From Specific Game Playing to General Game Playing

1. The victory of IBM's Deep Blue computer over the world champion chess player Gary Kasparov in 1997 is one of the most significant events in the history of Artificial Intelligence. However, Deep Blue is specially designed to play Chess, and it has no clue how to play another game like Tic-Tac-Toe. The most intuitive way is to represent a game as a state machine. For example the following graph represent part of the state machine of game Tic-Tac-Toe: * Each node repents a state of the game, and * One node is linked to another node if there is a move to transfer first node into the second. But the problem with the state machine representation is that when the game becomes more complex, the state space might be too large to represent. For example, Chess has around 10^30 states. Therefore Game Description Language (GDL) is introduced to represent a gamesuccinctly, using logical rules. A state is now represented by a set of propositions that are true, and the transitions between states are described by logical rules. The following is a fragment of Tic-Tac-Toe in GDL: (role xplayer) (role oplayer) (init (cell 1 1 b)) (init (cell 1 2 b)) (init (cell 1 3 b)) (init (cell 2 1 b)) (init (cell 2 2 b)) … (init (control xplayer)) (<= (next (cell ?m ?n x)) (does xplayer (mark ?m ?n)) (true (cell ?m ?n b))) … (<= (next (control xplayer)) (true (control oplayer))) … (<= (legal ?w (mark ?x ?y)) (true (cell ?x ?y b)) (true (control ?w))) (<= (legal xplayer noop) (true (control oplayer))) … (<= (goal xplayer 100) (line x)) (<= (goal xplayer 50) (not (line x)) (not (line o)) (not open)) (<= (goal xplayer 0) (line o)) … (<= terminal (line x)) … Strategic Reasoning with Game Description LanguageJi Ruan, Prof. W. van der Hoek, Prof. M. WooldridgeDepartment of Computer Science Http://www.csc.liv.ac.uk/~jruan From Specific Game Playing to General Game Playing Reasoning about games: ATL Alternating-time Temporal Logic (ATL)has been widely applied to reasoning about game-like multi-agent systems in recent years, and has proved to be a powerful and expressive tool for this purpose. The basic construct of ATL is the coalition modality, <<C>>Phi, where C is a collection of agents, meaning that coalition C can guarantee to achieve property Phi (or equivalently, C has a winning strategy for Phi.) Returning to the Tic-Tac-Toe, suppose its game model is G, in state w, we see that player X has a strategy such that it can eventually win. This property can be expressed in ATL by: G,w |= <<X>> E XWin So, the General Game Playing (GGP) Competition is introduced by AAAI in 2005 to test the ability to play games in general, rather than a specific game. The computer players are provided with previously unknown games just before the competition, and are required to play them autonomously. Game Representations: State Machine vs GDL In Our Research • We use ATL tools for reasoning about games represented in GDL. We show that GDL specifications can be viewed as specifications of an ATL model, and that ATL can thus be interpreted over GDL specifications. • We translate a GDL specification into an equivalent ATL specification, which is not significantly larger than the original GDL specification. And we prove the complexity of reasoning about GDL-specified games using ATL is EXPTIME-complete. • All well-defined games should have three basic properties: termination, winnability and playability. These properties can be easily represented as ATL specifications. But how to decide whether a GDL specification is well-defined? We try to transform a GDL specification into a representation accepted by model checker MOCHA, and use it to solve this question: • GDL-MOCHA model |= ATL Specification ? • Currently, GDL only allows for describing games with perfect information, e.g. chess. We try to extension GDL to enable the description of imperfect information games, e.g. poker. This will further connect to the epistemic extension of ATL.