[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