It is different. And it should be obvious.
If what I read on defining inductive reasoning was correct.
"'I think' I disagree."
Think more. Its good for you.
Now - to make sure I don't get drawn into the pingpong desired by another - time to unfollow this one for a while.
Several days from now there'll be some retorts - some nodes/square root spam - maybe some good posts by some !! Yes !
Chess will never be solved, here's why
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Jul 11, 2022
0
#3562
Listen, I read that Wiki article on mathematical induction for nearly 17.2 seconds. How could I be wrong over such a simple issue, when I studied it in such depth? 
I wrote that I think I disagree because I think your explanation's mistaken. You could never in a million years show I'm wrong and that mathematical induction isn't based squarely on general induction. You could always try though.
Naturally, the incentive to demonstrate a divergence from usual norms, by expressing thought in complex and meandering sentences, isn't to be taken as an obligation, since any judgement the reader may confer upon such endeavour may remain unmitigated by an empathy with the author's intentions; whether or not those intentions are consciously rational or merely the expression of an unconscious narrative which, when unpacked and set out clearly, may well demonstrate an inner and hidden antipathy towards ideas or ideals which the author may believe himself to espouse.
In short, madam, the writer may be confused.
#3644
"So your formula for the number of positions becomes:"
++ No. With width w = 4 and depth d = 39
Without transpositions:
1 + w + w² + w³ + ... + w^d = (w^(d + 1) - 1) / (w - 1)
With full permutations:
1 + w/1! + w²/2! + w³/3! + w^4/4! + ... + w^d/d! ~= e^w regardless of d
The truth lies in between
Optimissed wrote:
I'm asserting that anything other than an inductive understanding of a "chess solution" is impossible. I think that's completely incontrovertible.
Yes, we all think it's impossible with current technology, and the humankind might very well be extincted before it is possible, without a breakthrough in technology, or in our understanding of the game. You wondered why people seem fixated wiith a mathematical solution, then. To me it's not that, but exact solutions have been found for other games, so it's natural to take that as a reference. We already use inductive reasoning to play better and better, so it's completely fine, but it is not correct to call it an exact solution, a theorem, like someone (not you) does: chess is still too complex for that. Inductive reasoning cannot fully generalize, so its conclusions are, strictly speaking, only probable, while mathematical induction is a special case, where it can be proven that the generalization is full, i.e. exhaustive. That's still impossible for chess, without a complete mathematical representation of the game, as you noted.
tygxc wrote:
"So your formula for the number of positions becomes:"
++ No. With width w = 4 and depth d = 39, without transpositions:
1 + w + w² + w³ + ... + w^d = (w^(d + 1) - 1) / (w - 1)
o __ root node
/ | \ | White's moves
x x x x nodes at depth 1 in plies
| | | | Blak's moves
o o o o nodes at depth 2 in plies
After one full move the total number of nodes is 9, not 5 or 21, like in your formula. Anyway, d (that in your previous post was 34 and now has magically become 39), must be deduced from the total number of positions you intend to search, that is 10¹⁷. With d = 39, the total number of positions would be 4 × 10²³, per your formula. I am sure you are well aware of that.
makes sense tho
#3657
" o __ root node
/ | \ | White's moves
x x x x nodes at depth 1 in plies
| | | | Blak's moves
o o o o nodes at depth 2 in plies"
OK, without transpositions you count black replies too, getting upper bound:
2 * (1 + w + w² + w³ + ... + w^d) = 2*(w^(d+1) - 1) / (w - 1)
With full transpositions it becomes a lower bound:
2 * (1 + w/1! + w²/2! + w)³/3! + w^4/4! + ... + w^d/d!) =~ 2 * e^w
"d (that in your previous post was 34 and now has magically become 39"
++ No. d = 39 is the average number of moves in the ICCF WC.
d = 34 is the average number of moves past theory in the Madrid 2022 Candidates'
"the total number of positions would be 4 × 10²³, per your formula"
++ Let us take the 2 formulas above, w = 4, and d = 39
Without transpositions: upper bound 2*(4^40 - 1) / (8 - 1) = 8*10^23
With full transpositions: lower bound 2*e^4 = 100
Geometric mean on both: 9*10^12
So 10^17 is on the safe side.
