Tic-Tac-Toe Theory-An Escape From The Combinatorial Chaos

井字棋理论——摆脱组合混沌

• 批准号: 0701432
• 负责人: Jozsef Beck
• 金额: $ 14.82万
• 依托单位: Rutgers University New Brunswick
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2007
• 资助国家: 美国
• 起止时间: 2007-06-01 至 2013-11-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0701432&HistoricalAwards=false
• 关键词: Tic; Tac; Toe; Theory; Escape

Abstract: Tic-Tac-Toe Theory--an escape from the combinatorial chaosMathematics is spectacularly successful doing generalizations (Calculus, Modern Algebra, Algebraic Geometry, etc.); on the other hand, it could say surprisingly little about nontraditional complex systems like the economy or Chess. What is the fundamental problem with complex systems? The short answer is: the immense space of possibilities. The estimated total number of moves in Chess is more than ten-to-the-hundred, which is more than the number of particles in the observable universe. The majority of problems in Discrete Mathematics lead to cases studies of the same size. Statistical Mechanics faces the same problem: a mole of gas contains aboutten-to-23 particles; the underlying dynamic is incredibly complicated. The basic idea of Statistical Mechanics is to work with a priory probabilities; usually equiprobability in the phase space (it is counterintuitive: how does probability enter Newtonian mechanics?). The investigator develops aTic-Tac-Toe Theory which is a striking analog of the Statistical Mechanics approach: it can determine the exact value of infinitely many natural Game Numbers (the same paradox: how does probability enter Chess?). The investigator's book (700 pages) about the subject will come out in2007 at Cambridge University Press.Understanding complex systems is a major challenge for contemporary Mathematics (and Theoretical Computer Science, and Economy, etc.). Games are the perfect natural models. Also analyzing games is perhaps the best way to give the students a taste of mathematical discovery; it is the most natural mathematical experimentation. To integrate this new branch of game theory (the investigator's Tic-Tac-Toe Theory) into math education is an extremely promising aspect. Finally, understanding complex games may give unique insight into how human intelligence works.

摘要：Tic-Tac-Toe Theory--一种摆脱组合混乱的方法数学在推广方面非常成功（微积分、现代代数、代数几何等）;另一方面，它对经济或国际象棋等非传统复杂系统的描述却少得令人惊讶。复杂系统的根本问题是什么？简短的回答是：无限的可能性空间。国际象棋中的棋步总数估计超过10的100次方，这比可观测宇宙中的粒子数还要多。离散数学中的大多数问题都会导致相同大小的案例研究。 统计力学也面临着同样的问题：一摩尔气体含有大约10到23个粒子;其背后的动力学是极其复杂的。统计力学的基本思想是处理先验概率;通常是相空间中的等概率（这是违反直觉的：概率如何进入牛顿力学？）。 研究者发展了一种与统计力学方法惊人相似的井字棋理论：它可以确定无穷多个自然博弈数的确切值（同样的悖论：概率如何进入国际象棋？）。 研究者关于该主题的书（700页）将于2007年在剑桥大学出版社出版。理解复杂系统是当代数学（以及理论计算机科学和经济学等）的重大挑战。游戏是完美的自然模型。分析游戏也许是让学生体验数学发现的最好方式;这是最自然的数学实验。将博弈论的这一新的分支（研究者的井字游戏理论）融入数学教育是一个非常有前途的方面。最后，理解复杂的游戏可能会对人类智能的运作方式提供独特的见解。