Lattice point methods for combinatorial games

@article{Guo2011LatticePM,
  title={Lattice point methods for combinatorial games},
  author={Alan J. X. Guo and Ezra Miller},
  journal={Adv. Appl. Math.},
  year={2011},
  volume={46},
  pages={363-378}
}
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12 Citations
Background Citations
7
Methods Citations
3

12 Citations

Citation Type

Citation Type

Algorithms for lattice games
TLDR
This paper provides effective methods for the polyhedral formulation of impartial finite combinatorial games as lattice games using the theory of short rational generating functions.
Combinatorial games with a pass: a dynamical systems approach.
By treating combinatorial games as dynamical systems, we are able to address a longstanding open question in combinatorial game theory, namely, how the introduction of a "pass" move into a game…
Affine stratifications from finite misère quotients
TLDR
The motivating consequence of the main result is a special case of a conjecture due to Guo and the author on the existence of affine stratifications for (the set of winning positions of) any lattice game.
Presburger Arithmetic, Rational Generating Functions, and Quasi-Polynomials
TLDR
It is shown that every counting function obtained in a Presburger formula may be represented as, equivalently, either a piecewise quasi-polynomial or a rational generating function.
PRESBURGER ARITHMETIC, RATIONAL GENERATING FUNCTIONS, AND QUASI-POLYNOMIALS
TLDR
It is shown that every counting function obtained in a Presburger formula may be represented as, equivalently, either a piecewise quasi-polynomial or a rational generating function.
Lattice-valued matrix game with mixed strategies for intelligent decision support
Theory and applications of lattice point methods for binomial ideals
This survey of methods surrounding lattice point methods for binomial ideals begins with a leisurely treatment of the geometric combinatorics of binomial primary decomposition. It then proceeds to…
Combinatorics of JENGA
TLDR
This paper illustrates how to determine the Grundy value of a jenga tower by showing that it may be seen as a bidimensional vector addition game, and proposes a class of impartial rulesets, the clock nim games,jenga being an example of that class.
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Cellular Binomial Ideals
Without any restrictions on the base field, we compute the hull and provide an unmixed decomposition of a cellular binomial ideal. The latter had already been proved by Eisenbud and Sturmfels in…
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References

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Algorithms for lattice games
TLDR
This paper provides effective methods for the polyhedral formulation of impartial finite combinatorial games as lattice games using the theory of short rational generating functions.
An Investigation of Partizan Misere Games
TLDR
Partizan combinatorial \mis play games are investigated, by taking Plambeck's indistinguishability and \mis monoid theory for impartial positions and extending it to partizan ones, as well as examining the difficulties in constructing a category of \ Mis play games in a similar manner to Joyal's category of normal play games.
Short rational generating functions for lattice point problems
Abstract. We prove that for any fixed d the generating function of the projectionof the set of integer points in a rational d-dimensional polytope can be computed inpolynomial time. As a corollary, we…
Combinatorial Games: Tic-Tac-Toe Theory
Preface A summary of the book in a nutshell Part I. Weak Win and Strong Draw: 1. Win vs. weak win 2. The main result: exact solutions for infinite classes of games Part II. Basic Potential Technique…
The G-values of various games
A disjunctive combination of a finite set of two-person games Γ 1 , Γ 2 , …, Γ k may be defined thus: The players play alternately, each in turn making a move in one and only one of the individual…
Affine stratifications from finite misère quotients
TLDR
The motivating consequence of the main result is a special case of a conjecture due to Guo and the author on the existence of affine stratifications for (the set of winning positions of) any lattice game.
On Numbers and Games
ONAG, as the book is commonly known, is one of those rare publications that sprang to life in a moment of creative energy and has remained influential for over a quarter of a century. Originally…
Taming the wild in impartial combinatorial games
We introduce a misere quotient semigroup construction in impartial combinatorial game theory, and argue that it is the long-sought natural generalization of the normal-play Sprague-Grundy theory to…
Misère Games and Misère Quotients
These notes are based on a short course offered at the Weizmann Institute of Science in Rehovot, Israel, in November 2006. The notes include an introduction to impartial games, starting from the…
The complexity of generating functions for integer points in polyhedra and beyond
Motivated by the formula for the sum of the geometric series, we consider various classes  of sets S �¼ Zd of integer points for which an a priori �glong�h Laurent series or polynomial m�¸S xm can…
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