I derived the 10^17 by another way.
Per Tromp there are 10^44 legal positions, but none of his 56011 legal samples are sensible because of multiple underpromotions to pieces not previously captured.
Per Gourion there are 10^37 positions without promotions to pieces not previously captured.
I multiply this by 10 to accept positions with 3 or 4 queens.
I note that 1000 sampled positions are not sensible either and accept Tromp's estimate that only 1 in 10^6 is sensible. That leads to 10^32 sensible positions.
Then in analogy with the solution of Checkers I estimate that only 10^19 of these are reachable in the course of the solution. E.g. when working on 1 e4, all positions with a white pawn on e2 are no longer reachable.
Then I estimate that only 1% of these are relevant. E.g. when 1 e4 e5 is proven a draw, then it is not relevant if 1 e4 c5 draws as well or not. Likewise I consider 1 a4 and 1 e4 e5 2 Ba6 as not relevant either.
That leaves 10^17 legal, sensible, reachable, and relevant positions.
I give buy lopez a try
Jul 12, 2022
0
#3571
tygxc wrote:
...
none of his 56011 legal samples are sensible because of multiple underpromotions to pieces not previously captured
...
Breaking news - these days when you play chess you don't have to just use the pieces that come in the box.
#3661
" you don't have to just use the pieces that come in the box"
++ There has to be a good reason to underpromote to anything else but a queen.
Underpromotion to a knight sometimes makes sense to utilise its unique properties.
Underpromotion to a rook or a bishop sometimes makes sense to avoid stalemate.
Promoted rooks and / or bishops on the losing side make no sense: it were better queens.
Promoted rooks and / or bishops on both sides make no sense: it were better queens.
In real grandmaster / engine / correspondence games 3 or 4 queens occasionally happen, but multiple underpromotions to pieces not previously captured like in the 56011 samples never.
haiaku wrote:
Optimissed wrote:
I'm asserting that anything other than an inductive understanding of a "chess solution" is impossible. I think that's completely incontrovertible.
Yes, we all think it's impossible with current technology, and the humankind might very well be extincted before it is possible, without a breakthrough in technology, or in our understanding of the game. You wondered why people seem fixated wiith a mathematical solution, then. To me it's not that, but exact solutions have been found for other games, so it's natural to take that as a reference. We already use inductive reasoning to play better and better, so it's completely fine, but it is not correct to call it an exact solution, a theorem, like someone (not you) does: chess is still too complex for that. Inductive reasoning cannot fully generalize, so its conclusions are, strictly speaking, only probable, while mathematical induction is a special case, where it can be proven that the generalization is full, i.e. exhaustive. That's still impossible for chess, without a complete mathematical representation of the game, as you noted.
Just going over an old thread. I think I took issue with you, over your claim that mathematical induction is a special case which, somehow, can be exact.
tygxc wrote:
I derived the 10^17 by another way.
Per Tromp there are 10^44 legal positions, but none of his 56011 legal samples are sensible because of multiple underpromotions to pieces not previously captured.
Per Gourion there are 10^37 positions without promotions to pieces not previously captured.
I multiply this by 10 to accept positions with 3 or 4 queens.
I note that 1000 sampled positions are not sensible either and accept Tromp's estimate that only 1 in 10^6 is sensible. That leads to 10^32 sensible positions.
Then in analogy with the solution of Checkers I estimate that only 10^19 of these are reachable in the course of the solution. E.g. when working on 1 e4, all positions with a white pawn on e2 are no longer reachable.
