Subsidiary query: most games encompass PSPACE-complete problems, not NP. Could it be more likely your construction could resolve P vs. PSPACE (also an open problem if I remember right?) more easily than P vs. NP?

]]>Did this move to email collaboration?

As far my own contribution to Polymath9 being stalled, I have to apologize; I was hit by my day job pretty hard. I only just now had the time to work through this post. I will duly try to break Pavel’s definition.

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I thought about modifying the concept of Ramsey lift so that the small lifts with the parity construction is avoided. One idea is not to have a fixed division of coordinates between the players, but to allow all possible divisions into two equal size parts. Another is to allow players to play any coordinates. It seems that this could eliminate the parity construction because if bits in two coordinates can determine the function, in some division the two coordinates may be used by the same player to his/her advantage. Since allowing any division substantially restricts the class of available lifts, I tried to modify your doubly exponential construction of a “universal” lift. I succeeded to modify it so that every subset of is lifted to a 2-open set, but later I realized that it is also possible to do a single exponential lift in a different way. The new trick seems to be robust and thus may also exclude other conjectures.

The idea is that in the lift one of the players has to play the entire string , but if the player plays before the last move, the other player may cancel by playing a string that he prefers.

Here is a formal definition.

iff

1. there exist , and (the group of order 3) such that and for ,

2. or there exist such that , , , and for .

The projection sends to . Since for every , is uniquely determined by two coordinates of , the lift of any subset of is 2-open.

Let be a winning set for one of the players. The player who has a winning strategy can play the same strategy in until the other player plays a move of the form . If this is not the last move, the player with the winning strategy picks any and any consistent with the previous moves and plays .

A natural question is why not to allow players to play just instead of . This does not work because even if a player may choose the value of as he wishes, it may restrict him so much that he will not be able to win. While playing restricts him even more, it also restricts the other player.

]]>A condition that seems to be desirable but maybe not sufficient is as follows. We are looking for a definition of the form “ is a simple subset of if and only if .” Now is a subset of some set . So we could insist that the property is invariant under automorphisms of . (The automorphisms I am thinking of are ones of the form , where each is a permutation of , and also ones of the form , and also compositions of those two types.) That is, if is an automorphism of and is a simple subset of , then should be a simple subset of .

]]>hey, math/science is trial and error at heart. hypothesize, test, prove, repeat. wash, lather, rinse, repeat.

as they say “it all comes out in the wash eventually”.

if anyone around here gets totally frustrated/desperate and youve hit bottom & are finally willing to *try anything*, plz let me know, Ive had an idea relating to hypergraph decompositions vs P vs NP lying around for over a yr, no expert feedback so far. (& frankly think it might even tie in somewhat with the generalization of szemeredi’s thm to hypergraphs…) it has many elements of circumstantial evidence pointing in its favor, would be happy to discuss at length with anyone interested & esp those with strong math bkg, long/intense attn span/perseverence & huge/”outrageous” ambitions.

Here are two examples of definitions that carry over easily. The first is circuit complexity. We define a basic set to be a set that depends on just one coordinate, and then the circuit complexity of an arbitrary set is the length of the shortest sequence that ends with and has the property that every set in the sequence is a basic set or a union or intersection of two earlier sets in the sequence.

The second definition is that of a I-winning set. The shrinking-neighbourhoods game makes sense in any complexity structure and picks out a class of sets.

What do these two classes of sets have in common? Maybe it would help to give an example of a definition of a class of sets that does *not* generalize. Maybe one possibility would be the collection of subsets of of cardinality at least . That doesn’t feel like the kind of class I’m interested in, but why doesn’t “has cardinality at least half that of ” count as a valid definition? I don’t see an obvious answer to this question.

Or do I? Maybe a condition is that in some sense coordinates should not be special. So if you have a complexity structure like with lots of symmetry, then the definition of simplicity should be invariant under the maps . (That is, if a set is simple and you change the th coordinate of each element, then the resulting set is still simple.)

]]>Let’s consider the question just above in the case where all the specifications that define the payoff set are ones where and . Then we can think of the game as follows. Each move that Player I makes determines a condition that Player II must satisfy, which will be a specification of certain coordinates between and . For instance, if two of the sets that define are and , then the move for Player I tells Player II that he must ensure that to have any chance of winning. For each and each , let be the condition imposed on Player II by Player I playing the move . Then for each Player I can choose to impose the condition or the condition .

