,w,b]
for color, the two independent variables. I would not argue too hard if
you said the position values were [0,1,2,3,4] where 0 means “not on
the board”, but then you’d have to do something about the rule
“only one token in one position.” There would have to be a
different position for each token that is off the board.

To sum up this far:

While it seems that there is a state of a square on the board called
“occupied”, that state is not a property of the board alone;
this way of defining a state simply omits mention of a necessary element,
the thing doing the occupying. When we include that element, we can see
that “occupancy” is really just one state of a variable
relationship between two elements. A token can be away from a square
by any distance. If the distance is zero, we say the token
“occupies” the square, or that the square “is
occupied.” However, the latter terminology (which omits “by the
token”) makes it appear that being occupied is something the square
can do all by itself, an implication that shows what is wrong with this
way of speaking.

I’d appreciate a
clarification of the paragraph where I started to fail to follow, and I’d
also appreciate your comments on my analysis of the degrees of freedom
for perception, action, and in the environment for the board-game
example…

This means that only one
controlling agent controlling in both dimensions can place a token of any
possible color in any possible position on the board. Adding even one
more agent with two similar control systems means loss of range over
which reference conditions can freely be
varied.

This is where I lose
you.

Likewise, in any one position on
tbe board, a token having that position can range through the color
values r,w,b.

(or the square might be unoccupied)

So we can construct an
independent dimension at right angles to the dimension of position. We
now have two degrees of freedom for a token.

For the perception of where a token is on the board and what colour it
is, yes. Tokens can’t change their colours. I don’t know if this is a
niggle or a point.

If the situation were one in which token colours could be changed by some
action of a control system, as could their locations, then the colour and
location would be two degrees of freedom. I noted that in my analysis,
since one control system selects the colour of token to pick up, and the
action of that control system is followed by the action of another that
controls where it should be placed. Both are lower-level systems that
support a control system that controls a colour-location complex
perception. This upper level perception is a one df scalar, as is any
other controlled perception in classic HPCT.

Is that your point?

If N is one of the four
positions and C is one of the three colors, then (N,C) identifies the
state of a token in both dimensions.

Each token placed on the board can therefore have any one of 12 locations
in this two-dimensional space.

That’s true, you seem to be taking some implication for granted that I
don’t see. Tokens have no degrees of freedom any more than does the
board. It’s the description of the board occupancy, or from the token
point of view, the token’s colour and its placement, that has degrees of
freedom. The end result is the same, but getting at it from the token
viewpoint seems to me to be rather more complicated than getting at it
the way I did.

I think what you have done is substitute a 12-square (4 x 3) board and
replace the coloured tokens with identical pebbles. Why it becomes more
complicated when you do so is that the system doesn’t have 12 degrees of
freedom, because there are pattern constraints that limit where pebbles
can be placed on the board – only one per column. I didn’t want to get
into pattern constraints at all until a later message in the set, but
since you brought it up …

So far there are no constraints:
we could stack up any number of tokens in any way we liked, locating each
token with two numbers. Any number of controlling agents, each with a
control system for each dimension, could place tokens of any desired
color in any desired position.

Yes, but in this paragraph you begin to merge degrees of freedom for
perception with degrees of freedom of the external environment, while
disregarding the concept of degrees of freedom for action, which is the
location of the bottleneck in this example.

It’s important to keep these three separate. It’s particularly important
to keep distinct the concept of degrees of freedom for perception and the
degrees of freedom involved in manipulating the values of those
perceptual degrees of freedom. The key to my analysis is that the person
can use only one hand, and by conflating the degrees of freedom in the
three different conceptual areas, you miss that.

On re-reading, I’m wondering if this, rather than later, isn’t actually
the place where my understand of your writing begins to go awry. I don’t
think it is, but the possibility is there.

However, there is a constraint
in the form of another rule that says only one token can occupy a board
position at a time. One color can be chosen for each position, and once a
position is chosen it becomes unavailable as a location for any other
token of any color.

Yes. Whereas beforehand there were four degrees of freedom for
“colour at location”, with a wide range of possible values for
the number of tokens of any colour at each location, adding the
constraint that only one colour can occupy any one square means that
there are only three degrees of freedom left after one of them is fixed.
On the pebble board, without the constraints you could have put any
number of pebbles on any of the 12 squares, but with the constrain, once
a column is occupied, that degree of freedom is lost.

I used the rule of one token per square because I was thinking toward
future discussions. The array of values on the squares was intended as an
analogue of the values of samples in a waveform; you can’t have more than
one value for any one sample. To expand the analogy for a moment: on the
“pebble-board” imagine an infinitely long column with locations
not in labelled boxes, but identified by real numbers. On the column you
can still put only one pebble, its location being given by the real
number corresponding to the “box” on which it sits. The column
represents one degree of freedom, the boxes or real numbers representing
its range, and the waveform sample being represented by the position of
the pebble on the column.

This means that only one
controlling agent controlling in both dimensions can place a token of any
possible color in any possible position on the board. Adding even one
more agent with two similar control systems means loss of range over
which reference conditions can freely be varied.

This is where I lose you.

Four such agents is the
maximum possible, and the number of available reference conditions for
each one would shrink to one.

I suppose not understanding the previous sentence is the reason this one
makes no sense. So far as I can see, all four control systems can have
any reference values at all, without constraint. They just can’t all
satisfy their references by any legal actions in the game.

There would be no ability
on the part of any agent to vary the controlled variable, since only one
reference condition can be achieved by each one.

No, I really don’t follow. My last paragraph comment wouldn’t lead to
this. Four “agents” can control the four degrees of freedom
perfectly well. Five couldn’t.

Maybe you could restate this paragraph, which I cut to show where I get
lost.

In your game-board example, the
discrete nature of the variables makes “interactions” into
either-or situations.

That was a deliberate choice, for the very reason that it makes the
situation clear. There’s no question of nearly parallel perceptual
vectors, fractional degrees of freedom, exponentially increasing forces,
or anything like that. The didactic idea was like that of Shannon: deal
with the discrete case where the concepts are clear (or I hope they
become so if they are not), and then generalize to the continuous case
when the basic ideas are well understood.

Actually, I didn’t talk about interactions at all, since in the usual
sense there are none. There’s only one player, and the four control units
don’t disturb each other’s perceptions at all, since each perceives and
influences only the occupancy of one square on the board, each one
perceiving a different square. What they do is act through the same one
hand, and if controller 3 wants to pick up a red token while controller 2
wants to pick up a blue, they can’t both do it at the same time. That’s
why I quantized the time dimension into discrete moves as well as
specifying a discrete board with quantized values.

In a discrete system,
variables can have only the exact values specified, never any value that
is “almost” right. So either there is conflict or there is no
conflict; degrees of conflict don’t exist as they do in continuous
systems.

Precisely

And I’m glad to see you use the term “degrees of
conflict”.

I’d appreciate a clarification of the paragraph where I started to fail
to follow, and I’d also appreciate your comments on my analysis of the
degrees of freedom for perception, action, and in the environment for the
board-game example (as well as comments on the cow-distribution example I
posted a few days ago). I’d like to know you understand what I’m saying,
even if it isn’t the approach you would have taken, and especially if you
disagree with what I say. There are legitimately different approaches to
the analysis, but if they are legitimate, they should wind up with the
same result.

Thanks.

Martin

–

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AM

I should have said “loss of range over which the controlled variable
can be varied to match any possible reference signal.” Of course the
reference signals can be varied in any way you like. The following should
clarify this.

Degrees of freedom of action and degrees of freedom of perception are
clearly different, because the action can vary without a significant
change in the perception, and the perception can change without a
significant change in the action. This is possible because of
disturbances and other kinds of changes in the external part of the
loop.

An action is (almost always) a set of controlled variables of a lower
order than the perception controlled by them. If Fa is the function
turning the action-variables a into an effect on a controlled variable,
and Fd is the function turning the states of all disturbing variables d
into another effect on the controlled variable, then the state of the
controlled variable V is

V = Fa(a1…an) + Fd(d1 … dm).

