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Characterizing the ComputableDepth Zero Boolean Algebras
Asher M. Kach
University of Wisconsin - Madison
Notre Dame27 April 2006
Asher M. Kach (UW - Madison) Depth Zero BAs ND 27 April 2006 1 / 19
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Outline
1 Acknowledgements and Thanks...
2 BackgroundReviewThe Feiner Hierarchy
3 Ketonen InvariantsMeasuresMeasures and Boolean AlgebrasDepth Zero BAs
4 My WorkResultsCurrent and Future WorkApplications
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Boolean Algebras
DefinitionA Boolean algebra is a structure (A; +, ·,−, 0, 1) satisfying associativity,commutativity, absorption, distributivity, and complementation:
1a. x + (y + z) = (x + y) + z 1b. x · (y · z) = (x · y) · z2a. x + y = y + x 2b. x · y = y · x3a. x + (x · y) = x 3b. x · (x + y) = x4a. x · (y + z) = (x · y) + (x · z) 4b. x + (y · z) = (x + y) · (x + z)5a. x + (−x) = 1 5b. x · (−x) = 0
Example
The structure (P(ω);∪,∩, C , ∅, ω) is a Boolean algebra.
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Linear Orders
TheoremGiven a linear order L = (L;<) with least element, the set of clopensubsets is a Boolean algebra under union, intersection, andcomplementation.
RemarkIt is usually much easier to specify the structure of a Boolean algebraby describing a linear order that generates it. Note, though, thatnon-isomorphic linear orders can generate the same Boolean algebra.
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Computable Models
DefinitionA structure M = (M; · · · ) is computable if its universe can be identifiedwith a computable subset of ω in such a way that its atomic diagram iscomputable.
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Atoms
DefinitionAn element x of a Boolean algebra is an atom (0-atom) if
∀y(y ≤ x → y = 0 or x − y = 0).
More generally, an element x is an (n + 1)-atom if it is not an n-atombut
∀y(y ≤ x → y or x − y is a finite union of n-atoms).
ExampleLet L = (ω + 1;<).
Then {2} and {18} are atoms.
Then {1, 2} is not an atom as {1, 2} = {1} t {2}.Then [0, ω] and [5, ω] are 1-atoms.
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Cantor-Bendixson Rank
DefinitionGiven a linear order L, the Cantor-Bendixson derivative L′ of L is thesublinear order with universe
L′ = L− {x : x is isolated in L}.
Using transfinite induction, define the αth Cantor-Bendixson derivativeof L, denoted L(α), by
L(0) = L, L(α+1) =(L(α)
)′, and L(γ) =
⋂α<γ
L(α)
for limit ordinals γ.
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Cantor-Bendixson Example
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The Feiner Hierarchy
Definition
Recall that 0(ω) ={〈m, n〉 : m ∈ 0(n)
}.
Definition
A set S recursive in 0(ω) is said to be (a, b) in the Feiner Hierarchy,written S ∼ (a, b), if there is an index e such that
1 The function ϕ0(ω)
e is total,
2 The function ϕ0(ω)
e is the characteristic function of S, i.e. n ∈ S ifand only if ϕ0(ω)
e (n) = 1.
3 The computation ϕ0(ω)
e (n) uses no part of the oracle above 0(an+b).
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Measures
DefinitionA measure is a map σ from the clopen subsets of 2ω into ω1 inducedby a map σ̃ : 2<ω → ω1 satisfying
σ̃(τ) = max{
σ̃(τ a 0), σ̃(τ a 1)}
for all τ ∈ 2<ω
where
σ(τ) = max {σ̃(τ0), . . . , σ̃(τn) : τ = τ0 t · · · t τn} .
Example
The following map is (induces) a measure. 4
420
2
1
4
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Measures and Boolean Algebras
Theorem (Ketonen)There is a natural bijection between (isomorphism types) of thecountable uniform Boolean algebras and (homeomorphism types) ofmeasures.
Proof.Move from countable uniform Boolean algebras to linear orders to rankfunctions to measures.
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Depth Zero Boolean Algebras
DefinitionGiven a measure σ and an element x, define
∆σ(x) = {(σ(x0), . . . , σ(xn)) : x = x0 t · · · t xn} .
