On completely recursively enumerable classes and their key arrays

1956 ◽  
Vol 21 (3) ◽  
pp. 304-308 ◽  
Author(s):  
H. G. Rice
Keyword(s):  
Decision Procedure ◽  
Cardinal Number ◽  
Maximum Amount ◽  
Infinite Class ◽  
Finite Sets ◽  
Amount Of Information ◽  
Recursively Enumerable ◽  
Finite Set ◽  
Partial Recursive Functions ◽  
Set Up

The two results of this paper (a theorem and an example) are applications of a device described in section 1. Our notation is that of [4], with which we assume familiarity. It may be worth while to mention in particular the function Φ(n, x) which recursively enumerates the partial recursive functions of one variable, the Cantor enumerating functions J(x, y), K(x), L(x), and the classes F and Q of r.e. (recursively enumerable) and finite sets respectively.It is possible to “give” a finite set in a way which conveys the maximum amount of information; this may be called “giving explicitly”, and it requires that in addition to an effective enumeration or decision procedure for the set we give its cardinal number. It is sometimes desired to enumerate effectively an infinite class of finite sets, each given explicitly (e.g., [4] p. 360, or Dekker [1] p. 497), and we suggest here a device for doing this.We set up an effective one-to-one correspondence between the finite sets of non-negative integers and these integers themselves: the integer , corresponds to the set αi, = {a1, a2, …, an} and inversely. α0 is the empty set. Clearly i can be effectively computed from the elements of αi and its cardinal number.

Classifications of generalized index sets of open classes

1978 ◽  
Vol 43 (4) ◽  
pp. 694-714 ◽  
Author(s):  
Nancy Johnson
Keyword(s):  
Finite Collection ◽  
Finite Sets ◽  
Index Set ◽  
Recursively Enumerable ◽  
Ershov Hierarchy ◽  
Index Sets ◽  
Finite Set ◽  
The Difference ◽  
Original Classification

The Rice-Shapiro Theorem [4] says that the index set of a class of recursively enumerable (r.e.) sets is r.e. if and only if consists of all sets which extend an element of a canonically enumerable sequence of finite sets. If an index of a difference of r.e. (d.r.e.) sets is defined to be the pair of indices of the r.e. sets of which it is the difference, then the following generalization due to Hay [3] is obtained: The index set of a class of d.r.e. sets is d.r.e. if and only if is empty or consists of all sets which extend a single fixed finite set. In that paper Hay also classifies index sets of classes consisting of d.r.e. sets which extend one of a finite collection of finite sets. These sets turn out to be finite Boolean combinations of r.e. sets. The question then arises “What about the classification of the index set of a class consisting of d.r.e. sets which extend an element of a canonically enumerable sequence of finite sets?” The results in this paper come from an attempt to answer this question.Since classes of sets which are Boolean combinations of r.e. sets form a hierarchy (the finite Ershov hierarchy, see Ershov [1]) with the r.e. and d.r.e. sets respectively levels 1 and 2 of this hierarchy, we may define index sets of classes of level n sets. If is a class of level n sets which extend some element of a canonically enumerable sequence of finite sets and if we let co-, then we extend the original classification question to the classification of the index sets of the classes and co-.Now if the sequence of finite sets enumerates only finitely many sets or if only finitely many of the finite sets are minimal under inclusion, then it is a routine computation to verify that the index sets of and co- are in the finite Ershov hierarchy. Thus we are interested in the case in which infinitely many of the sequence of finite sets are minimal under inclusion. However if the infinite sequence is fairly simple, for instance{0}, {1}, {2}, … then the r.e. index set of co- is Σ20-complete as well as the index sets of and co- for all levels n > 2. Since the finite Ershov hierarchy does not exhaust ⊿20 there is a lot of “room” between these two extreme cases.

scholarly journals Fin-set: A syntactical definition of finite sets

2007 ◽  
pp. 155-163
Author(s):  
Slavisa Presic
Keyword(s):  
Cardinal Number ◽  
Cartesian Product ◽  
Finite Sets ◽  
Finite Set ◽  
Definition Of ◽  
Product Relation

We state Fin-set, by which one founds the notion of finite sets in a syntactical way. Any finite set {a1, a2,..., an} is defined as a well formed term of the form S(a1 + (a2 + (??? + (an?1 + an)???))), where + is a binary and S a unary operational symbol. Related to the operational symbol the term-substitutions (1) are introduced. Definition of finite sets is called syntactical because by two algorithms Set-alg and Calc one can effectively establish whether any given set-terms are equal or not equal. All other notions related to finite sets, like ?, ordered pair, Cartesian product, relation, function, cardinal number are defined as terms as well. Each of these definitions is recursive. For instance, ? is defined by x ? S(a1) iff x = a1 x ? S(a1 + ???+ an) iff x = a1 or x ? S(a2 + ???+ an) x/? ? (? denotes the empty set).

