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Introduction to Enumeration Reducibility

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Introduction to Enumeration Reducibility Introduction to Enumeration Reducibility Charles M. Harris Department Of Mathematics University of Leeds Computability Theory Seminar, Autumn 2009 (University Of Leeds) Introduction to Enumeration Reducibility CTS 2009 1 / 34 Outline 1 Introducing Enumeration Reducibility The Basic Definitions Examples Alternative definitions 2 Properties of De Definition of 0e Lattice Theoretic Properties (University Of Leeds) Introduction to Enumeration Reducibility CTS 2009 2 / 34 Outline 3 The Enumeration Jump Operator The Enumeration Jump (over sets) The Enumeration Jump Operator 4 Embedding DT into De Total Degrees Definition of the Embedding Quasi-minimal Enumeration Degrees (University Of Leeds) Introduction to Enumeration Reducibility CTS 2009 3 / 34 Outline 5 Further Properties of De Exact Pairs and Branching Density and Minimality Next Week (University Of Leeds) Introduction to Enumeration Reducibility CTS 2009 4 / 34 Subsection Guide 1 Introducing Enumeration Reducibility The Basic Definitions Examples Alternative definitions 2 Properties of De Definition of 0e Lattice Theoretic Properties (University Of Leeds) Introduction to Enumeration Reducibility CTS 2009 5 / 34 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 2 ! x 2 A iff 9u [ hx; ui 2 W & Du ⊆ B ] This is written A = ΦW (B). (University Of Leeds) Introduction to Enumeration Reducibility CTS 2009 6 / 34 Subsection Guide 1 Introducing Enumeration Reducibility The Basic Definitions Examples Alternative definitions 2 Properties of De Definition of 0e Lattice Theoretic Properties (University Of Leeds) Introduction to Enumeration Reducibility CTS 2009 7 / 34 Examples of sets W and X such that x 2 A iff 9D ( hx; Di 2 W & D ⊆ X ) A≤e A via the c.e. set W = f hn; fngi j n 2 ! g. If A is c.e. and B is any set, then A≤e B via the c.e. set W = f hn; ;i j n 2 A g. −1 If f is computable and A = f (B) (i.e. A≤m B), then A≤e B via the c.e. set W = f hn; ff (n)gi j n 2 ! g. (University Of Leeds) Introduction to Enumeration Reducibility CTS 2009 8 / 34 Examples of sets W and X such that x 2 A iff 9D ( hx; Di 2 W & D ⊆ X ) A≤e A via the c.e. set W = f hn; fngi j n 2 ! g. If A is c.e. and B is any set, then A≤e B via the c.e. set W = f hn; ;i j n 2 A g. −1 If f is computable and A = f (B) (i.e. A≤m B), then A≤e B via the c.e. set W = f hn; ff (n)gi j n 2 ! g. (University Of Leeds) Introduction to Enumeration Reducibility CTS 2009 8 / 34 Examples of sets W and X such that x 2 A iff 9D ( hx; Di 2 W & D ⊆ X ) A≤e A via the c.e. set W = f hn; fngi j n 2 ! g. If A is c.e. and B is any set, then A≤e B via the c.e. set W = f hn; ;i j n 2 A g. −1 If f is computable and A = f (B) (i.e. A≤m B), then A≤e B via the c.e. set W = f hn; ff (n)gi j n 2 ! g. (University Of Leeds) Introduction to Enumeration Reducibility CTS 2009 8 / 34 Examples of sets W and X such that x 2 A iff 9D ( hx; Di 2 W & D ⊆ X ) A≤e A via the c.e. set W = f hn; fngi j n 2 ! g. If A is c.e. and B is any set, then A≤e B via the c.e. set W = f hn; ;i j n 2 A g. −1 If f is computable and A = f (B) (i.e. A≤m B), then A≤e B via the c.e. set W = f hn; ff (n)gi j n 2 ! g. (University Of Leeds) Introduction to Enumeration Reducibility CTS 2009 8 / 34 A non trivial example Reminder A is regressive if there exists an enumeration fa0; a1 :::g 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 h(n; x) = f n(x) (so h is partial computable). Then A≤e B via W = f hy; fxgi j 9n [ h(n; x)# = y ] g : (University Of Leeds) Introduction to Enumeration Reducibility CTS 2009 9 / 34 A non trivial example Reminder A is regressive if there exists an enumeration fa0; a1 :::g 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 h(n; x) = f