An attempt is made to apply information-theoretic computational complexity to metamathematics. The paper studies the number of bits of instructions that must be a given to a computer for it to perform finite and infinite tasks, and also the amount of time that it takes the computer to perform these tasks. This is applied to measuring the difficulty of proving a given set of theorems, in terms of the number of bits of axioms that are assumed, and the size of the proofs needed to deduce the theorems from the axioms. 2 G. J. Chaitin Key Words and Phrases: complexity of sets, computational complexity, difficulty of theoremproving, entropy of sets, formal systems, Godel's incompleteness theorem, halting problem, information content of sets, information content of axioms, information theory, information time trade-offs, metamathematics, random strings, recursive functions, recursively enumerable sets, size of proofs, universal computers CR Categories: 5.21, 5.25, 5.27, 5.6 1. Introduct...

The machine learning program Progol was applied to the problem of forming the structure-activity relationship (SAR) for a set of compounds tested for carcinogenicity in rodent bioassays by the U.S. National Toxicology Program (NTP). Progol is the first inductive logic programming (ILP) algorithm to use a fully relational method for describing chemical structure in SARs, based on using atoms and their bond connectivities. Progol is well suited to forming SARs for carcinogenicity as it is designed to produce easily understandable rules (structural alerts) for sets of noncongeneric compounds. The Progol SAR method was tested by prediction of a set of compounds that have been widely predicted by other SAR methods (the compounds used in the NTP's first round of carcinogenesis predictions). For these compounds no method (human or machine) was significantly more accurate than Progol. Progol was the most accurate method that did not use data from biological tests on rodents (however, the difference in accuracy is not significant). tests for Salmonella mutagenicity. Using the full NTP database, the prediction accuracy of Progol was estimated to be 63 % (±3%) using 5-fold cross validation. A set of structural alerts for carcinogenesis was automatically generated and the chemical rationale for them investigatedthese structural alerts are statistically independent of the Salmonella mutagenicity. Carcinogenicity is predicted for the compounds used in the NTP's second round of carcinogenesis predictions. The results for prediction of carcinogenesis, taken together with the previous successful applications of predicting mutagenicity in nitroaromatic compounds, and inhibition of angiogenesis by suramin analogues, show that Progol has a role to play in understanding the SARs of cancer-related compounds. Environ Health Perspect 104(Suppl 5):1031-1040 (1996) The Progol predictions were based solely on chemical structure and the results of

A version of the Church-Turing Thesis states that every e#ectively realizable physical system can be defined by Turing Machines (`Thesis P'); in this formulation the Thesis appears an empirical, more than a logico-mathematical, proposition. We review the main approaches to computation beyond Turing definability (`hypercomputation'): supertask, non-well-founded, analog, quantum, and retrocausal computation. These models depend on infinite computation, explicitly or implicitly, and appear physically implausible; moreover, even if infinite computation were realizable, the Halting Problem would not be a#ected. Therefore, Thesis P is not essentially di#erent from the standard Church-Turing Thesis.

this document is based was supported in part by grant EQ41840 from the program Fonds pour la Formation de Chercheurs et l'aide à la Recherche (F.C.A.R.) of the Québec Ministry of Education and in part by grant OGP0023391 from the Natural Sciences and Engineering Research Council of Canada. 152 PROBLEM. Five more than three times a number is twenty; find the number. Solution. Suppose that there is a number such that three times the number plus five is twenty. Then, subtracting five from both "sides", three times the number is fifteen. Hence, the number is five. This proves: if five more than three times a number is twenty, then the number is five.

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Copyright c○2000 Gödel’s results have had a great impact in diverse fields such as philosophy, computer sciences and fundamentals of mathematics. The fact that the rule of mathematical induction is contradictory with the rest of clauses used by Gödel to prove his undecidability and incompleteness theorems is proved in this paper. This means that those theorems are invalid. In section 1, a study is carried out on the mathematical induction principle, even though it is not directly relevant to the problem, just to familiarize the reader with the operations that are used later; in section 2 the rule of mathematical induction is introduced, this rule has a metamathematical character; in section 3 the original proof of Gödel’s undecidability theorem is reproduced, and finally in section 4 the same proof is given, but now with the explicit and formal use of all the axioms; this is needed to be able to use logical resolution. It is shown that the inclusion of the mathematical induction rule causes a contradiction.