Computability in the class of Real Closed Fieldsdrh/mcs/ocasio_slides.pdf · OutlineTuring...
Transcript of Computability in the class of Real Closed Fieldsdrh/mcs/ocasio_slides.pdf · OutlineTuring...
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Outline Turing computable embeddings Categoricity in RCF
Computability in the class of Real Closed Fields
Vıctor A. Ocasio-Gonzalez
Department of MathematicsUniversity of Notre Dame
MCS @ UChicago
Oct. 1, 2013
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Outline Turing computable embeddings Categoricity in RCF
Outline
1 Turing computable embeddings
2 Categoricity in RCF
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Outline Turing computable embeddings Categoricity in RCF
Real Closed Fields
All structures have countable universe and all classes areclosed under isomorphism.
RCF =Mod(Th(R,+,∗,0,1,<))
RCF has many nice model theoretic properties
Complete, decidable, o-minimal and accepts quantifierelimination.Definable Skolemization which preserves the properties above.
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Outline Turing computable embeddings Categoricity in RCF
Real Closed Fields
All structures have countable universe and all classes areclosed under isomorphism.
RCF =Mod(Th(R,+,∗,0,1,<))RCF has many nice model theoretic properties
Complete, decidable, o-minimal and accepts quantifierelimination.Definable Skolemization which preserves the properties above.
![Page 5: Computability in the class of Real Closed Fieldsdrh/mcs/ocasio_slides.pdf · OutlineTuring computable embeddingsCategoricity in RCF TC embeddings De nition Let Kand K′ be two classes](https://reader033.fdocuments.net/reader033/viewer/2022042002/5e6e2e3f902e3966fe77086b/html5/thumbnails/5.jpg)
Outline Turing computable embeddings Categoricity in RCF
Skolemization of RCF
Let R ∈ RCF and X ⊆Rn be definable, say by ϕ(x, y). LetXa = {y ∶ (a, y) ∈X}. Then we have:
if Xa is empty, then fϕ(a) = 0
if Xa has a least element b, then fϕ(a) = bif the leftmost interval of Xa is (c, d), then fϕ(a) = d−c
2
if the leftmost interval of Xa is (−∞, d), then fϕ(a) = d − 1
if the leftmost interval of Xa is (c,∞), then fϕ(a) = c + 1
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Outline Turing computable embeddings Categoricity in RCF
TC embeddings
Definition
Let K and K′
be two classes of structures. A Turing computableembedding from K to K
′
is a Turing operator Φ = ϕe such that:
for each A ∈K, there is a A′ ∈K ′
such that ϕD(A)e = χD(A′)
for A,B ∈K correspond, respectively, to A′ ,B′ ∈K ′
thenA ≅ B iff A′ ≅ B′ .
Uniform procedure that respects isomorphism types.
For K and K′
as above we write that K ≤tc K′
, so that TCEinduces a preordering or classes.
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Outline Turing computable embeddings Categoricity in RCF
TC embeddings
Definition
Let K and K′
be two classes of structures. A Turing computableembedding from K to K
′
is a Turing operator Φ = ϕe such that:
for each A ∈K, there is a A′ ∈K ′
such that ϕD(A)e = χD(A′)
for A,B ∈K correspond, respectively, to A′ ,B′ ∈K ′
thenA ≅ B iff A′ ≅ B′ .
Uniform procedure that respects isomorphism types.
For K and K′
as above we write that K ≤tc K′
, so that TCEinduces a preordering or classes.
![Page 8: Computability in the class of Real Closed Fieldsdrh/mcs/ocasio_slides.pdf · OutlineTuring computable embeddingsCategoricity in RCF TC embeddings De nition Let Kand K′ be two classes](https://reader033.fdocuments.net/reader033/viewer/2022042002/5e6e2e3f902e3966fe77086b/html5/thumbnails/8.jpg)
Outline Turing computable embeddings Categoricity in RCF
TC embeddings
For all classes K, K ≤tc UG ≡tc LO ≡tc RCF .
Definition (Multiplicative Archimedean classes)
Let R ∈ RCF and let r, s ∈ R with r, s > 0. We say r <m s iff∀q ∈ Q rq < s
ARCF is a subclass of the class RCF where structures haveno infinite elements, i.e only multiplicative classes are [1]mand [2]m.
