Positional Games and QBF: The Corrective Encoding

Positional games are a mathematical class of two-player games comprising Tic-tac-toe and its generalizations. We propose a novel encoding of these games into Quantified Boolean Formulas (QBF) such that a game instance admits a winning strategy for first player if and only if the corresponding formula is true. Our approach improves over previous QBF encodings of games in multiple ways. First, it is generic and lets us encode other positional games, such as Hex. Second, structural properties of positional games together with a careful treatment of illegal moves let us generate more compact instances that can be solved faster by state-of-the-art QBF solvers. We establish the latter fact through extensive experiments. Finally, the compactness of our new encoding makes it feasible to translate realistic game problems. We identify a few such problems of historical significance and put them forward to the QBF community as milestones of increasing difficulty.Comment: Accepted for publication in the 23rd International Conference on Theory and Applications of Satisfiability Testing (SAT2020

Accelerating board games through Hardware/Software Codesign

Board games applications usually offer a great user experience when running on desktop computers. Powerful high-performance processors working without energy restrictions successfully deal with the exploration of large game trees, delivering strong play to satisfy demanding users. However, nowadays, more and more game players are running these games on smartphones and tablets, where the lower computational power and limited power budget yield a much weaker play. Recent systems-on-a-chip include programmable logic tightly coupled with general-purpose processors enabling the inclusion of custom accelerators for any application to improve both performance and energy efficiency. In this paper, we analyze the benefits of partitioning the artificial intelligence of board games into software and hardware. We have chosen as case studies three popular and complex board games, Reversi, Blokus, and Connect6. The designs analyzed include hardware accelerators for board processing, which improve performance and energy efficiency by an order of magnitude leading to much stronger and battery-aware applications. The results demonstrate that the use of hardware/software codesign to develop board games allows sustaining or even improving the user experience across platforms while keeping power and energy low

Plentiful Possibilities for Pen, Pencil, and Paper Play

Neller presented games such as Dots and Boxes, Sprouts, Jotto, Chomp, and Pentominoes in order to illustrate the diversity of existing pencil and paper games. Additionally, he presented his own pencil and paper game design, Paper Penguins, and discussed the game design process

Extensions de réseaux de connexité donnée

RésuméOn appellera G(i) un graphe d'ordre i. Partant d'un graphe G(i), on veut obtenir, en ajoutant p par p des sommets, une famille infinie des graphes G(n), qui conservent certaines propriés de G(i) (on parlera d'extension). On s'intéresse, ici, aux propriétés de k-connexités (réseaux qui supportent k-1 pannes) et de degré maximum Δ (le nombre de liaisons est limité à Δ).On considère d'abord le cas p = 1. On montre que pour certaines valeurs de Δ et k, il est impossible de trouver une extension conservant k et Δ sans admettre des modifications dans le graphe déjà existant. G(j) n'est pas forcément un sous-graphe de G(j + 1). Pour ces valeurs, on recherchera une borne théorique du nombre moyen minimum de remaillages par étape, notér0(Δ, k), où un remaillage est la suppression d'une liaison déjà existante.On prouve ensuite que cette borne est la meilleure possible. Plus exactment on montre le théorème suivant: quels que soient Δ et k, il existe un graphe initial et une extension infinie de graphes k-connexes et de degré maximum Δ obtenue avec un nombre moyen de remaillages égal à r0(Δ, k).La fin de ce travail sera consacrée à la généralisation à p quelconque. On montrera notamment que quels que soit k⩽Δ, il existe p tel qu'il existe une extension infinie d'un graphe initial par pas de p sans remaillage.AbstractLet us call G(i) a graph of order i. We want to construct an infinite family of graphs (G(n)) starting with a graph G(i) and adding new vertices p by p in such a way that all the graphs still verify some of the properties of G(i). Here we consider k-connected graphs (networks who stand k-1 node failures) with maximum degree Δ (the number of links at each vertex is bounded by Δ).First we consider the case p = 1. We show that, for some value of k and Δ, such an extension doesn't exists without allowing some modifications of the network. G(j) is not necessarily a subgraph of G(j+1). For these values we give a theorical bound for the average number of necessary relinkages at each step, denoted r0(Δ, k). A relinkage is a suppression of an existent edge. We show that this bound is the best possible. More exactly we prove the following theorem: let k and Δ be integers, k⩽Δ, there exist an initial graph and an infinite extension of k-connected graphs with maximum degree Δ with an average number of relinkages equal to r0(Δ, k) at each step.Eventually we generalize this result to the extension with any p. In particular we show that for all k⩽Δ, there exist p0(Δ, k) and an infinite extension with p0 new vertices at each step without relinkage

The parameterized complexity of positional games

We study the parameterized complexity of several positional games. Our main result is that Short Generalized Hex is W[1]-complete parameterized by the number of moves. This solves an open problem from Downey and Fellows’ influential list of open problems from 1999. Previously, the problem was thought of as a natural candidate for AW[*]-completeness. Our main tool is a new fragment of first-order logic where universally quantified variables only occur in inequalities. We show that model-checking on arbitrary relational structures for a formula in this fragment is W[1]-complete when parameterized by formula size. We also consider a general framework where a positional game is represented as a hypergraph and two players alternately pick vertices. In a Maker-Maker game, the first player to have picked all the vertices of some hyperedge wins the game. In a Maker-Breaker game, the first player wins if she picks all the vertices of some hyperedge, and the second player wins otherwise. In an Enforcer-Avoider game, the first player wins if the second player picks all the vertices of some hyperedge, and the second player wins otherwise. Short Maker-Maker, Short Maker-Breaker, and Short Enforcer-Avoider are respectively AW[*]-, W[1]-, and co-W[1]-complete parameterized by the number of moves. This suggests a rough parameterized complexity categorization into positional games that are complete for the first level of the W-hierarchy when the winning condition only depends on which vertices one player has been able to pick, but AW[*]-complete when it depends on which vertices both players have picked. However, some positional games with highly structured board and winning configurations are fixed-parameter tractable. We give another example of such a game, Short k-Connect, which is fixed-parameter tractable when parameterized by the number of moves