Abstract not found

Abstract not found

We survey the background and challenges of a number of open problems in the theory of relativization. Among the topics covered are pseudorandom generators, time hierarchies, the potential collapse of the polynomial hierarchy, and the existence of complete sets. Relativization (i.e., oracle) theory has seen its share of ups and downs. Extensive surveys of current knowledge [Ver94] and debates as to relativization theory's merits [Har85,All90, HCC + 92,For94] can be found in the literature. However, in a nutshell, one could rather fairly say that as ups and downs go, relativization theory is on the mat. Still, that is not to say that relativization theory has no interesting open issues left with which to challenge theoretical computer scientists. It does, and here are a few such issues. Problem 1: Show that with probability one, the polynomial hierarchy is proper. The above statement is, to say the least, elliptic. However, the problem is well-known in this formulation. The underlying...

Are minds subject to laws of physics? Are the laws of physics computable? Are conscious thought processes computable? Currently there is little agreement as to what are the right answers to these questions. Penrose ([41], p. 644) goes one step further and asserts that: a radical new theory is indeed needed, and I am suggesting, moreover, that this theory, when it is found, will be of an essentially non-computational character. The aim of this paper is three fold: 1) to examine the incompatibility between the hypothesis of strong determinism and computability, 2) to give new examples of uncomputable physical laws, and 3) to discuss the relevance of Gödel’s Incompleteness Theorem in refuting the claim that an algorithmic theory—like strong AI—can provide an adequate theory of mind. Finally, we question the adequacy of the theory of computation to discuss physical laws and thought processes. 1

Whether ’tis nobler in the mind to suffer The slings and arrows of outrageous fortune, Or to take arms against a sea of troubles And by opposing end them? Hamlet 3/1, by W. Shakespeare In this paper we propose a new perspective on the evolution and history of the idea of mathematical proof. Proofs will be studied at three levels: syntactical, semantical and pragmatical. Computer-assisted proofs will be give a special attention. Finally, in a highly speculative part, we will anticipate the evolution of proofs under the assumption that the quantum computer will materialize. We will argue that there is little ‘intrinsic ’ difference between traditional and ‘unconventional ’ types of proofs. 2 Mathematical Proofs: An Evolution in Eight Stages Theory is to practice as rigour is to vigour. D. E. Knuth Reason and experiment are two ways to acquire knowledge. For a long time mathematical

We discuss the question of if and how undecidability might be translatable into physics, in particular with respect to prediction and description, as well as to complementarity games. 1 1 Physics after the incompleteness theorems There is incompleteness in mathematics [22, 63, 65, 13, 9, 12, 51]. That means that there does not exist any reasonable (consistent) finite formal system from which all mathematical truth is derivable. And there exists a "huge" number [11] of mathematical assertions (e.g., the continuum hypothesis, the axiom of choice) which are independent of any particular formal system. That is, they as well as their negations are compatible with the formal system. Can such formal incompleteness be translated into physics or the natural sciences in general? Is there some question about the nature of things which is provable unknowable for rational thought? Is it conceivable that the natural phenomena, even if they occur deterministically, do not allow their complete d...

Complexity Theory---Extended Abstract #+ Cristian Calude # and Marius Zimand Abstract Strong variants of the Operator Speed-up Theorem, Operator Gap Theorem and Compression Theorem are obtained using an e#ective version of Baire Category Theorem. It is also shown that all complexity classes of recursive predicates have e#ective measure zero in the space of recursive predicates and, on the other hand, the class of predicates with almost everywhere complexity above an arbitrary recursive threshold has recursive measure one in the class of recursive predicates. Keywords: Complexity measure, Operator Speed-up Theorem, Operator Gap Theorem, Compression Theorem, e#ective Baire classification, e#ective measure. 1 Introduction The abstract complexity theory initiated by Blum [2] (see also Bridges [5], Calude [8], Hartmanis and Hopcroft [17], Machtey and Young [23], Seiferas [34]) has revealed fundamental properties of complexity measures. The striking importance of this theory relies in i...

Chaitin’s “heuristic principle”, the theorems of a finitely-specified theory cannot be significantly more complex than the theory itself was proved for an appropriate measure of complexity in [1]. The measure δ is a computable variation of the program-size complexity H: δ(x) = H(x) − |x|. The theorems of a finitely-specified, sound, consistent theory which is strong enough to include arithmetic have bounded δ-complexity, hence every sentence of the theory which is significantly more complex than the theory is unprovable. More precisely, according to Theorem 4.6 in [1], for any finitely-specified, sound, consistent theory strong enough to formalize arithmetic (like Zermelo-Fraenkel set theory with choice or Peano Arithmetic) and for any Gödel numbering g of its well-formed formulae, we can compute a bound N such that no sentence x with complexity δg(x)> N can be proved in the theory; this phenomenon is independent on the choice of the Gödel numbering. Question 1. Find other natural measures of complexity for which Chaitin’s “heuristic principle ” holds true.

Mathematics is in a dramatic and massive process of changing, mainly due to the advent of computers and computer science. Our aim is to present a pocket image of this phenomenon; a "case study" will give us the opportunity to describe some of these new ideas, problems, and techniques. Particularly, we will be concerned with foreseeable mutations in the interaction between deductive and experimental trends.

In 1930, Gödel [7] presented in Königsberg his famous Incompleteness Theorem, stating that some true mathematical statements are unprovable. Yet, this result gives us no idea about those independent (that is, true and unprovable) statements, about their frequency, the reason they are unprovable, and so on. Calude and Jürgensen [4] proved in 2005 Chaitin's heuristic principle for an appropriate measure: the theorems of a nitely-speci ed theory cannot be signi cantly more complex than the theory itself (see [5]). In this work, we investigate the existence of other measures, di erent from the original one, which satisfy this heuristic principle. At this end, we introduce the de nition of acceptable complexity measure of theorems. Résumé En 1930, Gödel [7] présente à Königsberg son célèbre Théorème d'Incomplétude, spéci ant que certaines a rmations mathématiques sont indémontrables. Cependant, ce résultat ne nous donne aucune indication à propos de ces a rmations indépendantes (c'està-dire vraies mais indémontrables), sur leur fréquence, les raisons de leur indémontrabilité, etc. Calude and Jürgensen [4] ont prouvé en 2005 le principe heuristique de Chaitin pour une mesure de complexité appropriée: les théorèmes d'une théorie niment axiomatisable ne peuvent être signi cativement plus complexes que la théorie elle-même (cf [5]). Dans ce rapport, nous étudions l'existence d'autres mesures, di érentes de la mesure originale utilisée dans [4], qui satisfassent ce principe heuristique. A cette n, nous introduisons la dé nition de mesure acceptable de complexité des théorèmes.