nake board. set of all occupancy-only-legal position (combinatorics formulat on that). so. occupancy rule separabable in some putative non circular construction of the rule set with any math. mean necessary (or sufficient too...., not sure yet I need to go just necessary, my experience is overshooting at least gets you in the ball park, STLT).
combination or arrangement number of 32 pieces on board or off board (need a ghost board maybe or 4 ranks of offboard slots).
self-explanation hypothesis for the wobble.
my problem is that cross combinatorics curiculum some math internal chunks got lost in translation between french we start with arrangment, and english (might just be math. pedagogy school) combination are the first notions to make forumas off. so oscillation behind the scene obscuring my remaining memory of all of that, but still I can read, and lump so one of those applies). maybe going backward to n-tuples might be clearer and always regenerate the formulas from there.... ok..
end of self-parenthesis.
then what.. long haul thread.. I have other pieces for it in musings past. but that just popped up.. I could keep on separating rule-set as I once thought, and it would make for less recursive definitions in the end.. and any language of mathematical nature could be skimming on top of that with no conflict... just they might appear as more explicit "projections" or feature reductions, consistent as they may be when stuck in them.
of course the details too.. 64x32 board and ghost board and for those on board the usual casseroles stringed along.... but nothing of game notion, only state of occupancy in all those slots (2D or not).
idea is to nest what is separable in the construction flow in going from maximal ambiant domain to more restricted one "separable/nest-able" rule-set at a time.. and narrow the counting that way... if that was only what we care about, but I also want all useful math. languages that those could allow as subsets later. including that which the chess engine (the full NN input ones) and our wet brains might be using internally.... that we haven't been very curious about, certitudes surrounded as we seem to like evolving around.
Best would be fully separable restrictions... but I am not sure I could do that at all steps.. (I or one). Being humble is to say this is an exercise, just having a bias for what I say I am looking for...
ahh, having slow thinking in written words as external memory to ponder back on whether we were faithful to the thinking. As the words are not the full thinking... needs iterations rephrasing and blatant contradictions etc... to get there... (belief going onto certitude, from experience, or persistent self sustaining delusion, I hope I am kidding!).
What of my fully exploded board occupancy point of view I had worked so hard on verbalizing before.. Did I mis-shoot in my first post already.. I should have spent some time on the occupancy thing, not assume it known...(entrant, undefined, atom symbol?).
Well before the occupancy filter (not 2 pieces can be at same point of the 2D board)... we might need to specify the ambient things where that can make constructive sense... or I would have fallen into loopy stuff already. Avert your eyes those who prefer integrity of full formed chess boards...
imagine 32 chess boards..... yes first have to start there to be clean (at least for me... I like to unfog things more than others I guess, but then combining back and poof... fog gone).. this post memo to that chunk... maybe I can skip for the other separations... maybe not.. will sleep on it I hope.. this kind of stuff wanders after writing, and not fully hammered rambling mode....
also, the board itself.. but that is solved.. how to usefully embedding in a robust mathematical embedding in R2, but we all "knew" that, with our soft matress chessboard we keep laying to flat table planes.... It is useful for our human chess thinking about whole board symetries. Only graph viewpoint without ambient tethering might make it hard for us to use our spatial intuition, and reduction of cases by symetry operations. (of the board, not the positions yet).
so that chunk.. there exist a representation that is both necessary to cover the core rules, and sufficient for our human brains ability to make groupings of many positions as similar problems (or exactly the same modulo square figure symmetry operation).
goes at step zero before this thread.... where to start? this is just a thought for yesterday.. memo to it. now going to touch wood, still abstract but closer to chess itself.. FLORES book.... mapping the middle to endgame plan level logic to the game examples for each chapter.. starting with 1.. the shallow view.. how is this related here.. in the grand scope of things (which is the fairy tales I like to not get bored with daily, in my small universe remaining), many levels of reasoning or "logic" can exist.
so, preparing from all angles.. beating the humongous bush around... never know which end of the many dimensions will be most productive.... and anyway this is how my brain works, it hops around... it appears like not multitasking at small productivity time scales.. it is too fast multitasking, and usually not observable from the outside.. it would need language speed to be visible. I am only approximating this with my verbiage here..
got my necessary and sufficient cross-wired.. (words!).
graph might be necessary for handling the linguistic rules of chess, but is not sufficient to handle the implied spatial rules that need not linguistic explicit say.
while the R2 represensentation does both. I may not need the word sufficient. as we were always growing from necessary and minimal mathematical aesthetics, we only have to discuss target, and the rest is about necessary.
