Space in Chess  TEXT  HOME 
Apparent Space
NonApparent Space
Compression
NonNonApparency
Real Space
Simulation
Using the game of chess as an example we are presented initially with the physical level of the game as played mechanically on a board with 8 squares on each side. The chess board itself and the symbols or pieces represent a form of the game that we might term the apparent space of the game. Without further rules the symbols might all be considered of equal value and might occupy any open square.
Without rules, chess is merely an artistic expression whose interpretation is left to individual perspective. With rules, chess becomes a game whose objective changes as play unfolds, but whose conclusion is the capture of one or other King symbol. Initially the game proceeds from a state of lower, to higher entropy in the midgame, beginning with symbols on prescribed positions and all empty spaces continuously filling the center of the board.
The apparent space of the chess game is what we can all physically see and represents also a single episode or round of play. The board is an explicit representation of all possibilities, of all positions and while at first would appear to be a closed and simple universe, it is in time unbound and made much more complex.
At any point in time there are with rules a warping of the apparent space into a nonapparent or collapsed space of possibility into which the game is actually played. Not only may impossible positions be eliminated in this nonapparent space, but the whole notion of the grid itself may be eliminated as we understand the difference between fixed and dynamic rules.
The fixed rules of the game are the rules that do not change and are in force throughout the entire game. The grid itself may be represented as a rule mathematically so ultimately we are left with only rules. Within these rules, there are some that change and some that do not. The dynamic or changing rules are those that bound the play within the scope of a single round. They are products of rule interactions. So we have a counter intuitive situation where we see immediately simple rules through time naturally increasing complexity, or entropy, on the board.
The result of these considerations is to see the apparent space of the chess game as itself an abstraction, a distraction really to the ultimate nature of the game which is the interaction of simple rules over time. In the theory of games, chess may on the basis of its rules be compared or evaluated to other games as special in its capacity to endure.
Once we have collapsed the game of chess into mathematical expressions or rules, and see the board itself as merely another rule and abstraction, then we have moved into the nonapparent space. The board itself disappears and we are left with a collection of primary symbols that do not change, those of the 'pieces', and some new symbols we introduce to represent moves. The pattern here is to separate the apparent from the nonapparent through encryption or the creation of new symbols to collapse complexity or space into a lower energy or entropy form. In extreme repetition and outside of the game of chess this habit is refered to formally as data compression.
Let me describe for a moment compression, symbols, encryption and something very difficult to touch; information. When one compresses data which at bottom is a collection of symbols in a specific sequence or 'list', one engages in substitution of symbol collections with new symbols. For example in the permute space of symbols A B C D, the alphabetical order of these symbols as a single specie may be given the symbol A. Instead of writing ABCD explicitly, we may instead write simply A. This provides a compression ratio of 4:1, or that 4 symbols have been compressed into 1. However, to be fair, we should explain that this efficiency is only possible while ignoring the method of encryption or the rule that A = ABCD that in itself requires 6 symbols. So in fact the compression ratio is not 4:1, but 4:7 which indicates not only substitution, but including the description of the compression method, some inflation. Important here to understand is not only the datum being compressed, but a ruleset that defines a compression method we will call henceforth a shared compression library, and to understand that while one side compresses, another inflates that includes the total description of how to restore the datum object to its original form. So there are 4 things to consider; the object, the description file (the crumbs in the maze), the shared library, and the product. Similar to understanding how energy is conserved in thermodynamic systems, we can in adiabatic fashion understand that through compression using heat or noise to drive the process, that information overall is not in amount changed.
Effective compression can only be evaluated considering all 4 parts, and any efficiency gains as a reduction of symbols will be countered with the inflation of description.
To continue, we imagine a datum as object or input being compressed and made smaller, while the compression product as a residual tar through a shared library (an artifacted set of assumptions), is made larger, but slower than the rate of compression. With datum objects being larger than 4 symbols, we can see efficiency gains where the complexity of an object may be evacuated and made more orderly and cold.
There is a point however if we continue to compress where the gains in total are no longer possible and we find that in substitution, we are merely transferring symbols from one bag into another. From the object, into the description. This is a point where we are no longer engaging in compression, but mere encryption or translation of symbols where there is no net change in entropy. This point or equilibrium or better, limit; represents a boundary that defines the effective information in the object. Stated simply, the compression limit quantifies how much information exists in the object that may not be further compressed, without engaging in encryption. If we were to change the context of the product for example transferring it to another network and inflating with heat into the object, we would be transferring information from one system to another. What that information is cannot be touched or revealed directly with any clear understanding, but we may feel the result as a change between the systems as a transfer of entropy or complexity.
Back to chess, we may continue to discriminate the apparent space of possbililty from the nonapparent space through encryption of; pieces, squares and the rules themselves into symbols. Collections of these symbols may form either a fixed language or local dialect reflecting the horizon of play. New symbols may be invented to represent patterns of play that are found in the language of chess as openings, defenses all the way to include entire games in reference. Once possessing these symbols, we may consider compressing further but not so far as to introduce more ambiguity, or to lose any essential information which disregarded or lost will fracture defense or give the opponent an avenue of attack. By transferring the game play into a nonapparent space we perceive and experience the truth of the game. Without introducing the concept of time yet, we may understand a balance of the apparent with the nonapparent and a limit that must not be crossed without becoming more confused.
