34944417
Logic of Many-Sorted Theories
Certain axiomatic systems involve more than one category of fundamental objects; for example, points, lines, and planes in geometry; individuals, classes of individuals, etc. in the theory of types or in predicate calculi of orders higher than one. It is natural to use variables of different kinds with their ranges respectively restricted to different categories of objects, and to assume as substructure the usual quantification theory (the restricted predicate calculus) for each of the various kinds of variables together with the usual theory of truth functions for the formulas of the system. An axiomatic theory set up in this manner will be called many-sorted. We shall refer to the theory of truth functions and quantifiers in it as its (many-sorted) elementary logic, and call the primitive symbols and axioms (including axiom schemata) the proper primitive symbols and proper axioms of the system. Our purpose in this paper is to investigate the many-sorted systems and their elementary logics. Among the proper primitive symbols of a many-sorted system T n ( n = 2, …, ω ) there may be included symbols of some or all of the following kinds: (1) predicates denoting the properties and relations treated in the system; (2) functors denoting the functions treated in the system; (3) constant names for certain objects of the system. We may either take as primitive or define a predicate denoting the identity relation in T n .
10510054
Words in the History of a Turing Machine with a Fixed Input
The usual decision (halting or printing) problems for Turing machines (T.M.) are relative to an infinite class of them or an infinite class of inputs or1 a single T.M. We consider now a new type of problem (the consequence problem) con-neeted with a single input on a single T.M. It is known that in a definite sense everything which an erasing T.3{. can do can also be done by a nonerasing T.M., e.g., the halting problem is undeeidable in both bases. It turns out that for the consequence problem erasing does make an essential difference: decidable without erasing, but undecidable Mth erasing. From the undeeidability of the consequence problem for a T.M., it also follows that the same problem is undeeidable for certain simple monogenie normal systems. Consider a fixed finite alphabet A including a symbol b for the blank squares on a tape. For definiteness, we may assume that A = {b, ,, a}. Assume some familiar formulation of Turing machines. Definition. The word at any moment t in the history of a T.M. is the string consisting of the contents of the squares in the minimum block on the tape at t that includes all the marked squares and the square scanned at the initial moment (the origin). THEOREM 1. For arty fixed (finite) word at the initial moment, we can find a T.M. such that the set of words in its subsequent history is not ~cursive. Ullian shows [1] that there exists some nonrecursive but recursively enumerable set S such that there is a general recursive function f such that S = {f(0), f(f(O)), f(f(f(O))),-.. }. We can now give a T.M. which first erases the finitely many marks on the initial tape and returns to the origin, then it puts down the representation of 0 on tape, calculates the value of f(0), erasing everything else except the representation of f(0), calculates f2(0). In particular, we can do it in such a way that the symbol a is used, after the initial tape content is erased, only to mark f(0), f2(0), etc; it is erased before we evaluate f(f (0)) from f(0). Moreover, the string of a followed by the representation of f(0) always begins at the origin. Clearly the set of words in the history of this T.M. cannot be recursive because otherwise the set S would be recursive, whether a 'gd(0) ' belongs to the …
1953614
Alternative Proof of a Theorem of Kleene
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.
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.
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
39099940
Efficient error recovery techniques in a novel multimedia streaming framework with peer-paired collaboration
Recently, active research has gone to application-level multicast that tries to bring all the merits of unicast and IP multicast together while avoiding their drawbacks. How to efficiently recover from errors in such a system is a difficult question yet to be answered. The current research focus is primarily on developing local error recovery techniques, but these techniques would not work in case of shared losses, which often occur in application-level multicast. In this paper, a novel multimedia streaming framework called peer-paired pyramid streaming (P/sup 3/S) is proposed. It is basically a hybrid client/server and peer-to-peer structure. In P/sup 3/S, clients are hierarchically organized and those at the same level are coupled and peer-pairs are then formed. Based on P/sup 3/S, we introduce some controlled delay between packets that are vulnerable to shared losses to combat with the shared losses. The proposed technique is supported by both theoretical and experimental results presented in this paper.
6016307
Research and Implementation of Knowledge-Enhanced Information Services
Information isolation has been identified as a big challenge in IT Service Management (ITSM). Existing ITSM practices mostly rely on configuration information and are geared towards individual applications and processes. However, information available in complicated IT infrastructure goes beyond data from the configuration management domain. How to efficiently extract and integrate the hidden knowledge from a wide variety of information sources is a major pain point for ITSM. In this paper, a threading strategy (TS) with KPI mark and knowledge-enhanced information services is proposed to improve ITSM quality. The essential contribution of this work is to organize the highly complex IT service information with KPI mark and to build a knowledge repository for accumulateing and reusing experts' knowledge. In addition, a prototype called BIANCHIN is implemented to explore this knowledge-enhanced information services framework. Finally, a real business application of Cisco VoIP system is used as a case study for evaluating the effectiveness and efficiency of the knowledge-enhanced information services framework.
6306890
APPLE: A Novel P2P Based e-Learning Environment
With the rapid development of information technology, dramatic changes have been taken place in the fundamental ways that people acquire and disseminate knowledge. Various e-Learning systems have been designed to help people take full advantage of the benefit brought by the technology. The newly emerged Peer to Peer (P2P) and grid computing will serve as the key driven forces to bring revolutionary impact on the future society. This paper evaluates the potential contribution of the P2P and grid technology into the e-Learning system and introduces a novel P2P based e-Learning environment, called APPLE. With the help of Gnutella-like P2P network, APPLE provides live broadcasting services to share services of education resources. We build a virtual classroom service based on WSRF.NET, which exposes the service as a grid service that could be accessed by all the users among the grid.
17568245
Security metrics for software systems
Security metrics for software products provide quantitative measurement for the degree of trustworthiness for software systems. This paper proposes a new approach to define software security metrics based on vulnerabilities included in the software systems and their impacts on software quality. We use the Common Vulnerabilities and Exposures (CVE), an industry standard for vulnerability and exposure names, and the Common Vulnerability Scoring System (CVSS), a vulnerability scoring system designed to provide an open and standardized method for rating software vulnerabilities, in our metric definition and calculation. Examples are provided in the paper, which show that our definition of security metrics is consistent with the common practice and real-world experience about software quality in trustworthiness.
21956342
Architectural design and implementation of highly available and scalable medical system with IBM Websphere middleware
Medical systems require the end-to-end high availability and scalability. There are many components in a medical system, and each individual component contributes to the overall system availability and scalability. We present the formulas to assess the overall system availability and capacity planning, and we architect approaches and technologies to make all components highly available in all tiers by clustering, workload management, and replication. We discuss how to unify all medical systems and design the uniform framework for different kinds of medical systems to hook up with this framework so that we can maintain a highly available and highly scalable system with ease. We use the n-tier distributed component based architecture and J2EE technologies as well as Web services to interact with .NET and others. We discuss how to program in the framework for high availability and scalability and how to upgrade application software without service interruption to achieve continuous availability. We show how to add or reduce hardware capacity in each tier and how to maintain hardware and software without service interruption as well as how to balance the capacity and availability among tiers.