Σ1 definitions with parameters

1986 ◽  
Vol 51 (2) ◽  
pp. 453-461
Author(s):  
T. A. Slaman
Keyword(s):  
Transitive Closure ◽  
Admissible Set ◽  
Selection Scheme ◽  
Ordinal Numbers ◽  
Recursively Enumerable ◽  
Total Function ◽  
Existential Quantification

AbstractLet p be a set. A function Φ is uniformly Σ1(p) in every admissible set if there is a Σ1 formula ϕ in the parameter p so that ϕ defines Φ in every Σ1-admissible set which includes p. A theorem of Van de Wiele states that if Φ is a total function from sets to sets then Φ is uniformly Σ1 in every admissible set if and only if it is E-recursive. A function is ESp-recursive if it can be generated from the schemes for E-recursion together with a selection scheme over the transitive closure of p. The selection scheme is exactly what is needed to insure that the ESP-recursively enumerable predicates are closed under existential quantification over the transitive closure of p. Two theorems are established: a) If the transitive closure of p is countable then a total function on sets is ESp-recursive if and only if it is uniformly Σ1(p) in every admissible set. b) For any p, if Φ is a function on the ordinal numbers then Φ is ESP-recursive if and only if it is uniformly Σ1(p) in every admissible set.

α-degrees of α-theories

1972 ◽  
Vol 37 (4) ◽  
pp. 677-682 ◽  
Author(s):  
George Metakides
Keyword(s):  
Initial Segment ◽  
Recursion Theory ◽  
The Other ◽  
Infinite Length ◽  
Infinitary Logic ◽  
Admissible Set ◽  
Recursively Enumerable ◽  
Limit Ordinal ◽  
The Subject ◽  
Further Development

Let α be a limit ordinal with the property that any “recursive” function whose domain is a proper initial segment of α has its range bounded by α. α is then called admissible (in a sense to be made precise later) and a recursion theory can be developed on it (α-recursion theory) by providing the generalized notions of α-recursively enumerable, α-recursive and α-finite. Takeuti [12] was the first to study recursive functions of ordinals, the subject owing its further development to Kripke [7], Platek [8], Kreisel [6], and Sacks [9].Infinitary logic on the other hand (i.e., the study of languages which allow expressions of infinite length) was quite extensively studied by Scott [11], Tarski, Kreisel, Karp [5] and others. Kreisel suggested in the late '50's that these languages (even which allows countable expressions but only finite quantification) were too large and that one should only allow expressions which are, in some generalized sense, finite. This made the application of generalized recursion theory to the logic of infinitary languages appear natural. In 1967 Barwise [1] was the first to present a complete formalization of the restriction of to an admissible fragment (A a countable admissible set) and to prove that completeness and compactness hold for it. [2] is an excellent reference for a detailed exposition of admissible languages.

Yet another hierarchy theorem

2000 ◽  
Vol 65 (2) ◽  
pp. 627-640 ◽  
Author(s):  
Max Kubierschky
Keyword(s):  
Fixed Point ◽  
Transitive Closure ◽  
Binary Relations ◽  
Finite Structures ◽  
Existential Quantification

Abstractn + 1 nested k-ary fixed point operators are more expressive than n. This holds on finite structures for all sublogics of partial fixed point logic PFP that can express conjunction, existential quantification and deterministic transitive closure of binary relations using at most k-ary fixed point operators and that are closed against subformulas. Among those are a lot of popular fixed point logics.

Characterization of recursively enumerable sets

1972 ◽  
Vol 37 (3) ◽  
pp. 507-511 ◽  
Author(s):  
Jesse B. Wright
Keyword(s):  
Transitive Closure ◽  
Successor Function ◽  
Function Composition ◽  
Recursive Functions ◽  
Recursively Enumerable ◽  
Abstract Setting ◽  
Recursively Enumerable Sets ◽  
Bracket Operation

AbstractLet N, O and S denote the set of nonnegative integers, the graph of the constant 0 function and the graph of the successor function respectively. For sets P, Q, R ⊆ N2 operations of transposition, composition, and bracketing are defined as follows: P∪ = {〈x, y〉 ∣ 〈y, x〉 ∈ P}, PQ = {〈x, z〉 ∣ ∃y〈x, y〉 ∈ P & 〈y, z〉 ∈ Q}, and [P, Q, R] = ⋃n ∈ M(Pn Q Rn).Theorem. The class of recursively enumerable subsets of N2 is the smallest class of sets with O and S as members and closed under transposition, composition, and bracketing.This result is derived from a characterization by Julia Robinson of the class of general recursive functions of one variable in terms of function composition and “definition by general recursion.” A key step in the proof is to show that if a function F is defined by general recursion from functions A, M, P and R then F = [P∪, A∪M, R].The above definitions of the transposition, composition, and bracketing operations on subsets of N2 can be generalized to subsets of X2 for an arbitrary set X. In this abstract setting it is possible to show that the bracket operation can be defined in terms of K, L, transposition, composition, intersection, and reflexive transitive closure where K: X → X and L: X → X are functions for decoding pairs.

scholarly journals On Some Properties of Positive and Strongly Positive Arithmetical Sets