With multiple control systems all using the same set of action variables,
it is possible for each of k systems to make Vk match an arbitrary
reference value Rk, as long as k is less than or equal to n. So the
degrees of freedom of control are limited by the degrees of freedom of
action. Note that the number of disturbances is not a limit to the
degrees of freedom of control.

The most important thing about this little derivation, with which you are
quite familiar, is that when all the degrees of freedom are just used up,
it is still possible for each control systems to bring its own input to
any reference condition within the physically possible range without
preventing any other control system in the set from doing the same. This
says there is a solution to the set of equations Vk = Rk for all possible
Rk.

This is not like your example of the game board. In that example, only
one control system can freely place a token in any of the 12 states
possible. If there are two control systems, the first to act is free to
create any state, but the second one can then create only 9 of the
states, since one position is no longer accessible. If there are four
control systems, the last one to act can set its reference signal to any
of the 12 states (as you noted) but its controlled variable can reach
only three of them. Contrast that with my derivation above that shows all
k control systems still being able to select and reach all reference
states within the same range that one system alone could reach.

Note that we almost automatically introduce new degrees of freedom.
“The first one to act” brings in the concept of qualitative
temporal ordering, with a huge increase in the number of control systems
that can at least momentarily achieve their reference conditions (think
of how scores are made in a pinball machine). Of course that doesn’t add
any degrees of freedom to those that existed before, but it brings them
to notice and provides new things for new higher-level systems to
control. I say that there are 11 major classes of degrees of freedom that
can be controlled by using our limited outputs in ways that bring up new
kinds of variables to control – ways in which the environment can be
controlled that were there all along (I presume) but which don’t get
counted until they’re mentioned. We are limited in how many of them can
be controlled simultaneously, but the “bottlenecks” of which
you speak don’t mean that n outputs can’t be used to control more than n
inputs. It just means you can’t act independently on more than that many
at one time.

As I said a few posts ago, most environmental variables change very
slowly, so with n independent outputs we can control most variables, far
more than n of them, as fast as we need to even though we have to
time-share among them. The apparent degrees of freedom of our joints and
muscles are only a part of the story; when we bring in time-sharing, we
see that the same output units can be applied to different parts of the
environment so as to affect completely different sets of variables on
different occasions, greatly multiplying the degrees of freedom of action
by allowing the same joint-angle actions to have very different
environmental consequences. I can reach to my left and press a button
that turns on a light; I can then (while the light stays in the state I
want) reach to my right and press a button that opens the spillway of a
dam. Surely these controlled variables represent two quite different and
independent degrees of freedom.

Best,

Bill P.

t A token on the board can have a color r,w or b and a
position 0,1,2,3, but to say a token is in hand means that it can’t have
any position on the board. If “in hand – on board” were really
a degree of freedom, you would be able to specify either of those
conditions, and at the same time specify any position on the board for
that token.

[MartinT]

[Martin Taylor 2007.12.13.11.41]

This is another test message, since I've been getting two copies of everything. I think the problem was at my end, and I think I've fixed it. So, rather than answer all of Bill's long message (Bill Powers, apparently Wed, 12 Dec 2007 11:58:41 -0700), I will just answer one point.

Martin Taylor (12:15 AM 12/12/2007 -0500) --

I didn't specify how many tokens the player starts with. Let's say there are N, so until we know something about the constraints, any one of them could be in-hand, or on square 0, 1, 2, or 3. That would be N degrees of freedom, each with a range of 5 possible values.

There is a logical contradiction here, that led me to leave out "absent" as one of the states of a token. The problem is that these degrees of freedom are not independent, so they are not all legitimate degrees of freedom.

"Absent" is NOT one of the degrees of freedom. It is a member of the set of values that constitute the range of each of the N degrees of freedom for the N variables "location of token number xx".

In any case, to use "absent" as a replacement for "in-hand" is incorrect, since "absent" carries an implication of an indefinite additional range of variation for the location of the token, which includes the possibility of "non-existent", whereas "in-hand" is a definite location of the token in the game.

Much of what immediately follows in Bill's message seems to take off from this fundamental misunderstanding of the difference between a degree of freedom and the range of values over which the variable identified with that degree of freedom can vary. In the case in point, the variable is an attribute of the token -- its location. In other parts of the board game analysis, the variable in question is an attribute of a square on the board, or the reference value for the controller of the hand.

I think we have to start back at an even simpler point. I thought I had reached the ultimate in simplicity with this trivial one-person board game (in my first draft I had two people placing chess pices on a chess board), but apparently I was wrong. I'll read the rest of Bill's message, and see whether it includes valid comments that don't follow from misunderstanding the difference between a varioable and a degree of freedom, and of the difference between a degree of freedom and the range of the associated variable.

There's an earlier misunderstanding that may already have been fixed, and that is the notion that a "variable" is eqivalent to a degree of freedom. It isn't. A degree of freedom implies the existence of a variable, but the reverse is not true.

Incidentally, I may be wrong, but I have the impression that when messages get very long, it's a sign that the discussion is getting lost in a fog of cross purposes and misunderstandings, so I'll try to be shorter in future.

Leesten vairy carefully...I shall say thees aonly waance [Michelle of the Resistance]. I hope the reference is familiar. If it isn't, you have missed one of the funniest shows ever on TV.

I hope I receive it only once.

Martin

[MartinT]

Martin Taylor (12:15 AM 12/12/2007 -0500) --

responding to Bill Powers, apparently Wed, 12 Dec 2007 11:58:41 -0700

True also that the board itself has no degrees of freedom. If I used that language it was shorthand for saying that there are just so many "degrees of freedom for the perception of the occupancy of the board squares by tokens of different colours". It's shorter to say "board degrees of freedom".

Yes, but very misleading when you're constructing an explanation. Several times you have said there are degrees of freedom in the environment where there are no variables to vary. There is a similar problem with "occupancy," which has characteristics in common with the smile of the Cheshire Cat.

I don't agree with that characterization. The "occupancy" is an attribute of each square of the board. It is a variable with the possible values {0, r, w, and b}. The player can perceive the current occupancy of each square, and can influence it if the current value is 0. There's nothing Cheshire Cat-like about it. It's a simple observable value of an attribute.

If a token with one of three colors can be either on the board in one of four positions, or not on the board, we then have the 12 combinations I defined, plus what appears to be one more degree of freedom: on the board or off the board ("in hand").

It's not a degree of freedom. It's an element of the range of a degree of freedom. The attribute (variable) in question is an attribute of the token: its location.

True degrees of freedom allow for free choice of a state without regard to other degrees of freedom.

That close enough to true that there's no point yet in going into the reasons it isn't quite true, since we haven't reached a clear understanding of the distinction between a degree of freedom and the range of its possible values.

A true third degree of freedom would be, e.g., shape: round or square.

Yep. That's a possible attribute of an object, and one that can be perceived. If you have the right tools, you can even influence it.

The real problem here is one that arises often in designing switching circuits -- failure to account for all logical states of each variable.

No it isn't. It's the failure to recognize that if a variable "x" can have values 1,2,3 or 4, the variable doesn't have four degrees of freedom. A scalar variable has one degree of freedom, no matter how many values it might have, even aleph 1 ( as would be the case if x could be any real number in a finite range). Bill's following commentary depends on a failure to realize this distinctionso I'll omit it.

This means that only one controlling agent controlling in both dimensions can place a token of any possible color in any possible position on the board. Adding even one more agent with two similar control systems means loss of range over which reference conditions can freely be varied.

This is where I lose you.

I should have said "loss of range over which the controlled variable can be varied to match any possible reference signal." Of course the reference signals can be varied in any way you like. The following should clarify this.

Degrees of freedom of action and degrees of freedom of perception are clearly different,

Yes they are, but not for the following reason -- or not only for the following reason -- and not when we consider a single standard type control system.

because the action can vary without a significant change in the perception, and the perception can change without a significant change in the action. This is possible because of disturbances and other kinds of changes in the external part of the loop.