we did not prove, though that the R2 rep. was necessary, but were satisfied it was sufficient and could not imagine something less restrictive toward graph specification insufficient non Euclidian space... (being relaxed with what is a space.... testing testing 1,2,1,2). So for all useful purposes it was necessary to go to R2 and sweat out the representation in practical terms (it was not me, I don't do that or finish many ideas, but I was part of discussion... and pushing from the get go to start with the representation as given... I learned about foundational thinking, and I am tyring this sauce here to some extent, still thinking all sort of maths. are possible within explicit and labeled undefined and defined... we can all work with hypotheses levels and keep those in mind to see what gives.... but it is nice to find out, that some aspects were actually not just sufficient (my goto) but also necessary.. for your target of human chess thinking....
me realizing I might be using my counting formula interpretation problem workaround right here in having the 32 boards....
easy to construct cartesian product and count there, and then make contingencies on class of them we consider equilvalent and such.. like do I really want to differentiate pawn by its arbitrary indexing, while all that matters (ex-machina point of view in the construction, as this is a rigged exercise, all that matter is not shooting oneself in the foot in the end....), is where are the pawns.. (not minding their musical share in between turns, but hey, no turns yet, I said no game notion yet, just having fun in with those toys).
I am not going to a fit a theorem with the appropriate givens, but maybe not far from it.... That is the luxury of constructing palaces,, we just don't want it crumble under own weight.. hence the need for foundation aesthetics being a useful bias, one tool or measure instrument to have nearby... (not to fly off the handle in any directions, either, such grey.). I say this is art. but not much appreciation for it out there... .bouhou...
shrinking rambling to concise memo.. sorry. might be true but still unreadable....
Point of this simple permutation (Green being tracked).
Graph permutation set. R2 "natural" representation or embedding. nodes to regular grid of points. edges to the geometric segments that the R2 points mapped nodes pair part of the edge set create. The pair of nodes is mapped to a segment (interval of many many points). We don't care much about the amplitude of the space between the grid point, just its regularity, but boards with full small squares look nicer than just grid pont centers.. so we fill the squares, also point wooden pieces or plastic, don't exist, or then would fall on their side (nerd joke, but rather true, and funny too). Abstractions. Me not the first: Nimzo uses points too, when nobody is really paying attention while reading (I guess), as he would use rook for queen when enthusiastic about the mobility of the rook whether used by queen or rook... I can't read books linearly, but I might have jumped into excerpts carefully aimed for me by some mentor... I like the points it helped me here... with the graph stuff.
so.. above is the gist of the thinking about our beloved particular way of layout the chess graph that we need for at least the king to walk as it does... (not needing to define the kind mobility, we are dealing with entrants here, just motivated the adjacency, I could also just enumerate it, not my cup of tea, I would rather ramble I guess. than trip over one link.... in the shapeless object enumeration. I would gladly find a work around with more math. objects that could if crancked give the set. and why I like R2, myself, even if it was not needed. I got internalized tools from there that would simplify my chess-math life... Why I am excited by this foundational thing, that makes it ok for me to wallow in my long trained intuition in euclidian spaces....and maybe even non-linear transformations, maybe.
so.. mapping the nodes. and the links.. then looking at all permutation that preserve adjacency (permutation of the arbitray indexing of the node sets). what is so particular about our boards in R2, that might relate to those perumations.. are they all ok for chess rules to apply.. I am not sure what is printed.. if the mobility figure in the FIDE scripture, is a geometric figure, but the board is never defined.. I assume so. As I have seen here or there, or even in some engine view of chess moves, that all move are a priori distinct and might as well be on any permuted graph... but this figure up there.. is key I think.. to evoke what happens if not chosing the spatially conform board that we know. I am happy with the foundational exercise finally. who knew we took for granted so many things.
cool that given a representation of nodes as a regular grid, the linkage has to be like our board. and minizes the sum of sum of all segments length representing the edge set. above shows (or suggest) that any departure is adding cumulative length to that sum.. sorry.. not more details that I might have skipped. mostly memo here.
somewhere in there (or did I delete it with the wall of words, even bigger that left above). I said spatial chess rules "implied".
The words I chose, again, made me look like I was fumbling logic. I meant implicit. more assumptions than logical implication, but since they are not linguistically explicit, I have to induce that they are made assumptions, to get to our target chess board that we know.
