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Symmetric Enumeration Reducibility SYMMETRIC ENUMERATION REDUCIBILITY CHARLES M. HARRIS DEPARTMENT OF MATHEMATICS, LEEDS, UK. Abstract. Symmetric Enumeration reducibility ( ≤se ) is a subrelation of Enu- meration reducibility (≤e) in which both the positive and negative informa- tion content of sets is compared. In contrast with Turing reducibility (≤T) however, the positive and negative parts of this relation are separate. A basic classification of ≤se in terms of standard reducibilities is carried out and it is shown that the natural embedding of the Turing degrees into the Enumera- tion degrees easily translates to this context. A generalisation of the relativised Arithmetical Hierarchy is achieved by replacing the relation c.e. in by ≤e and ≤T by ≤se in the underlying framework of the latter. 1. Introduction The structure of the relativised Arithmetical Hierarchy is dictated by the relation computably enumerable in (c.e. in) in the sense that a set belongs to level n + 1 of the hierarchy if either it or its complement is c.e. in some set belonging to level n. This of course means that each level is the union of a positive (Σ) class and a negative (Π) class. The intersection (∆) of these classes comprises, by definition, those sets B such that B is Turing reducible to (i.e. B and B are both c.e. in) a set in the antecedent level. A natural question to ask is whether one can refine the framework of the rela- tivised Arithmetical Hierarchy to the context of Enumeration Reducibility. In other words, can one define a similar hierarchy in which the levels are dictated by the relation enumeration reducible to (≤e) and which is identical to the Arithmetical Hierarchy when relativised to graphs of characteristic functions (i.e. Turing embed- ded sets, see Section 4). One approach to this question, is to find an appropriate relation to define the intersection (∆) levels of the structure (in the place of ≤T ). This is the initial motivation behind the introduction of Symmetric Enumeration Reducibility since the latter allows us to define a hierarchy which is a generalisation of the relativised Arithmetical Hierarchy in the required sense. 2. Preliminaries Background Notation. We let N denote the set of natural numbers and A, B, . denote subsets of N. Lower case letters n, x, . and f, g, . represent numbers and functions (from N to N) respectively, whereas A, B,... represent classes of sets. A denotes the complement of A. The set { n · x + m | x ∈ A } is written nA + m and 2A ∪ 2B +1 is written A ⊕ B. We use h , i to denote the standard diagonal coding function defined by hx, yi = 1/2(x2 + y2 + 2xy + 3x + y). The characteristic Date: March 31, 2006. 1 2 CHARLES M. HARRIS, UNIVERSITY OF LEEDS function of A is written cA, and for any function f, its graph is written F (and so CA stands for the graph of cA). We assume the availability of effective enumerations of (oracle) Turing machines ϕ0, ϕ1,..., computably enumerable (c.e) sets W0,W1,..., 0 and finite sets D0,D1,... Note that, to simplify notation, we usually use D, D etc. to denote both the finite sets themselves and their indices. For example if i, j are the indices of D, D0 then hD, D0i is shorthand for hi, ji. Basic Reducibilities. We assume the standard multitape Turing machine model for computing partial functions and we suppose an oracle Turing machine to be equipped with a function oracle. We say that the set A is Turing reducible to the set B (A≤T B) if there is an oracle machine ϕ that computes cA when equipped B with oracle cB (written cA ' ϕ ). A is said to be computably enumerable in B (A c.e. in B) if, A is the range of some function f computable in B or, equivalently, B if A = { x | ϕ (x) ↓ } for some oracle Turing machine ϕ. KB denotes the set B (n) (0) (n+1) { x | ϕx (x) ↓ } and KB denotes the iterated form of KB: KB = B and KB = (n) (n) K (n) . K and K are shorthand for K∅ and K∅ . For Turing reductions we use KB Q(ϕ, x, B) to denote the set of oracle queries made by ϕB on input x. Likewise we use Q+(ϕ, x, B) to denote the set of positive queries of the computation and Q−(ϕ, x, B) to denote the set of negative queries. Note that, for fixed x (and ϕ) these sets are c.e. in B and, if ϕB(x) ↓ they are computable in B. We say that A is many one reducible to B (A≤m B) if there is a computable function f such that A = f −1(B). Furthermore, if f is one-one, A is said to be one-one reducible to B (A≤1 B). We say that A is enumeration reducible to B (A≤e B) if there exists a computably enumerable set W such that, for all x, x ∈ A iff ∃D( hx, Di ∈ W & D ⊆ B ) and in this case we also say that A≤e B via W. Similarly, the e-th enumeration operator Φe is defined such that, for any set A, Φe(B) = { x | ∃D( hx, Di ∈ We & D ⊆ B ) } . A is said to be positive reducible to B,(A ≤p B) if there exists a computable function f such that, for all x ≥ 0, x ∈ A ⇔ ∃y(y ∈ Df(x) & Dy ⊆ B). We say that A is wtt-reducible to B (A≤wtt B), if there exists a Turing machine ϕ and B computable function f such that cA ' ϕ and such that for all x ≥ 0, Q(ϕe, x, B) ⊆ {0, . , f(x)}. degr(A) denotes the degree of A under the reducibility ≤r , i.e. the class { B | B ≡r A }. We use ar, br,... to denote the degrees derived according to this definition and hDr, ≤i to denote the corresponding degree structure (with ≤ the ordering induced by ≤r ). Subscripts are dropped if the context is clear. A is said to be r-hard for a class C if X ≤r A for all X in C and A is said to be r- complete for C if A also belongs to C. We use the shorthand Comp(A), Enum(A) and Ce(A) to denote the classes { E | E R A } such that (respectively) R is ≤T , ≤e or “ c.e. in ”. Accordingly, we use Comp and Ce to denote the classes of com- putable and c.e. sets. Also we will employ the abbreviations r-reduction, r-degree etc. when appropriate. String Notation. A string is a partial function σ : N → {0, 1} with finite do- main. λ denotes the empty string and |σ| the length of σ (i.e. the cardinality of its domain). For (s, i) ∈ { (+, 1) , (−, 0) }, we use σs to denote the set { n | σ(n)↓ = i } SYMMETRIC ENUMERATION REDUCIBILITY 3 s s s and (σA) to denote the set { n | n ∈ A & σ(n)↓ = i } (and so σ = (σN) for s ∈ {+, −}). If the domain of σ is an initial segment of N, σ is said to be an initial segment. Note that this means that, if |σ| = n + 1 the domain of σ is {0, . , n}. 0 We use the shorthand σ = σ b(i) to denote the extension of σ of length |σ| + 1, such that σ0(|σ|) = i. 3. Introduction to Symmetric Enumeration Reducibility Enumeration reducibility compares the positive information content of two sets. Symmetric enumeration reducibility, as we will see, compares both positive and negative information content. We will now introduce this reducibility and consider how it relates to other standard reducibilities. Firstly, however we draw the reader’s attention to the fact that Alan Selman defined this reducibility in Section 4 of [Sel72] and exhibited some of its basic properties. In particular Selman noted the inclusion ≤m ⊆≤se ⊆≤T and proved Theorem 3.6 (below) relative to the pair (≤tt , ≤se ). Definition 3.1. For any sets A and B, A is defined to be symmetric enumeration reducible to B (A≤se B) if A≤e B and A≤e B. Notation 1. For any set A, s-Enum(A) denotes the class { E | E≤se A }. Note 3.2. Clearly ≤se inherits reflexivity and transitivity from ≤e . It thus gives rise to a degree structure (hDse, ≤i). The least upper bound of any two degrees ase, bse (written ase ∪ bse) always exists: it is the degree of A ⊕ B for any A ∈ ase and B ∈ bse . Therefore hDse, ≤i is an upper semi lattice. The zero element (0se) of hDse, ≤i is Comp. Each of these properties is easily checked. Lemma 3.3. For any sets A and B, if A≤se B then A≤e B and A≤T B. In other words, T ≤se ⊆ ≤e ≤T . Moreover, this inclusion is proper. Proof. Since ≤se is a subrelation of ≤e by definition, in order to prove the inclu- sion it suffices to note that, for any sets A and B, A≤e B implies that A c.e. in B. Also, CK ≤r K for r ∈ {e, T} whereas CK se K (since this would imply K ≤e K). Thus the inclusion is proper. Theorem 3.4. ≤p ⊆ ≤se . Proof. Let A and B be any sets such that A≤p B and suppose that f is a com- putable function that witnesses this reduction in the sense that, for all x ≥ 0, x ∈ A ⇔ ∃y(y ∈ Df(x) & Dy ⊆ B). Now define the two c.e.
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