9203049
A New Theory of Element and Number
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.
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.
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.
1953614
Alternative Proof of a Theorem of Kleene
27793638
Wiener Indices of Balanced Binary Trees
We study a new family of trees for computation of the Wiener indices. We introduce general tree transformations and derive formulas for computing the Wiener indices when a tree is modified. We present several algorithms to explore the Wiener indices of our family of trees. The experiments support new conjectures about the Wiener indices.
4332805
Stable egress route selection for interdomain traffic engineering: model and analysis
We present a general model of interdomain route selection to study interdomain traffic engineering. In this model, the routing of multiple destinations can be coordinated. Thus the model can capture general traffic engineering behaviors such as load balancing and link capacity constraints. We first identify potential routing instability and inefficiency of interdomain traffic engineering. We then derive a sufficient condition to guarantee convergence. We also show that the constraints on local policies imposed by business considerations in the Internet can guarantee stability without global coordination. Using realistic Internet topology, we evaluate the extent to which routing instability of interdomain traffic engineering can happen when the constraints are violated
49376
Pipelined van Emde Boas Tree: Algorithms, Analysis, and Applications
Priority queues are essential for various network processing applications, including per-flow queueing with quality-of-service (QoS) guarantees, management of large fast packet buffers, and management of statistics counters. In this paper, we propose a new data structure for implementing high-performance priority queues based on a pipelined version of the van Emde Boas tree. We show that we can achieve O(1) amortized time operations using our architecture, but we can achieve this algorithmic efficiency using only O (log log u) number of pipelined stages, where u is the size of the universe used to represent the priority keys.
18226102
On Synchronizing with Web Service Evolution
To catch up with today's fast changing business markets, web services has never slowed down its paces for evolution. As a consequence, service consumers have to employ in-time upgrades to guarantee continuous business integrity and avoid unnecessary runtime errors in their IT systems. However, a new web service release can involve hundreds of changes, and thus it is non-trivial for the service consumer to rapidly track and adopt relevant changes. This paper proposes a framework for facilitating the service consumer to keep synchronized with the web service evolution. On the service side, a Service Invocation Monitor is installed to monitor the interaction history of the client, a Service Delta Analyzer exports the service delta into well-formatted document, and a Release Note Customizeris configured to customize the full release note produced using the monitored interaction history and exported service delta. On the client side, the Consumer Code Customizer is used by the consumer developer to highlight the code fragments to be changed and provide facilitation linkages between the code fragments and the customized release note. An example is shown to demonstrate the usefulness of our proposal.
37867207
A Low-Latency Peer-to-Peer Approach for Massively Multiplayer Games
This paper focuses on the latency reduction problem in massively multiplayer games (MMGs). As the client-server (CS) architecture in use today in most commercial MMGs applications are exposing its weakness in scalability as the number of players increases, researchers start to consider the peer-to-peer (P2P) model that has inherent high scalability for MMGs. However, existing P2P models generally lead to high latency that significantly detracts from the playing experience. To improve this, we present a novel communication model intended to reduce the latency of network communication on game states. The model, based on DHT protocol, can select the best candidate node to perform server's role for a game zone. In contrast, existing P2P models designate 'peer servers' randomly without considering latency effects. The simulation shows that our approach obtains much lower latency than existing P2P models. The performance of our approach is shown to be even better than that of CS models.