In some sense, if I accept that the target is what we all share, then, the implicit assumptions I induced, are implied by the whole shebang.. (or thing). something like that, be my excuse joker phrase, in all circumstances.
just "implicit assumption" ... not "implied", though.. the rest is me saving face for my previous use of words... also making a fuss so next time I will be more careful with the word approximations (often poor and rambling inducing, mechanically).
An illegal move[45] is a move not made according to a piece's possible defined movements [46] or made according to its possible movements but such that its own king is left or placed in check.[47]
it seems to be that one can isolate legality aspects, dimensions or layer accroding to some ruleset subsets.
mobility. check. game ending of static board status like stalemate or mate all those being static testable with one move to consider (or not for ending).
so I kind of have bumped into that question again looking at static mates, or mates in 0. just staring at them....
what makes a mate. how wild can they be. what are the dimensions or low level core rules or their geometric higher level consequences (in tune with our intuition of line movers and space fillers and fence bulders and other tetris combinations).
can we also dissect the set of mates. now.. the question of "practical", or likely in one life time (sigh!), versus grotesque, or theoretical etc... arises. but even in more simple abstrac view of chess ruleset layers, it also arises.
continuation legal versus static legal. So static legal, would have a lot of grotesque ones, for sure, but in term of chess psycholgoy and generalizatoin training and pattern deifitions it might just be unclooging the sinuses, toward less dynamics headbanging. anyway we can always reduce, after research on that what else is there, to the exiting called out static mate patterns fom history of chess and its literature, where named or simply called out as patterns in some book or web document.
anyway. EGTB also do not get bugged down into legal continuation problems. It is only retrograding from static endings (I think, wondering though about N fold in TB, I know about 50 move being an added layer on top of the more general TB, so I happily do not care about it, fluff it is). Using immediate mobiliity rules. cranked backward.
it is likely to have crancked it in all directions within 7 men. We could wonder what is the space of mates that are at the source though. They might not consider pawns of the mated color on own first rank ever. hardly a retrograde check back to initial position, more like forward mobility from rank 2 of apwns initial setup, of pawns never getting there. but we might also, for the generalization bottom formalism, also let everything loose about initial setups defintions
i mean yes. 960 setups and all pawn walls on rank2 (from respective backranks, damn it) initial setup they are taken from would not allow that.. but what if we just allowed anything .. ..
for thiking about best formalism. it might be more psychologically enconomical to do the research on the simplest most static problem of all possible non colliding and mating meetings positoins. so if the thing works there... it will work in any restriction.
I come from that sort of mathemtical modelling where going outside of the object of interest would allow easier creative work toward building the formalism and appropriate equtaions that would adress the problem there with less headaches (for some people, i guess). and then find that the originial problem is a particular case. but that having limitied ourselves in the complex particularity would not have allowed our simple minds to find the tools. or with too much hard work.
so i am biased by many seen examples of solving the smaller scientifc problem by modelling a bigger problem and finder psychological easy solutions there. having separated or divided the work into the representation of the smalle problem into the bigger one (it is only bigger if counting the points, but there might already be big guns that mean we don'T need to counts all the points, it is already known that anyway certain non counting properties are already reducting the effective problem that is shared, the tangling that is without asperities our small mind can count or spread apart in the smaller more dense problem. chunking by systematic expansion. (we don't need to count the expansion if there is a rule of exansion, but then we might see the relations that we are compacted in our visual field now laid bare next to each other.
it is a question of formalism discerning power. (and thinking outside the formalisms in probably ugly back and forth between levels. ). I don't find it uglly. I revel in modelling research . learning about the thing to model.. while doing so.. I might be difficult to convery that it is feasible. for chess problems. I have to keep trying.
That is why i am working of a way to share what are basic simple NN doing like with the one hidden layer MLP and the XOR. got bug down into doubts about my memory shortcut between XOR and euclidian topologoy of the donut in 3 D, or its cross-section in 2D (XOR being minimally 2D). elswehre. with equations. not words. and with 2D picture representations..
chess boards are not the only place or "space" where to look for xor logic or it combination with other logic gates. or donuts topopoloty detection. i.e being able to classify something that is not linearly separable in some statistical learning language. if i am remember well. (SVM would also explode the input space so it could find that linear hyper?plance separatrix for the output classes to predict on new inputs.) i was not thinking of that even above. it is everywhere. optimal controll theory. (with function spaces, what else...). i need help in finding document mathematical formalism going up the dimensions like that for easier solution construction that can then be easily reduced and using a representation parallelogram of sorts, not lose any logical kitten.
