Mathematical Sciences: Some Recursion Theoretic Problems

数学科学：一些递归理论问题

• 批准号: 9622290
• 负责人: Leo Harrington
• 金额: $ 14.4万
• 依托单位: University of California-Berkeley
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1996
• 资助国家: 美国
• 起止时间: 1996-08-01 至 2000-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9622290&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Some; Recursion; Theoretic

DMS-96-22290 Harrington, U. Cal-Berkeley This project studies the recursively enumerable (re) sets of integers, in a way which is independent of their enumeration and independent of how integers are viewed as coding information. It is the study of re sets purely in relation to each other. The main ingredients of this study are: automorphisms of the collection of re sets - the symmetries that exist among re sets; and invariant sets - the walls that prevent symmetries. These two ingredients naturally go hand-in-hand. The main method used in this field is called `priority arguments'. This is a technique for performing computable constructions that ensure appropriate non-computable properties. A current guiding problem is: if an invariant set has non-trivial minimum information content then each of its members must have maximum information content. It is known that settling this would involve an extension of current automorphism techniques. A set is "re" (recursively enumerable) if it can be computably listed. For example, the set of all possible theorems of mathematics (deduced from the usual axioms) is re. This project studies the re sets themselves. Penrose, in his book The Emperor's New Mind, suggests an analogy between computing and consciousness. This projects seeks possibilities in the more strained analogy between re sets (independent of their enumeration) and the timeless aspects of consciousness. There are existing results which can be seen as indicative of this direction. The main method used in this field is called "priority arguments". This is a method whereby a common purpose can be achieved by establishing an extensive field of "world views", all working towards the common goal in their own separate ways, most of which are conflicting. One interesting feature of the method is that all reasonable world views must be allowed, including those that deny the common purpose (those world views do their jobs by working towards their own ne gations). Success in this project would strengthen the potentially profound implications of the analogy linking consciousness with a priority argument -like universe.

DMS-96-22290 Harrington，U。Cal-Berkeley该项目以独立于枚举和独立于整数被视为编码信息的方式，研究了整数的递归（RE）集合。这是对RE集的研究，纯粹是相互关联的。这项研究的主要成分是：RE集合集合的自动形态 - RE集合之间存在的对称性；和不变的集合 - 防止对称性的墙壁。这两种成分自然齐头并进。 该字段中使用的主要方法称为“优先参数”。这是一种用于执行可计算结构的技术，可确保适当的不可兼容的属性。当前的指导问题是：如果不变集具有非平凡的最低信息内容，则其每个成员必须具有最大的信息内容。众所周知，解决这将涉及当前的自动形态技术的扩展。 如果可以计算列出，则一组为“ re”（递归枚举）。例如，数学的所有可能定理（从通常的公理推论）的集合是RE。该项目研究了RE设置。彭罗斯（Penrose）在他的《皇帝的新思想》一书中提出了计算与意识之间的类比。该项目以更紧张的类比（独立于枚举）和意识的永恒方面寻求可能性。现有结果可以看作是指这个方向的。该字段中使用的主要方法称为“优先参数”。这是一种方法，可以通过建立广泛的“世界观”领域来实现共同的目的，所有这些领域都以自己的独立方式朝着共同的目标朝着共同的目标方向努力，其中大多数是矛盾的。该方法的一个有趣的特征是，必须允许所有合理的世界观，包括那些否认共同目的的人（这些世界观通过朝着自己的努力努力来完成工作）。该项目的成功将加强将意识与优先论证般的宇宙联系起来的类比的潜在深刻含义。