26896458
The Axiomatization of Arithmetic
1953614
Alternative Proof of a Theorem of Kleene
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.
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 …
38679232
Short Definitions of the Ordinals
In this paper, a simple inductive characterization of the ordinal numbers is stated and developed. The characterization forms the basis for a set of axioms for ordinal theory and also for several short explicit definitions of the ordinals. The axioms are shown to be sufficient for ordinal theory, and, subject to suitable existence assumptions, each of the definitions is shown to imply the axioms. The present results apply to the familiar von Neumann version of the ordinals, but the methods used are easily adapted to other versions. ? 1. Axioms for ordinal theory. According to the usual interpretation, each ordinal is the set of all lesser ordinals. Thus zero is the empty set A, a successor ordinal x' is x U {x} for some ordinal x, and a limit ordinal is a set w of ordinals that is equal to its own sum set Zw. In fact, the class On of all ordinals might be characterized informally as the smallest class Z such that (1) AEZ, (2) (y) (yEZ D y'EZ), and (3) (w) (1w = w c Z D wEZ). More simply, On can be characterized as the smallest Z such that
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.
3839837
Creating Vulnerability Signatures Using Weakest Preconditions
Signature-based tools such as network intrusion detection systems are widely used to protect critical systems. Automatic signature generation techniques are needed to enable these tools due to the speed at which new vulnerabilities are discovered. In particular, we need automatic techniques which generate sound signatures - signatures which will not mistakenly block legitimate traffic or raise false alarms. In addition, we need signatures to have few false negatives and will catch many different exploit variants. We investigate new techniques for automatically generating sound vulnerability signatures with fewer false negatives than previous research using program binary analysis. The key problem to reducing false negatives is to consider as many as possible different program paths an exploit may take. Previous work considered each possible program path an exploit may take separately, thus generating signatures that are exponential in the size of the number of branches considered. In the exact same scenario, we show how to reduce the overall signature size and the generation time from exponential to polynomial. We do this without requiring any additional assumptions, or relaxing any properties. This efficiency gain allows us to consider many more program paths, which results in reducing the false negatives of generated signatures. We achieve these results by creating algorithms for generating vulnerability signatures that are based on computing weakest preconditions (WP). The weakest precondition for a program path to a vulnerability is a function which matches all exploits that may exploit the vulnerability along that path. We have implemented our techniques and generated signatures for several binary programs. Our results demonstrate that our WP-based algorithm generates more succinct signatures than previous approaches which were based on forward symbolic execution.
206827796
Application of Support Vector Machine to Mobile Communications in Telephone Traffic Load of Monthly Busy Hour Prediction
Telephone traffic of busy hour is one of indicators of load capacity of telecommunication network, which has a significant meaning to dilate and modify the network. A good performance of predicting the monthly busy hour traffic load is cared about by the mobile operators. As a promising learning theory,support vector machine (SVM) has been studied and applied in a wide area, such as financial markets and weather forecast. In this paper, we use SVM to forecast monthly busy hour traffic load of two regions in Xinjiang. A good result has been achieved via an improved grid search method for the search of hyper-parameter of SVM.
18424476
Proving theorems by pattern recognition I
Certain preliminary results on doing mathematics by machines (“mechanical mathematics”) were reported in an earlier paper [20]. The writer suggested developing inferential analysis as a branch of applied logic and as a sister discipline of numerical analysis. This analogy rests on the basic distinction of pure existence proofs, elegant procedures which in theory always terminate, and efficient procedures which are more complex to describe but can more feasibly be carried out in practice. In contrast with pure logic, the chief emphasis of inferential analysis is on the efficiency of algorithms, which is usually attained by paying a great deal more attention to the detailed structures of the problems and their solutions, to take advantage of possible systematic short cuts. The possibilities of much more elaborate calculations by machines provide an incentive to studying a group of rather minute questions which were formerly regarded as of small theoretical interest. When the range of actual human computation was narrow, there seemed little point in obtaining faster procedures which were still far beyond what was feasible. Furthermore, on account of the versatility of machines, it now appears that as more progress is made, strategies in the search for proofs, or what are often called heuristic methods, will also gradually become part of the subject matter of inferential analysis. An analogous situation in numerical analysis would be, for example, to make the machine choose to apply different tricks such as taking the Fourier transform to obtain a solution of some differential equation.
14517479
The Logic of Automata—Part I
We are concerned in this paper with the use of logical systems and techniques in the analysis of the structure and behavior of automata. In Section 2 we discuss automata in general. A new kind of automaton is introduced, the growing automaton, of which Turing machines and self-duplicating automata are special cases. Thereafter we limit the discussion to fixed, deterministic automata and define their basic features. We give methods of analyzing these automata in terms of their states. Four kinds of state tables--complete tables, admissibility trees, characterizing tables, and output tables--are used for this purpose. These methods provide a decision procedure for determining whether or not two automaton junctions behave the same. Finally, a class of well-formed automaton nets is defined, and it is shown how to pass from nets to state tables and vice versa. A coded normal form for nets is given. In Section 3* we show how the information contained in the state tables can be expressed in matrix form. The (i, j) element of a transition matrix gives those inputs which cause state S~ to produce state Sj. Various theorems are proved about these matrices and a corresponding normal form (the decoded normal form or matrix form) for nets is introduced. In Section 4* we first show how to decompose a net into one or more subnets which contain cycles but which are not themselves interconnected cyclically. We then discuss the relation of cycles in nets to the use of truth functions and quantifiers for describing nets. We conclude by relating nerve nets to other automaton nets.
32406655
A Parallel Algorithm for Learning Bayesian Networks
Computing the expected statistics is the main bottleneck in learning Bayesian networks in large-scale problem domains. This paper presents a parallel learning algorithm, PL-SEM, for learning Bayesian networks, based on an existing structural EM algorithm (SEM). Since the computation of the expected statistics is in the parametric learning part of the SEM algorithm, PLSEM exploits a parallel EM algorithm to compute the expected statistics. The parallel EM algorithm parallelizes the E-step and M-step. At the E-step, PLSEM parallel computes the expected statistics of each sample; and at the M-step, with the conditional independence of Bayesian networks and the expected statistics computed at the E-step, PL-SEM exploits the decomposition property of the likelihood function under the completed data to parallel estimate each local likelihood function. PL-SEM effectively computes the expected statistics, and greatly reduces the time complexity of learning Bayesian networks.