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Outline Turing computable embeddings Categoricity in RCF
TC embeddings
For all classes K, K ≤tc UG ≡tc LO ≡tc RCF .
Definition (Multiplicative Archimedean classes)
Let R ∈ RCF and let r, s ∈ R with r, s > 0. We say r <m s iff∀q ∈ Q rq < s
ARCF is a subclass of the class RCF where structures haveno infinite elements, i.e only multiplicative classes are [1]mand [2]m.
![Page 10: Computability in the class of Real Closed Fieldsdrh/mcs/ocasio_slides.pdf · OutlineTuring computable embeddingsCategoricity in RCF TC embeddings De nition Let Kand K′ be two classes](https://reader033.fdocuments.net/reader033/viewer/2022042002/5e6e2e3f902e3966fe77086b/html5/thumbnails/10.jpg)
Outline Turing computable embeddings Categoricity in RCF
TC embeddings
For all classes K, K ≤tc UG ≡tc LO ≡tc RCF .
Definition (Multiplicative Archimedean classes)
Let R ∈ RCF and let r, s ∈ R with r, s > 0. We say r <m s iff∀q ∈ Q rq < s
ARCF is a subclass of the class RCF where structures haveno infinite elements, i.e only multiplicative classes are [1]mand [2]m.
![Page 11: Computability in the class of Real Closed Fieldsdrh/mcs/ocasio_slides.pdf · OutlineTuring computable embeddingsCategoricity in RCF TC embeddings De nition Let Kand K′ be two classes](https://reader033.fdocuments.net/reader033/viewer/2022042002/5e6e2e3f902e3966fe77086b/html5/thumbnails/11.jpg)
Outline Turing computable embeddings Categoricity in RCF
Daisy graphs and ARCF
Definition
A daisy graph is an undirected graph G with a distinguished vertex,say x0, and a set of edges E, such that every vertex x ≠ x0 in theuniverse of G is part of a unique loop containing x0.
Every S ⊆ ω can be represented as a daisy graph by having aloop of size 2n + 3 if n ∈ S and a loop is size 2n + 4 otherwise.
So, A ∈DG is a collection of daisy graphs each representing adistinct subset of ω.
ARCF ≤tc DG ApG ≰tc DG
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Outline Turing computable embeddings Categoricity in RCF
Daisy graphs and ARCF
Definition
A daisy graph is an undirected graph G with a distinguished vertex,say x0, and a set of edges E, such that every vertex x ≠ x0 in theuniverse of G is part of a unique loop containing x0.
Every S ⊆ ω can be represented as a daisy graph by having aloop of size 2n + 3 if n ∈ S and a loop is size 2n + 4 otherwise.
So, A ∈DG is a collection of daisy graphs each representing adistinct subset of ω.
ARCF ≤tc DG ApG ≰tc DG
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Outline Turing computable embeddings Categoricity in RCF
Daisy graphs and ARCF
Definition
A daisy graph is an undirected graph G with a distinguished vertex,say x0, and a set of edges E, such that every vertex x ≠ x0 in theuniverse of G is part of a unique loop containing x0.
Every S ⊆ ω can be represented as a daisy graph by having aloop of size 2n + 3 if n ∈ S and a loop is size 2n + 4 otherwise.
So, A ∈DG is a collection of daisy graphs each representing adistinct subset of ω.
ARCF ≤tc DG
ApG ≰tc DG
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Outline Turing computable embeddings Categoricity in RCF
Daisy graphs and ARCF
Definition
A daisy graph is an undirected graph G with a distinguished vertex,say x0, and a set of edges E, such that every vertex x ≠ x0 in theuniverse of G is part of a unique loop containing x0.
Every S ⊆ ω can be represented as a daisy graph by having aloop of size 2n + 3 if n ∈ S and a loop is size 2n + 4 otherwise.
So, A ∈DG is a collection of daisy graphs each representing adistinct subset of ω.
ARCF ≤tc DG ApG ≰tc DG
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Outline Turing computable embeddings Categoricity in RCF
DG ≤tc ARCF
The ‘reverse’ the procedure above to get DG ≤tc ARCF willnot work
If we are to succeed in DG ≤tc ARCF , we must do so byassociating families of sets to families of algebraicallyindependent reals.
