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Informationtheoretic Limitations of Formal Systems
 JOURNAL OF THE ACM
, 1974
"... An attempt is made to apply informationtheoretic 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 ..."
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An attempt is made to apply informationtheoretic 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.
Hypercomputation and the Physical ChurchTuring Thesis
, 2003
"... A version of the ChurchTuring 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 logicomathematical, proposition. We review the main approaches to computation beyond Turing ..."
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Cited by 21 (0 self)
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A version of the ChurchTuring 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 logicomathematical, proposition. We review the main approaches to computation beyond Turing definability (`hypercomputation'): supertask, nonwellfounded, 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 ChurchTuring Thesis.
Prediction of rodent carcinogenicity bioassays from molecular structure using inductive logic programming. Environmental Health Perspectives
, 1996
"... The machine learning program Progol was applied to the problem of forming the structureactivity 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 ..."
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Cited by 18 (7 self)
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The machine learning program Progol was applied to the problem of forming the structureactivity 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 5fold 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 cancerrelated compounds. Environ Health Perspect 104(Suppl 5):10311040 (1996) The Progol predictions were based solely on chemical structure and the results of
A formalisation of nonfinite computation
, 1997
"... Abstract. Abstract Recent work in the field of relativitistic spacetimes suggests that it may be possible for a machine to perform an infinite number of operations in a finite time. Investigation of these machines has three motivations. First, because the machines may be physically possible, they h ..."
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Abstract. Abstract Recent work in the field of relativitistic spacetimes suggests that it may be possible for a machine to perform an infinite number of operations in a finite time. Investigation of these machines has three motivations. First, because the machines may be physically possible, they have implications for the general question of what problems are theoretically solvable. Second, the physical laws that govern their operation give rise to a rich mathematical structure. Third, by investigating general relativistic computation, we get a clearer picture of properties peculiar to the special case of Turing computation. A mathematical formalisation of the operation of the machines is presented, and shown to correspond to their physical operation. It is proved that there is no satisfactory way to give a finite description for the machines. Although Gödel sentences exist if attention is restricted to a finite set of machines, further results [Ear94,Hog96] about the computational power of the machines and their equivalence to the Kleene arithmetical hierarchy are shown to depend upon arbitrary assumptions. This gives rise to a nonfinite version of the ChurchTuring thesis. The case of nonfinite computation is used to arrive at an abstract principle of computation that is independent of physics. A dissertation submitted to the University of Cambridge
Seldin, Jonathan P., "Reasoning in Elementary Mathematics".
, 151
"... 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 PROB ..."
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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.
GÖDEL’S THEOREM IS INVALID
, 2005
"... 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 th ..."
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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.