1953614
Alternative Proof of a Theorem of Kleene
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.
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 …
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.
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.
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
42170888
LVQ Based Distributed Video Coding with LDPC in Pixel Domain
Presently, distributed source coding (DSC) and distributed video coding (DVC) are given high attention in sensor network and other multimedia transmission. That is due to their contribution to easy encoding and robust multimedia communication. In this paper, we propose a new DVC framework where two LVQ (lattice vector quantization) and a rate-variable LDPC (low-density parity-check) are exploited. In both LVQ and LDPC, we use the interpolated from decoded frames as side information to help coding. Because of LVQ's exploitation to dependence of source and the better error-correcting capacity of LDPC, our system achieves more than 1 dB improvement in PSNR than the referenced. Meanwhile, the property of low-complexity encoding is still preserved.
12292877
Weighted Pushdown Systems and Trust-Management Systems
The authorization problem is to decide whether, according to a security policy, some principal should be allowed access to a resource. In the trust-management system SPKI/SDSI, the security policy is given by a set of certificates, and proofs of authorization take the form of certificate chains. The certificate-chain-discovery problem is to discover a proof of authorization for a given request. Certificate-chain-discovery algorithms for SPKI/SDSI have been investigated by several researchers. We consider a variant of the certificate-chain discovery problem where the certificates are distributed over a number of servers, which then have to cooperate to identify the proof of authorization for a given request. We propose two protocols for this purpose. These protocols are based on distributed model-checking algorithms for weighted pushdown systems (WPDSs). These protocols can also handle cases where certificates are labeled with weights and where multiple certificate chains must be combined to form a proof of authorization. We have implemented these protocols in a prototype and report preliminary results of our evaluation.
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.
1947084
Fair Payment Protocols for E-Commerce
It has been widely accepted that fairness is a critical property for electronic commerce. Fair payment protocol is designed to guarantee fairness in a payment process over asynchronous network. Fairness means that when the protocol terminates, either both parties get their expected items, or neither does. In this paper we first present a new generic offline fair payment protocol with fairness, timeliness and invisibility of TTP. Then we introduce the property of abuse-freeness into electronic payment and implement a fair abuse-free payment protocol.
14158906
A New Certified Email Protocol
As a continuative effort of our WICS'05 paper, a new certified Email protocol is proposed. Email is a standard communication method nowadays but certified email is poorly developed and has very limited application. A certified email protocol assures non-repudiability which means that if the email is successfully sent, the sender cannot deny s/he has actually sent the mail and the recipient cannot deny s/he has actually received it. Our protocol uses an efficient RSA-based convertible signature with non- interactive partial signature proof method and additively split signing secret key.