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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 18 (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...
Complexity issues for vacillatory function identification
 Information and Computation
, 1995
"... It was previously shown by Barzdin and Podnieks that one does not increase the power of learning programs for functions by allowing learning algorithms to converge to a finite set of correct programs instead of requiring them to converge to a single correct program. In this paper we define some new, ..."
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Cited by 12 (9 self)
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It was previously shown by Barzdin and Podnieks that one does not increase the power of learning programs for functions by allowing learning algorithms to converge to a finite set of correct programs instead of requiring them to converge to a single correct program. In this paper we define some new, subtle, but natural concepts of mind change complexity for function learning and show that, if one bounds this complexity for learning algorithms, then, by contrast with Barzdin and Podnieks result, there are interesting and sometimes complicated tradeoffs between these complexity bounds, bounds on the number of final correct programs, and learning power. CR Classification Number: I.2.6 (Learning – Induction). 1
On the Impact of Forgetting on Learning Machines
 Journal of the ACM
, 1993
"... this paper contributes toward the goal of understanding how a computer can be programmed to learn by isolating features of incremental learning algorithms that theoretically enhance their learning potential. In particular, we examine the effects of imposing a limit on the amount of information that ..."
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Cited by 10 (3 self)
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this paper contributes toward the goal of understanding how a computer can be programmed to learn by isolating features of incremental learning algorithms that theoretically enhance their learning potential. In particular, we examine the effects of imposing a limit on the amount of information that learning algorithm can hold in its memory as it attempts to This work was facilitated by an international agreement under NSF Grant 9119540.
Learning in the presence of partial explanations
 Information and Computation
, 1991
"... The effect of a partial explanation as additional information in the learning process is investigated. A scientist performs experiments to gather experimental data about some phenomenon, and then, tries to construct an explanation (or theory) for the phenomenon. A plausible model for the practice ..."
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Cited by 4 (2 self)
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The effect of a partial explanation as additional information in the learning process is investigated. A scientist performs experiments to gather experimental data about some phenomenon, and then, tries to construct an explanation (or theory) for the phenomenon. A plausible model for the practice of science is an inductive inference machine (scientist) learning a program (explanation) from graph (set of experiments) of a recursive function (phenomenon). It is argued that this model of science is not an adequate one, as scientists, in addition to performing experiments, make use of some approximate partial explanation based on the “state of the art ” knowledge about that phenomenon. An attempt has been made to model this partial explanation as an additional information in the scientific process. It is shown that inference capability of machines is improved in the presence of such a partial explanation. The quality of this additional information is modeled using certain “density ” notions. It is shown that additional information about a “better ” quality partial explanation enhances the inference capa
Vacillatory learning of nearly minimal size grammars
 Journal of Computer and System Sciences
, 1994
"... In Gold’s influential language learning paradigm a learning machine converges in the limit to one correct grammar. In an attempt to generalize Gold’s paradigm, Case considered the question whether people might converge to vacillating between up to (some integer) n> 1 distinct, but equivalent, cor ..."
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Cited by 4 (4 self)
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In Gold’s influential language learning paradigm a learning machine converges in the limit to one correct grammar. In an attempt to generalize Gold’s paradigm, Case considered the question whether people might converge to vacillating between up to (some integer) n> 1 distinct, but equivalent, correct grammars. He showed that larger classes of languages can be algorithmically learned (in the limit) by converging to up to n + 1 rather than up to n correct grammars. He also argued that, for “small ” n> 1, it is plausible that people might sometimes converge to vacillating between up to n grammars. The insistence on small n was motivated by the consideration that, for “large ” n, at least one of n grammars would be too large to fit in peoples ’ heads. Of course, even for Gold’s n = 1 case, the single grammar converged to in the limit may be infeasibly large. An interesting complexity restriction to make, then, on the final grammar(s) converged to in the limit is that they all have small size. In this paper we study some of the tradeoffs in learning power involved in making a welldefined version of this restriction. We show and exploit as a tool the desirable property that the learning power under our
Characterizing language identification by standardizing operations
 Journal of Computer and System Sciences
, 1994
"... Notions from formal language learning theory are characterized in terms of standardizing operations on classes of recursively enumerable languages. Algorithmic identification in the limit of grammars from text presentation of recursively enumerable languages is a central paradigm of language learnin ..."
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Cited by 1 (0 self)
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Notions from formal language learning theory are characterized in terms of standardizing operations on classes of recursively enumerable languages. Algorithmic identification in the limit of grammars from text presentation of recursively enumerable languages is a central paradigm of language learning. A mapping, F, from the set of all grammars into the set of all grammars is a standardizing operation on a class of recursively enumerable languages L just in case F maps any grammar for any language L ∈ L to a canonical grammar for L. Investigating connections between these two notions is the subject of this paper. 1 1
NotSoNearlyMinimalSize Program Inference ∗ (Preliminary Report)
"... Freivalds defined an acceptable programming system independent criterion for learning programs for functions in which the final programs were required to be both correct and “nearly ” minimal size, i.e, within a computable function of being purely minimal size. Kinber showed that this parsimony requ ..."
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Freivalds defined an acceptable programming system independent criterion for learning programs for functions in which the final programs were required to be both correct and “nearly ” minimal size, i.e, within a computable function of being purely minimal size. Kinber showed that this parsimony requirement on final programs severely limits learning power. Nonetheless, in, for example, scientific inference, parsimony is considered highly desirable. A limcomputable function is (by definition) one computable by a procedure allowed to change its mind finitely many times about its output. Investigated is the possibility of assuaging somewhat the limitation on learning power resulting from requiring parsimonious final programs by use of criteria which require the final, correct programs to be “notsonearly” minimal size, e.g., to be within a limcomputable function of actual minimal size. It is interestingly shown that some parsimony in the final program is thereby retained, yet learning power strictly increases. Also considered are limcomputable functions as above but for which notations for constructive ordinals are used to bound the number of mind changes allowed regarding the output. This is a variant of an idea introduced by Freivalds and Smith. For this ordinal complexity bounded version of limcomputability, the power of the resultant learning criteria form strict infinite hierarchies intermediate between the computable and the limcomputable cases. Many open questions are also presented. 1