You state a fact here, but it's irrelevant to the degrees of freedom. Let's consider an elementary control unit (ECU).

                        reference
                            >
                            V
                 ^----p-----0------e------
                 p e
                 > >
            perceptual output
              input function
             function |
               /|\ o
              / | \ /|\

···

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
                   Lower level ECUs,
                which are part of the
               evironmental feedback path
                  of the higher one

          ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
             / | \ / | \
           components of effects on
          the controlled <------- the world
            perceptual <------- outside the
             variable etc organism
        / | \
          disturbances

In this control system, the reference, perception, error, and output are all single-valued functions. Each has one degree of freedom. But those degrees of freedom are not independent, since e = r-p, and o(t) = f(e(t)). At any moment, one cannot determine o from e, so if one looks at the internal part of the ECU, there are three degrees of freedom. They are different, and p is the degree of freedom for perception and o is the degree of freedom for action.

Now it's quite true, as you say, that action might not influence perception, either because it happens to be exact in its opposition to the disturbance, and that perception can vary without action. But that's a side issue. The main point is that internal to the ECU, the single-values signals can vary independently.

An action is (almost always) a set of controlled variables of a lower order than the perception controlled by them. If Fa is the function turning the action-variables a into an effect on a controlled variable, and Fd is the function turning the states of all disturbing variables d into another effect on the controlled variable, then the state of the controlled variable V is

V = Fa(a1..an) + Fd(d1 .. dm).

With multiple control systems all using the same set of action variables, it is possible for each of k systems to make Vk match an arbitrary reference value Rk, as long as k is less than or equal to n. So the degrees of freedom of control are limited by the degrees of freedom of action.

Well, I'm glad that you agree with me that the action pathway from output to the extenal world is one possible bottleneck. I've been trying to make that point in several messages to which your comment was that I was insisting that the limit was in the environment. But the bottleneck isn't always there.

The most important thing about this little derivation, with which you are quite familiar, is that when all the degrees of freedom are just used up, it is still possible for each control systems to bring its own input to any reference condition within the physically possible range without preventing any other control system in the set from doing the same. This says there is a solution to the set of equations Vk = Rk for all possible Rk.

This is not like your example of the game board. In that example, only one control system can freely place a token in any of the 12 states possible. If there are two control systems, the first to act is free to create any state, but the second one can then create only 9 of the states, since one position is no longer accessible. If there are four control systems, the last one to act can set its reference signal to any of the 12 states (as you noted) but its controlled variable can reach only three of them.

Yep. In this case, the bottleneck is indeed in the environment.

Contrast that with my derivation above that shows all k control systems still being able to select and reach all reference states within the same range that one system alone could reach.

Your derivation operates in a world in which that is in fact true. It's a different environment.

Note that we almost automatically introduce new degrees of freedom.

The degrees of freedom you introduce are attributes of something new, seen by somebody like a captian, who could manipulate the sequence. It's not relevant to the issue.

I say that there are 11 major classes of degrees of freedom that can be controlled by using our limited outputs in ways that bring up new kinds of variables to control

They aren't "classes of degrees of freedom". That's like saying "classes of number". You are talking about different variables, to which degrees of freedom may be association.

-- ways in which the environment can be controlled that were there all along (I presume) but which don't get counted until they're mentioned. We are limited in how many of them can be controlled simultaneously, but the "bottlenecks" of which you speak don't mean that n outputs can't be used to control more than n inputs. It just means you can't act independently on more than that many at one time.

Exactly my point. There's NEVER been an argument about that. And moreover, that n cannot be larger than the n at any lower level in the hierarchy.Nor can it be larger than the fewest degrees of freedom available anywhere in the combined environmental feedback paths of the n systems. The skeleto-muscular system puts an upper bound on n, but it's not usually the effective bound.

Time division multiplexing all has to do with the degrees of freedom implied by the bandwidths of the signals at different levels, and in particular the bandwidths of any disturbances. Time multiplexing doesn't always work, but if the bandwidths are appropriate, it can.

I find this all very frustrating, but you do seem to be moving in what I see as the right direction.

As I said a few posts ago, most environmental variables change very slowly, so with n independent outputs we can control most variables, far more than n of them, as fast as we need to even though we have to time-share among them.

Precisely. I knew you knew it, and I wondered why you resisted making the statement quantitative.

The apparent degrees of freedom of our joints and muscles are only a part of the story; when we bring in time-sharing, we see that the same output units can be applied to different parts of the environment so as to affect completely different sets of variables on different occasions, greatly multiplying the degrees of freedom of action by allowing the same joint-angle actions to have very different environmental consequences. I can reach to my left and press a button that turns on a light; I can then (while the light stays in the state I want) reach to my right and press a button that opens the spillway of a dam. Surely these controlled variables represent two quite different and independent degrees of freedom.

I suppose I should say "two cheers". Only two, because we still have a way to go -- we haven't even started in on tolerance, yet -- and I don't understand why, knowing the above, you chose to argue the identical points I made quantitatively about how analyzing the available degrees of freedom made clear when time-sharing (time-division multiplexing) will and won't work. Maybe it was the misunderstanding about mixing up the degrees of freedom with the range of the variable associated with a degree of freedom. I hope at least that is cleared up now.

Martin

[Bill_Powers1]

Incidentally, I may be wrong,
but I have the impression that when messages get very long, it’s a sign
that the discussion is getting lost in a fog of cross purposes and
misunderstandings, so I’ll try to be shorter in future.
[From Bill Powers (2007.12.13.2113 MST)]

Martin Taylor 2007.12.13.11.41 –

I’ve been having the same thought. I will be as brief as I can here,
which isn’t very brief.

The discussion of degrees of freedom has taken us off on a tangent from
the original subject, which was conflict.

Conflict came up because conflict between control systems results in
impairment of control, so avoiding or resolving conflicts is
important.

We don’t need to talk about degrees of freedom to talk about conflicts.
All we have to do is define what a conflict is. We can define it either
qualitatively or quantitatively.

A qualitative definition would be “A conflict exists if the
achievement of a reference condition by one control system prevents
another control system from achieving its reference condition, and vice
versa.” In this view, a conflict either exists or it doesn’t exist,
because reference conditions are either achieved or not achieved (are or
are not matched by the relevant perception).

A quantitative definition would be “Conflict exists when two control
systems interact to a degree that impairs their abilities to
control.” Just what constitutes “impairment” depends on
the situation. The quantitative definition is the most useful one because
it covers degrees of conflict.

Conflicts can arise in a number of ways and among more than two systems.
The actions of each system can interfere with the actions of the other
systems. The actions of each system can, without interfering with the
actions of the others, disturb the controlled variables of the others.
The controlled variables of the systems may be defined by non-orthogonal
perceptual input functions so it is impossible for one perception to be
changed by any means without altering the others.

The controlled perceptions in control systems are in general derived from
multiple variables of lower order. Each perception can be described as a
function of some set of variables:

p[i] = F[i](x1, x2, …xn).

If the x-variables in each function are a different subset of all such
variables, there will be minimal conflict. Conflicts can arise when the
subsets interact and effectively overlap, so the same x contributes to
two or more different perceptions. Generally the non-overlapping x’s can
be adjusted to make it unnecessary for any x in the overlapping set to
have more than one value at a time. Conflicts may temporarily exist but
can be eliminated by reorganizing the way the systems act on the
lower-order world.

As the size of overlapping sets of x’s increases, the adjustments of the
non-overlapping x’s needed to avoid conflict are likely to become more
and more extreme, and it becomes more important that the different
perceptual input functions be orthogonal in the shared x’s. Some degree
of interaction can be tolerated, but above some minimum “composite
loop gain” (tracing loops through more than one system) the positive
feedback becomes great enough to produce a runaway condition and the
conflict becomes acute. Control systems are driven to limits of operation
and control is lost. If the control systems have high loop gains, the
amount of interaction needed to result in runaway conflict can be quite
small.

