From: xeno@iastate.edu (Gary L Snethen)
Subject: Re: "Solvability" in Chess
Organization: Iowa State University, Ames IA
Date: Mon, 28 Sep 1992 00:11:18 GMT

In <1992Sep27.161451.20136@cbnewsl.cb.att.com> ed@cbnewsl.cb.att.com (edward.m.hummel) writes:

>In article <xeno.717575523@pv34ad.vincent.iastate.edu> xeno@iastate.edu (Gary L Snethen) writes:

>Not according to the rules.  Take a look at FIDE rules 10.5 and 10.8.
>They clearly state that a condition for the draw is the registration of 
>a claim.  Unless a claim is made, the game continues and no draw is
>scored.

I incorrectly stated my point.  My meaning is that the rules of chess (FIDE
and USCF) offer the draw-option under these conditions, not that they
dictate the game MUST be drawn.

But I propose that we will always either take the draw option or continue
out of the loop using the following reasoning:

One of the assumptions we make is that our opponent, when given the chance to
force a win, will, and if not, will force a draw if possible...

When in a loop:

If all moves lead to a forced loss, claim a draw.
   reasoning:  We cannot assume our opponent will make a mistake.  Our best
               option is therefore to claim a draw.

If any move leads to a forced win, take the move.
   reasoning:  We can force a win, and winning is the goal of the game.

If all moves lead to draws or losses, claim a draw.
   reasoning:  We cannot assume our opponent will make a mistake.  All possible
               continuations of the game lead to draw, so there is nothing left
               to GAIN...  we have achieved our maximal outcome.

>>If you are interested in 'rigor' then all you need do is add these two rules:
>>(1) ANYTIME a position has been repeated three times, the game is a draw.
>>and/OR (yes, OR...  either rule suffices)
>>(2) If 50 moves pass with no capture, the game is a draw.

>Yep, that would fix the problem.  But such is indeed an addition to
>the current rules of chess.  My argument is based on the current rules.

Ok...  I've removed the rules...  But I maintain that under these conditions,
we may 'claim a draw' (as opposed to being FORCED to draw).

>Consider the game 1. Nf3 Nf6 2. Ng1 Ng8 3.Nf3 Nf6 4.Ng1 Ng8 5.Nf3 Nf6
>and so on, forever; with neither player choosing to claim a draw.  I 
>am asking for a proof that when collapsing the game tree to determine the
>value of the initial position, this sequence must have the value of a draw.
>As far as I can see, such a sequence (or the node corresponding to
>one of the positions in the sequence) has indeterminable value.  This
>does not mean that chess is unsolvable.

I would evaluate the position according to the algorithm mentioned earlier.

>>Even if so, I think I can easily gain a consensus amoung the players here
>>on the net that if a game lasts forever (ie., has infinite moves whether in a
>>loop or not), then that game is a draw since neither side has a hope to
>>win unless the other side makes a mistake.

>Consensus has nothing to do with it.  Either there is a proof or there
>isn't.  If there isn't a known proof, then people shouldn't claim that
>there is a known proof.

Consensus has everything to do with it...  We must have rules before we
may prove or disprove the solvability of chess...  and since the rules
of FIDE and USCF (to my knowledge) do not cover what happens in an
'infinite' game, we should set this rule down before we can continue with
the analysis.  Since humans are interested in finite-length games of chess
(when playing each other), I think we could arrive at a consensus that
any position that is repeated infinitely constitutes a forced draw.

But I'll try to contain my analysis to the case where this rule does not
apply, since I think I can show solvability on your ground.

---Xeno

Newsgroups: sci.logic,sci.math,rec.games.chess