Thanks to some prodding from Sara, I've implemented some new functionality to the dialogues site: interactively searching for strategies. As usual, one specifies an initial formula and a ruleset. The site then develops the full dialogue tree (that is, the tree that represents all possible moves consistent with the chosen ruleset) to the first Opponent node that has multiple children. One then chooses one of the children (i.e.., one chooses a move for Proponent). The site then continues to develop the dialogue tree, ignoring the unselected children of that Opponent node, until it encounters another Opponent node that has multiple children. One again makes a choice, and so forth. This process ends either when one finds that the choices that one has made for Proponent amount to a winning strategy, or when one finds that there is no winning strategy that is consistent with the choices made so far.

Some functionality is still not implemented. During the interactive search for a strategy, one makes choices about how Proponent should respond. There may not be a winning strategy consistent with those choices. If so, the website simply informs you of this, without letting you go back and choose some alternative course. Even if a winning strategy is found, the website does not permit one to choose alternative paths (thereby possibly discovering a different winning strategy). This functionality actually does exist in the interactive "command line" interface (for the brave, see `PLAY-STRATEGY-SEARCH-GAME` in the source), but it has not yet been translated to an HTML presentation yet.

With the site one can also now specify heuristics to tack on to a selected ruleset. Strictly speaking, there's no difference between heuristic rules and any other structural rule (in fact, heuristic rules are implemented as structural rules). Playing a game with a ruleset S and heuristic set H just means playing according to the ruleset S ∪ H.

At the moment, only one heuristic rule is presented to you: forbid Proponent from making identical moves (same move, same stance, same reference, same assertion). This can help significantly cut down the set of possibilities available to Proponent. Sara and I have conjectured that adding this heuristic rule to the D and E rulesets does not affect their logical strength: Proponent has a winning D (respectively E) strategy iff Proponent has a winning D (respectively E) winning strategy in which Proponent never repeats a move.

"Sara and I have conjectured that adding this heuristic rule to the D and E rulesets does not affect their logical strength: Proponent has a winning D (respectively E) strategy iff Proponent has a winning D (respectively E) winning strategy in which Proponent never repeats a move."

ReplyDeleteBut we were wrong, as a number of counterexamples discovered yesterday showed.