Enumeration Reducibility Properties of ≤se DT and Dse Jump Operators Some Properties of Dse Symmetric Enumeration Reducibility Charles M. Harris Department Of Mathematics University of Leeds Computability Theory Seminar, Autumn 2005 Enumeration Reducibility Properties of ≤se DT and Dse Jump Operators Some Properties of Dse Outline 1 Enumeration Reducibility The Basic Definitions Examples Alternative definitions 2 Properties of ≤se Defining ≤se Classifying ≤se Basic Structure of Dse Enumeration Reducibility Properties of ≤se DT and Dse Jump Operators Some Properties of Dse Outline 1 Enumeration Reducibility The Basic Definitions Examples Alternative definitions 2 Properties of ≤se Defining ≤se Classifying ≤se Basic Structure of Dse Enumeration Reducibility Properties of ≤se DT and Dse Jump Operators Some Properties of Dse Outline 3 DT and Dse Characteristic Degrees Definition of the Embedding Quasi-minimal Degrees 4 Jump Operators The Enumeration Jump The Symmetric Enumeration Jump Enumeration Reducibility Properties of ≤se DT and Dse Jump Operators Some Properties of Dse Outline 3 DT and Dse Characteristic Degrees Definition of the Embedding Quasi-minimal Degrees 4 Jump Operators The Enumeration Jump The Symmetric Enumeration Jump Enumeration Reducibility Properties of ≤se DT and Dse Jump Operators Some Properties of Dse Outline 5 Some Properties of Dse The c.e. and co-c.e. degrees Minimal and Exact Pairs Next Week Enumeration Reducibility Properties of ≤se DT and Dse Jump Operators Some Properties of Dse Subsection Guide 1 Enumeration Reducibility The Basic Definitions Examples Alternative definitions 2 Properties of ≤se Defining ≤se Classifying ≤se Basic Structure of Dse Enumeration Reducibility Properties of ≤se DT and Dse Jump Operators Some Properties of Dse What is enumeration reducibility? Definition (Intuitive) A≤e B if there exists an effective procedure that, given any enumeration of B, computes an enumeration A. Definition (Formal) A≤e B if there exists a c.e. set W such that for all x ∈ ω x ∈ A iff ∃u [ hx, ui ∈ W & Du ⊆ B ] This is written A = ΦW (B). Enumeration Reducibility Properties of ≤se DT and Dse Jump Operators Some Properties of Dse What is enumeration reducibility? Definition (Intuitive) A≤e B if there exists an effective procedure that, given any enumeration of B, computes an enumeration A. Definition (Formal) A≤e B if there exists a c.e. set W such that for all x ∈ ω x ∈ A iff ∃u [ hx, ui ∈ W & Du ⊆ B ] This is written A = ΦW (B). Enumeration Reducibility Properties of ≤se DT and Dse Jump Operators Some Properties of Dse What is enumeration reducibility? Definition (Intuitive) A≤e B if there exists an effective procedure that, given any enumeration of B, computes an enumeration A. Definition (Formal) A≤e B if there exists a c.e. set W such that for all x ∈ ω x ∈ A iff ∃u [ hx, ui ∈ W & Du ⊆ B ] This is written A = ΦW (B). Enumeration Reducibility Properties of ≤se DT and Dse Jump Operators Some Properties of Dse Subsection Guide 1 Enumeration Reducibility The Basic Definitions Examples Alternative definitions 2 Properties of ≤se Defining ≤se Classifying ≤se Basic Structure of Dse Enumeration Reducibility Properties of ≤se DT and Dse Jump Operators Some Properties of Dse Examples of sets W and X such that x ∈ A iff ∃D ( hx, Di ∈ W & D ⊆ X ) A≤e A via the c.e. set W = { hn, {n}i | n ∈ ω }. If A is c.e. and B is any set, then A≤e B via the c.e. set W = { hn, ∅i | n ∈ A }. −1 If f is computable and A = f (B) (i.e. A≤m B), then A≤e B via the c.e. set W = { hn, {f (n)}i | n ∈ ω }. Enumeration Reducibility Properties of ≤se DT and Dse Jump Operators Some Properties of Dse Examples of sets W and X such that x ∈ A iff ∃D ( hx, Di ∈ W & D ⊆ X ) A≤e A via the c.e. set W = { hn, {n}i | n ∈ ω }. If A is c.e. and B is any set, then A≤e B via the c.e. set W = { hn, ∅i | n ∈ A }. −1 If f is computable and A = f (B) (i.e. A≤m B), then A≤e B via the c.e. set W = { hn, {f (n)}i | n ∈ ω }. Enumeration Reducibility Properties of ≤se DT and Dse Jump Operators Some Properties of Dse Examples of sets W and X such that x ∈ A iff ∃D ( hx, Di ∈ W & D ⊆ X ) A≤e A via the c.e. set W = { hn, {n}i | n ∈ ω }. If A is c.e. and B is any set, then A≤e B via the c.e. set W = { hn, ∅i | n ∈ A }. −1 If f is computable