Since we have at this point only considered single rounds or episodes of play, we may now introduce time and consider extended consequences, groups of turns, and what we will term nonnonapparency. For simplicity and a certain amount of trendiness the fastidious will adore we shall replace nonapparent with 'virtual' to indicate layers of reality separated by compression, and heat.
Not to be confused with a simple double negative that returns the original, we instead imply a further virtualization that not only collapses the game space in physical dimension and representation, but further collapses the game in time; to a limit. Here we may consider not only the physical board, but virtual boards representing possible game states for the next move, but also for future moves. As one increases the game horizon in relative time, we then consider states upon states like chains whose metal are the fixed rules. We find these possible configurations confer different advantages and while impossible to consider all, we find that many of these virtual boards do interact and interfere with occassional consequence and cancellation. Here, we can appreciate the existence of isomerism or equivalences reducing many equivalent game states into fewer consequent states. The encryption of game rules and the collapse of apparent into a smaller space containing only those possibilities that effectively make some change in advantage, approaches the information of significance in the game that will likely determine its outcome.
Realizing all the states however will not win the game because of ambiguity, mistakes, noise and the ultimate existence of transferable utility and information. While not formally considered a game of transferable utility, the existence of information in the game and the input of noise from the Aleph make the game consistently indeterminate.
While machine programs may render a scoring method that within a time horizon reports asymmetry in game advantage between players, these collected methods as genetically enhanced will approach a fitness limit whose horizon itself is a direct product of information reliability. Fundamental in example is the assumption that an opponent would even elect a position of advantage. This enters into the hairy waters of machine morality and whether machines themselves can ever possess their own indeterminacy. Rather than disturb the supremacy and entitlement of any reader in this matter, we shall instead suggest an alternate machine motivation, which is to win in as few steps as possible or as we describe below, aggression.
During the game, the conditions of checkmate and stalemate as conclusions to the game are some number of steps away from occuring. This distance may be either determined without ambiguity and obvious, or it may be indeterminate. As an example we imagine the artifacted game condition of 2 kings in opposite corners, with no other pieces on the board. In this concoction the game conclusion itself is indeterminate as check and stalemate are not possible. Any computers involved will either report infinite distance to mate, or draw; depending on the program.
Lets introduce the concept of isomerism or equivalence of form as applied to space. The above condition of naked kings while seen in apparent space as occuring on a board area of 8 by 8 squares, it may be collapsed into a 4 by 4 with the same potential of complexity. The game of chess played with naked kings is perfectly isomeric between boards of 64 and 16 squares. Those kings can chase each other all day long in either case until the players, getting rather snappy on the clock decide to finish and go drink some vodka.
However it should be pointed out that the condition itself of naked kings is purely synthetic and highly unlikely to occur since the game of chess includes symbol transformation, in the language of chess called promotion. Specifically, the transformation of pawns into any other piece, usually queens, and sometimes horses. Because pawns may transform, and including the assumption that the game should proceed through shortest steps to mate, the condition or synthetic game state of naked kings now called NK is itself only possible through mutual cooperation of both players. I propose the NK example to illustrate how machines and scoring systems in reporting advantage or mate distance might erroneously always assume perfect animosity and aggression between players. Aggression now mechanically equated with the desire to mate in as few steps as possible and animosity the persistent desire to win through balance.
With the additional rules of a time limit, and switching game clocks that record the time each player spends making game decisions, there are other motivations to consider besides pure aggression. With the necessity in chess competitions to produce a clear winner there is the comparison of time balance if the time limit is reached without a check or stalemate. In this case, there is a strategy to win that is not a product of pure aggression, but to minimize the time required for decision making and the attrition of pieces, if a mate will not be possible that may be best termed animosity and perhaps the capacity to delay or avoid a mate if a win based on balance is possible.
So back to space, we may see that the space of possibility and interaction of virtual game states can be warped dramatically by assumptions about motivations, and the clarity of mutual game intentions. While a player is engaged, there is an internal conflict between game outcomes and when to time the transition of strategy from aggression to animosity or more normally in game evolution animosity to aggression. To that extent, the player itself must admit its' strategy or pattern of response may change and is itself indeterminate. Since the players cannot predict their own responses there is a great deal of the game space that is dark because the players themselves cannot predict their own behaviour with any reliability let alone the reliability of their opponent.
The residual after all the possible game states have converged is the real or actual space of the game. There is a great amount of permutations and areas of infinity that may not be considered without wasting precious computation time, and also vast areas whose value in simulation is near or approaches zero.
Since we see both internal and virtual levels of competition between players, we perceive that despite the apparency of the game, despite its nonapparency, and its reality, what the players are doing is competitive simulation. Both sides may at any point in time visualize a potential state or outcome, but this requires an opponent to cooperate and behave consistently if this outcome will be expressed. Instead, the real game space is bounded by limits of information regarding players respective plans and intentions. Because each player must simulate the others' intentions in defense and offense, there is in total only a strategy necessary to improve the possession of a single unit of transfer in the game; information. The evaluation of information in the game of chess is not done at the level of the board, but beyond the nonapparent into the compressed real space where the separation of players itself becomes a mirage. The conclusion of the game arriving through a bias of information. The transition of play from animosity to aggression expresses the confidence a player has about intentions and outcomes. In game evolution a mate is only possible if a mistake occurs. Between machines with infinite time to compute all outcomes, the only source of indeterminacy is the expression of strategy and the disambiguity of a single game outcome.
Contact  contact@jonathanleonard.com 
Revised 
10/02/12
