Svenja Huntemann Research Statement 2

Transcription

1 Svenja Huntemann esearch Statement Combinatorial game theory studies games of pure strategy, such as Chess or Go, often using connections to many other mathematical areas. My work focuses on combinatorial games coming from objects studied in combinatorial commutative algebra and other combinatorial objects, such as designs, posets, and hypergraphs. A combinatorial game, denoted in small caps, is a 2-player game without any chance devices and with perfect information. Examples of such games are Chess, Checkers, Go, and Nine Men's Morris. Non-examples are Snakes And adders (chance) and Battleship (incomplete information). The two players are called eft () and ight (). The two most commonly considered winning conditions are normal play, where the last player to make a move wins, and misère play, where the last player to make a move loses. At the core of combinatorial game theory are the operations of sum and inverse, as well as properties such as game value and temperature. These have important applications in game playing algorithms, but are not easy to determine in general. estricting to a specic class of games usually leads to good results. A game introduced early in combinatorial game theory is Snort [2], sometimes also known as Cats And Dogs. In this game, which can be played on any nite graph, eft and ight claim vertices. Two vertices belonging to opposite players may never be adjacent. The following diagram gives two positions in Snort. The one to the left is a legal position, while the one on the right is illegal. A second commonly seen game is Domineering [1], a game played on a grid on which the players place dominoes, thus claiming two orthogonally adjacent squares. eft places dominoes vertically, while ight places them horizontally. In many combinatorial games, including Snort and Domineering, pieces are being placed on a graph, without being moved or removed after. We consider games of this type with an additional condition: Denition 1. A strong placement (SP-)game is a combinatorial game in which pieces are placed on empty, connected spaces of the board (a nite graph), and never moved or removed. Furthermore, any sequence of moves leading to a legal position has to be legal, eectively meaning that the order of moves taken does not matter.

2 Svenja Huntemann esearch Statement 2 Besides Snort and Domineering, examples of well-studied games in this category are NoGo [4], Arc Kayles [14], and many Partizan Octals [7, 13]. Several other games, such as Nim and Hex, may not immediately meet the denition, but are indeed equivalent to SP-games. We call a position with a single piece played, whether this has been legal or not, a basic position. Any position in an SP-game is the union of a nite number of basic positions. For example, positions in Snort played on a path of three vertices break up as follows: = = An (abstract) simplicial complex on a nite vertex set V is a set of subsets of V such that whenever A and B A, then B. The elements of are called faces, and its maximal elements are facets. In [6] and [5] we show that SP-games are in one-to-one correspondence with simplicial complexes whose vertex sets have been bipartitioned. A brief description of this correspondence follows. Given an SP-game, which is the set of rules played on a board B, we construct the legal complex,b by representing each legal basic position by a vertex and letting a set of vertices form a face whenever the union of the corresponding basic position is a legal position. The vertex set is naturally bipartitioned into sets (eft basic positions) and (ight basic positions). We can similarly also construct the illegal complex Γ,B in which the facets correspond to the minimal illegal positions. As an example, the legal and illegal complex of Snort played on a path of 3 vertices are given below, with vertices belonging to or labelled. Snort,P 3 = Γ Snort,P 3 = Conversely, given a simplicial complex with vertex set bipartitioned into sets and, we construct an SP-game as follows: the board will be the 1-skeleton of