Then I estimate that only 1% of these are relevant. E.g. when 1 e4 e5 is proven a draw, then it is not relevant if 1 e4 c5 draws as well or not. Likewise I consider 1 a4 and 1 e4 e5 2 Ba6 as not relevant either.
That leaves 10^17 legal, sensible, reachable, and relevant positions.
To the poster above this: making repeated one line posts one after the other is spamming.
"I derived by another way" is the problem.
You routinely take orders of magnitude away based on counting sets that overlap, while discounting that you are double counting. Reduce your argument to its logical conclusion. If you managed to "reduce" to 10^6, and you still had your Tromp card in it's holster, would you then make a claim that there's only 1 position? Of course not, because the set of 10^6 positions were already eliminated piecemeal in your other "loose" reduction estimates. But somehow, you completely overlook this when you are reducing 10^23 to 10^17.
It's been mentioned to him, by a few people. I don't expect he'll do it again.
#3661
"You routinely take orders of magnitude away based on counting sets that overlap"
++ No, the sets do not overlap.
The set of legal positions counts 10^44 elements.
The subset of legal and sensible positions counts 10^32 elements.
The subset of legal and sensible positions reachable during the solution counts 10^19 elements.
The subset of legal and sensible and reachable and relevant positions counts 10^17 elements.
tygxc wrote:
#3661
"You routinely take orders of magnitude away based on counting sets that overlap"
++ No, the sets do not overlap.
The set of legal positions counts 10^44 elements.
The subset of legal and sensible positions counts 10^32 elements.
The subset of legal and sensible positions reachable during the solution counts 10^19 elements.
The subset of legal and sensible and reachable and relevant positions counts 10^17 elements.
You have no fundamental basis for assessments of "reasonable" or "relevant", either.
#3664
"You have no fundamental basis for assessments of "reasonable" or "relevant", either."
++ Yes I do.
Sensible: results from a game with > 50% accuracy.
Reachable in the course of the solution: every capture or pawn move is irreversible and thus renders huge numbers of positions not reachable.
Relevant: for weakly solving chess only one strategy is needed to reach the game-theoretical value, it is not relevant if other strategies do or do not reach the game-theoretical value as well.
Aug 1, 2022
0
#3579
What does "50% accuracy" mean?
How is it determined without a 32 piece tablebase?
You do not have answers to this question that make it more than a guess.
#3667
"What does "50% accuracy" mean?"
++ It means that you feed the proof game into the engine and read the accuracy of the game. If that accuracy is < 50% then it is sure that the play is not optimal.
"How is it determined without a 32 piece tablebase?"
++ By feeding the proof game into the engine.
"You do not have answers to this question that make it more than a guess."
++ Yes, I do. If the accuracy is < 50% then that makes it sure it is no optimal play.
Inductive reasoning needs to be distinguished from mathematical induction.
The conflict between the two is enough that the words induction and inductive aren't good for this context. They would tend to serve themselves rather than the discussion.
Acceptance of the unknown. Lots of wriggling and bobbing and weaving ... by so many. Trying to make the unknown a known.
With so many unknowns always still there at the end. Undented.
I think I disagree. https://www.google.com/url?sa=t&rct=j&q=&esrc=s&source=web&cd=&cad=rja&uact=8&ved=2ahUKEwiEgbzjjPL4AhVxSkEAHQOGC0wQFnoECB0QAw&url=https%3A%2F%2Fen.wikipedia.org%2Fwiki%2FMathematical_induction%23%3A~%3Atext%3DMathematical%2520induction%2520is%2520a%2520mathematical%2C3)%252C%2520...%2520.&usg=AOvVaw26pDBSfPkj6xv-4c2gJb5I
Mathematical induction gives the base case (that the basis of a repetitive sequence is achievable) and then the repetition .... that if a mathematical mechanism is possible, then it can be repeated in similar mathematical conditions.
I believe this is no different from "any old" induction.