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On reducibility to complex or sparse sets
 Journal of the ACM
, 1975
"... ABSTRACT. Sets which are efficiently reducible (in Karp's sense) to arbitrarily complex sets are shown to be polynomial computable. Analogously, sets efficiently reducible to arbitrarily sparse sets are polynomial computable. A key lemma for both proofs shows that any set which is not polynomia ..."
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ABSTRACT. Sets which are efficiently reducible (in Karp's sense) to arbitrarily complex sets are shown to be polynomial computable. Analogously, sets efficiently reducible to arbitrarily sparse sets are polynomial computable. A key lemma for both proofs shows that any set which is not polynomial computable has an infinite recursive subset of its domain, on which every algorithm runs slowly on almost all arguments. KEY WORDS AND PHRASES. &quot; polynomial time reducibility, a.e. complexity, manyone reducibility, complexity core CR CATEGORmS: 5.25, 5.26 1.
On learning limiting programs
 International Journal of Foundations of Computer Science
, 1992
"... Machine learning of limit programs (i.e., programs allowed finitely many mind changes about their legitimate outputs) for computable functions is studied. Learning of iterated limit programs is also studied. To partially motivate these studies, it is shown that, in some cases, interesting global pr ..."
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Cited by 11 (5 self)
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Machine learning of limit programs (i.e., programs allowed finitely many mind changes about their legitimate outputs) for computable functions is studied. Learning of iterated limit programs is also studied. To partially motivate these studies, it is shown that, in some cases, interesting global properties of computable functions can be proved from suitable (n + 1)iterated limit programs for them which can not be proved from any niterated limit programs for them. It is shown that learning power is increased when (n + 1)iterated limit programs rather than niterated limit programs are to be learned. Many tradeoff results are obtained regarding learning power, number (possibly zero) of limits taken, program size constraints and information, and number of errors tolerated in final programs learned.
Results on MemoryLimited UShaped Learning
"... Abstract. Ushaped learning is a learning behaviour in which the learner first learns a given target behaviour, then unlearns it and finally relearns it. Such a behaviour, observed by psychologists, for example, in the learning of pasttenses of English verbs, has been widely discussed among psychol ..."
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Cited by 6 (1 self)
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Abstract. Ushaped learning is a learning behaviour in which the learner first learns a given target behaviour, then unlearns it and finally relearns it. Such a behaviour, observed by psychologists, for example, in the learning of pasttenses of English verbs, has been widely discussed among psychologists and cognitive scientists as a fundamental example of the nonmonotonicity of learning. Previous theory literature has studied whether or not Ushaped learning, in the context of Gold’s formal model of learning languages from positive data, is necessary for learning some tasks. It is clear that human learning involves memory limitations. In the present paper we consider, then, the question of the necessity of Ushaped learning for some learning models featuring memory limitations. Our results show that the question of the necessity of Ushaped learning in this memorylimited setting depends on delicate tradeoffs between the learner’s ability to remember its own previous conjecture, to store some values in its longterm memory, to make queries about whether or not items occur in previously seen data and on the learner’s choice of hypotheses space. 1
Log Space Machines with Multiple Oracle Tapes
, 1978
"... As an alternative to previously studied models for spacebounded relative compu tation, an oracle Turing machine with a space bound on its worktape and an arbitrary number of oracle tapes is considered. Basic properties of the resulting reducibilities are examined. ..."
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Cited by 4 (2 self)
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As an alternative to previously studied models for spacebounded relative compu tation, an oracle Turing machine with a space bound on its worktape and an arbitrary number of oracle tapes is considered. Basic properties of the resulting reducibilities are examined.
"Helping": Several Formalizations
 J. Symbolic Logic
, 1975
"... this paper, we first define "complexity sequences" and present two results which motivate their use in one possible definition of "helping". We then present several other definitions which naturally suggest themselves. Finally, we show that these definitions are all essentially e ..."
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Cited by 2 (2 self)
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this paper, we first define "complexity sequences" and present two results which motivate their use in one possible definition of "helping". We then present several other definitions which naturally suggest themselves. Finally, we show that these definitions are all essentially equivalent
Effectivizing Inseparability
, 1991
"... Smullyan's notion of effectively inseparable pairs of sets is not the best effective /constructive analog of Kleene's notion of pairs of sets inseparable by a recursive set. We present a corrected notion of effectively inseparable pairs of sets, prove a characterization of our notion, and ..."
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Cited by 2 (0 self)
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Smullyan's notion of effectively inseparable pairs of sets is not the best effective /constructive analog of Kleene's notion of pairs of sets inseparable by a recursive set. We present a corrected notion of effectively inseparable pairs of sets, prove a characterization of our notion, and show that the pairs of index sets effectively inseparable in Smullyan's sense are the same as those effectively inseparable in ours. In fact we characterize the pairs of index sets effectively inseparable in either sense thereby generalizing Rice's Theorem. For subrecursive index sets we have sufficient conditions for various inseparabilities to hold. For inseparability by sets in the same subrecursive class we have a characterization. The latter essentially generalizes Kozen's (and Royer's later) Subrecursive Rice Theorem, and the proof of each result about subrecursive index sets is presented "Rogers style" with care to observe subrecursive restrictions. There are pairs of sets effectively inseparab...
POLYNOMIAL AND ABSTRACT SUBRECURSIVE CLASSES
, 1974
"... We define polynomial time computable operator. Our definition generalizes Cook's definition to arbitrary function inputs. Polynomial classes are defined in terms of these operators; the properties of these classes are investigated. Honest polynomial classes are generated by runnina time. They p ..."
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We define polynomial time computable operator. Our definition generalizes Cook's definition to arbitrary function inputs. Polynomial classes are defined in terms of these operators; the properties of these classes are investigated. Honest polynomial classes are generated by runnina time. They posses a modified RitchieCobham property. A polynomial class is a complexity class iff it is honest. Starting from the observation that many results about subrecursive classes hold for all reducibility relations (e.g. primitive recursive in, elementary recursive in), which were studied so far, we define abstract subrecursive reducibility relation. Many results hold for all abstract subrecursive reducibilities.
Machine Learning of Higher Order Programs ∗
, 2007
"... A generator program for a computable function (by definition) generates an infinite sequence of programs all but finitely many of which compute that function. Machine learning of generator programs for computable functions is studied. To partially motivate these studies, it is shown that, in some ca ..."
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A generator program for a computable function (by definition) generates an infinite sequence of programs all but finitely many of which compute that function. Machine learning of generator programs for computable functions is studied. To partially motivate these studies, it is shown that, in some cases, interesting global properties for computable functions can be proved from suitable generator programs which can not be proved from any ordinary programs for them. The power (for variants of various learning criteria from the literature) of learning generator programs is compared with the power of learning ordinary programs. The learning power in these cases is also compared to that of learning limiting programs, i.e., programs allowed finitely many mind changes about their correct outputs.
Inductive Inference of ... vs. ...Definitions for Computable Functions
, 2002
"... This document contains excerpts from [Sur01] ..."
A Comparison of Polynomial Time Redicibilities
 Theoretical Computer Science
, 1975
"... Various forms of polynomial time reducibility are compared. Among the forms examined are manyone, bounded truth table, truth table and Turing reducibility. The effect of introduclng nondeterminism into reduction procedures is also examined. ..."
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Various forms of polynomial time reducibility are compared. Among the forms examined are manyone, bounded truth table, truth table and Turing reducibility. The effect of introduclng nondeterminism into reduction procedures is also examined.