Completely free control by all systems means that p[i] = r[i] for all
systems. The condition of free control at one level can be defined
as

F[i](x1, x2, …xn) = r[i]

for all values of r[i].

There is one limiting case in which all of the systems derive their
perceptions from the same subset of x’s, and there are no non-overlapping
x’s. In this case, conflict can be avoided only to the extent that all
the F[i] input functions are nearly orthogonal. In this case, there is a
limit on the number of different equations that can be satisfied
perfectly at once; it cannot be greater than the number of x’s. As the
number of equations increases above that limit, some or all of the
control systems must experience increasing error signals, defined
as

e[i] = r[i] - F[i](x1, x2, …xn).

These increasing error signals will produce increasing outputs that
further increase the error signals, and at some point the positive
feedback will exceed unity and the transition to maximum conflict (or
self-sustained oscillation) will occur.

I don’t think I have given you anything to disagree with here.

I have just realized that in the “ThreeSystems” demonstration,
transitions to runaway conflict quite commonly occur. They show up when
the plot of total error suddenly jumps to a very high value and
reorganizations become almost continuous. The plots of the variables
typically go to extreme and chaotic forms while this is happening. After
some time, the magnitude of total error drops just as suddenly back to
its former value and reorganization continues in the usual way. These
episodes tend to occur the most after the total error has become
relatively small. I have never seen the systems fail to recover from this
condition.

Best,

Bill P.

···

[rsmarken]

[From Rick Marken (2007.12.14.0910)]

Bill Powers (2007.12.13.2113 MST)--

Martin Taylor 2007.12.13.11.41 --

I've got to admit that this conversation is way above my head but,
from what I can understand of it, I'd have to give the nod to Bill,
for just plain ol' simplicity.

I'm posting because I saw this little headline in that liberal rag,
the NYTimes and I think it's relevant to the topic of conflict and
degrees of freedom (df):

Consumer Prices Rise 0.8% in November

The pick-up in prices will complicate efforts by the Fed as it tries
to stave off a slowdown in economic growth.

The "complication" is, I think, a nice example of a conflict. The Fed
is trying to control two perceptions, of growth and inflation. It
controls both by controlling the amount of money in circulation (by
adjusting interest rates): let's call that variable "funds". So the
Fed is trying to control two variables -- growth and inflation -- by
varying only one variable (one degree of freedom): funds.

The Fed tries to keep inflation down by decreasing funds (increasing
interest rates) and it tries to keep growth up by increasing funds
(lowering interest rates). So the growth control and inflation
control systems in the Fed are often working at cross purposes
because, in order to control these two variables, the the funds
variable must often be in two different states simultaneously which,
as Martin points out, is impossible when there is only one df. This is
the conflict (complication) the Fed currently faces. It wants to
decrease funds to curb inflation and increase funds to present
recession (0 or negative growth).

This conflict could be easily solved if the Fed would go "up a level",
look at the data on the relationship between variations in funds and
variations in inflation and growth, and see that variations in the
funds actually has no effect on either inflation or growth. Then the
Fed could relax and stop trying to control inflation and growth. This
would make the funds df available for more productive things, like
reducing income inequality through micro-loans.

Best

Rick

···

---
Richard S. Marken PhD
rsmarken@gmail.com

[MartinT]

[Martin Taylor 2007.12.14.10.18]

[From Bill Powers (2007.12.13.2113 MST)]

Martin Taylor 2007.12.13.11.41 --

Incidentally, I may be wrong, but I have the impression that when messages get very long, it's a sign that the discussion is getting lost in a fog of cross purposes and misunderstandings, so I'll try to be shorter in future.

I've been having the same thought. I will be as brief as I can here, which isn't very brief.

The discussion of degrees of freedom has taken us off on a tangent from the original subject, which was conflict.

I disagree. My reason for getting into the discussion of degrees of freedom was so that we could discuss conflict in a general way, without the ambiguities inherent (as you often point out) in purely verbal descriptions, and without relying on linear algebra or simulation for the behavioural part. I think the discussion has been getting closer to returning to that goal.

The DF discussion became tedious because we had unexpectedly different views as to the nature of the degrees of freedom of a system. To me, the problem since that discovery has seemed to be very much like that of the first European explorers trying to find a way through the mountains to the Pacific. One valley leads to an impasse, so the explorers backtrack and try another side valley. When that fails, they backtrack even further and try a different starting valley. The objective, however, remains the same: to cross the mountains and reach the ocean they know to be on the other side of the mountains, and to be worth reaching.

I started with the presumption that most people on this list would clearly understand the implications of degrees of freedomd, at least in the static sense of "If I can move east-west, north-south, or in a combination of those directions, I have two degrees of freedom." I was not so sanguine about everyone understanding the way that degrees of freedom depend on time and bandwidth, but I thought a few sentences to explain it, with maybe an example or two, would suffice to get the point across.

I was apparently too optimistic. Apart from Bill Powers, only Erling has made any comment. Erling asked excellent questions, but in only one message, and nobody else has asked any questions or made any comment. So I wonder whether the issue of how ccontrol systems interact in a social context is simply uninteresting, whether it's all a bit much, technically, or whether most people do understand it and I'm only rehashing boring old stuff.

Conflict came up because conflict between control systems results in impairment of control, so avoiding or resolving conflicts is important.

That is indeed true. It's a very general statement, and it immediately suggests that one should ask "Under what circumstances does conflict between control systems arise, and what are the ways its effects can be mitigated."

That's the question I began to address by introducing the degrees of freedom. Bill P. made the comment that very seldom does conflict have anything to do with degrees of freedom, a response that puzzled me greatly, because it seemed (and seems) to me that almost all, if not all, conflict arises because of a reestriction in the available degrees of freedom somewhere in the feedback circuits of the conflicted systems. At that point, Bill had defined conflict as happening between two control systems when they reach a limit where both are exerting maximum output against each other, whereas the degrees-of-freedom approach allows one easily to deal with conflicts involving an arbitrary number of control systems, whether the conflicts become overt or not.

We don't need to talk about degrees of freedom to talk about conflicts.

True, we don't have to. But if we want to advance the theoretical (and practical) understanding of the interactions among many control systems, it certainly helps if we do. Every different viewpoint helps, if it is correct. Just as we don't _need_ to talk about degrees of freedom, we don't _need_ to stick to the well used ways of talking about conflicts. But we _can_ do both.

When I participate in CSGnet discussions, my hope is to advance the science of PCT, and I think this is one area in which a noticeable advance is possible.

All we have to do is define what a conflict is. We can define it either qualitatively or quantitatively.

And we can define it in terms of the effects that occur under conditions of conflict, or in tems of the conditions that lead to those effects. In what follows, you use the former, and then describe mechanisms that have frequently been described over the years. That's useful. It's also useful to start at the other end and use an approach that I don't remember having been worked through on CSGnet.

The approach you are using is the same that you have used for as long as I can remember. It's either verbal or depends on the systems using continuous-valued signals and (for the mathematics to work) having linear components. At levels above the category level, the systems work with signals of discontinuous values (something is or is not peceived as belonging to a particular category), and in a real living system it is unlikely that the linear condition applies at anu level at all. Certainly the hierarchy would be irrelevant if the perceptual input functions were linear, since all a set of linear perceptual input functions do is rotate the input vector, and all possible rotations can be peformed in a single level.

A qualitative definition would be "A conflict exists if the achievement of a reference condition by one control system prevents another control system from achieving its reference condition, and vice versa." In this view, a conflict either exists or it doesn't exist, because reference conditions are either achieved or not achieved (are or are not matched by the relevant perception).

An equivalent, but more inclusive qualitative definition would be: "A conflict exists if, in order to achieve their reference conditions, a set of N control systems is constrained to use fewer than N degrees of freedom at some point in their set of feedback loops."

A quantitative definition would be "Conflict exists when two control systems interact to a degree that impairs their abilities to control." Just what constitutes "impairment" depends on the situation. The quantitative definition is the most useful one because it covers degrees of conflict.

