Kleene's Second Recursion Theorem provides a means for transforming any program p into a program e(p) which first creates a quiescent self copy and then runs p on that self copy together with any externally given input. e(p), in effect, has complete (low level) self knowledge, and p represents how e(p) uses its self knowledge (and its knowledge of the external world). Infinite regress is not required since e(p) creates its self copy outside of itself. One mechanism to achieve this creation is a self replication trick isomorphic to that employed by single-celled organisms. Another is for e(p) to look in a mirror to see which program it is. In 1974 the author published an infinitary generalization of Kleene's theorem which he called the Operator Recursion Theorem. It provides a means for obtaining an (algorithmically) growing collection of programs which, in effect, share a common (also growing) mirror from which they can obtain complete low level models of themselves and the other prog...

AsetAof nonnegative integers is computably enumerable (c.e.), also called recursively enumerable (r.e.), if there is a computable method to list its elements. Let E denote the structure of the computably enumerable sets under inclusion, E =({We}e∈ω,⊆). Most previously known automorphisms Φ of the structure E of

A set X of nonnegative integers is computably enumerable (c.e.), also called recursively enumerable (r.e.), if there is a computable method to list its elements. Let E denote the structure of the computably enumerable sets under inclusion, E = (fW e g e2! ; `). We previously exhibited a first order E-definable property Q(X) such that Q(X) guarantees that X is not Turing complete (i.e., does not code complete information about c.e. sets). Here we show first that Q(X) implies that X has a certain "slowness " property whereby the elements must enter X slowly (under a certain precise complexity measure of speed of computation) even though X may have high information content. Second we prove that every X with this slowness property is computable in some member of any nontrivial orbit, namely for any noncomputable A 2 E there exists B in the orbit of A such that X T B under relative Turing computability ( T ). We produce B using the \Delta 0 3 -automorphism method we introduced earli...

We study the existence of effective winning strategies in certain infinite games, so called enumeration games. Originally, these were introduced by Lachlan (1970) in his study of the lattice of recursively enumerable sets. We argue that they provide a general and interesting framework for computable games and may also be well suited for modelling reactive systems. Our results are obtained by reductions of enumeration games to regular games. For the latter effective winning strategies exist by a classical result of Buchi and Landweber. This provides more perspicuous proofs for several of Lachlan's results as well as a key for new results. It also shows a way of how strategies for regular games can be scaled up such that they apply to much more general games. 1 Introduction Infinite games have been studied for a long time in many areas of mathematical logic. In recent years they also appeared in computer science as a framework for modelling reactive systems (see [Tho95] for a recent sur...

this paper.

A set A ` ! is computably enumerable (c.e.), also called recursively enumerable, (r.e.), or simply enumerable, if there is a computable algorithm to list its members. Let E denote the structure of the c.e. sets under inclusion. Starting with Post [1944] there has been much interest in relating the denable (especially E-denable) properties of a c.e. set A to its iinformation contentj, namely its Turing degree, deg(A), under T , the usual Turing reducibility. [Turing 1939]. Recently, Harrington and Soare answered a question arising from Post's program by constructing a nonemptly E-denable property Q(A) which guarantees that A is incomplete (A !T K). The property Q(A) is of the form (9C)[A ae m C & Q \Gamma (A; C)], where A ae m C abbreviates that iA is a major subset of Cj, and Q \Gamma (A; C) contains the main ingredient for incompleteness. A dynamic property P (A), such as prompt simplicity, is one which is dened by considering how fast elements elements enter A relat...

. We study the problems of efficiently learning infinite branches for finite state trees and winninig strategies for closed finitestate games using membership, and branch or strategy queries, respectively. We show that generally no efficient branch learning algorithm exists but we provide such algorithms for several natural cases, in particular for deadend free finite-state trees, the class of trees such that the set of infinite branches has positive measure, and several classes of modulo trees. Furthermore, we find a way to apply Angluin's results about the identification of deterministic finite automata from queries, which yields positive and negative strategy learning results, in particular, we show that the class of deadend free closed finite-state games is efficiently strategy learnable from membership and strategy queries. 1 Introduction In [6, 10,13] an inductive inference approach to learning process controllers has been studied. Hereby, the control problem is modeled as a clo...

The purpose of this article is to summarize some of the results on the algebraic structure of the computably enumerable (c.e.) sets since 1987 when the subject was covered in Soare 1987 , particularly Chapters X, XI, and XV. We study the c.e. sets as a partial ordering under inclusion, (E; `). We do not study the partial ordering of the c.e. degrees under Turing reducibility, although a number of the results here relate the algebraic structure of a c.e. set A to its (Turing) degree in the sense of the information content of A. We consider here various properties of E: (1) deønable properties; (2) automorphisms; (3) invariant properties; (4) decidability and undecidability results; miscellaneous results. This is not intended to be a comprehensive survey of all results in the subject since 1987, but we give a number of references in the bibliography to other results. 1 A Brief History of C.E. Sets G#del 1934 introduced the deønition of a (general) recursive function, and Church 1936 p...

Post 1944 began studying properties of a computably enumerable (c.e.) set A such as simple, h-simple, and hh-simple, with the intent of finding a property guaranteeing incompleteness of A. From observations of Post 1943 and Myhill 1956, attention focused by the 1950's on properties definable in the inclusion ordering of c.e. subsets of!, namely E = (fWngn2! ; ae). In the 1950's and 1960's Tennenbaum, Martin, Yates, Sacks, Lachlan, Shoenfield and others produced a number of elegant results relating E-definable properties of A, like maximal, hh-simple, atomless, to the information content (usually the

this paper is to introduce the \Delta