Theorem
There is a perfect, computable, binary tree T whose continuummany paths represent algebraically independent reals over Q.
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Outline Turing computable embeddings Categoricity in RCF
DG ≤tc ARCF
The ‘reverse’ the procedure above to get DG ≤tc ARCF willnot work
If we are to succeed in DG ≤tc ARCF , we must do so byassociating families of sets to families of algebraicallyindependent reals.
Theorem
There is a perfect, computable, binary tree T whose continuummany paths represent algebraically independent reals over Q.
![Page 17: Computability in the class of Real Closed Fieldsdrh/mcs/ocasio_slides.pdf · OutlineTuring computable embeddingsCategoricity in RCF TC embeddings De nition Let Kand K′ be two classes](https://reader033.fdocuments.net/reader033/viewer/2022042002/5e6e2e3f902e3966fe77086b/html5/thumbnails/17.jpg)
Outline Turing computable embeddings Categoricity in RCF
DG ≤tc ARCF
The ‘reverse’ the procedure above to get DG ≤tc ARCF willnot work
If we are to succeed in DG ≤tc ARCF , we must do so byassociating families of sets to families of algebraicallyindependent reals.
Theorem
There is a perfect, computable, binary tree T whose continuummany paths represent algebraically independent reals over Q.
![Page 18: Computability in the class of Real Closed Fieldsdrh/mcs/ocasio_slides.pdf · OutlineTuring computable embeddingsCategoricity in RCF TC embeddings De nition Let Kand K′ be two classes](https://reader033.fdocuments.net/reader033/viewer/2022042002/5e6e2e3f902e3966fe77086b/html5/thumbnails/18.jpg)
Outline Turing computable embeddings Categoricity in RCF
DG ≤tc ARCF
Theorem
There is a perfect, computable, binary tree T whose continuummany paths represent algebraically independent reals over Q.
Recall: A tree T is perfect if for every σ ∈ T there is σ ⪯ τ suchthat τ∧0, τ∧1 ∈ T .
Note: T ∶ 2<ω → Q ×Q, where σ ↦ (q, q∗) which we interpret asan interval Iσ = [q, q∗]
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Outline Turing computable embeddings Categoricity in RCF
DG ≤tc ARCF
Theorem
There is a perfect, computable, binary tree T whose continuummany paths represent algebraically independent reals over Q.
Recall: A tree T is perfect if for every σ ∈ T there is σ ⪯ τ suchthat τ∧0, τ∧1 ∈ T .Note: T ∶ 2<ω → Q ×Q, where σ ↦ (q, q∗) which we interpret asan interval Iσ = [q, q∗]
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Outline Turing computable embeddings Categoricity in RCF
DG ≤tc ARCF
Theorem
There is a perfect, computable, binary tree T whose continuummany paths represent algebraically independent reals over Q.
1 T (∅) = [0,1]2 If σ ⪯ τ , then Iτ ⊆ Iσ.
3 If length(σ) = n, then diameter(Iσ) ≤ 2−n, wherediameter(a, b), for an interval (a, b), is defined to be b − a.
4 If σ, τ are incomparable and both of length n, Iσ ∩ Iτ = ∅5 For f ∈ 2ω, let rf be the unique real in ⋂σ⊆f Iσ. Then for
distinct f1, . . . , fn ∈ 2ω, rf1 , . . . , rfn are algebraicallyindependent.
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Outline Turing computable embeddings Categoricity in RCF
Further work on TCE
Definition
For all 0 ≤ n, let V RCFn be all structures with n + 2 distinctmultiplicative classes.
Obs. ARCF = V RCF0
Theorem
For all 0 ≤ n < ω, V RCFn ≤tc V RCFn+1
Theorem
ARCF <tc V RCF1
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Outline Turing computable embeddings Categoricity in RCF
Further work on TCE
Definition
For all 0 ≤ n, let V RCFn be all structures with n + 2 distinctmultiplicative classes.