Yes. However, in this, there is that lovely vagueness "depends on the situation". In what follows, you refer to perceptions being functions of many lower-level variables, in which the set of lower-level variables may overlap to some degree among different controlled perceptions (p[i] = F[i](x1, x2, ....xn)). Using the degrees-of-freedom approach, one would use the same technique, but would take each of the x as x(t) and consider its bandwidth and any time constraints, to assess how many degrees of freedom were actually involved. That would be the number this variable contributes to the degrees of freedom needed by the N control systems in the definition above. The vague phrase "depends on the situation" is replaced by a quantifiable statement.

...
The controlled perceptions in control systems are in general derived from multiple variables of lower order. Each perception can be described as a function of some set of variables:

p[i] = F[i](x1, x2, ....xn).

...
As the number of equations increases above that limit, some or all of the control systems must experience increasing error signals, defined as

e[i] = r[i] - F[i](x1, x2, ....xn).

These increasing error signals will produce increasing outputs that further increase the error signals, and at some point the positive feedback will exceed unity and the transition to maximum conflict (or self-sustained oscillation) will occur.

I don't think I have given you anything to disagree with here.

No. Nothing at all. In most of this discussion I have not disagreed with what you have said regarding your own analyses. My issues have been with your treatment of the degrees of freedom.

As a side note, though, you said "sustained oscillation" above, which implies that at least one of the systems is non-linear. Non-linear systems can do some mighty weird things.

I want to continue with the degrees of freedom discussion, in the hope that we can arrive at a description and analysis that is both agreed, and simple enough that anyone can use it, at least heuristically, to consider cases in which multiple control systems in some way interact either internal to an organism or in a common environment.

Martin

[Bill_Powers1]

The DF discussion became tedious
because we had unexpectedly different views as to the nature of the
degrees of freedom of a system.
[From Bill Powers (2007.12.14.0500 MST)]
Martin Taylor 2007.12.14.10.18 -
I’m afraid that this remark is not destined to draw a brief
response.
I think this is a misapprehension on your part. We have the same concept
of degrees of freedom as far as your definitions go. If you will adopt
that premise, you might see that where you thought I was talking about
degrees of freedom, I was talking about the number of states, and where I
as talking about the number of states, you thought I really meant degrees
of freedom. In fact there is no one way of defining degrees of freedom:
it’s a matter of perception. That is the main point I’ve be both working
out for myself and trying to communicate.
For example, when you spoke of the game board’s squares, you said that
each one had two states: occupied or unoccupied. That would mean that the
set of four squares can be seen as having four degrees of freedom with
two states each, or one degree of freedom with 16 possible states.
You could also have spoken of the left two squares (which together have 4
states) or the right squares (also having 4 states), so now we have two
degrees of freedom with 4 states of each. The degrees of freedom of which
you speak depend on how you’re perceiving the board. The total number of
states remains the same in this case, but not in all cases as I will
show.

I pointed out that the tokens have 2 degrees of freedom: a location
relative to the squares on the board, and the selection of a color. I did
debate defining “on the board – not on the board” as an
independent degree of freedom, but concluded that this was not possible
because it led to a contradiction. The exact number of states implied is
a little uncertain, because the location “not on the board”
appears to be a fifth possible position relative to the squares – except
for the rule saying no two tokens can occupy one position. If there are
more than 5 tokens, “not on the board” can’t be a single
location, so this is not a fifth state of the position relative to the
squares OR a degree of freedom. It’s an indeterminate position. That part
of my comments was about the number of states of the tokens relative to
the board, not the degrees of freedom.

I also pointed out that in speaking of “occupancy” of a square,
you were actually making an oblique reference to elements that were not
part of the board: the token doing the occupying. So
“occupancy” could not be a state of the board alone; you also
had to know where the tokens were. Occupancy is a relationship between
two things, not a state of a single thing. You seemed to agree that the
board by itself has no dimensions of variation, so has no degrees of
freedom. But somehow you took this to mean I didn’t understand degrees of
freedom. Do you understand what I was talking about, now?

As I’ve been exploring these ideas, the idea of logical relationships
occurred to me (and I mentioned them) as a possible alternative way of
measuring states within degrees of freedom. Now I think I see how to
follow that line.

Let us say that there are four squares, and to eliminate problems of
introducing extraneous elements, let’s just say that they can have marks
or no marks on them, forgetting about colors for now. So the pattern of
marks would seem to be limited to 16 possibilities. It would seem that no
matter what perceptions you build out of the arrangements of those marks,
there can never be more than 16 possible states of the board.

But now instead of looking at the marks numerically, let’s look at them
logically. Any logical expression can be expanded into
“minterms”. A minterm is made of all the variables or their
negations connected by AND. When all the minterms and their negations are
ORed together, the result has the same logical state as the original
expression. An expression in two variables A and B has 16 minterms. In
general a logical function (with AND, OR, and NOT) of n variables has
2^2^n distinct minterms, each of which can be true or false. For two
variables that is 2^(2^2) = 2^4 = 16; for three variables it is 2^(2^3) =
2^8 = 256; and for four variables it is 2^(2^4) = 2^16 = 65536. So now,
perceiving the board through logical input functions, we can distinguish
65536 different states.

I am talking here about the number of states, not the number of degrees
of freedom, which is still apparently four (although they are different
from numerical degrees of freedom). But the meaning of these four degrees
of freedom has vastly changed. If these are output degrees of freedom,
they can be connected not just to 16 different consequences, but to 65536
consequences. Each consequence could be the state of an independent
degree of freedom in the environment (for example, which of 65536
different squares has a state of 0 or 1). What had appeared to be a
bottleneck now is at least much less of one.

I started with the
presumption that most people on this list would clearly understand the
implications of degrees of freedomd, at least in the static sense of
“If I can move east-west, north-south, or in a combination of those
directions, I have two degrees of freedom.” I was apparently too
optimistic.

Perhaps you were also overly optimistic in assuming that you had
considered all the facets of degrees of freedom. I believe you have
overlooked the fact that at the output of a control system, the n output
signals can be connected to the environment in many different ways and
places, affecting not just n degrees of freedom but far more than n at
any one time. As I pointed out above, if we look at the number of
logically different ways in which four output variables can affect the
environment, we find a number many times as great as the number of ways
the same variables can affect the environment in terms of vector
magnitudes. Logic is, of course, one of the levels of perception and
control that I have defined for HPCT. So it would seem that at the logic
level, we are not limited to the number of different states of a
perception that we can control by the joint configurations of an arm. In
fact, if variables in all 27 degrees of freedom of the arm and hand
joints could be independently adjusted, 2^2^27 or 2^134217728 logically
different output states could be produced with one arm and hand. The
actual number is probably only about 2^2^20, which is “only”
2^1048576, still immensely more than the number of elementary particles
in the universe. Given the right auxiliary devices, one hand could
control the states of any number of independent binary variables that you
could imagine, far more than you could perceive as magnitudes. And this
is considering only binary variables.

I am not confusing the number of states with the number of degrees of
freedom. Each different state of an output can have an effect on a
different degree of freedom of the environment, as I indicated with my
example of using the position of an arm to allowing pressing two
different buttons that have two completely different kinds of effects on
the world. If the states are binary, we can compute how many logically
different degrees of freedom can be affected. If the states are
continuous variables, then we can subdivide the magnitudes into ranges,
and each range can adjust the state of a different degree of freedom in
the environment (although not all simultaneously).

When you introduce the time dimension, exactly the same considerations
arise, and more. Location along the time axis is not the only way in
which variables can change. The ordering of temporal relationships can be
changed, and ordering can be varied (over some range) independently of
variations in time of occurrance. Logic can be introduced in this
dimension, too, so far more possible states are generated, and again
through the use of suitable features in the environment, each different
STATE of the temporal variables can be connected to variables in a
different DEGREE OF FREEDOM of the environment.

