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18
Infinitary Self Reference in Learning Theory
, 1994
"... 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 ..."
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Cited by 19 (6 self)
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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 singlecelled 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...
Codable Sets and Orbits of Computably Enumerable Sets
 J. Symbolic Logic
, 1995
"... 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 ..."
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Cited by 10 (5 self)
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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 Edefinable 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...
Inclusion Problems in Parallel Learning and Games
, 1994
"... In a recent paper Kinber, Smith, Velauthapillai, and Wiehagen introduced a new notion of "parallel learning". They call a set S of functions (m; n)learnable if there is a learning machine which for any ntuple of pairwise distinct functions from S learns at least m functions correctly ..."
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Cited by 6 (6 self)
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In a recent paper Kinber, Smith, Velauthapillai, and Wiehagen introduced a new notion of "parallel learning". They call a set S of functions (m; n)learnable if there is a learning machine which for any ntuple of pairwise distinct functions from S learns at least m functions correctly from examples of their behavior after seeing some finite amount of input. One of the basic open questions in this area is the "inclusion problem", i.e., the question for which m; n; h; k, every (m; n)learnable class is also (h; k)learnable. In this paper we develop a general approach to solve this problem. The idea is to associate with each m; n; h; k in a uniform way a finite 2player game such that the first player has a winning strategy in this game iff every (m; n)learnable class is (h; k)learnable. In this way we take the recursion theoretic disguise off the problem and isolate its combinatorial core. We also explicitly characterize the "strength" of each particular noninclusion by ...
Effective Strategies for Enumeration Games
 Proceedings of Computer Science Logic CSL '95
, 1996
"... 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 ..."
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Cited by 3 (3 self)
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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...
Structural Measures for Games and Process Control in the Branch Learning Model
 Proceedings of the Third European Conference on Computational Learning Theory, volume 1208 of LNAI
, 1997
"... Process control problems can be modeled as closed recursive games. Learning strategies for such games is equivalent to the concept of learning infinite recursive branches for recursive trees. We use this branch learning model to measure the difficulty of learning and synthesizing process controllers ..."
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Process control problems can be modeled as closed recursive games. Learning strategies for such games is equivalent to the concept of learning infinite recursive branches for recursive trees. We use this branch learning model to measure the difficulty of learning and synthesizing process controllers. We also measure the difference between several process learning criteria, and their difference to controller synthesis. As measure we use the information content (i.e. the Turing degree) of the oracle which a machine need to get the desired power. The investigated learning criteria are finite, EX , BC , Weak BC  and online learning. Finite, EX  and BC style learning are well known from inductive inference, while weak BC  and online learning came up with the new notion of branch (i.e. process) learning. For all considered criteria  including synthesis  we also solve the questions of their trivial degrees, their omniscient degrees and with some restrictions their inference degree...
Short lists with short programs in short time
"... Abstract—Given a machine U, a cshort program for x is a string p such that U(p) = x and the length of p is bounded by c + (the length of a shortest program for x). We show that for any universal machine, it is possible to compute in polynomial time on input x a list of polynomial size guaranteed t ..."
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Abstract—Given a machine U, a cshort program for x is a string p such that U(p) = x and the length of p is bounded by c + (the length of a shortest program for x). We show that for any universal machine, it is possible to compute in polynomial time on input x a list of polynomial size guaranteed to contain a O(logx)short program for x. We also show that there exist computable functions that map every x to a list of size O(x  2) containing a O(1)short program for x and this is essentially optimal because we prove that such a list must have size Ω(x  2). Finally we show that for some machines, computable lists containing a shortest program must have length Ω(2 x ).
An Overview of the Computably Enumerable Sets
"... 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 ..."
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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.
A General Framework for Priority Arguments
 THE BULLETIN OF SYMBOLIC LOGIC
, 1995
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Dynamic Properties of Computably Enumerable Sets
 In Computability, Enumerability, Unsolvability, volume 224 of London Math. Soc. Lecture Note Ser
, 1995
"... 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 t ..."
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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 Edenable) 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 Edenable 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...