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) .