1. Playing Game with Algorithms: Algorithmic Combinatorial Game Theory OUTLINE 1.Introduction 2.Combinatorial Game Theory 3.Algorithms for Two-Player Games 4.Algorithms for Puzzles

2. Introduction  One-player puuzzles Often NP-complete Ex: Minesweeper ( 踩地雷 )  Two-player gaems Often PSPACE-complete  Othello ( 黑白棋 ) or EXPTIME-complete  Chess, Go  對人類而言的難易度.. 對電腦而言並不一定 完全相等

3. Turing machine 1. 一條無限長的 type ，分成無限多個小格， 分成左邊跟右邊，內容只能是有限個字元 集，包含空白。 2. 有一個讀寫頭，顧名思義，此頭可以讀 type 上的內容，也可寫入，也可以像左移 或是像右移動。 3. 有一個狀態暫存器 可以表示 machine 的狀 態，狀態的表示只有有限多種狀態，其中 包含起始狀態。 4. 有一個動作表，會告訴 machine 該如何動 作， ex: 寫 or 讀 or 左 or 右，也會告知 machine 目前的狀態，機器在每一刻的動 作，取決於當時機器的狀態與所掃瞄的小 方格上的符號 。

4. PSPACE-complete  PSPACE: The set of decision problems that can be solved by a Turing machine using a polynomial amount of memory, and unlimited time.  PSPACE-complete: if it is in PSPACE, and every problem in PSPACE can be reduced to it in polynomial time. The problems in PSPACE- complete can be thought of as the hardest problems in PSPACE

5. EXPTIME-complete  EXPSPACE:The set of all decision problems solvable by a deterministic Turing machine in O (2p(n)) time, where p(n) is a polynomial function of n  EXPTIME-complete is also a set of decision problems. A decision problem is in EXPTIME- complete if it is in EXPTIME, and every problem in EXPTIME has a polynomial-time many-one reduction to it.. EXPTIME-complete might be thought of as the hardest problems in EXPTIME.

6. Combinatorial Game Theory  Combinatorial Game Theory is typically involves two players, called Left and Right.  Left wins; Right wins; first wins; second wins.  Two player games can be described by a rooted tree

7. Conway ’ s Surreal Numbers  {L|R} L <= R Ex:{|}=0; {0|}=1  Game: {L|R} No constraints L and R All games equal to 0 is second player to move win  Other outcome?

8. Sprague- Grundy Theory  Nim is a game played with several heaps.  Nim game with a single heap of size n (*n)  called nimber  K piles of sizes n 1,n 2,n 3,….n k 1 pile of size n XOR (n 1,n 2,n 3,..n k ) n

9.  Every impartial two-player perfect- information game has the same calue as a single-pile Nim game,*n. n called G-value, …….etc Strategy Stealing Using contradiction

10. Algorithms for Two-Player Games  Many nonloopy two-player games are PSPACE-complete and also EXPTIME- complete

11. Hex  Nash proved that the first player to move can win.

12. Checkers  子僅能向斜前方前進，或是跳過對方的子並 取走該子  PSPACE-HARD  And EXPTIME-complete