this article, I have attempted to organize and describe this literature, including an occasional opinion about the most fruitful directions, but no technical details. In the first half of this century, work on the power of formal systems led to the formalization of the notion of algorithm and the realization that certain problems are algorithmically unsolvable. At around this time, forerunners of the programmable computing machine were beginning to appear. As mathematicians contemplated the practical capabilities and limitations of such devices, computational complexity theory emerged from the theory of algorithmic unsolvability. Early on, a particular type of computational task became evident, where one is seeking an object which lies

Consider the following type of problem: There is an unknown function, f : R n ! R m , there is also a black-box that on query x (2 R n ) returns f(x). Is there an algorithm that, using probes to the black-box, can figure out analytic information about f? (For an example: "Is f a polynomial? ", "Is f a second order differentiable at x = (0; 0; : : : ; 0)?" etc.). Clearly, for examples as these, if we bound the number of probes an algorithm has to settle for, no algorithm can carry the task. On the other hand, if one allows an infinite iteration of a `probe compute and guess' process, then, (quite surprisingly) for many such questions, there are algorithms that are guaranteed to be correct in all but finitely many of their guesses. We call such questions Decidable In the Limit, (DIL). We analyze the class of DIL problems and provide a necessary and sufficient condition for the membership of a decision problem in this class. We offer an algorithm for any DIL problem, and apply it to several types of learning tasks. We introduce a an extension of the usual Inductive Inference learning model - Inductive Inference with a Cheating Teacher. In this model the teacher may choose to present to the learner, not only a language belonging to the agreed - upon family of languages, but also an arbitrary language outside this family. In such a case we require that the learner will be able to eventually detect the faulty choice made by the teacher. We show that such strong type of learning is possible, and there exist learning algorithms that will fail only on arbitrarily small sets of faulty languages. Furthermore, if an a-priori probability distribution P , according to which f is being chosen, is available to the algorithm, then it can be strengthened into a finite A prelimi...

I have moved back to the University of Chicago and so has the web page for this column. See above for new URL and contact informaion. This issue Scott Aaronson writes quite an interesting (and opinionated) column on whether the P = NP question is independent of the usual axiom systems. Enjoy!

Abstract. This paper introduces new notions of asymptotic proofs, PT(polynomial-time)extensions, PTM(polynomial-time Turing machine)-!-consistency, etc. on formal theories of arithmetic including PA (Peano Arithmetic). An asymptotic proof is a set of in nitely many formal proofs, which is introduced to de ne and characterize a property, PTM-!-consistency, of a formal theory. Informally speaking, PTM-!-consistency is a polynomial-time bounded version (in asymptotic proofs) of!-consistency, and characterized in two manners: (1) (in the light of the extension of PTM to TM) the resource unbounded version of PTM-!-consistency is equivalent to!-consistency, and (2) (in the light of asymptotic proofs by PTM) aPTM-!-inconsistent theory includes an axiom that only a super-polynomial-time Turing machine can prove asymptotically over PA, under some assumptions. This paper shows that P6=NP (more generally, any super-polynomial-time lower bound in PSPACE) is unprovable in a PTM-!-consistent theory T, where T is a consistent PT-extension of PA (although this paper does not show that P6=NP is unprovable in PA, since PA has not been proven to be PTM-!-consistent). This result implies that to prove P6=NP by any technique requires a PTM-!-inconsistent theory, which should include an axiom that only a super-polynomialtime machine can prove asymptotically over PA (or implies a super-polynomial-time computational