The Edge-Balanced Index Set of Power-Cycle Nested Graph C6m × Pm6

2013 ◽  
Vol 416-417 ◽  
pp. 2128-2131
Author(s):  
Yu Ge Zheng ◽  
Qing Wen Zhang
Keyword(s):  
Decomposition Method ◽  
Single Point ◽  
Index Set ◽  
Power Cycle ◽  
Index Sets ◽  
Constructive Proofs ◽  
Cycle Graph

Based on the research of Power-cycle Nested Graph, the decomposition method of single point sector has come up. By the use of the process of clawed nested-cycle graph and nested-cycle sub-graph, the edge-balanced index sets of the power-cycle nested graphare solved whenand. Besides, the constructive proofs of the computational formulas are also completed.

Index sets and degrees of unsolvability

1982 ◽  
Vol 47 (2) ◽  
pp. 241-248 ◽  
Author(s):  
Michael Stob
Keyword(s):  
Density Theorem ◽  
Index Set ◽  
Index Sets ◽  
Priority Method ◽  
Ingenious Method ◽  
Priority Argument ◽  
Degrees Of Unsolvability ◽  
Standard Techniques

The characterization of classes of r.e. sets by their index sets has proved valuable in producing new results about the r.e. sets and degrees. The classic example is Yates' proof [5, Theorem 7] of Sacks' density theorem for r.e. degrees using his classification of {e: We ≤TD) as Σ3(D) whenever D is r.e. Theorem 1 of this paper is a refinement of this index set theorem of Yates which has already proved to have interesting consequences about the r.e. degrees. This theorem was originally announced by Kallibekov [1, Theorem 1]. Kallibekov there proposed a new and ingenious method for doing priority arguments which has also since been used by Kinber [2]. Unfortunately his proof to this particular theorem contains an error. We have a totally different proof using standard techniques which is of independent interest.The proof to Theorem 1 is an infinite injury priority argument. In §1 therefore we give a short summary of the infinite injury priority method. We draw heavily on the exposition of Soare [4] where a complete description of the method is given along with many examples. In §2 we prove the main theorem and also give what we think are the most interesting corollaries to this theorem announced by Kallibekov. In §3 we prove a theorem about Σ3 sets of indices of r.e. sets. This theorem is a strengthening of a theorem of Kinber [2, Theorem 1] which was proved using a modification of Kallibekov's technique. As application, we use our theorem to show that an r.e. set A has supersets of every r.e. degree iff A is not simple.

Index sets of finite classes of recursively enumerable sets

1969 ◽  
Vol 34 (1) ◽  
pp. 39-44 ◽  
Author(s):  
Louise Hay
Keyword(s):  
Index Set ◽  
Recursive Functions ◽  
Recursively Enumerable ◽  
Index Sets ◽  
Recursively Enumerable Sets ◽  
Partial Recursive Functions ◽  
Recursive Isomorphism ◽  
Standard Enumeration

Let q0, q1,… be a standard enumeration of all partial recursive functions of one variable. For each i, let wi = range qi and for any recursively enumerable (r.e.) set α, let θα = {n | wn = α}. If A is a class of r.e. sets, let θA = the index set of A = {n | wn ∈ A}. It is the purpose of this paper to classify the possible recursive isomorphism types of index sets of finite classes of r.e. sets. The main theorem will also provide an answer to the question left open in [2] concerning the possible double isomorphism types of pairs (θα, θβ) where α ⊂ β.

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.

THE COMPLEXITY OF INDEX SETS OF CLASSES OF COMPUTABLY ENUMERABLE EQUIVALENCE RELATIONS

2016 ◽  
Vol 81 (4) ◽  
pp. 1375-1395 ◽  
Author(s):  
URI ANDREWS ◽  
ANDREA SORBI
Keyword(s):  
Open Problem ◽  
Equivalence Classes ◽  
Equivalence Relations ◽  
Index Set ◽  
Index Sets ◽  
Image Position ◽  
Computably Enumerable

AbstractLet$ \le _c $be computable the reducibility on computably enumerable equivalence relations (or ceers). We show that for every ceerRwith infinitely many equivalence classes, the index sets$\left\{ {i:R_i \le _c R} \right\}$(withRnonuniversal),$\left\{ {i:R_i \ge _c R} \right\}$, and$\left\{ {i:R_i \equiv _c R} \right\}$are${\rm{\Sigma }}_3^0$complete, whereas in caseRhas only finitely many equivalence classes, we have that$\left\{ {i:R_i \le _c R} \right\}$is${\rm{\Pi }}_2^0$complete, and$\left\{ {i:R \ge _c R} \right\}$(withRhaving at least two distinct equivalence classes) is${\rm{\Sigma }}_2^0$complete. Next, solving an open problem from [1], we prove that the index set of the effectively inseparable ceers is${\rm{\Pi }}_4^0$complete. Finally, we prove that the 1-reducibility preordering on c.e. sets is a${\rm{\Sigma }}_3^0$complete preordering relation, a fact that is used to show that the preordering relation$ \le _c $on ceers is a${\rm{\Sigma }}_3^0$complete preordering relation.