3 Svenja Huntemann esearch Statement 3 (the vertices and edges); eft may only play on vertices belonging to and ight only on those belonging to ; and any vertices claimed need to form a face in. Then is the legal complex. The construction is similar if we wanted to be the illegal complex. Note that this also allows us to think of an SP-game as being played on a simplicial complex, either corresponding to the legal complex or the illegal complex. The previous construction means that the game's ruleset will highly depend on the board though, which is undesirable. The authors in [12] and [11] introduced the concept of invariance for the class of subtraction games, and we have dened it similarly for SP-games: An invariant SP-game (or isp-game) is an SP-game where (a) every basic position is legal, and (b) if B 1 and B 2 are isomorphic boards, then a move on B 1 is legal if and only if its isomorphic image on B 2 is legal. This denition in some sense forces the ruleset to be uniform across the entire board. It turns out that every simplicial complex is the legal complex of some isp-game. Theorem 2 ([5]). Given any simplicial complex with vertex set partitioned into and, we can construct an isp-game (, B) such that =,B and the sets of eft, respectively ight, positions is, respectively. The game tree of a game is a diagram constructed inductively in which from every position an edge points to every option, with the edge oriented to the left for eft options and similarly for ight options. I was able to show that whenever two SPgames have isomorphic legal complexes (including the bipartition of vertices), their game trees are also isomorphic. This is a very strong statement since isomorphic game trees imply that two games are in the same equivalence class, independent of the winning condition considered. Further, games in the same equivalence class can have dierent game trees. The following theorem then gives us that in most circumstances when studying SP-games, it is sucient to study isp-games. Theorem 3 ([5]). Given an SP-game G, there exists an isp-game G so that G and G have isomorphic game trees. The equivalence class a game belongs to is called its game value. A xed game or class of games is said to be universal if it takes on all possible game values under normal play. A question of interest in combinatorial game theory is whether a game is universal, and if it is not, which values it takes on. This was recently positively answered for a non-sp-game [3], and has also received attention for Domineering (see for example [10, 16]) and for Snort (see [2]) as examples of SP-games. Knowing that every simplicial complex with the vertex set bipartitioned is the legal complex of some SP-game helps with this problem: it is sucient to construct a complex where the corresponding game(s) have the value to be considered, rather than having

4 Svenja Huntemann esearch Statement 4 to construct the game (a usually much more dicult task). I have used the previous results to show that many common values under normal play come from SP-games, including all numbers and nimbers. Finally, the temperature of a game in some sense tells us how urgent it is to make a certain move. A goal of computer scientists in combinatorial game theory is nding heuristics to evaluate the temperature, which points to a good, hopefully the best, move. Of mathematical interest is the boiling point of a game, which is the maximum temperature no matter which board one plays on. We have recently been able to prove the rst upper bound on boiling points for combinatorial games which holds in general [9]. We also give a technique for applying this bound, which is particularly suitable for SP-games. For example, Berlekamp conjectured that the boiling point of Domineering is 2 (see [8, 15]) and we have been able to prove for a large class of boards that it is at most 5, the rst known upper bound. I am interested in further studying SP-games as well as similar classes of combinatorial games coming from other geometric and combinatorial objects: Combinatorial Designs: We can also think about playing on combinatorial designs, such as nite projective planes, triple systems, and more generally block designs, by letting players claim points. Play can then be dened in many dierent ways, from claimed points having to be contained in a block, to any three points of a player forming an independence set. Posets: Similarly, one can play on a poset by having players alternatingly picking covers of the previous element until a maximal element has been reached. Eectively, this means that play is to form a chain. Other methods of play could include pieces claimed having to form antichains. For both of these classes of games, we are interested in several questions: Game Values: First and foremost we are interested in whether all game values, the equivalence classes of combinatorial games, can be achieved by either of these classes of games under normal play (the last player to make a move wins). One of the major research problems in combinatorial game theory is to nd a class of games for which this is the case, as we would be able to map all games to this class. For SP-games, the structure of the associated simplicial complex gives an indication of which game values are possible, and this will likely also be the case for the other classes of games under consideration. Temperature: Berlekamp conjectured that the boiling point of Domineering is 2 (see [8, 15]). Since the legal complex encodes all information about an SP-game, studying the structure of the legal complexes of Domineering should provide us with new approaches to tighten this bound. Similarly for the other classes of games, the related combinatorial structure should allow us to improve this bound, given

