Jump upper semilattice embeddings Local Structures ASL Summer Meeting ”Logic Colloquium ’06”. Embeddability and Decidability in the Turing Degrees Antonio Montalb´ an. University of Chicago Nijmegen, Netherlands, 27 July- 2 Aug. of 2006 Antonio Montalb´ an. University of Chicago Embeddability and Decidability in the Turing Degrees
Jump upper semilattice embeddings Local Structures 1 Jump upper semilattice embeddings Background JUSL Embeddings Other Embeddability results 2 Local Structures High/Low Hierarchy Ordering of the classes Fragments of the theory Antonio Montalb´ an. University of Chicago Embeddability and Decidability in the Turing Degrees
Background Jump upper semilattice embeddings JUSL Embeddings Local Structures Other Embeddability results Basic definitions Given sets A , B ⊆ N we say that A is computable in B , and we write A � T B , if there is a computable procedure that can tell whether an element is in A or not using B as an oracle . (Note: Instead of N we could’ve chosen 2 <ω , ω <ω , or V ( ω ),...) This defines a quasi-ordering on P ( N ). We say that A is Turing equivalent to B , and we write A ≡ T B if A � T B and B � T A . [Kleene Post 54] We let D = ( P ( D ) / ≡ T ), and D = ( D , � T ). Question: How does D look like? Antonio Montalb´ an. University of Chicago Embeddability and Decidability in the Turing Degrees
Background Jump upper semilattice embeddings JUSL Embeddings Local Structures Other Embeddability results Some simple observations about D There is a least degree 0 . The degree of the computable sets. D has the countable predecessor property , i.e., every element has at countably many elements below it. Because there are countably many programs one can write. Each Turing degree contains countably many sets. So, D has size 2 ℵ 0 . Because P ( N ) has size 2 ℵ 0 , and each equivalence class is countable. Antonio Montalb´ an. University of Chicago Embeddability and Decidability in the Turing Degrees
Background Jump upper semilattice embeddings JUSL Embeddings Local Structures Other Embeddability results Operations on D Turing Join Every pair of elements a , b of D has a least upper bound (or join ), that we denote by a ∪ b . So, D is an upper semilattice. Given A , B ⊆ N , we let A ⊕ B = { 2 n : n ∈ A } ∪ { 2 n + 1 : n ∈ B } . Clearly A � T A ⊕ B and B � T A ⊕ B , and if both A � T C and B � T C then A ⊕ B � T C . Turing Jump Given A ⊆ N , we let A ′ be the Turing jump of A , that is, A ′ = { programs that HALT, when run with oracle A } . For a ∈ D , let a ′ be the degree of the Turing jump of any set in a a < T a ′ If a � T b then a ′ � T b ′ . Antonio Montalb´ an. University of Chicago Embeddability and Decidability in the Turing Degrees
Background Jump upper semilattice embeddings JUSL Embeddings Local Structures Other Embeddability results Operations on D . Definition A jump upper semilattice (JUSL) is structure ( A , � , ∨ , j ) such that ( A , � ) is a partial ordering. For every x , y ∈ A , x ∨ y is the l.u.b. of x and y , x < j ( x ), and if x � y , then j ( x ) � j ( y ). D = ( D , � T , ∨ , ′ ) is a JUSL. Antonio Montalb´ an. University of Chicago Embeddability and Decidability in the Turing Degrees
Background Jump upper semilattice embeddings JUSL Embeddings Local Structures Other Embeddability results Questions one may ask Are there incomparable degrees? YES Are there infinitely many degrees such that non of them can be computed from all the other ones toghether? YES What about ℵ 1 many? YES Is there a descending sequence of degrees a 0 , � T a 1 � T .... ? YES Could we also get such a sequence with a ′ n +1 = a n ? YES A more general question: Which structures can be embedded into D ? Antonio Montalb´ an. University of Chicago Embeddability and Decidability in the Turing Degrees
Background Jump upper semilattice embeddings JUSL Embeddings Local Structures Other Embeddability results Embedding structures into D Theorem: The following structures can be embedded into the Turing degrees. Every countable upper semilattice. [Kleene, Post ’54] Every partial ordering of size ℵ 1 with the countable predecessor property (c.p.p.). [Sacks ’61] (It’s open whether this is true for size 2 ℵ 0 .) Every upper semilattice of size ℵ 1 with the c.p.p. Moreover, the embedding can be onto an initial segment. [Abraham, Shore ’86] Every ctble. jump partial ordering ( A , � , ′ ) . [Hinman, Slaman ’91] Theorem (M.) Every ctble. jump upper semilattice ( A , � , ∨ , ′ ) is embeddable in D . Antonio Montalb´ an. University of Chicago Embeddability and Decidability in the Turing Degrees