On Edge-Balance Index Sets of a Nested Graph with Infinite Paths and Power Circles

2013 ◽  
Vol 340 ◽  
pp. 561-566
Author(s):  
Hong Juan Tian ◽  
Yu Ge Zheng
Keyword(s):  
Index Set ◽  
Index Sets ◽  
Class Graph ◽  
Balance Index

We generalize the concept of edge-balanced labeling to the concept of edge-balance index set of graphs. In this article, we define all class of the nested graph with infinite for the integer and investigate the edge-balance index set of a class graph. In particular, we completely determine the edge-balance index sets of the class graph for the integer and solve formula proof and graphic tectonic methods.

A noninitial segment of index sets

1974 ◽  
Vol 39 (2) ◽  
pp. 209-224 ◽  
Author(s):  
Louise Hay
Keyword(s):  
Partial Order ◽  
Initial Segment ◽  
Finite Sets ◽  
Index Set ◽  
Difference Hierarchy ◽  
Recursively Enumerable ◽  
Index Sets ◽  
The Difference ◽  
The One ◽  
Standard Enumeration

Let {Wk}k ≥ 0 be a standard enumeration of all recursively enumerable (r.e.) sets. If A is any class of r.e. sets, let θA denote the index set of A, i.e., θA = {k ∣ Wk ∈ A}. The one-one degrees of index sets form a partial order ℐ which is a proper subordering of the partial order of all one-one degrees. Denote by ⌀ the one-one degree of the empty set, and, if b is the one-one degree of θB, denote by the one-one degree of . Let . Let {Ym}m≥0 be the sequence of index sets of nonempty finite classes of finite sets (classified in [5] and independently, in [2]) and denote by am the one-one degree of Ym. As shown in [2], these degrees are complete at each level of the difference hierarchy generated by the r.e. sets. It was proved in [3] that, for each m ≥ 0,(a) am+1 and ām+1 are incomparable immediate successors of am and ām, and(b) .For m = 0, since Y0 = θ{⌀}, it follows from (a) that(c) .Hence it follows that(d) {⌀, , ao, ā0, a1, ā1 is an initial segment of ℐ.

Rice Theorems For D.R.E. Sets

1975 ◽  
Vol 27 (2) ◽  
pp. 352-365 ◽  
Author(s):  
Louise Hay
Keyword(s):  
Finite Sets ◽  
Index Set ◽  
Recursively Enumerable ◽  
Index Sets

Two of the basic theorems in the classification of index sets of classes of recursively enumerable (r.e.) sets are the following:(i) The index set of a class C of r.e. sets is recursive if and only if C is empty or contains all r.e. sets; and(ii) the index set of a class C or r.e. sets is recursively enumerable if and only if C is empty or consists of all r.e. sets which extend some element of a canonically enumerable class of finite sets.The first theorem is due to Rice [7, p. 364, Corollary B]. The second was conjectured by Rice [7, p. 361] and proved independently by McNaughton, Shapiro, and Myhill [6].

Kleene index sets and functional m-degrees

1983 ◽  
Vol 48 (3) ◽  
pp. 829-840 ◽  
Author(s):  
Jeanleah Mohrherr
Keyword(s):  
Partial Order ◽  
Turing Degree ◽  
Index Set ◽  
Natural Embedding ◽  
Recursive Functions ◽  
Index Sets ◽  
Partial Recursive Functions

AbstractA many-one degree is functional if it contains the index set of some class of partial recursive functions but does not contain an index set of a class of r.e. sets. We give a natural embedding of the r.e. m-degrees into the functional degrees of 0′. There are many functional degrees in 0′ in the sense that every partial-order can be embedded. By generalizing, we show there are many functional degrees in every complete Turing degree.There is a natural tie between the studies of index sets and partial-many-one reducibility. Every partial-many-one degree contains one or two index sets.

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