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. " polynomial time reducibility, a.e. complexity, many-one reducibility, complexity core CR CATEGORmS: 5.25, 5.26 1.

Machine learning of limit programs (i.e., programs allowed finitely many mind changes about their legitimate outputs) for computable functions is stud-ied. 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 pro-grams for them which can not be proved from any n-iterated limit programs for them. It is shown that learning power is increased when (n + 1)-iterated limit programs rather than n-iterated limit programs are to be learned. Many trade-off 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.

Abstract. U-shaped 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 past-tenses of English verbs, has been widely discussed among psychologists and cognitive scientists as a fundamental example of the non-monotonicity of learning. Previous theory literature has studied whether or not U-shaped 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 U-shaped learning for some learning models featuring memory limitations. Our results show that the question of the necessity of Ushaped learning in this memory-limited 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

As an alternative to previously studied models for space-bounded 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.

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

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...

Various forms of polynomial time reducibility are compared. Among the forms examined are many-one, bounded truth table, truth table and Turing reducibility. The effect of introduclng nondeterminism into reduction procedures is also examined.

This document contains excerpts from [Sur01]