Background Jump upper semilattice embeddings JUSL Embeddings Local Structures Other Embeddability results Idea of the proof Definition: A JUSL J is h-embeddable if there is a map H : J → P ( N ) s.t., for all x , y ∈ P , if x < J y then H ( x ) ′ � T H ( y ). uniformity condition : J � T H ( y ), and � x � J y H ( x ) � T H ( y ); Obs: Every well-founded JUSL is h-embeddable, by taking x �→ 0 rk( x ) . Theorem Every ctble JUSL which is h-embeddable, is embeddable into D . Proof: Forcing Construction. Lemma Every ctble JUSL embeds into one which is h-embeddable. Proof: Uses Fra¨ ıss´ e limits and non-standard ordinals. Antonio Montalb´ an. University of Chicago Embeddability and Decidability in the Turing Degrees
Background Jump upper semilattice embeddings JUSL Embeddings Local Structures Other Embeddability results Corollary Emedabbility results are usually related to the decidability of existential theories. Corollary ∃ − Th ( D , � T , ∨ , ′ ) is decidable. Note: ∃ − Th ( D , � T , ∨ , ′ ) is the set of existential forumulas, in the language of JUSL, true about D An ∃ -formula about ( D , � T , ∨ , ′ ) is true iff is does not Proof: contradict the axioms of jump upper semilattice. Antonio Montalb´ an. University of Chicago Embeddability and Decidability in the Turing Degrees
Background Jump upper semilattice embeddings JUSL Embeddings Local Structures Other Embeddability results History of Decidability Results. Th ( D , � T ) is undecidable. [Lachlan ’68] ∃ − Th ( D , � T ) is decidable. [Kleene, Post ’54] Question: Which fragments of Th ( D , � T , ∨ , ′ ) are decidable? ∃∀∃ − Th ( D , � T ) is undecidable. [Shmerl] ∀∃ − Th ( D , � T , ∨ ) is decidable. [Jockusch, Slaman ’93] ∃ − Th ( D , � T , ′ ) is decidable. [Hinman, Slaman ’91] ∃ − Th ( D , � T , ∨ , ′ ) is decidable. [M. 03] ∀∃ − Th ( D , � T , ∨ , ′ ) is undecidable. [Slaman, Shore ’05] . ∃ − Th ( D , � T , ∨ , ′ , 0) is decidable. [Lerman, in preparation] Question: Is ∀∃ − Th ( D , � T , ′ ) decidable? Antonio Montalb´ an. University of Chicago Embeddability and Decidability in the Turing Degrees
Background Jump upper semilattice embeddings JUSL Embeddings Local Structures Other Embeddability results Other Embeddability results. Definition: A jump upper semilattice with 0 (JUSL w/0) is a structure J = � J , � J , ∪ , j , 0 � such that • � J , � J , ∪ , j � is a JUSL, and • 0 is the least element of � J , � J � . Q: Which JUSL w/0 can be embedded into D ? Q: What about among the ones which have only finitely many generators? Antonio Montalb´ an. University of Chicago Embeddability and Decidability in the Turing Degrees
� � � � � � � � � � � � � � � Background Jump upper semilattice embeddings JUSL Embeddings Local Structures Other Embeddability results other results. . . . . . . Theorem (M. 03) . . . . Not every JUSLw/0 even with one 0 (5) . x (4) . ��� � � � generator is embeddable in D . 0 (4) ∪ x (3) � � � 0 (4) � � There are 2 ℵ 0 JUSLw/0 with x (3) Proof: � � � � � � a generator x satisfying x � 0 ′′ . 0 (3) ∪ x ′′ ��� 0 (3) � � x ′′ ��� � � � � Theorem (Hinman, Slaman 91; M.03) 0 ′′ ∪ x ′ ���� � � � Every JPOw/0 with one genrator is 0 ′′ x ′ ���� � � � � � realized in D . 0 ′ ∪ x ����� � � � � � 0 ′ x � � � � � Question: What about JPOw/0 and � � � � � � � � � � � 0 with two generators? Antonio Montalb´ an. University of Chicago Embeddability and Decidability in the Turing Degrees
Background Jump upper semilattice embeddings JUSL Embeddings Local Structures Other Embeddability results Other results. Let κ be a cardinal, ℵ 0 < κ � 2 ℵ 0 . Q: Is every JUSL with the c.p.p. and size κ embeddable in D ? Proposition If κ = 2 ℵ 0 , then the answer is NO . Proposition If Martin’s axiom holds at κ , the answer is YES . Corollary For κ = ℵ 1 , it is independent of ZFC. Proof: It is FALSE under CH, but TRUE under MA( ℵ 1 ). Antonio Montalb´ an. University of Chicago Embeddability and Decidability in the Turing Degrees
High/Low Hierarchy Jump upper semilattice embeddings Ordering of the classes Local Structures Fragments of the theory 1 Jump upper semilattice embeddings Background JUSL Embeddings Other Embeddability results 2 Local Structures High/Low Hierarchy Ordering of the classes Fragments of the theory Antonio Montalb´ an. University of Chicago Embeddability and Decidability in the Turing Degrees
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