This paper surveys investigations into how strong these commonalities are. More concretely, we are concerned with: What do NP-complete sets look like? To what extent are the properties of particular NP-complete sets, e.g., SAT, shared by all NP-complete sets? If there are are structural differences between NP-complete sets, what are they and what explains the differences? We make these questions, and the analogous questions for other complexity classes, more precise below. We need first to formalize NP-completeness. There are a number of competing definitions of NP-completeness. (See [Har78a, p. 7] for a discussion.) The most common, and the one we use, is based on the notion of m-reduction, also known as polynomial-time manyone reduction and Karp reduction. A set A is m-reducible to B if and only if there is a (total) polynomial-time computable function f such that for all x, x 2 A () f(x) 2 B: (1) 1

This paper investigates some delicate tradeoffs between the generality of an algorithmic learning device and the quality of the programs it learns successfully. There are results to the effect that, thanks to small increases in generality of a learning device, the computational complexity of some successfully learned programs is provably unalterably suboptimal. There are also results in which the complexity of successfully learned programs is asymptotically optimal and the learning device is general, but, still thanks to the generality, some of those optimal, learned programs are provably unalterably information deficient — in some cases, deficient as to safe, Preprint submitted to Elsevier Science 11 March 2007 algorithmic extractability/provability of the fact that they are even approximately optimal. For these results, the safe, algorithmic methods of information extraction will be by proofs in arbitrary, true, computably axiomatizable extensions of Peano Arithmetic. Key words: Computational learning theory; Applications of computability theory

This paper investigates some delicate tradeoffs between the generality of an algorithmic learning device and the quality of the programs it learns successfully. There are results to the effect that, thanks to small increases in generality of a learning device, the computational complexity of some successfully learned programs is provably unalterably suboptimal. There are also results in which the complexity of successfully learned programs is asymptotically optimal and the learning device is general, but, still thanks to the generality, some of those optimal, learned programs are provably unalterably information deficient — in some cases, deficient as to safe, algorithmic extractability/provability of the fact that they are even approximately optimal. For these results, the safe, algorithmic methods of information extraction will be by proofs in arbitrary, true, computably axiomatizable extensions of Peano Arithmetic.