5 Svenja Huntemann esearch Statement 5 that they completely encode the game. Misère Play: Under misère play the situation is generally much more complicated than under normal play. One of the advantages of our approach of representing positions through other objects is that they are independent of the winning condition, and likely these tools will be very useful in the harder case as well. Two challenges often found in misère play are determining the sum of two games or the inverse of a game, both relatively easy under normal play. For SP-games, we are able to determine sums using operations on the simplicial complexes, and likely similar operations exist for the other object. Similarly as under normal play, we would also like to study the possible game values and temperatures. Enumerating Positions: A recent trend in combinatorial game theory has been to enumerate the positions in a game. For SP-games this often relates to enumeration problems in graph theory, for example independence sets, and in combinatorial commutative algebra, as it equates to counting the faces of the related simplicial complexes. For the other classes, enumerating positions should also relate to enumeration problems of the structures played on. In summary, my past research has provided a strong new tool to studying a class of combinatorial games. Future work will look at nding similar methods for other classes of games by providing connections with other combinatorial areas, and applying these methods to open and interesting questions in combinatorial game theory. eferences [1] E.. Berlekamp. Blockbusting and domineering. J. Combin. Theory Ser. A, 49(1):67 116, [2] E.. Berlekamp, J.H. Conway, and.k. Guy. Winning ways for your mathematical plays. Vol. 1. A K Peters td., Wellesley, MA, second edition, [3] A. Carvalho and C. Pereira dos Santos. A nontrivial surjective map onto the short Conway group. In Urban arsson, editor, Games of No Chance 5, volume 70 of Mathematical Sciences esearch Institute Publications, pages Cambridge University Press, [4] C. Chou, O. Teytaud, and S. Yen. evisiting Monte-Carlo Tree Search on a Normal Form Game: NoGo. In C. Di Chio, S. Cagnoni, C. Cotta, M Ebner, A. Ekárt, A. Esparcia-Alcázar, J. Merelo, F. Neri, M. Preuss, H. ichter, J. Togelius, and G. Yannakakis, editors, Applications of Evolutionary Computation, volume 6624, pages Springer Berlin/Heidelberg, 2011.

6 Svenja Huntemann esearch Statement 6 [5] S. Faridi, S. Huntemann, and.j. Nowakowski. Simplicial Complexes are Game Complexes. Preprint [6] S. Faridi, S. Huntemann, and.j. Nowakowski. Games and Complexes I: Transformation via Ideals. In Urban arsson, editor, Games of No Chance 5, volume 70 of Mathematical Sciences esearch Institute Publications, pages Cambridge University Press, [7] A. S. Fraenkel and A. Kotzig. Partizan octal games: partizan subtraction games. Internat. J. Game Theory, 16(2):145154, [8] ichard K. Guy. Unsolved problems in combinatorial games. In Games of no chance (Berkeley, CA, 1994), volume 29 of Math. Sci. es. Inst. Publ., pages Cambridge Univ. Press, Cambridge, [9] S. Huntemann,.J. Nowakowski, and C. Pereira dos Santos. Boiling Point. Preprint [10] Y. Kim. New values in domineering. Theoret. Comput. Sci., 156(1-2):263280, [11] Urban arsson. The -operator and invariant subtraction games. Theoret. Comput. Sci., 422:5258, [12] Urban arsson, P. Hegarty, and A.S. Fraenkel. Invariant and dual subtraction games resolving the Duchêne-igo conjecture. Theoret. Comput. Sci., 412:729735, [13] G. A. Mesdal. Partizan splittles. In M. H. Albert and. J. Nowakowski, editors, Games of No Chance 3, number 56 in Math. Sci. es. Inst. Publ., pages Cambridge Univ. Press, [14] Thomas J. Schaefer. On the complexity of some two-person perfect-information games. J. Comput. System Sci., 16(2):185225, [15] Ajeet Shankar and Manu Sridharan. New temperatures in Domineering. Integers, 5(1):G4, 13, [16] Jos W. H. M. Uiterwijk and Michael Barton. New results for Domineering from combinatorial game theory endgame databases. Theoret. Comput. Sci., 592:7286, 2015.