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The Computably Enumerable Sets: a Partial Survey with Questions

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The Computably Enumerable Sets: a Partial Survey with Questions Ᏹ Orbits Complete Sets D-Maximal Sets Low Sets The Computably Enumerable Sets: a Partial Survey with Questions Peter Cholak University of Notre Dame Department of Mathematics [email protected] June 2012 Ᏹ Orbits Complete Sets D-Maximal Sets Low Sets The Computably Enumerable Sets, Ᏹ • We is the domain of the eth Turing machine. • The structure Ᏹ is (fWe : e 2 !g; ⊆), the c.e. (r.e.) sets under inclusion. Definability in Ᏹ: • A set A is computable iff A and A are both computable enumerable. • A set A is finite iff every subset of A is computable. • 0; !; [, \, and t (disjoint union) are definable from ⊆ in Ᏹ. Ᏹ Orbits Complete Sets D-Maximal Sets Low Sets Definable Sets • A set A is simple iff for every (c.e.) set B if A \ B is empty then B is finite. • A set M is maximal iff for every (c.e.) set B either there is a finite set F such that B ⊆ M [ F or ! ⊆ M [ B [ F. Ᏹ Orbits Complete Sets D-Maximal Sets Low Sets Soare’s 1974 Result Theorem (Soare) The maximal sets form an orbit of Ᏹ. Definition X is automorphic to Y , X ≈ Y , iff there is an automorphism of Ᏹ such that Φ(X) = Y . Is every orbit equally “nice” as the maximal sets? Ᏹ Orbits Complete Sets D-Maximal Sets Low Sets CK The Scott Rank of Ᏹ is !1 + 1 Theorem (Cholak, Downey, Harrington) There is an c.e. set A such that the set IA = fi : A is automorphic to Wig 1 is Σ1-complete. Corollary The complexity of the L!1;! formula describing the above orbit is as high as possible. Ᏹ Orbits Complete Sets D-Maximal Sets Low Sets Orbits of Simple Sets Let Φ be an automorphism of Ᏹ and f be such that Φ(We) = Wf (e). We use f to describe Φ. So Φ is effective iff f is also. Theorem (Cholak and Harrington) 0 Two simple sets are automorphic iff they are ∆6 automorphic. Theorem (Harrington 2012) The complexity of the L!1;! formula describing the orbit of any simple set is very low (close to 6). Ᏹ Orbits Complete Sets D-Maximal Sets Low Sets More Orbits Theorem 0 For all α ≥ 8, there is a properly ∆α orbit (an orbit which is 0 an orbit under ∆α automorphisms). Conjecture We can build the above orbits to have arbitrary complexity in terms of the L!1;! formula describing the orbit (close to α). Ᏹ Orbits Complete Sets D-Maximal Sets Low Sets Turing Complete Sets Definition A set c.e. A is Turing complete iff for every c.e. set B, B is Turing reducible to A, B ≤T A. Question (Completeness) Which c.e. sets are automorphic to complete sets? The orbits in the previous results all contain complete sets. Question Is the above question an arithmetical question? Ᏹ Orbits Complete Sets D-Maximal Sets Low Sets Prompt Sets Definition A is promptly simple iff there is computable function p such that for all e, if We is infinite then there is a x and s with x 2 (We;at s \ Ap(s)). Definition A c.e. set A is prompt iff there is computable function p such that for all e, if We is infinite then there is a x and s with x 2 We;at s and As u x ≠ Ap(s) u x. Theorem (Cholak, Downey, Stob) All prompt simple sets are automorphic to a complete set. Ᏹ Orbits Complete Sets D-Maximal Sets Low Sets Almost Prompt Sets Definition X = (We1 − We2 ) [ (We3 − We4 ) [ : : : (We2n−1 − We2n ) iff X is 2n-c.e. and X is 2n + 1-c.e. iff X = Y [ We, where Y is 2n-c.e. Definition n Let Xe be the eth n-c.e. set. A is almost prompt iff there is a computable nondecreasing function p(s) such that for all e n n and n if Xe = A then (9x)(9s)[x 2 Xe;s and x 2 Ap(s)]. Lemma Prompt implies almost prompt. So every Turing complete set is almost prompt. Theorem (Harrington, Soare) All almost prompt sets are automorphic to a complete set. Ᏹ Orbits Complete Sets D-Maximal Sets Low Sets Tardy Sets Definition D is 2-tardy iff for every computable nondecreasing function 2 2 p(s) there is an e such that Xe = D and (8x)(8s)[if x 2 Xe;s then x 62 Dp(s)] Theorem (Harrington, Soare) There are realizable Ᏹ definable properties Q(D) and P(D; C) such that • Q(D) implies that D is 2-tardy (so not Turing complete), • if there is a C such that P(D; C) and D is 2-tardy then Q(D) (and D is high), Ᏹ Orbits Complete