Note that if there exist distinct and bits such that the conditions are inconsistent, then Player I trivially has a winning strategy.

However, the converse is far from true, which is what makes the game potentially interesting. Even if Player I does not have a set of incompatible conditions like that, she may be able to exploit what Player II does. For example, if is the set of all sequences such that or , then Player I cannot find an inconsistent set of conditions using different , but can win the game by waiting until Player II chooses either or and only then playing .

Hmm, that was quite a useful example actually, since it shows that a set can be very simple indeed (it depends on just three coordinates) but that the player with a winning strategy may not be able to force a win until very near the end of the game. Here Player I has a winning strategy but if Player II waits until his last two moves before playing either of or , then Player I can’t get to the point where the outcome is decided until her final move. This seems to kill off the idea I had above about -winning sets. If a set as simple as this one, which lives in a complexity structure as simple as , is not -winning for any , then that does seem to suggest that -winningness for small is not a very good definition of simplicity.

]]>Before leaping in and trying to guess a definition of “strategically simple” it might be a good idea to think about the complexity of strategies for 2-open games played in . That is, we are given a set of specifications of the form , and Player I is trying to end up with a sequence for which at least one of these specifications holds, while Player II is trying to stop her. Are there sets of pairs for which it is very difficult to decide who wins the game? My guess is yes, but that is just a guess at this stage.

]]>Now let’s consider a big difference between the lifts Martin defined (and some that I have defined in the finite case) and the lift that Pavel defines. Suppose we play the auxiliary game and take itself as the payoff set. One might expect, given that is substantially simpler than , that the strategy for the player who wins to be much simpler for than it is for . And indeed that is the case for Martin’s lifts. But it is not at all the case for Pavel’s lift: the game is pretty well unaltered. (I don’t quite mean that: if one of the players declares the extra bit early on, then the other player can quickly force the outcome of the game, but assuming “sensible” play, the extra bits have no real effect on how the game is played.)

This is an example of a curious phenomenon that can occur in complexity spaces: that a set may be “topologically simple” but “strategically complicated”. I don’t have a good example to hand, so let me settle for an explanation of why I believe it should be reasonably easy to construct one. Consider the set in some complexity structure . One might think that there was a blindingly obvious strategy for Player I, which is simply to play the move and then the move . But what if there are no sequences in with and ? Then starting with the move would be a mistake, since Player II could respond with .

But can’t Player II do that *anyway*? No, not necessarily, since there might be no sequences with and . In that case, Player I might have a winning strategy, but only if she begins with the move .

I think that’s enough to demonstrate that there is a disconnect between topological simplicity and simplicity from the point of view of the shrinking-neighbourhoods game.

Since that probably counts as a single unit of observation, I’ll post it and again continue in a separate comment.

]]>Throughout the project, I’ve been trying to keep the analogies with Martin’s theorem as close as possible. Perhaps it would be helpful if I explained why I abandoned one obvious idea, which is to have a game where Player I picks the first coordinate, Player II the second, and so on. (Instead, the players get to pick any unpicked coordinates in their halves.) Then we have a game tree where the nodes are labelled by sequences .

Let’s stick with the case where we have all 01-sequences of length , so the tree is just a binary tree. Then the set corresponds to a set of leaves of this tree that contains exactly one leaf out of each pair of the form . Now we want that set to be “simple” in some sense (the sense being something like that it lifts to a very simple set in a not too much larger tree). But the problem with this is that if our definition of “simple” is invariant under automorphisms of the tree, then the simplicity of this set would imply the simplicity of a huge number of other sets: indeed, all sets such that if then . And from such sets you can make all sets with just a few Boolean operations.

It was because of that observation that I felt forced to abandon the tree approach. In the infinite case, you can get away with trees because every coordinate is only finitely far along and therefore “very near the beginning”. But that phenomenon has no analogue in the finite case. The shrinking-neighbourhoods game felt like the minimal modification I could make to the tree set-up that would ensure that all the sets were simple.

I’ll continue this discussion in a new comment (the idea being to try to keep to one observation per comment).

]]>That is a possibility that has already occurred to me and that I am considering seriously (not that I’ve asked Pavel yet what he thinks about the idea). But if anybody else has been thinking about these ideas and would like to be involved in some future more traditional collaboration, I don’t want to exclude people. Also, I still like the idea of occasionally posting if there are any interesting developments, whether or not they are sufficiently concrete to be publishable.

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