The Computably Enumerable Sets: Recent Results and Future Directions
نویسنده
چکیده
We survey some of the recent results on the structure of the computably enumerable (c.e.) sets under inclusion. Our main interest is on collections of c.e. sets which are closed under automorphic images, such as the orbit of a c.e. set, and their (Turing) degree theoretic and dynamic properties. We take an algebraic viewpoint rather than the traditional dynamic viewpoint.
منابع مشابه
Definable encodings in the computably enumerable sets
The purpose of this communication is to announce some recent results on the computably enumerable sets. There are two disjoint sets of results; the first involves invariant classes and the second involves automorphisms of the computably enumerable sets. What these results have in common is that the guts of the proofs of these theorems uses a new form of definable coding for the computably enume...
متن کاملKolmogorov complexity and computably enumerable sets
We study the computably enumerable sets in terms of the: (a) Kolmogorov complexity of their initial segments; (b) Kolmogorov complexity of finite programs when they are used as oracles. We present an extended discussion of the existing research on this topic, along with recent developments and open problems. Besides this survey, our main original result is the following characterization of the ...
متن کاملDefinability, automorphisms, and dynamic properties of computably enumerable sets
We announce and explain recent results on the computably enumerable (c.e.) sets, especially their definability properties (as sets in the spirit of Cantor), their automorphisms (in the spirit of FelixKlein’sErlanger Programm), their dynamic properties, expressed in terms of how quickly elements enter them relative to elements entering other sets, and theMartin Invariance Conjecture on their Tur...
متن کاملUniversal computably enumerable sets and initial segment prefix-free complexity
We show that there are Turing complete computably enumerable sets of arbitrarily low non-trivial initial segment prefix-free complexity. In particular, given any computably enumerable set A with non-trivial prefixfree initial segment complexity, there exists a Turing complete computably enumerable set B with complexity strictly less than the complexity of A. On the other hand it is known that s...
متن کاملThe global structure of computably enumerable sets
We will work in the structure of the computably enumerable sets. The language is just inclusion, ⊆. This structure is called E. Our quest is to partially survey our current understanding of the global structure of E and the relationship between E and the computably enumerable degrees and to pose questions whose answers should provide further insight.
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
دوره شماره
صفحات -
تاریخ انتشار 2004