A correcting algorithm is one that receives an endless stream of corrections to its initial input data and terminates when all the corrections received have been taken into account. We give a characterization of correcting algorithms based on the theory of data-accumulating algorithms. In particular, it is shown that any correcting algorithm exhibits superunitary behavior in a parallel computation setting if and only if the static counterpart of that correcting algorithm manifests a strictly superunitary speedup. Since both classes of correcting and data-accumulating algorithms are included in the more general class of real-time algorithms, we show in fact that many problems from this class manifest superunitary behavior. Moreover, we give an example of a real-time parallel computation that pertains to neither of the two classes studied (namely, correcting and data-accumulating algorithms), but still manifests superunitary behavior. Because of the aforementioned results, the usual measures of performance for parallel algorithms (that is, speedup and efficiency) lose much of their ability to convey effectively the nature of the phenomenon taking place in the real-time case. We propose therefore a more expressive measure that captures all the relevant parameters of the computation. Our proposal is made in terms of a graphical representation. We state as an open problem the investigation of such a measure, including nding an analytical form for it.

In 1927 Heisenberg discovered that the "more precisely the position is determined, the less precisely the momentum is known in this instant, and vice versa". Four years later

Abstract. For more than 2000 years, from Pythagoras and Euclid to Hilbert and Bourbaki, mathematical proofs were essentially based on axiomatic-deductive reasoning. In the last decades, the increasing length and complexity of many mathematical proofs led to the expansion of some empirical, experimental, psychological and social aspects, yesterday only marginal, but now changing radically the very essence of proof. In this paper, we try to organize this evolution, to distinguish its different steps and aspects, and to evaluate its advantages and shortcomings. Axiomatic-deductive proofs are not a posteriori work, a luxury we can marginalize nor are computer-assisted proofs bad mathematics. There is hope for integration! 1

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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

In 1927, Heisenberg discovered that the “more precisely the position is determined, the less precisely the momentum is known in this instant, and vice versa. ” Four years later Gödel showed that a finitely specified, consistent formal system which is large enough

Atrocity presents a unique challenge both to media and to philosophy. Atrocity, not suicide, it is suggested, is the key issue for modern philosophy. Atrocity annihilates all attempts to give life meaning and destroys forever a subject’s possibility of seeking justice as well as the individual’s relation with the world. For media, concerned with assembling the contingent into socially approved meanings, atrocity remains an irreducible anomaly. The suffering of victims never really becomes news more than momentarily, while the issue never goes away from the public agenda, but remains there unresolved. The difficulties, it is suggested, come from the philosophical challenges posed. But, along with celebrity, atrocity is a major tool of modern politics, employing the same

The object of mathematical rigour is to sanction and

In 1927 Heisenberg discovered that the “more precisely the position is determined, the less precisely the momentum is known in this instant, and vice versa”. Four years later Gödel showed that a finitely specified, consistent formal system which is large enough to include arithmetic is incomplete. As both results express some kind of impossibility it is natural to ask whether there is any relation between them, and, indeed, this question has been repeatedly asked for a long time. The main interest seems to have been in possible implications of incompleteness to physics. In this note we will take interest in the converse implication and will offer a positive answer to the question: Does uncertainty imply incompleteness? We will show that algorithmic randomness is equivalent to a “formal uncertainty principle ” which implies Chaitin’s information-theoretic incompleteness. We also show that the derived uncertainty relation, for many computers, is physical. This fact supports the conjecture that uncertainty implies randomness not only in mathematics, but also in physics. 1

We discuss the evolution of radiation and Bekenstein-Hawking entropies in expanding isotropic universes. We establish a general relation which shows why it is inevitable that there is currently a huge difference in the numerical values of these two entropies. Some anthropic constraints on their values are given and other aspects of the cosmological ’entropy gap ’ problem are discussed. The coincidence of the classical and quantum entropies for black holes with Hawking lifetime equal to the age of the universe, and hence of radius equal to the proton size, is shown to be identical to the condition that we obseve the universe at the main sequence lifetime. 1.