Sets D-Maximal Sets Low Sets n-Tardy Sets Definition D is n-tardy iff for every computable nondecreasing function n p(s) there is an e such that Xe = D and n (8x)(8s)[if x 2 Xe;s then x 62 Dp(s)]. Theorem (Cholak, Gerdes, Lange) There are realizable Ᏹ definable properties Qn(D) such that Qn(D) implies that D is properly n-tardy (so not Turing complete). Question If D is not automorphic to a complete set must D satisfy some Qn? Ᏹ Orbits Complete Sets D-Maximal Sets Low Sets Degrees of n-Tardy sets Splits of 2-tardys are 3-tardy but Theorem (Cholak, Gerdes, Lange) There is a 3-tardy that is not computable in any 2-tardy. Question Is there an n + 1-tardy set that is not computed by any n-tardy set? Is there a very tardy sets which is not computed by any n-tardy? Ᏹ Orbits Complete Sets D-Maximal Sets Low Sets Questions about Tardiness Question How do the following sets of degrees compare: • the highn hemimaximal degrees, • the tardy degrees, • for each n, fd : there is a n-tardy D such that d ≤T Dg, • fd : there is a 2-tardy D such that Q(D) and d ≤T Dg, • fd : there is a A 2 d which is not automorphic to a complete setg. Theorem (Harrington, Soare) There is a maximal 2-tardy set. Question Is there a nonhigh 2-tardy set which is automorphic to a complete set? Ᏹ Orbits Complete Sets D-Maximal Sets Low Sets D-Maximal Sets Definition (The sets disjoint from A) D(A) = fB : 9W (B ⊆ A [ W and W \ A =∗ ;)g under inclusion. Let ᏱD(A) be Ᏹ modulo D(A). Definition 0 A is D-hhsimple iff ᏱD(A) is a Σ3 Boolean algebra. A is D-maximal iff ᏱD(A) is the trivial Boolean algebra iff for all c.e. sets B there is a c.e. set D disjoint from A such that either B ⊂ A [ D or B [ D [ A = !. Lemma Maximal sets are D-maximal. Plus there are many of examples of D-maximal sets. Question (D-Maximal Completeness) Does the orbit of every D-maximal set contain a complete set? Ᏹ Orbits Complete Sets D-Maximal Sets Low Sets D-hhsimple and Simple Theorem (Maass 84) If A is D-hhsimple and simple (i.e., hhsimple) if ᏱD(A) 0 Ᏹ then A≈ Ab. ∆3 D(A)b Theorem (Cholak, Harrington) If A is hhsimple then A≈ Ab iff ᏱD(A) 0 Ᏹ . ∆3 D(A)b All such orbits contain complete sets. Ᏹ Orbits Complete Sets D-Maximal Sets Low Sets Complexity Restrictions Theorem (Cholak, Harrington) If A is D-hhsimple and A and Ab are in the same orbit then ᏱD(A) 0 Ᏹ . ∆3 D(A)b Does not provide an answer to the following: Question (D-maximal Completeness) Which D-maximal sets are automorphic to complete sets? Question Is the above question an arithmetical question? Question Can we classify the D-maximal sets? Ᏹ Orbits Complete Sets D-Maximal Sets Low Sets Beginning the Classification D-maximal sets Definition For a D-maximal set, a list of c.e. sets fXigi2! generates D(A) iff for all D if D is disjoint from A then there is a n ∗ S such that D ⊆ i≤n Xi. (This list need not be computable.) Lemma (Cholak, Gerdes, Lange) • f;g generates D(A) iff A is maximal. • For any computable set R, fRg generates D(A) iff A is maximal on R. • For any noncomputable c.e. set W , fW g generates D(A) iff A is a trivial split of a maximal set. • In all other cases, the list of generators is infinite. Ᏹ Orbits Complete Sets D-Maximal Sets Low Sets Disjoint Generators Lemma If infinitely many computable sets are used to (partially) generate D(A) we can assume that (partial) list is pairwise disjoint. Lemma (Cholak, Gerdes, Lange) Assume an infinite pairwise disjoint list generates D(A). Then either • All the generators are computable. • All but one of the generators is computable. • None of the generators are computable. and, in all these cases, A is automorphic to a complete set. Ᏹ Orbits Complete Sets D-Maximal Sets Low Sets Other Generators Theorem (Cholak, Gerdes, Lange) There are four more cases of possible generators and all these cases break up into infinitely many orbits. It is unknown if any of these orbits contain complete sets. Ᏹ Orbits Complete Sets D-Maximal Sets Low Sets Nested Noncomputable Generators and Disjoint Computable Generators Theorem (Cholak, Gerdes, Lange) • There are D-maximal sets which are generated by noncomputable sets fWigi2! and pairwise disjoint computable sets fRigi2! such that for all i ≥ j, ∗ ∗ Wi \Rj ≠ ; and Wi \ Ri ⊂ Wi+1.
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