43577142
Undecidable Sentences Generated by Semantic Paradoxes
In applying the method of arithmetization to a proof of the completeness of the predicate calculus, Bernays has obtained a result which, when applied to set theories formulated in the predicate calculus, may be stated thus. 1.1. By adding an arithmetic sentence Con( S ) (expressing the consistency of a set theory S ) as a new axiom to the elementary number theory Z μ (HB II, p. 293), we can prove in the resulting system arithmetic translations of all theorems of S . It then follows that things definable or expressible in S have images in a simple extension of Z μ , if S is consistent. Since S can be a “strong” system, this fact has interesting consequences. Some of these are discussed by me and some are discussed by Kreisel. Kreisel finds an undecidable sentence of set theory by combining 1.1 and the Cantor diagonal argument. I shall prove below, using similar methods, a few further results, concerned with the notions of truth and designation. The method of numbering sets which I use (see 3.1 below) is different from Kreisel's. While the method used here is formally more elegant, Kreisel's method is much more efficient if we wish actually to calculate the numerical values.
29040872
A Variant to Turing's Theory of Computing Machines
The principal purpose of this paper is to offer a theory which is closely related to Turing's [1] but is more economical in the basic operations. I t will be proved that a theoretically simple basic machine can be imagined and specified such tha t all partial recursive functions (and hence all solvable computation problems) can be computed by it and tha t only four basic types of instruction are employed for the programs: shift left one space, shift right one space, mark a blank space, conditional transfer. In particular, erasing is dispensable, one symbol for marking is sufficient, and one kind of transfer is enough. The reduction is somewhat similar to the realization of, for instance, the definability of conjunction and implication in terms of negation and disjunction, or of the definability of all these in terms of Sheffer's stroke function. As a result, i t becomes less direct to prove that certain things can be done by the machines, but a little easier to prove that certain things cannot be done. This serf-contained theory will b e presented, as far as possible, in a language which is familiar to those who are engaged in the use and construction of largescale computers. Turing's theory of computable functions antedated but has not much influenced the extensive actual construction of digital computers. These two aspects of theory and practice have been developed almost entirely independently of each other. The main reason is undoubtedly that logicians are interested in questions radically different from those with which the applied mathematicians and electrical engineers are primarily concerned. I t cannot, however, fail to strike one as rather strange that often the same concepts are expressed by very different terms in the two developments. One is even inclined to ask whether a rapprochement might not produce some good effect. This paper wiU, it is believed, be of use to those who wish to compare and connect the two approaches. Extensive use of Turing [1] and Kleene [2] (Chapter XI I I ) will be made, although the exposition is sufficient in itself.
29402189
Non-Standard Models for Formal Logics
showed that there must be a model in which that portion of the model which is supposed to represent the positive integers of the formal logic is not in fact isomorphic to the positive integers; indeed it is not even well ordered by what is supposed to be the relation of <. For the purposes of the present paper, we do not need a precise definition of what is meant by a standard model of a formal logic. The non-standard models which we shall discuss will be flagrantly non-standard, as for instance a model of the sort whose existence is proved by Henkin. It will suffice if we and our readers are in agreement that a model of a formal logic is not a standard model if either: (a) The relation in the model which represents the equality relation in the formal logic is not the equality relation for objects of the model. (b) That portion of the model which is supposed to represent the positive integers of the formal logic is not well ordered by the relation <. (c) That portion of the model which is supposed to represent the ordinal numbers of the formal logic is not well ordered by the relation <. Our first result is that there is no standard model for the system known as "Quine's New Foundations" (see [2] and [3]). This is proved for the logic originally proposed by Quine (see [2]), which is a system of such apparent weakness that so far no one has succeeded in proving the axiom of infinity in the system. Clearly, if there is no standard model for this apparently weak system, there cannot be a standard model for any apparently stronger system which one might form by adding additional axioms, such as the axiom of infinity. A natural reaction to this result would be to suspect that there is no model of any sort for Quine's New Foundations; that is to suspect that it is inconsistent. An alternative (and much less likely) reaction would be to suspect that the property of not having a standard model is perhaps a not uncommon property of strong formal logics, and to suspect that perhaps several of the familiar strong formal logics have no standard model. Unlikely as this might appear at first sight, we will present considerable evidence in favor of the idea. In particular we shall show that it is a property of each of the familiar formal logics that if it is w-consistent then one cannot prove in the logic itself that if the logic is consistent it must have a standard model; in fact, one cannot prove this statement about the logic even in certain stronger logics if these stronger logics are simply consistent.
