Depending on how you define 'maximum amount of control' you may find yourself playing for something other than checkmate, since the game ends for both of you in a mate. For example, if we define 'amount of control of the board' by the number of moves open to you, divided by the number of moves open to your opponent - or perhaps the sum of that quantity over all positions throughout the game - then you will be playing for a drawn position in which you have as much freedom to move as possible and your opponent has as little freedom to move as possible. This also assumes that you don't control the pieces by directly manipulating the screen image, and that you don't intervene in Deep Blue's computational processes.

The game that you learnt while interacting with Deep Blue would depend on the utility function you brought to the experience, and on the range of interactions you permitted yourself. Of course there is a relationship between the game of chess and the state transition diagram for Deep Blue, but you cannot infer the former from the latter alone.

By what virtue is a chess game a chess game and not two people playing with statues? The rules by which the chess computer operates parallel the rules by which chess operates

I don't think it's quite that simple. If a couple of four-year-olds encounter a chess set, and start moving the pieces around on the board, they might happen to take turns and make only legal "moves" until they got bored. I don't think they'd be playing chess. Similarly, if a couple of incompetent adults encounter a chess set and try to play chess, but because they aren't very smart or paying very close attention, about a quarter of their moves aren't actually legal, they're playing chess - they're just making mistakes in so doing.

Wow, great minds think alike. ;-)

(No, I didn't see your reply before posting.)

A computation is "about" something else if and to the extent that there exists mutual information between the computation and the something else.

Mutual information is defined for two random variables, and random variables are mappings from a common sample space to the variables' domains. What are the mappings for two "things"? Mutual information doesn't just "exist", it is given by mappings which have to be somehow specified, and which can in general be specified to yield an arbitrary result.

This doesn't fall into the trap of "joke interrelations" where e.g. you apply complicated, convoluted transformations to molecular motions to hammer them into a mapping to addition. The reason is that by applying such a complicated (and probably ever-expanding) interpretation, the physical process is no longer telling you something about the answer; rather, the source of the output, by means of specifying the convoluted transformation, is you, and every result originates in you, not the physical process.

When you distinguish between the mappings "originating" in the interpreter versus in the "physical process itself", you are judging the relevance of output of mutual information calculation in the same motion ("no true Scotsman"). Mutual information doesn't compute your answer, deciding whether the mapping came from an approved source does.

Are you saying, then, that the meaning of a computation depends on what the user thinks or the programmer intends, rather than being intrinsic to the computation?