I get the impression that there is some background reason for which you
want to make degrees of freedom a basic limiting factor in PCT. Perhaps
this has to do with your idea that the second law of thermodynamics is
fundamental to the principles of negative feedback control. I can see why
you would need to preserve a simple view of what constitutes a degree of
freedom and of what limitations are imposed by degrees of freedom, but
goal-oriented reasoning is not usually very reliable, and I doubt that
the thermodynamic argument is worth preserving for its own sake.

This is no doubt far from the last time we will collide on these issues,
but I’ve really said all I have to say on this subject for now, and
simply repeating our positions will not lead to anything new. I suggest
that we shelve this discussion until some remarkable new consideration
occurs to one of us,

Best.

Bill P.

[Bill_Powers1]

This is the conflict
(complication) the Fed currently faces. It wants to

decrease funds to curb inflation and increase funds to pre[v]ent

recession (0 or negative growth).
[From Bill Powers (2007.12.15.1149 MST)]

Rick Marken (2007.12.14.0910) –

Nice catch. We desperately need a model.

Best,

Bill P>

[MartinT]

[Martin Taylor 2007.12.16.00.47]

[From Bill Powers (2007.12.14.0500 MST)]

This is no doubt far from the last time we will collide on these issues, but I've really said all I have to say on this subject for now, and simply repeating our positions will not lead to anything new. I suggest that we shelve this discussion until some remarkable new consideration occurs to one of us,

I am loath to do that, as I perceive substantial convergence, and there is material in your latest post on which we can build. There is no reason why we should continue to "collide". We need to reach a workable understanding of the issues. Your post seems to me to represent a step in this direction.

For example, you have nicely illustrated how the one degree of freedom controlled by a high level system can be manifest in a number of degrees of freedom controlled at lower levels, in the same way that a system controlling x+y+z can be manifest in three lower-level systems, each of which controls its own one df perception, one for x, one for y, and one for z. As far as the higher-level controller is concerned, it doesn't perceive the 3-D matrix of varying values of x, y, and z. It perceives only x+y+z.

I still don't know whether our apparent disagreements are from different uses of language or from different appreciations of the underlying phenomena, but I get the impression that common understanding should not be too hard to achieve.

I get the impression that there is some background reason for which you want to make degrees of freedom a basic limiting factor in PCT.

Yes, there is a reason why I want to make sure the concepts are clear and the ways to apply them are understood. It is that using degrees of freedom appropriately, one can draw a lot of conclusions very easily that are not easily seen from the viewpoints we generally use. One item of evidence for this is that the value of tolerance zones was not apparent to you when I brought them up as one way of permitting systems of many control systems to remain stable in the face of resource limitation. Another is in the example of two people trying to pass through a door wide enough for one.

I don't "want to make degrees of freedom a basic limiting factor in PCT." They just are, in the same way that loop gain is a limiting factor for the behaviour of a feedback loop, in that a positive loop gain greater than unity leads to an exponential explosion. It's just a simple fact. I don't say you "want to make G = +1 a limiting factor in the behaviour of a loop". It just is. I don't "want to make phase shift a limiting factor in the high frequency gain of a control loop". It just is. Same deal for degrees of freedom. M into N won't go, if M > N. If M and N refer to degrees of freedom and number of control loops, we are immediately into conflict analysis. That's the background reason.

I'll comment further on your message tomorrow, probably.

Martin

[Bill_Powers1]

[From Bill Powers
(2007.12.14.0500 MST)]

I suggest that we shelve this discussion until some remarkable new
consideration occurs to one of us,

We need to reach a workable understanding of the issues. Your post seems
to me to represent a step in this direction.
For example, you have nicely
illustrated how the one degree of freedom controlled by a high level
system can be manifest in a number of degrees of freedom controlled at
lower levels, in the same way that a system controlling x+y+z can be
manifest in three lower-level systems, each of which controls its own one
df perception, one for x, one for y, and one for z. As far as the
higher-level controller is concerned, it doesn’t perceive the 3-D matrix
of varying values of x, y, and z. It perceives only
x+y+z.
[From Bill Powers (2007.12.16.0703 MST)]

Martin Taylor 2007.12.16.00.47 –

OK, we can go on …

I am thinking of another example that seems different from
yours.

We perceive the world as two two-dimensional retinal images, neither one
alone containing depth information. These images have two degrees of
freedom, but when signals representing the two images are combined in the
right kind of network, we don’t just get an algebraic function of x and y
with the same two degrees of freedom (we could have two such algebraic
functions at one time); we get a new degree of freedom. It was there all
along, of course, implicit in the two separate retinal images. But it was
neither perceivable nor controllable at the lower level. We can now
actually have three independent simultaneous equations in x, y, and z,
where before we could have only two simultaneous equations in x and
y,

This looks to me like a case in which “going up a level” does
reveal a new degree of freedom rather than just another function of the
same variables at the lower level. We do not see the extra degree of
freedom until we adopt the point of view of the higher system. Then we
realize that this kind of dimension was always present in the lower-level
world, but was unrecognized and not under control at those
levels.

I don’t know what makes the difference between gaining a new degree of
freedom (in the sense of newly perceivable, though implicit in the lower
level information) and simply adding another function of existing
variables without adding a degree of freedom. Working that out would
probably be useful.

Somewhere in my stored materials is a notebook with a passage transcribed
from a paper by James Clerk Maxwell on “Thompson and Tait’s
Philosophy,” the passage being titled “The ignoration of
coordinates.” Maxell gives an example of bellringers holding ropes
which pass upward through holes in the ceiling to the belfry, where they
operate invisible bells. Through feeling the resistances of the bells,
the ringers can set them tolling or quiet them, and so achieve the sounds
they wish to produce. But they have no knowledge of how many
“coordinates” there are in the belfry – there could be many
more than they sense through pulling on the ropes, and the actual ones
could be different from those that the ringers deduce. The ringers,
Maxwell says, have no choice but to ignore the coordinates they cannot
sense.

What I would add is that by increasing the ability to perceive the
environment in more and more ways, it may be possible to discover new
coordinates (degrees of freedom) and learn to control in new ways, so
that coordinates previously ignored become visible and new kinds of
control can be achieved – and so that the number of different variables
that can be controlled simultaneously and independently becomes greater
than it was before.

Adding a second eye just like the first one but in a different place
makes it possible to perceive and control in a new dimension, in a way
that just adding the same number of new rods and cones to one eye would
not accomplish.

I wonder how many dimensions a spider can see.

Best,

Bill P.

[MartinT]

Re: Degrees of freedom, conflict and
tolerance
[Martin Taylor 2007.12.16.11.10]

[From Bill Powers (2007.12.14.0500
MST)]

At the end of your message, you say:

I’ve really said all I have to say on
this subject for now, and simply repeating our positions will not lead
to anything new. I suggest that we shelve this discussion until some
remarkable new consideration occurs to one of us,

For the last several days, I’ve been trying one new approach
after another. I’m surprised you think of it as simply repeating
my position. I thought I was using the PCT approach of “many
means to the same end”. I’ll keep doing it for a while, whenever
I can think of a different approach or if I think of something that
changes my understanding. But for now, I’ll deal with your message,
trying to trace where we agree, which I suspect is in most
places.

Martin Taylor 2007.12.14.10.18 -

The DF discussion became tedious because
we had unexpectedly different views as to the nature of the degrees of
freedom of a system.
In fact there is no one way of defining
degrees of freedom: it’s a matter of perception. That is the main
point I’ve be both working out for myself and trying to
communicate.

I’m not sure I agree with that, except in the sense that
everything is a matter of perception. However, let’s leave that aside
and see where it leads us.

For example, when you spoke of the game
board’s squares, you said that each one had two states: occupied or
unoccupied. That would mean that the set of four squares can be seen
as having four degrees of freedom with two states each, or one
degree of freedom with 16 possible states.

Yes. Good. We do agree on that. The key is “can be seen
as”, which I read as “could be the perception provided by
the input function of a control system”. Thinking more generally
of a larger grid, you could have a control system perceiving
“location” as a scalar variable, and that control system
could send reference values for two lower level control systems, one
of which controls x, the other y, each of which is a scalar
variable.