More generally, define
∆α+1σ(x) = {(∆ασ(x0), . . . ,∆ασ(xn)) : x = x0 t · · · t xn} .
DefinitionA measure σ is depth zero if
σ(x) = σ(y) =⇒ ∆σ(x) = ∆σ(y).
RemarkNote that the reverse implication always holds.
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Depth Example
ExampleNote that for the measure
4
420
2
1
4zy
x
we have
∆σ(x) is the set of all strings of 0, 1, 2, and 4 containing at leastone 4
∆σ(y) is the set of all strings of 0 and 2 containing at least one 2
∆σ(z) is the set of all strings of 1, and 4 containing at least one 4
In particular, σ is not depth zero since σ(x) = σ(z) but ∆σ(x) 6= ∆σ(z).
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Depth Zero BAs
Proposition (K)Given a bounded set S ⊂ ω1 with maximal element, there are exactlytwo depth zero measures with range S.
Proof.For example, with S = {0, 1} and S = {0, 1, 2}, the Boolean algebrasare given by the measures
1
0 0
0 0001 1 1 1
111
1
1 1
1
1
2
2
2
22
1 1
2
2
2 2
2 2 2 1 11 1 2
001 1
11 2 2
2 1 1 2
1
1
1
1
1
1
0
0
0
0
0
T(0,1)
T(0,1)
T(0,1)
T(0,1)
T(0,1)
2
2
2
2
2
2
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Results (I)
Theorem (K)If S ∼ (2, 0), then BS is computable.
Proof.Begin with the measure σω+1. When it appears that an integer n isn’t inS, attempt to build an n-atom (e.g. the linear order wn + 1) rather thanan (n − 1)-atom (e.g. the linear order wn−1 + 1) at the appropriateplace. Carefully modifying the proof of the following theorem,
Theorem (Thurber)
There is a procedure, uniform in n and a ∆02n+1 index for the open
diagram of a linear order A with distinguished first element, whichbuilds a ∆0
1 copy of the linear order ωn · A.
it is possible to relativize and nest the construction controlled by a Σ02
predicate that builds either FIN or ω + 1 to yield BS.
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Results (II)
Theorem (K)If BS is computable, then S ∼ (2, 3).
Proof.A number n ≥ 1 is in S if and only if
BS |= ∃x∀m∃x1, . . . , xm( m∧i=1
xi ≤ x
)∧
m∧i 6=j
xixj = 0
∧
(m∧
i=1
Atn−1(xi)
)∧∀y[y ≤ x =⇒ ¬Atn(y)
]Since the formula Atn(x) stating that x is an n-atom is expressible asan infinitary Π0
2n+1 formula, counting quantifiers the above is Σ02n+3.
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Current and Future Work...
ConjectureWith S ⊆ ω + 1, the depth zero Boolean algebras BS are computable ifand only if S ∼ (2, 3).
Conjecture
More generally with S ⊂ ωCK1 , the depth zero Boolean algebras are
computable if and only if S ∼ (2, 3).
Other future projects include analyzing both the finite depth Booleanalgebras and the rank one, depth ω Boolean algebras.
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The LOW-n Problem
Definition (The LOW-n Problem)The LOW-n Problem is the statement that every LOW-n Booleanalgebra is computable.
RemarkIf an iff statement can be achieved, then the LOW-n Problem would besolved (positively) for the depth zero Boolean algebras.
TheoremThe LOW-n Problem is true for n = 1 (Downey and Jockusch), n = 2(Thurber), and n = 4 (Knight and Stob).
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References
Ash, C.J. and Knight, J.F.Computable Structures and the Hyperarithmetical Hierarchy.Studies in Logic, Vol. 144, 2000.
Ketonen, Jussi.The structure of countable Boolean algebras.Annals of Mathematics, 108(1):41–89, 1978.
Knight, Julia F. and Stob, Michael.Computable Boolean algebras.Journal of Symbolic Logic, 65(4):1605-1623, 2000.
Pierce, R. S.Countable Boolean algebras.Handbook of Boolean algebras, Vol. 3:775-876, 1989.
Thurber, John J.Recursive and r.e. quotient Boolean algebras.Archive for Mathematical Logic, 33(2):121–129, 1994.
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