Deciding the Winner of an Arbitrary Finite Poset Game is PSPACE-Complete

A poset game is a two-player game played over a partially ordered set (poset) in which the players alternate choosing an element of the poset, removing it and all elements greater than it. The first player unable to select an element of the poset loses. Polynomial time algorithms exist for certain restricted classes of poset games, such as the game of Nim. However, until recently the complexity of arbitrary finite poset games was only known to exist somewhere between NC^1 and PSPACE. We resolve this discrepancy by showing that deciding the winner of an arbitrary finite poset game is PSPACE-complete. To this end, we give an explicit reduction from Node Kayles, a PSPACE-complete game in which players vie to chose an independent set in a graph

Blocking Wythoff Nim

The 2-player impartial game of Wythoff Nim is played on two piles of tokens. A move consists in removing any number of tokens from precisely one of the piles or the same number of tokens from both piles. The winner is the player who removes the last token. We study this game with a blocking maneuver, that is, for each move, before the next player moves the previous player may declare at most a predetermined number, $k - 1 \ge 0$, of the options as forbidden. When the next player has moved, any blocking maneuver is forgotten and does not have any further impact on the game. We resolve the winning strategy of this game for $k = 2$ and $k = 3$ and, supported by computer simulations, state conjectures of the asymptotic `behavior' of the $P$-positions for the respective games when $4 \le k \le 20$.Comment: 14 pages, 1 Figur

2-pile Nim with a Restricted Number of Move-size Imitations

We study a variation of the combinatorial game of 2-pile Nim. Move as in 2-pile Nim but with the following constraint: Suppose the previous player has just removed say $x>0$ tokens from the shorter pile (either pile in case they have the same height). If the next player now removes $x$ tokens from the larger pile, then he imitates his opponent. For a predetermined natural number $p$, by the rules of the game, neither player is allowed to imitate his opponent on more than $p-1$ consecutive moves. We prove that the strategy of this game resembles closely that of a variant of Wythoff Nim--a variant with a blocking manoeuvre on $p-1$ diagonal positions. In fact, we show a slightly more general result in which we have relaxed the notion of what an imitation is.Comment: 18 pages, with an appendix by Peter Hegart

Impartial coloring games

Coloring games are combinatorial games where the players alternate painting uncolored vertices of a graph one of $k > 0$ colors. Each different ruleset specifies that game's coloring constraints. This paper investigates six impartial rulesets (five new), derived from previously-studied graph coloring schemes, including proper map coloring, oriented coloring, 2-distance coloring, weak coloring, and sequential coloring. For each, we study the outcome classes for special cases and general computational complexity. In some cases we pay special attention to the Grundy function

Restrictions of mm-Wythoff Nim and pp-complementary Beatty Sequences

Fix a positive integer $m$. The game of \emph{$m$-Wythoff Nim} (A.S. Fraenkel, 1982) is a well-known extension of \emph{Wythoff Nim}, a.k.a 'Corner the Queen'. Its set of $P$-positions may be represented by a pair of increasing sequences of non-negative integers. It is well-known that these sequences are so-called \emph{complementary homogeneous} \emph{Beatty sequences}, that is they satisfy Beatty's theorem. For a positive integer $p$, we generalize the solution of $m$-Wythoff Nim to a pair of \emph{$p$-complementary}---each positive integer occurs exactly $p$ times---homogeneous Beatty sequences a = (a_n)_{n\in \M} and b = (b_n)_{n\in \M}, which, for all $n$, satisfies $b_n - a_n = mn$. By the latter property, we show that $a$ and $b$ are unique among \emph{all} pairs of non-decreasing $p$-complementary sequences. We prove that such pairs can be partitioned into $p$ pairs of complementary Beatty sequences. Our main results are that \{\{a_n,b_n\}\mid n\in \M\} represents the solution to three new '$p$-restrictions' of $m$-Wythoff Nim---of which one has a \emph{blocking maneuver} on the \emph{rook-type} options. C. Kimberling has shown that the solution of Wythoff Nim satisfies the \emph{complementary equation} $x_{x_n}=y_n - 1$. We generalize this formula to a certain '$p$-complementary equation' satisfied by our pair $a$ and $b$. We also show that one may obtain our new pair of sequences by three so-called \emph{Minimal EXclusive} algorithms. We conclude with an Appendix by Aviezri Fraenkel.Comment: 22 pages, 2 figures, Games of No Chance 4, Appendix by Aviezri Fraenke

Master index

Pla general, del mural ceràmic que decora una de les parets del vestíbul de la Facultat de Química de la UB. El mural representa diversos símbols relacionats amb la química

Numeration Systems: a Link between Number Theory and Formal Language Theory

We survey facts mostly emerging from the seminal results of Alan Cobham obtained in the late sixties and early seventies. We do not attempt to be exhaustive but try instead to give some personal interpretations and some research directions. We discuss the notion of numeration systems, recognizable sets of integers and automatic sequences. We briefly sketch some results about transcendence related to the representation of real numbers. We conclude with some applications to combinatorial game theory and verification of infinite-state systems and present a list of open problems.Comment: 21 pages, 3 figures, invited talk DLT'201