Thinking only of the two lower-level control systems, there are
two degrees of freedom, but if we notice that they are both serving
the “location” control system, we see that if its perception
is at its reference, BOTH the lower level ones have been fixed, and
there is no second degree of freedom.

So, whether you see one or two degrees of freedom depends on how
much of the setup you look at. often, if you look at more of the
system, you see more constraints, so fewer degrees of freedom
overall.

You could also have spoken of the
left two squares (which together have 4 states) or the right squares
(also having 4 states), so now we have two degrees of freedom with 4
states of each. The degrees of freedom of which you speak depend on
how you’re perceiving the board.

Yes, we agree. But I think it might be clearer to say not
“how you’re perceiving the board” but “what perceptual
functions are being used to peceive the board”. “You”
is an ill-defined set of control systems, and I think probably refers
to your “Observer” function.

The total number of states remains
the same in this case, but not in all cases as I will show.
I pointed out that the tokens have 2
degrees of freedom: a location relative to the squares on the board,
and the selection of a color.

A token doesn’t have a degree of freedom for colour, unless you
are allowing the player to paint it. I don’t think this really
matters, except when it comes to determining the degrees of freedom in
different parts of the set of control loops. Then it matters.

I also pointed out that in speaking of
“occupancy” of a square, you were actually making an oblique
reference to elements that were not part of the board: the token doing
the occupying. So “occupancy” could not be a state of the
board alone; you also had to know where the tokens were.

Occupancy is an attribute of the square, just as location is an
attribute of the token. The two are not independent, which is where
the relationship comes in.

Occupancy is a relationship between
two things, not a state of a single thing. You seemed to agree that
the board by itself has no dimensions of variation, so has no degrees
of freedom.

That’s right. The board is fixed.

But somehow you took this to mean I
didn’t understand degrees of freedom. Do you understand what I was
talking about, now?

I don’t know. Apparently I didn’t. Perhaps I do, a little better,
now. Let me see if I can paraphrase in a way that you would accept as
truly representing what you were talking about. I’ll try not to let my
own view intrude, perhaps without success

1. A degree of freedom represents a variable that can take more
   than one state.

2. The Board has no such varables. It is a fixed entity.

3. A square on the board has no such variables. It is
   fixed.

4. A token can be on any of the four squares. Therefore it has a
   variable attribute “location” that is represented by one
   degree of freedom.

4a. A token can be on or off the board. That is a variable
attribute, but may not be represented by a degree of freedom, because
“on the board” incorporates the “location” degree
of freedom, and because “off the board” does not specify any
particular place.

5. The concept [I introduce that term, which you did not use, because I think it is what you meant – correct me if I’m wrong] of
   “token” is instantiated by a token of variable colour.
   Colour is therefore a variable attribute of “token” and
   represents a degree of freedom.

6. The location of a token is a reference to a square, and the
   occupancy of a square is a reference to a token. Both specify a
   relationship. I don’t know whether you attribute any degrees of
   freedom to that relationship.

If this is reasonably close, we can proceed.

I’ll await your response on this before continuing with the rest
of your message.

···

---

I will, however, add a bit of my own. The key is your: “In
fact there is no one way of defining degrees of freedom: it’s a matter
of perception.”

I would separate the two parts of this statement. What is a
matter of perception is not the definition of degrees of freedom, but
the particular variables that are or are not free to vary. My
conceptual starting point in this discussion was to note that each
perceptual input function specifies one degree of freedom, as does the
output signal of each control system.

That the output signal affects possibly complicated functions of
many variables in the lower-level control systems and the esternal
environment in no way changes the fact that it is associated with a
single degree of freedom; likewise that the perceptual input function
may have many different inputs does not change the fact that the
perceptual signal represents one degree of freedom.

If N control systems are active, there are N output signals, but
it is not necessarily true that there are N independent output
pathways to accommodate those N independent output signals. Likewise,
it is not necessarily true that the N independent perceptual input
functions use altogether as many as N independent input signals form
lower-levels or from the environment.

That’s my conceptual foundation. Maybe there’s something there
with which you will disagree, but I can’t see what.

---

If you think I have correctly represented your meaning in my
numbered points, and if you accept my conceptual basis, I think we
must be very close to coming to a mutual understanding and will be
able to advance the science more securely.

Martin

[MartinT]

[Martin Taylor 2007.12.16.13.30]

Between hour-long bouts of snow shovelling!

[From Bill Powers (2007.12.16.0703 MST)]

Martin Taylor 2007.12.16.00.47 --

[From Bill Powers (2007.12.14.0500 MST)]

OK, we can go on ...

Thank you.

I am thinking of another example that seems different from yours.

We perceive the world as two two-dimensional retinal images, neither one alone containing depth information. These images have two degrees of freedom, but when signals representing the two images are combined in the right kind of network, we don't just get an algebraic function of x and y with the same two degrees of freedom (we could have two such algebraic functions at one time); we get a new degree of freedom. It was there all along, of course, implicit in the two separate retinal images. But it was neither perceivable nor controllable at the lower level. We can now actually have three independent simultaneous equations in x, y, and z, where before we could have only two simultaneous equations in x and y,

To make this very short, for fear of splitting the thread into two when you respond to my following message:

    If the two retinal images were uncorrelated, what would you see in the third dimension?
    If the two retinal images are identical, what do you see in the third dimension?

If necessary, I'll comment of what follows along with answering your (or anyone else's) response to my [Martin Taylor 2007.12.16.11.10] message.

Martin

[Bill_Powers1]

Thinking only of the two
lower-level control systems, there are two degrees of freedom, but if we
notice that they are both serving the “location” control
system, we see that if its perception is at its reference, BOTH the lower
level ones have been fixed, and there is no second degree of
freedom.
From Bill Powers (2007.12.17.0110 MST)]

Martin Taylor 2007.12.16.11.10 –

That is a matter of definition. I don’t know what the word
“serving” means above; the system that controls for color of a
token (by searching for a token with the intended color and picking it
up, to clarify a point you raise below) does not have to have anything to
do with the system that positions whatever token is picked up. These two
systems can control their perceptions independently, with any of the 12
combination of reference conditions at least until the first token has
been placed.

So, whether you see
one or two degrees of freedom depends on how much of the setup you look
at. often, if you look at more of the system, you see more constraints,
so fewer degrees of freedom overall.

I don’t know what you mean by that.

You could
also have spoken of the left two squares (which together have 4 states)
or the right squares (also having 4 states), so now we have two degrees
of freedom with 4 states of each. The degrees of freedom of which you
speak depend on how you’re perceiving the
board.

Yes, we agree. But I think it
might be clearer to say not “how you’re perceiving the board”
but “what perceptual functions are being used to perceive the
board”.

Yes, or “at what level of organization the board is being
perceived”. I agree that “you” is ambiguous.

I pointed out that
the tokens have 2 degrees of freedom: a location relative to the squares
on the board, and the selection of a color.

A token doesn’t have a degree of freedom for colour, unless you are
allowing the player to paint it. I don’t think this really matters,
except when it comes to determining the degrees of freedom in different
parts of the set of control loops.

I used the term “selection of a color” to mean selecting a
token that has the desired color, before placing it. The player has two
variables to control: the color of token selected for placement, and the
location where it is to be placed.

I also pointed out
that in speaking of “occupancy” of a square, you were actually
making an oblique reference to elements that were not part of the board:
the token doing the occupying. So “occupancy” could not be a
state of the board alone; you also had to know where the tokens
were.

Occupancy is an attribute of the square, just as location is an attribute
of the token. The two are not independent, which is where the
relationship comes in.

I prefer to say that occupancy or location is an attribute of a
relationship in which the square is one element and the token is the
other element.

Occupancy is
a relationship between two things, not a state of a single thing. You
seemed to agree that the board by itself has no dimensions of variation,
so has no degrees of freedom.