42852449
A Formal System of Logic
The main purpose of this paper is to present a formal system P in which we enjoy a smooth-running technique and which countenances a universe of classes which is symmetrical as between large and small. More exactly, P is a system which differs from the inconsistent system of [ 1 ] only in the introduction of a rather natural new restrictive condition on the defining formulas of the elements (sets, membership-eligible classes). It will be proved that if the weaker system of [ 2 ] is consistent, then P is also consistent. After the discovery of paradoxes, it may be recalled, Russell and Zermelo in the same year proposed two different ways of safeguarding logic against contradictions (see [ 3 ], [ 4 ]). Since then various simplifications and refinements of these systems have been made. However, in the resulting systems of Zermelo set theory, generation of classes still tends to be laborious and uncertain; and in the systems of Russell's theory of types, complications in the matter of reduplication of classes and meaningfulness of formulas remain. In [ 2 ], Quine introduced a system which seems to be free from all these complications. But later it was found out that in it there appears to be an unavoidable difficulty connected with mathematical induction. Indeed, we encounter the curious situation that although we can prove in it the existence of a class V of all classes, and we can also prove particular existence theorems for each of infinitely many classes, nobody has so far contrived to prove in it that V is an infinite class or that there exists an infinite class at all.
11313688
Existence of Classes and Value Specification of Variables
In mathematics, when we want to introduce classes which fulfill certain conditions, we usually prove beforehand that classes fulfilling such conditions do exist, and that such classes are uniquely determined by the conditions. The statements which state such unicity and existence of classes are in mathematical logic consequences of the principles of extensionality and class existence. In order to illustrate how these principles enable us to introduce classes into systems of mathematical logic, let us consider the manner in which Godel introduces classes in his book on set theory. For instance, before introducing the definition of the non-ordered pair of two classes Godel puts down as its justification the following two axioms: By A4, for every two classes y and z there exists at least one non-ordered pair w of them; and by A3, w is uniquely determined in A4.
1953614
Alternative Proof of a Theorem of Kleene
5494803
Wireless Network Dynamic Topology Routing Protocol Based on Aggregation Tree Model
As the scale of the network increases, especially wireless and mobile networks comes into fashion, the problem of network dynamic topology management turns into a focus topic in the network world. When network information changes, frequent route computing and network state updates can cause computing and traffic overhead, respectively. Therefore, minimum of influence to network structure has been identified as one of the key issues in designing dynamic network routing protocols. A novel protocol, which is based on Aggregation Tree Routing Model (ATRM) as well as an effective ball-and-string algorithm, is introduced in this paper which aims to minimize the alteration of network topology and to speed up the minimum cost route computation. This routing protocol is applicable to all kinds of networks, including Internet, ad hoc and wireless sensor networks. The simulations show that our protocol not only results in significantly lower overhead, but also gives high levels of dynamic network topology management performance while keeping practical.
15500084
Understanding the Prediction of Transmembrane Proteins by Support Vector Machine using Association Rule Mining
With the efforts to understand protein structure, many computational approaches have been made recently. Among them, the support vector machine (SVM) methods have been recently applied and showed successful performance compared with other machine learning schemes. However, despite the high performance, the SVM approaches suffer from the problem of understandability since it is a black-box model. To overcome this limitation, this study attempted to combine the SVM with the association rule based classifier which can present the meaningful explanation about the prediction. To perform this task, a new association rule based classifier (PCPAR) was devised based on the existing classifier, CPAR, to handle the sequential data. PCPAR creates the patterns by merging the generated rules and then classifies the sequential data based on the pattern match. The experimental result presents the following: with sequential data, the PCPAR scheme shows better performance with respect to the accuracy and the number of generated patterns than CPAR method whether applied alone or combined with SVM. The combined scheme of SVMPCPAR generates more compact patterns than the combined scheme of SVM with decision tree, SVM DT, with similar performance. These patterns are easily understandable and biologically meaningful
10926021
Agora: Grid Community in Vega Grid
In grid environment, how to organize, discover and use grid resources is one of the basic research problems. In this paper, we introduce a new concept called agora, which refers grid community in our Vega grid project. We propose a principle on agora as one of the basic design principles of Vega grid, which is the community principle. We give a detailed definition of the agora as a 4-tuple(S, O, C, P), representing the set of Subject, Object, Context, and Policy respectively. Our research methodology is comparative studies, drawing an analog between the grid and other platform systems such as OS, CPU, and MPI. We also study some properties about the agora, for example, grid law of inertia. This agora approach could be used to unify the grid invariability and dynamic variability effectively, develop the appropriate technology to support the dynamic and open properties of the grid, and deal with the complexity in the grid more efficiently.
15067408
Adapting LDA Model to Discover Author-Topic Relations for Email Analysis
Analyzing the author and topic relations in email corpus is an important issue in both social network analysis and text mining. The Author-Topic model is a statistical model that identifies the author-topic relations. However, in its inference process, it ignores the information at the document level, i.e., the co-occurrence of words within documents are not taken into account in deriving topics. This may not be suitable for email analysis. We propose to adapt the Latent Dirichlet Allocation model for analyzing email corpus. This method takes into account both the author-document relations and the document-topic relations. We use the Author-Topic model as the baseline method and propose measures to compare our method against the Author-Topic model. We did empirical analysis based on experimental results on both simulated data sets and the real Enron email data set to show that our method obtains better performance than the Author-Topic model.
30738286
The algorithms for dynamic scheduling