Relation between the Usual Order and the Enumeration Orders of Elements of r.e. Sets

In section 2, we introduce some new concepts to compare Turing machines and other concepts to compare r.e. sets based on their enumeration orders. In this section, a relation named “uniformity” is defined to compare r.e. sets and Turing machines. We prove that this relation is equivalence. In section 3, we found some relationships between “uniformity” and both one-reducibility and Turing reducibility. Some results in sections 2 and 3 motivated us to extend uniformity to type-2 uniformity in section 4. In this section, we obtained some results based on type-2 uniformity. Finally, in section 6, we introduce a novel structure named by Turing Output Binary Search Tree and show that it is a proper tool to show some ideas about enumeration orders of elements of r.e. sets. We used the standard notions from recursion theory [6]. The set 1,2, … of natural numbers is denoted by . The th recursively enumerable set is denoted by .

In this section, we define some concepts to compare different Turing machines and their generated sets upon their enumeration orders.

Let be an infinite non-empty recursively enumerable set. There are some total computable functions :

such that 1 , 2 , … (If the set is finite then 1 , 2 , … , for and should be a partial function). In this paper, we call function a listing of . Definition 2.1 A listing of an r.e. set is a bijective and surjective computable function :

.

We can assign a listing to each Turing machine. Indeed a listing shows the enumeration order of output elements of the related Turing machine. Proof: It is evident that uniformity on listings is an equivalence relation. This deduced directly from definition. Since the reflexivity and symmetric properties hold for “” on listings, we can deduce easily that these properties hold for “” on sets. Now it is sufficient to prove that the transitivity property holds for this relation on sets. Consider three sets , , such that ~ and ~ . Then there exist two listings of and of such that ~ . According to lemma 2.3 there is a listing of such that ~ . Since transitivity holds for “~” on listings, we can deduce that ~ , so ~ .

Let be an r.e. subset of . We say

is the “uniformity equivalence class” of . We illustrate the above concepts by the following example. In the following, we introduce two sets that are not uniform. Henceforth, we consider

Lemma 2.7 Two sets and are not uniform.

Proof: For the sake of a contradiction, assume that these two sets are uniform. The identity function : is a listing of . Then, according to lemma 2.3, there exists a listing of such that ~ . Therefore, for all , , . But this cannot be true. This is a contradiction. □ 3 One-reducibility, Turing-reducibility and uniformity

In this section, we want to explore some relationships between both one-reducibility & Turing-reducibility and uniformity on sets. First, we investigate one-reducibility equivalence classes.

Lemma 3.1 Consider a non-decidable r.e. set such that there is not any Turing machine to obtain minimum element of it. Two sets and 1 are not uniform.

Proof: For the sake of a contradiction, assume that these two sets are uniform. Consider a listing of such that 1 1. Then there exists a listing of such that ~ . Since 1 is the minimum element of then 1 should be the minimum element of . This shows that we could compute the minimum element of . It leads us to Contradiction. □ Lemma 3.2 Two sets 2 : and 1 do not belong to the same onereducibility equivalence class.

Proof: For the sake of a contradiction, assume that these two sets belong to the same onereducibility equivalence class. In the sense of the definition of one-reducibility, two sets A and B are of equal cardinality. A is an infinite set and in contrast, the cardinality of B is 1. This is a contradiction. □ Proposition 3.3 If two r.e. sets belong to same “one-reducibility equivalence class”, then they do not belong necessarily to same “uniformity equivalence class”.

Proof: Consider a non-decidable r.e. set such that there is not any Turing machine for obtain minimum element of it. In the lemma 3.1, we proved that and 1 are not uniform, whereas these sets belong to on-reducibility equivalence class .

Proposition 3.4 If two r.e. sets belong to the same “uniformity equivalence class”, then they do not belong necessarily to the same “one-reducibility equivalence class”.

In the example 2.5, we showed that two sets 2 : and 1 are uniform and in the lemma 3.2, we showed that they do not belong to the

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