That’s right. The board is fixed.

Well, doesn’t that say that “occupancy” is not a state of the
board itself? It’s very hard to have a conversation if we can’t even
agree on what the words mean.

Let me see if I can
paraphrase in a way that you would accept as truly representing what you
were talking about. I’ll try not to let my own view intrude, perhaps
without success

1. A degree of freedom represents a variable that can take more than one
   state.

That just seems to define a variable: something that can exist in more
than one state. If something is a variable, it can take more than one
state. To me, a degree of freedom refers to a relationship between
different variables. If it is possible for two variables to be set to any
pair of values at the same time, as we can place a point (x,y) in
Cartesian coordinates anywhere in the plane, the relationship has two
degrees of freedom. If x + y = constant, that relationship has one degree
of freedom since specifying the value of one variable also specifies the
value of the other.

2. The Board has no
   such variables. It is a fixed entity.

3. A square on the board has no such variables. It is
   fixed.

Yes. The board is like the basis vectors: it is the Cartesian coordinate
system with no points plotted in it (the location of the squares relative
to each other has not been specified).

4. A token can be
   on any of the four squares. Therefore it has a variable attribute
   “location” that is represented by one degree of
   freedom.

Yes, but that attribute is an attribute not of the physical token itself
but of the relation of the token to a square. You could pick up the token
and slide the right square under it. The token and the square are equal
partners in the relationship.
Tokens in the starting collection have an attribute of color as well.
This color, as you pointed out, stays with the token. However, it is
possible to vary the color of the token that is selected for
placement by changing the selection prior to placement. Changing the
selection of color in no way affects the placement and vice versa, so by
my defintion there are two degrees of freedom. Specifying one does not
specify the other.

4a. A token can be
on or off the board. That is a variable attribute, but may not be
represented by a degree of freedom, because “on the board”
incorporates the “location” degree of freedom, and because
“off the board” does not specify any particular
place.

That is relevant only if there is a rule saying that no two tokens can be
in the same place, as in your discussion of the game. If n tokens can be
in the same place, then n tokens can simultaneously be “off the
board” which consists of all of the area that is not on the board.
In that case we can say that there are five possible values for the
relationship of a token to the board without contradicting a rule of the
game.

5. The concept [I introduce that term, which you did not use, because I think it is what you meant – correct me if I’m wrong] of “token” is
   instantiated by a token of variable colour. Colour is therefore a
   variable attribute of “token” and represents a degree of
   freedom.

Yes. It’s a degree of freedom in the choice of a token.

6. The location of
   a token is a reference to a square, and the occupancy of a square is a
   reference to a token. Both specify a relationship. I don’t know whether
   you attribute any degrees of freedom to that
   relationship.
   I would separate the two parts
   of this statement. What is a matter of perception is not the definition
   of degrees of freedom, but the particular variables that are or are not
   free to vary.

Location is a relative measure, so location is already a relationship.
“Occupancy” simply refers to a specific state of that
relationship: a relative distance of zero.

I will, however, add a bit of my own. The key is your:
“In fact there is no one way of defining degrees of freedom: it’s a
matter of perception.”

Yes, I’ll go along with that. When there is only one variable, of course,
there is always one degree of freedom. Multiple degrees of freedom
involve multiple simultaneous independent relationships between
variables.

My conceptual
starting point in this discussion was to note that each perceptual input
function specifies one degree of freedom, as does the output signal of
each control system.

That the output signal affects possibly complicated functions of many
variables in the lower-level control systems and the esternal environment
in no way changes the fact that it is associated with a single degree of
freedom; likewise that the perceptual input function may have many
different inputs does not change the fact that the perceptual signal
represents one degree of freedom.

I agree. Degrees of freedom have to be counted all at the same level, or
you will get too many.

If N control
systems are active, there are N output signals, but it is not necessarily
true that there are N independent output pathways to accommodate those N
independent output signals.

Yes, but it is also not true that the number of independent pathways has
to be N or less. It is independence of RELATIONSHIPS that defines degrees
of freedom, not independence of pathways, so if one output can be aimed
to affect one variable while not disturbing another, one output variable
can affect many environmental variables in ways that are independent of
each other. This entails multiplexing, of course, but it shows that the
number of degrees of freedom in the world being controlled can be far
greater than the number of outputs available to control them – as long
as the disturbances don’t change too fast.

Likewise, it is not
necessarily true that the N independent perceptual input functions use
altogether as many as N independent input signals from lower-levels or
from the environment.

That’s my
conceptual foundation. Maybe there’s something there with which you will
disagree, but I can’t see what.

First I have to figure out what you said. I think you said that N
independent perceptual input functions probably do not use all N
independent inputs from lower levels. If that’s what the above says, I
agree with it, though I would add that there are probably a lot more than
N independent inputs available from lower systems. My demos in which all
degrees of freedom are used up is a worst case; if reorganization can
achieve control under those conditions, it can certainly do so when there
are more available degrees of freedom than are needed (which I consider
to be by far the most likely case).

Best,

Bill P.

[_Erling_Jorgensen]

[From Erling Jorgensen (2007.12.17 1355 EST)]

Martin Taylor 2007.12.16.11.10

Bill Powers (2007.12.17.0110 MST)

I've been trying to follow this discussion of degrees of freedom,
though it gets muddled for me at times. Every so often, a term like
"bandwidth" comes along, & I have yet to fully wrap my brain around
that notion in a way that matters & that allows me to use it correctly
in a subsequent discussion. (Some of us do not have much mathematical
or engineering background, alas...)

The definitions of terms definitely help. For instance --

1. A degree of freedom represents a variable that can take more than
one state.

That just seems to define a variable: something that can exist in
more than one state. If something is a variable, it can take more
than one state. To me, a degree of freedom refers to a relationship
between different variables.

A question arises for me, with Bill's continuing comment:

If it is possible for two variables to
be set to any pair of values at the same time, as we can place a
point (x,y) in Cartesian coordinates anywhere in the plane, the
relationship has two degrees of freedom. If x + y = constant, that
relationship has one degree of freedom since specifying the value of
one variable also specifies the value of the other.

I am thinking of the situation where a color is exactly specified
by the weightings of three more primary colors. (I realize there is
some slack in the choice of the primary colors, an instance of the
concept of "rotation" of the coordintates, if I have this right.)

So the weights of the three primary colors (whatever they may be)
constitute three degrees of freedom. But if a particular weighted
sum combination of the three is desired, for instance in specifying
a background color on a computer screen, then choosing the weights
of two of them no longer leaves any freedom to vary the weight of
the third.

In this situation, it appears to me as though the third degree of
freedom from the weight of that third color has been "transferred"
to the higher level combination. There remain three degrees of
freedom, but their distribution has altered. Is this correct?

This seems to relate to a later comment of Bill's in this post:

I agree. Degrees of freedom have to be counted all at the same level,
or you will get too many.

My other question has to do with temporal order as a degree of
freedom. Both of you have referred to the possibility of multiplexing.
Bill mentions:

It is independence of RELATIONSHIPS that defines
degrees of freedom, not independence of pathways, so if one output
can be aimed to affect one variable while not disturbing another, one
output variable can affect many environmental variables in ways that
are independent of each other. This entails multiplexing, of course,
but it shows that the number of degrees of freedom in the world being
controlled can be far greater than the number of outputs available to
control them -- as long as the disturbances don't change too fast.

Consider the following example. I can choose different objects to
point at with my finger, & so in the language of this discussion, the
objects are different "values" of the pointing degree of freedom. If I
add a temporal degree of freedom, I can point to all of them (in turn).
The speed at which I can point (is that an output "bandwidth"?) would
seem to become the unit of the different "values" of that temporal
degree of freedom.

Are we still just dealing with two degrees of freedom here, one for
pointing & one for time, or does time effectively multiply the
available degrees of freedom? Your words, Bill, seem to imply the
latter.

Thanks for the care with which you are each conducting this discussion.

All the best,
Erling