Enumeration Order complexity Equivalency

Throughout this article we develop and change the definitions and the ideas in [1], in order to consider the efficiency of functions and complexity time problems. The central idea here is effective enumeration and listing, and efficiency of function which is defined between two sets proposed in basic definitions. More in detail, it might be that h and were co-order but the velocity of them be different. Introduction: Every recursive set could be ordered increasingly and every c.e set which could be ordered increasingly is a recursive set , so increasing order type is related tightly to recursive sets. In [1] the posing question is: What about the other c.e sets? Which order types could be associated to an arbitrary c.e set? In that article trying to answer the above question, we define some equivalence relations and we investigate different properties of that. The material in [1] could be considered In Computability field setting. We could develop the same ideas and approximately in a parallel way in Complexity Theory. So, we define different concepts to cope with some difficulties which arise. Here, we repeat the definition 1, 2 in [1], with some essential modifications. Definition 1: A listing of an infinite c.e set ⊆ N is a bijective computable function : N → . Definition 2: 1. Two listings h, are co-order, h~ , if h < h ⟺ < for all , ∈ N. 2. Two c.e subsets A and B of N with equal cardinality are co-order, A~B, if there exist listings h of and of such that h~ . Definition 3: Let h enumerate . In the case that Turing machine computes h, h be the essential steps to halt Turing machine . we have: h = h ! "! #h $h h%& ! " We apply h to compare the time of two different listing. Definition 4: We know listing h strictly more rapid than listing if for any ∈ N we have: h < Definition 5: We know listing h more rapid than listing if there exists ∈ N such that: ∀ > ) h +,< ) * +,Definition 6: Let be a c.e set. . / represents time complexity of enumerating the set . By supposing 00 as the set of all listing of , It is defined as follows: . / ∈ 12 3 ∃h ∈ 00 ! $h h% h h%! % 12 3 5 ! $ %$h Definition 7: Let be a c.e set. By supposing 00 as the set of all listing of , 6 . / is defined as follows: 6 . / ∈ 12 3 ∃h ∈ 00 ! $h h% h h%! % 12 3 − 5 ! $ %$h Example1: Let be the set of all prime numbers, as we know there are infinite numbers of algorithms to produce prime numbers. Since = 89. / ∈ 8, there is a deterministic algorithm for this problem in polynomial time. Consequently, there is a deterministic Turing machine which enumerates in polynomial time, in other words . / ∈ 1 : . Example 2: Note that = ; is a 68 − $ "& problem ̧so there is a non deterministic Turing machine which it enumerates in polynomial time, equivalently 6 . / ∈ 1 : . Remark: It is notable to know that there are non recursive c.e sets like such that . / ∈ 12 =3, and there are non recursive c.e set like such that, 6 . / ∈ 1 : . Definition 8: The c.e set is P co-order if there are sets and = ∈ N such that . / ∈ 1 : and ~ . Definition 9: The c.e set is NP co 1 : and ~ . Theorem 1: Any P co-order set is Proof: Straight forward. Theorem 2: Any recursive set is P co Two equivalence relations PU and NPU: Definition 10: is non deterministic polynomial reducible to function , such that there is a Turing machine time, such that: > ∈ ? > ∈ Definition 11: Two sets and and <*@ . Definition 12: Two sets , ⊆ N Definition 13: Two sets , ⊆ N Conclusion 1: A ABC B D A AE B Conclusion2: Let and belong to two different Turing classes. Conclusion3: Suppose that for two different FGHIJKLK. A@M iff 8 = 68 Theorem 3: Let and be two different subsets of -order if there are sets and = ∈ N such that