Statistical analysis of classification

1966 ◽  
Vol 24 ◽  
pp. 188-189
Author(s):  
T. J. Deeming
Keyword(s):  
Multivariate Analysis ◽  
Statistical Analysis ◽  
Classification Scheme ◽  
Analytical Procedure ◽  
Narrow Band ◽  
Scientific Research ◽  
Maximum Amount ◽  
Amount Of Information

If we make a set of measurements, such as narrow-band or multicolour photo-electric measurements, which are designed to improve a scheme of classification, and in particular if they are designed to extend the number of dimensions of classification, i.e. the number of classification parameters, then some important problems of analytical procedure arise. First, it is important not to reproduce the errors of the classification scheme which we are trying to improve. Second, when trying to extend the number of dimensions of classification we have little or nothing with which to test the validity of the new parameters.Problems similar to these have occurred in other areas of scientific research (notably psychology and education) and the branch of Statistics called Multivariate Analysis has been developed to deal with them. The techniques of this subject are largely unknown to astronomers, but, if carefully applied, they should at the very least ensure that the astronomer gets the maximum amount of information out of his data and does not waste his time looking for information which is not there. More optimistically, these techniques are potentially capable of indicating the number of classification parameters necessary and giving specific formulas for computing them, as well as pinpointing those particular measurements which are most crucial for determining the classification parameters.

scholarly journals Ordinal synchronization mark sequence and its steganography for a multi-link network covert channel

PLoS ONE ◽  
2021 ◽  
Vol 16 (6) ◽  
pp. e0252813
Author(s):  
Songyin Fu ◽  
Rangding Wang ◽  
Li Dong ◽  
Diqun Yan
Keyword(s):  
Hash Function ◽  
High Capacity ◽  
Superior Performance ◽  
Covert Channel ◽  
Correlation Test ◽  
Switching Patterns ◽  
Finite Set ◽  
Set Up ◽  
Network Covert Channel ◽  
Real Traffic

A multi-link network covert channel (MLCC) such as Cloak exhibits a high capacity and robustness and can achieve lossless modulation of the protocol data units. However, the mechanism of Cloak involving an arrangement of packets over the links (APL) is limited by its passive synchronization schemes, which results in intermittent obstructions in transmitting APL packets and anomalous link switching patterns. In this work, we propose a novel ordinal synchronization mark sequence (OSMS) for a Cloak framework based MLCC to ensure that the marked APL packets are orderly distinguishable. Specifically, a unidirectional function is used to generate the OSMS randomly before realizing covert modulation. Subsequently, we formulate the generation relation of the marks according to their order and embed each mark into the APL packets by using a one-way hash function such that the mark cannot be cracked during the transmission of the APL packet. Finally, we set up a retrieval function of the finite set at the covert receiver to extract the marks and determine their orders, and the APL packets are reorganized to realize covert demodulation. The results of experiments performed on real traffic indicated that the MLCC embedded with OSMS could avoid the passive synchronization schemes and exhibited superior performance in terms of reliability, throughput, and undetectability compared with the renowned Cloak method, especially under a malicious network interference scenario. Furthermore, our approach could effectively resist the inter-link correlation test, which are highly effective in testing the Cloak framework.

scholarly journals On minimal log discrepancies and kollár components

2021 ◽  
pp. 1-20
Author(s):  
Joaquín Moraga
Keyword(s):  
Blow Up ◽  
Chain Condition ◽  
Fano Varieties ◽  
Finite Sets ◽  
Fixed Dimension ◽  
Log Canonical ◽  
Finite Set ◽  
Canonical Singularities ◽  
Ascending Chain Condition

Abstract In this article, we prove a local implication of boundedness of Fano varieties. More precisely, we prove that $d$ -dimensional $a$ -log canonical singularities with standard coefficients, which admit an $\epsilon$ -plt blow-up, have minimal log discrepancies belonging to a finite set which only depends on $d,\,a$ and $\epsilon$ . This result gives a natural geometric stratification of the possible mld's in a fixed dimension by finite sets. As an application, we prove the ascending chain condition for minimal log discrepancies of exceptional singularities. We also introduce an invariant for klt singularities related to the total discrepancy of Kollár components.