2018 ◽  
pp. 52-55
Author(s):  
Seda Manukian
Keyword(s):  
Transitive Closure ◽  
Proved Theorem ◽  
Recursively Enumerable Set ◽  
One Dimensional ◽  
Logical Description ◽  
Recursively Enumerable ◽  
Logical Operations ◽  
Arithmetical Formula

The notions of positive and strongly positive arithmetical sets are defined as in [1]-[4] (see, for example, [2], p. 33). It is proved (Theorem 1) that any arithmetical set is positive if and only if it can be defined by an arithmetical formula containing only logical operations ∃, &,∨ and the elementary subformulas having the forms 𝑥𝑥=0 or 𝑥𝑥=𝑦𝑦+1, where 𝑥𝑥and 𝑦𝑦 are variables.Corollary:the logical description of the class of positive sets is obtained from the logical description of the class of strongly positive sets replacing the list of operations &,∨ by the list ∃, &,∨. It is proved (Theorem 2) that for any one-dimensional recursively enumerable set 𝑀𝑀 there exists 6- dimensional strongly positive set 𝐻𝐻 such that 𝑥𝑥 ∈𝑀𝑀 holds if and only if (1, 2𝑥𝑥, 0, 0, 1, 0)∈𝐻𝐻+, where 𝐻𝐻+ is the transitive closure of 𝐻𝐻.

Femtocell Selection Scheme for Reducing Unnecessary Handover and Enhancing Downlink QoS in Cognitive Femtocell Networks

2017 ◽  
Author(s):  
Hoang Nhu Dong ◽  
Hoang Nam Nguyen ◽  
Hoang Trong Minh ◽  
Takahiko Saba
Keyword(s):  
Cellular Networks ◽  
Estimation Methods ◽  
Mobility Prediction ◽  
Capacity Estimation ◽  
Femtocell Networks ◽  
Selection Scheme ◽  
Indoor Communications ◽  
Performance Results ◽  
Cognitive Femtocell

Femtocell networks have been proposed for indoor communications as the extension of cellular networks for enhancing coverage performance. Because femtocells have small coverage radius, typically from 15 to 30 meters, a femtocell user (FU) walking at low speed can still make several femtocell-to-femtocell handovers during its connection. When performing a femtocell-to-femtocell handover, femtocell selection used to select the target handover femtocell has to be able not only to reduce unnecessary handovers and but also to support FU’s quality of service (QoS). In the paper, we propose a femtocell selection scheme for femtocell-tofemtocell handover, named Mobility Prediction and Capacity Estimation based scheme (MPCE-based scheme), which has the advantages of the mobility prediction and femtocell’s available capacity estimation methods. Performance results obtained by computer simulation show that the proposed MPCE-based scheme can reduce unnecessary femtocell-tofemtocell handovers, maintain low data delay and improve the throughput of femtocell users. DOI: 10.32913/rd-ict.vol3.no14.536

A Stable Routing Selection Scheme in the Internet

2012 ◽  
Vol 35 (12) ◽  
pp. 2668
Author(s):  
Qi LI ◽  
Ming-Wei XU ◽  
Jian-Ping WU
Keyword(s):  
The Internet ◽  
Selection Scheme ◽  
Stable Routing

Relay Selection-aware Non-orthogonal Multiple Access Networks: Direct and Relaying Mode

2020 ◽  
Vol 13 (3) ◽  
pp. 348-354
Author(s):  
Dinh-Thuan Do ◽  
Minh-Sang V. Nguyen
Keyword(s):  
Relay Selection ◽  
Multiple Access ◽  
Decode And Forward ◽  
Selection Scheme ◽  
Fixed Power ◽  
Transmission Quality ◽  
Improved Performance ◽  
Outage Probabilities ◽  
Future Work ◽  
The Impact

Objective: In this paper, Decode-and-Forward (DF) mode is deployed in the Relay Selection (RS) scheme to provide better performance in cooperative downlink Non-orthogonal Multiple Access (NOMA) networks. In particular, evaluation regarding the impact of the number of multiple relays on outage performance is presented. Methods: As main parameter affecting cooperative NOMA performance, we consider the scenario of the fixed power allocations and the varying number of relays. In addition, the expressions of outage probabilities are the main metric to examine separated NOMA users. By matching related results between simulation and analytical methods, the exactness of derived formula can be verified. Results: The intuitive main results show that in such cooperative NOMA networks, the higher the number of relays equipped, the better the system performance can be achieved. Conclusion: DF mode is confirmed as a reasonable selection scheme to improve the transmission quality in NOMA. In future work, we will introduce new relay selections to achieve improved performance.

The Natural Numbers

2018 ◽  
Author(s):  
Øystein Linnebo
Keyword(s):  
Natural Number ◽  
Peano Arithmetic ◽  
Number Sequence ◽  
Natural Numbers ◽  
Ordinal Numbers ◽  
Cardinal Numbers

How are the natural numbers individuated? That is, what is our most basic way of singling out a natural number for reference in language or in thought? According to Frege and many of his followers, the natural numbers are cardinal numbers, individuated by the cardinalities of the collections that they number. Another answer regards the natural numbers as ordinal numbers, individuated by their positions in the natural number sequence. Some reasons to favor the second answer are presented. This answer is therefore developed in more detail, involving a form of abstraction on numerals. Based on this answer, a justification for the axioms of Dedekind–Peano arithmetic is developed.

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