Obs. ARCF = V RCF0
Theorem
For all 0 ≤ n < ω, V RCFn ≤tc V RCFn+1
Theorem
ARCF <tc V RCF1
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Outline Turing computable embeddings Categoricity in RCF
ARCF <tc V RCF1
Theorem (Pull-Back Theorem, Knight, Miller, Vanden Boom)
If K ≤tc K′
via some Φ, then for any computable infinitarysentence ϕ
′
in the language of K′
we can find a computableinfinitary sentence ϕ in the language of K such that for all A ∈K,A ⊧ ϕ iff Φ(A) ⊧ ϕ′ . Moreover, if ϕ
′
is Σα (Πα) then so is ϕ.
A ≅ B ∈ ARCF iff A and B satisfy the same Σc2 sentences.
We find V and V ′ ∈ V RCF1 non-isomorphic satisfying thesame Σc
2 sentences.
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Outline Turing computable embeddings Categoricity in RCF
ARCF <tc V RCF1
V = RC(Q(g, a0, . . . , an, . . .)) and V ′ = RC(V(b)), where
b = Σiaigqi
(qi)i<ω decreasing sequence converging to an irrational.
Lemma
For any Π02 set S, we can uniformly produce a sequence of
structures (Fn)n<ω such that Fn ≅ V ′ if n ∈ S and Fn ≅ Votherwise.
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Outline Turing computable embeddings Categoricity in RCF
Relative Categoricity
Definition (Relatively ∆0γ-categorical)
A structure A is relatively ∆0γ-categorical if for all structures
B ≅ A, there is some isomorphism F ∶ A→ B such that F is∆0γ(B).
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Outline Turing computable embeddings Categoricity in RCF
Relative Categoricity
A structure A is relatively ∆0γ categorical iff A has a formally Σc
γ
Scott family.
Definition
A formally Σcγ Scott family for a structure A is a set Φ of
formulas, with fixed parameters c from A, such that:
for each tuple a of elements of A, there is a formulaϕ(x, c) ∈ Φ, such that A ⊧ ϕ(a, c)if two tuples a and b from A satisfy the same formula from Φ,then there is an automorphism of A mapping a to b
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Outline Turing computable embeddings Categoricity in RCF
Motivation
Theorem (Corollary from work of Nurtazin)
Let R be a computable RCF, then R is computably categorical ifand only if R has finite transcendence degree.
Theorem (Calvert)
If R is a computable archimedean RCF, then R is ∆02-categorical.
Moral: The complexity is in the infinite elements.
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Outline Turing computable embeddings Categoricity in RCF
Two Results
Theorem (Ash)
Suppose α is a computable ordinal, with ωδ+n ≤ α < ωδ+n+1, δ iseither 0 or a limit ordinal, and n < ω. Then α is ∆0
δ+2n-stable butnot ∆0
β-stable for β < δ + 2n.
Theorem
Let α be a computable well-order and let Rα be the RCFconstructed around α. Then Rα is relatively ∆0
γ-categorical and
not ∆0β-categorical for β < γ, where n < ω and
γ = { 2n + 1, if ωn ≤ α < ωn+1
δ + 2n, if ωδ+n ≤ α < ωδ+n+1
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Outline Turing computable embeddings Categoricity in RCF
Relative categoricity
LO ∶λ0(x) ≡ (∀y)(x ≤ y)
RCF ∶x ≈m y ≡⩔
n∈Nx < yn & ⩔
n∈Ny < xn
λ∗0(x) ≡ INF (x) & (∀y)(x ≈m y ∨ x <m y)
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Outline Turing computable embeddings Categoricity in RCF
Relative categoricity
LO ∶λ0(x) ≡ (∀y)(x ≤ y)
RCF ∶x ≈m y ≡⩔
n∈Nx < yn & ⩔
n∈Ny < xn
λ∗0(x) ≡ INF (x) & (∀y)(x ≈m y ∨ x <m y)
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Outline Turing computable embeddings Categoricity in RCF
Scott family for Rα
For all β < α, we take all formulas of the form:
ϕ(x) ≡ (∃y1)⋯(∃yi)(λ∗β1(y1) & ⋯ & λ∗βi(yi) & ψ(x, y)),where ψ(x, y) is quantifier free.
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Outline Turing computable embeddings Categoricity in RCF
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