Rice Theorems for ∑n-1 Sets

1977 ◽  
Vol 29 (4) ◽  
pp. 794-805 ◽  
Author(s):  
Nancy Johnson
Keyword(s):  
Finite Sets ◽  
Index Set ◽  
Recursively Enumerable ◽  
Recursively Enumerable Sets ◽  
The Difference

In [3] Hay proves generalizations of Rice's Theorem and the Rice-Shapiro Theorem for differences of recursively enumerable sets (d.r.e. sets). The original Rice Theorem [5, p. 3G4, Corollary B] says that the index set of a class C of r.e. sets is recursive if and only if C is empty or C contains all r.e. sets. The Rice-Shapiro Theorem conjectured by Rice [5] and proved independently by McNaughton, Shapiro, and Myhill [4] says that the index set of a class C of r.e. sets is r.e. if and only if C is empty or C consists of all r.e. sets which extend some element of a canonically enumerable class of finite sets. Since a d.r.e. set is a difference of r.e. sets, a d.r.e. set has an index associated with it, namely, the pair of indices of the r.e. sets of which it is the difference.

scholarly journals Chaotic Quantum Key Distribution

Cryptography ◽  
2020 ◽  
Vol 4 (3) ◽  
pp. 24
Author(s):  
Noah Cowper ◽  
Harry Shaw ◽  
David Thayer
Keyword(s):  
Quantum Computing ◽  
Quantum Key Distribution ◽  
Single Photon ◽  
Photon Emission ◽  
Key Distribution ◽  
Maximum Amount ◽  
Single Photons ◽  
Single Photon Emission ◽  
Amount Of Information ◽  
Communication Scheme

The ability to send information securely is a vital aspect of today’s society, and with the developments in quantum computing, new ways to communicate have to be researched. We explored a novel application of quantum key distribution (QKD) and synchronized chaos which was utilized to mask a transmitted message. This communication scheme is not hampered by the ability to send single photons and consequently is not vulnerable to number splitting attacks like other QKD schemes that rely on single photon emission. This was shown by an eavesdropper gaining a maximum amount of information on the key during the first setup and listening to the key reconciliation to gain more information. We proved that there is a maximum amount of information an eavesdropper can gain during the communication, and this is insufficient to decode the message.

scholarly journals On the sum and the difference of finite sets of integers

1976 ◽  
Vol 15 (2) ◽  
pp. 245-251
Author(s):  
Reinhard A. Razen
Keyword(s):  
Finite Sets ◽  
Finite Set ◽  
The Difference

Let A = {ai} be a finite set of integers and let p and m denote the cardinalities of A + A = {ai+aj} and A - A {ai–aj}, respectively. In the paper relations are established between p and m; in particular, if max {ai–ai-1} = 2 those sets are characterized for which p = m holds.

Probabilistic Semi–Simple Splicing System and Its Characteristics

2013 ◽  
Vol 62 (3) ◽  
Author(s):  
Mathuri Selvarajoo ◽  
Fong Wan Heng ◽  
Nor Haniza Sarmin ◽  
Sherzod Turaev
Keyword(s):  
Regular Languages ◽  
Computational Power ◽  
Dna Molecules ◽  
Finite Sets ◽  
Generative Power ◽  
Recursively Enumerable ◽  
Splicing Systems ◽  
Recursively Enumerable Languages

The concept of splicing system was first introduced by Head in 1987. This model has been introduced to investigate the recombinant behavior of DNA molecules. Splicing systems with finite sets of axioms only generate regular languages. Hence, different restrictions have been considered to increase the computational power up to the recursively enumerable languages. Recently, probabilistic splicing systems have been introduced where probabilities are initially associated with the axioms, and the probability of a generated string is computed by multiplying the probabilities of all occurrences of the initial strings in the computation of the string. In this paper, some properties of probabilistic semi-simple splicing systems, which are special types of probabilistic splicing systems, are investigated. We prove that probabilistic semi-simple splicing systems can also increase the generative power of the generated languages.

scholarly journals Unicity theorems for meromorphic or entire functions II

1995 ◽  
Vol 52 (2) ◽  
pp. 215-224 ◽  
Author(s):  
Hong-Xun Yi
Keyword(s):  
Meromorphic Functions ◽  
Entire Functions ◽  
Inverse Image ◽  
Finite Sets ◽  
Finite Set ◽  
Special Case

In 1976, Gross posed the question “can one find two (or possibly even one) finite sets Sj (j = 1, 2) such that any two entire functions f and g satisfying Ef(Sj) = Eg(Sj) for j = 1,2 must be identical?”, where Ef(Sj) stands for the inverse image of Sj under f. In this paper, we show that there exists a finite set S with 11 elements such that for any two non-constant meromorphic functions f and g the conditions Ef(S) = Eg(S) and Ef({∞}) = Eg({∞}) imply f ≡ g. As a special case this also answers the question posed by Gross.

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