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.
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.
21405683
Set-Theoretical Basis for Real Numbers
twenty axioms used by Tarski which are sufficient for the arithmetic of real numbers and are to the effect that real numbers form a complete ordered field.2 Indeed, we cannot prove in L all Tarski's twenty axioms in their full generality. One of them, stating in effect that every bounded class of real numbers possesses a least upper bound, can only be proved as a metatheorem which states that every bounded nameable class of real numbers possesses a least upper bound. However, all the other nineteen axioms can be proved in L without any modification. This result may be of some interest because the axioms of L are considerably weaker than those commonly employed for the same purpose. In L variables need to take as values only classes each of whose members has no more than two members. In other words, only classes each with no more than two members are to be elements. On the other hand, it is usual to assume for the purpose of natural arithmetic that all finite classes are elements, and, for the purpose of real arithmetic, that all enumerable classes are elements. 2. The system L. In developing L, elementary logic (viz., the theory of truth functions and quantifiers) is assumed from the beginning. The single additional primitive is the two place predicate e with which we can generate formulas of L from variables and logical operators in the usual manner. The usual definitions (e.g., D9-D35 of [21) of the theory of classes and relations are assumed. In particular, let tx denote the unit class of x, x u y denote the sum class of x and y, and V denote the class of all elements. The axioms of L are given by the following three statements: Mi. tx U by f V. M2. x = y D . x e z D y e z. M3. If q is any formula of L in which y is not free, then . (3y)(x)(x e yW
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.
1953614
Alternative Proof of a Theorem of Kleene
7625765
Solving QoS-driven Web service dynamic composition as fuzzy constraint satisfaction
The creation of value-added composite web service raises an opportunity to the formation of online B2B collaborations. However, web services are usually overlapping in functionality. How to make a choice based on non-functional factors becomes a problem that needs to be solved. In this paper, it argues that the selection of component services should be considered in a global manner and the user's QoS preferences is more appropriate to be expressed in a fuzzy way. It advocates using fuzzy set to express the user's QoS preference on a specific QoS criteria and using fuzzy expression to express the user's trade-off among QoS criteria's. The service selection problem is formalized as a fuzzy constraint satisfaction problem and deep-first branch-and-bound method is chosen to search for a solution with some adjustments to web service composition. It also develops an uncritical fuzzy consistency checking algorithm as an aid to faster search. Experiments are conducted to prove the effectiveness of the approach.
44314
A Novel Fault Diagnosis of Analog Circuit Algorithm Based on Incomplete Wavelet Packet Transform and Improved Balanced Binary-Tree SVMs
Aiming at to the characteristics of fault diagnosis of analog circuit with tolerances, noise, circuit nonlinearities and small sample set, this paper presents a novel algorithm for the existent problems based on incomplete wavelet packet transform for feature extraction and improved balanced binary-tree support vector machines (BBSVMs) for multi-classification. Firstly, analyzing characters of the optimal wavelet packet transform and the incomplete wavelet packet transform, the conclusion that the latter is perfect for fault diagnosis of analog circuit is obtained. Secondly, in order to perform multi-class classification, binary-tree-based SVMs has been studied. By introducing the separability measure that based on the space distribution of pattern classes, an improved balanced binary-tree structure is constructed. The proposed algorithm for analog circuits with tolerances and noise is implemented, and simulation results show us that compared with several existent fault diagnosis methods, the current algorithm has highest classification speed and higher classification accuracy, which make fault diagnosis of analogy circuit on-line promising.
16346838
Ant-MST: An Ant-Based Minimum Spanning Tree for Gene Expression Data Clustering
We have proposed an ant-based clustering algorithm for document clustering based on the travelling salesperson scenario. In this paper, we presented an approach called Ant-MST for gene expression data clustering based on both ant-based clustering and minimum spanning trees (MST). The ant-based clustering algorithm is firstly used to construct a fully connected network of nodes. Each node represents one gene, and every edge is associated with a certain level of pheromone intensity describing the co-expression level between two genes. Then MST is used to break the linkages in order to generate clusters. Comparing to other MST-based clustering approaches, our proposed method uses pheromone intensity to measure the similarity between two genes instead of using Euclidean distance or correlation distance. Pheromone intensities associated with every edge in a fully-connected network records the collective memory of the ants. Self-organizing behavior could be easily discovered through pheromone intensities. Experimental results on three gene expression datasets show that our approach in general outperforms the classical clustering methods such as K-means and agglomerate hierarchical clustering.
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.
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.