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

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

A Mathematical Approach to Gomoku

This goal of this thesis is to design and implement a light weighted AI for playing Gomoku with high level intelligence. Our work is built upon an innovative algebraic monomial theory to help assess values for each possible move and estimate chances for the AI to win at each move. With the help of the monomial theory, we are able to convert winning configurations into monomials of variables that represent the underlying board positions. In the existing approaches to building an AI for playing Gomoku, one common challenge is about how to represent the present configuration of the game along with the history of the moves of the two players. Compared with the usual 2D matrix of the board positions, our monomials can make the AI easily understand the current state and the history of the game, and they also allow the AI to compute the potential values for future moves from the current state and the history of moves made by the players. In addition, when we adopt the Monte Carlo Tree Search to probe for a possible winning strategy for the AI, those monomials help reduce the search space, in addition to help estimate rates for exploration of the historical moves and exploitation of the future moves. Based on the proposed algebraic monomial theory, we have implemented a lightweight powerful AI that is capable of playing Gomoku at highly competitive level. At this stage, our AI can win top rated AIs (up to top 7) from the most recent Gomocup rating

Searching by learning: Exploring artificial general intelligence on small board games by deep reinforcement learning

In deep reinforcement learning, searching and learning techniques are two important components. They can be used independently and in combination to deal with different problems in AI. These results have inspired research into artificial general intelligence (AGI).We study table based classic Q-learning on the General Game Playing (GGP) system, showing that classic Q-learning works on GGP, although convergence is slow, and it is computationally expensive to learn complex games.This dissertation uses an AlphaZero-like self-play framework to explore AGI on small games. By tuning different hyper-parameters, the role, effects and contributions of searching and learning are studied. A further experiment shows that search techniques can contribute as experts to generate better training examples to speed up the start phase of training.In order to extend the AlphaZero-likeself-play approach to single player complex games, the Morpion Solitaire game is implemented by combining Ranked Reward method. Our first AlphaZero-based approach is able to achieve a near human best record.Overall, in this thesis, both searching and learning techniques are studied (by themselves and in combination) in GGP and AlphaZero-like self-play systems. We do so for the purpose of making steps towards artificial general intelligence, towards systems that exhibit intelligent behavior in more than one domain. China Scholarship CouncilAlgorithms and the Foundations of Software technolog