and A = f (B) (i.e. A≤m B), then A≤e B via the c.e. set W = { hn, {f (n)}i | n ∈ ω }. Enumeration Reducibility Properties of ≤se DT and Dse Jump Operators Some Properties of Dse Examples of sets W and X such that x ∈ A iff ∃D ( hx, Di ∈ W & D ⊆ X ) A≤e A via the c.e. set W = { hn, {n}i | n ∈ ω }. If A is c.e. and B is any set, then A≤e B via the c.e. set W = { hn, ∅i | n ∈ A }. −1 If f is computable and A = f (B) (i.e. A≤m B), then A≤e B via the c.e. set W = { hn, {f (n)}i | n ∈ ω }. Enumeration Reducibility Properties of ≤se DT and Dse Jump Operators Some Properties of Dse A non trivial example Reminder A is regressive if there exists an enumeration {a0, a1 ...} of A and a partial computable function f such that f (0)↓ = a0 and f (an+1)↓ = an for all n ≥ 0. Example If A is regressive and B ⊆ A is infinite then A≤e B. In effect, suppose that partial computable f regresses A. Let n h(n, x) = f (x) (so h is partial computable). Then A≤e B via W = { hy, {x}i | ∃n [ h(n, x)↓ = y ] } . Enumeration Reducibility Properties of ≤se DT and Dse Jump Operators Some Properties of Dse A non trivial example Reminder A is regressive if there exists an enumeration {a0, a1 ...} of A and a partial computable function f such that f (0)↓ = a0 and f (an+1)↓ = an for all n ≥ 0. Example If A is regressive and B ⊆ A is infinite then A≤e B. In effect, suppose that partial computable f regresses A. Let n h(n, x) = f (x) (so h is partial computable). Then A≤e B via W = { hy, {x}i | ∃n [ h(n, x)↓ = y ] } . Enumeration Reducibility Properties of ≤se DT and Dse Jump Operators Some Properties of Dse Subsection Guide 1 Enumeration Reducibility The Basic Definitions Examples Alternative definitions 2 Properties of ≤se Defining ≤se Classifying ≤se Basic Structure of Dse Enumeration Reducibility Properties of ≤se DT and Dse Jump Operators Some Properties of Dse Enumeration reducibility - alternative views Selman’s Definition X X A≤e B iff ∀X [ B ∈ Σ1 ⇒ A ∈ Σ1 ]. Scott’s Definition A≤e B iff there exists some closed LAMBDA term u such that A = [[(u)B ]]. A Turing machine definition A≤e B iff A is positive non deterministic Turing reducible to B. Enumeration Reducibility Properties of ≤se DT and Dse Jump Operators Some Properties of Dse Enumeration reducibility - alternative views Selman’s Definition X X A≤e B iff ∀X [ B ∈ Σ1 ⇒ A ∈ Σ1 ]. Scott’s Definition A≤e B iff there exists some closed LAMBDA term u such that A = [[(u)B ]]. A Turing machine definition A≤e B iff A is positive non deterministic Turing reducible to B. Enumeration Reducibility Properties of ≤se DT and Dse Jump Operators Some Properties of Dse Enumeration reducibility - alternative views Selman’s Definition X X A≤e B iff ∀X [ B ∈ Σ1 ⇒ A ∈ Σ1 ]. Scott’s Definition A≤e B iff there exists some closed LAMBDA term u such that A = [[(u)B ]]. A Turing machine definition A≤e B iff A is positive non deterministic Turing reducible to B. Enumeration Reducibility Properties of ≤se DT and Dse Jump Operators Some Properties of Dse Enumeration reducibility - alternative views Selman’s Definition X X A≤e B iff ∀X [ B ∈ Σ1 ⇒ A ∈ Σ1 ]. Scott’s Definition A≤e B iff there exists some closed LAMBDA term u such that A = [[(u)B ]]. A Turing machine definition A≤e B iff A is positive non deterministic Turing reducible to B. Enumeration Reducibility Properties of ≤se DT and Dse Jump Operators Some Properties of Dse Subsection Guide 1 Enumeration Reducibility The Basic Definitions Examples Alternative definitions 2 Properties of ≤se Defining ≤se Classifying ≤se Basic Structure of Dse Enumeration Reducibility Properties of ≤se DT and Dse Jump Operators Some Properties of Dse What is Symmetric Enumeration Reducibility. Definition A≤se B iff A≤e B and A≤e B. Remark If B 6= ∅, N and A≤se B due to e-reductions A = Φa(B) and A = Φb(B) then A≤se B via W = { hn, D ∪ {d}i | hn, Di ∈ Φa } S 0 { hn, D ∪ {d }i | hn, Di ∈ Φb } where d ∈ B and d 0 ∈ B . Enumeration Reducibility Properties of ≤se DT and Dse Jump Operators Some Properties of Dse What is Symmetric Enumeration Reducibility. Definition A≤se B iff A≤e B and A≤e B. Remark If B 6= ∅, N and A≤se B due to e-reductions A = Φa(B) and A = Φb(B) then A≤se B via W = { hn, D ∪ {d}i | hn, Di ∈ Φa } S 0 { hn, D ∪ {d }i | hn, Di ∈ Φb } where d ∈ B and d 0 ∈ B .