n(x) (so h is partial computable). Then A≤e B via W = f hy; fxgi j 9n [ h(n; x)# = y ] g : (University Of Leeds) Introduction to Enumeration Reducibility CTS 2009 9 / 34 Subsection Guide 1 Introducing Enumeration Reducibility The Basic Definitions Examples Alternative definitions 2 Properties of De Definition of 0e Lattice Theoretic Properties (University Of Leeds) Introduction to Enumeration Reducibility CTS 2009 10 / 34 Enumeration reducibility - alternative views Selman’s Definition X X A≤e B iff 8X [ B 2 Σ1 ) A 2 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. (University Of Leeds) Introduction to Enumeration Reducibility CTS 2009 11 / 34 Enumeration reducibility - alternative views Selman’s Definition X X A≤e B iff 8X [ B 2 Σ1 ) A 2 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. (University Of Leeds) Introduction to Enumeration Reducibility CTS 2009 11 / 34 Enumeration reducibility - alternative views Selman’s Definition X X A≤e B iff 8X [ B 2 Σ1 ) A 2 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. (University Of Leeds) Introduction to Enumeration Reducibility CTS 2009 11 / 34 Enumeration reducibility - alternative views Selman’s Definition X X A≤e B iff 8X [ B 2 Σ1 ) A 2 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. (University Of Leeds) Introduction to Enumeration Reducibility CTS 2009 11 / 34 Subsection Guide 1 Introducing Enumeration Reducibility The Basic Definitions Examples Alternative definitions 2 Properties of De Definition of 0e Lattice Theoretic Properties (University Of Leeds) Introduction to Enumeration Reducibility CTS 2009 12 / 34 The Degree of the Computably Enumerable Sets. If A is any set and W is c.e. then W ≤e A Hence 0e comprises the class of c.e. sets (University Of Leeds) Introduction to Enumeration Reducibility CTS 2009 13 / 34 The Degree of the Computably Enumerable Sets. If A is any set and W is c.e. then W ≤e A Hence 0e comprises the class of c.e. sets (University Of Leeds) Introduction to Enumeration Reducibility CTS 2009 13 / 34 Subsection Guide 1 Introducing Enumeration Reducibility The Basic Definitions Examples Alternative definitions 2 Properties of De Definition of 0e Lattice Theoretic Properties (University Of Leeds) Introduction to Enumeration Reducibility CTS 2009 14 / 34 Lattice Theoretic Properties There exists a pair of sets A,B with no g.l.b. under≤e . However every such pair A,B, always has l.u.b. A ⊕ B. Thus De is an upper semi lattice. (University Of Leeds) Introduction to Enumeration Reducibility CTS 2009 15 / 34 Lattice Theoretic Properties There exists a pair of sets A,B with no g.l.b. under≤e . However every such pair A,B, always has l.u.b. A ⊕ B. Thus De is an upper semi lattice. (University Of Leeds) Introduction to Enumeration Reducibility CTS 2009 15 / 34 Lattice Theoretic Properties There exists a pair of sets A,B with no g.l.b. under≤e . However every such pair A,B, always has l.u.b. A ⊕ B. Thus De is an upper semi lattice. (University Of Leeds) Introduction to Enumeration Reducibility CTS 2009 15 / 34 Subsection Guide 3 The Enumeration Jump Operator The Enumeration Jump (over sets) The Enumeration Jump Operator 4 Embedding DT into De Total Degrees Definition of the Embedding Quasi-minimal Enumeration Degrees (University Of Leeds) Introduction to Enumeration Reducibility CTS 2009 16 / 34 The Enumeration Jump (over sets) Notation For any set A, KA =def f x j x 2 Φx (A) g , JA =def KA ⊕ KA , (0) (k+1) JA =def JA = A and JA =def J (k) . JA Lemma For any sets A and B, KA is 1-complete for Enum(A), A≤e B iff A≤1 KB iff KA ≤1 KB . Forward (University Of Leeds) Introduction to Enumeration Reducibility CTS 2009 17 / 34 The Enumeration Jump (over sets) Notation For any set A, KA =def f x j x 2 Φx (A) g , JA =def KA ⊕ KA , (0) (k+1) JA =def JA = A and JA =def J (k) .
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