On the back-and-forth relation on Boolean Algebras. Antonio Montalb - PowerPoint PPT Presentation

On the back-and-forth relation on Boolean Algebras. Antonio Montalb´ an. U. of Chicago AMS - NZMS joint meeting, December 2007 Joint work with Kenneth Harris (University of Michigan). Antonio Montalb´ an. U. of Chicago On the back-and-forth relation on Boolean Algebras.

Boolean Algebras Definition A Boolean algebra, BA, is a structure B = ( B , ≤ , 0 , 1 , ∨ , ∧ , ¬ ), where ( B , ≤ ) is a partial ordering, 0 is the least element and 1 the greatest, x ∨ y is the least upper bound of x and y , x ∧ y is the greatest lower bound of x and y , ¬ x ∨ x = 1 and ¬ x ∧ x = 0 Example: ( P ( X ) , ⊆ , ∅ , X , ∪ , ∩ , X \ · ) We will only consider countable BAs and assume B ⊆ ω . A BA B is X-computable if X can compute B and all the operations in B . Antonio Montalb´ an. U. of Chicago On the back-and-forth relation on Boolean Algebras.

Low Boolean Algebras Theorem: [Downey, Jockusch 94] Every low Boolean Algebra has a computable copy. i.e. If X is low and B is X -computable, then there is a computable BA isomorphic to B . Theorem: [Thurber 95] Every low 2 Boolean Algebra has a computable copy. Theorem: [Knight, Stob 00] Every low 4 Boolean Algebra has a computable copy. Open Question: Does every low n Boolean Algebra have a computable copy? Antonio Montalb´ an. U. of Chicago On the back-and-forth relation on Boolean Algebras.

Boolean Algebra Predicates • 1-predicates atom( x ) • 2-predicates atomless( x ) infinite( x ) • 3-predicates atomic( x ) 1-atom( x ) atominf( x ) • 4-predicates ∼ -inf( x ) I ( ω + η )( x ) infatomicless( x ) 1-atomless( x ) nomaxatomless( x ) n -predicates have n alternations of quantifiers Antonio Montalb´ an. U. of Chicago On the back-and-forth relation on Boolean Algebras.

Definition For n = 0 , 1 , 2 , 3 , 4, a BA B is n-approximable if 0 ( n ) can compute B and all its m -predicates for m ≤ n . Note: B is 0-approximable ⇐ ⇒ B is computable. Note : B is low n = ⇒ B is n -approximable. Lemma : [Downey, Jockusch 94; Thurber 95; Knight, Stob 00] For n = 0 , 1 , 2 , 3, every ( n + 1)-approximable BA has an n -approximable copy. So: B low 4 = ⇒ 4-approx = ⇒ 3-approx copy = ⇒ 2-approx copy = ⇒ 1-approx copy = ⇒ 0-approx copy = ⇒ computable copy. Antonio Montalb´ an. U. of Chicago On the back-and-forth relation on Boolean Algebras.

General Idea A and B are n -equivalent iff 0 ( n ) cannot distinguish them. Def: Let A ≤ n B ⇐ ⇒ given C that’s isomorphic to either A or B , deciding whether C ∼ = A is Σ 0 n -hard. We will write A ≡ n B iff both A ≤ n B and B ≤ n A . Antonio Montalb´ an. U. of Chicago On the back-and-forth relation on Boolean Algebras.

Back-and-Forth relations Notation: a 1 , ..., a k is a partition of a BA B if a 0 ∨ ... ∨ a k = 1 and ∀ i � = j ( a i ∧ a j = 0). We write B ↾ a for the BA whose domain is { x ∈ B : x ≤ a } . Theorem [Ash, Knight] TFAE 1 A ≤ n B . 2 All the infinitary Σ n sentences true in B are true in A . 3 for every partition ( b i ) i ≤ k of B , there is a partition ( a i ) i ≤ k of A such that ∀ i ≤ k B ↾ b i ≤ n − 1 A ↾ a i . Antonio Montalb´ an. U. of Chicago On the back-and-forth relation on Boolean Algebras.

The bf-types Obs: ≡ n is an equivalence relation on the class of BAs. We call the equivalence classes n-bf-types . We study the following family of ordered monoids ( BAs / ≡ n , ≤ n , ⊕ ) where A ⊕ B is the product BA with coordinatewise operations, together with the projections ( · ) n − 1 : BAs / ≡ n → BAs / ≡ n − 1 . Antonio Montalb´ an. U. of Chicago On the back-and-forth relation on Boolean Algebras.

The invariants For each n we define a set INV n of finite objects, and an invariant map T n : BAs → INV n such that A ≡ n B ⇐ ⇒ T n ( A ) = T n ( B ) Moreover, on INV n we define ≤ n and + so that ( BAs / ≡ n , ≤ n , ⊕ ) ∼ = ( INV n , ≤ n , +) , Antonio Montalb´ an. U. of Chicago On the back-and-forth relation on Boolean Algebras.

Indecomposable Boolean Algebras Definition A BA A is n-indecomposable if for every partition a 1 , ..., a k of A , there is an i ≤ k such that A ≡ n A ↾ a i . Theorem 1 Every BA is a finite product of n-indecomposable BAs. 2 There are finitely many ≡ n -equivalence classes among the n-indecomposable BAs. Let BF n = { T n ( B ) : B is n -indecomposable } ⊂ INV n . BF n is a finite generator of ( INV n , ≤ n , +). n 1 2 3 4 5 6 ... | BF n | 2 3 5 9 27 1578 ... Antonio Montalb´ an. U. of Chicago On the back-and-forth relation on Boolean Algebras.

Boolean Algebra Predicates Definition For each α ∈ BF n we define a relation R α ( · ) on B : R α ( x ) ⇐ ⇒ T n ( B ↾ x ) ≥ n α . Observation For n = 0 , 1 , 2 , 3 , 4, the ( ≤ n )-predicates are boolean combinations of the R α for α ∈ BF ≤ n , and vice versa. Lemma The relations R α for α ∈ BF n can be defined by computable infinitary Π n formulas of BAs. Antonio Montalb´ an. U. of Chicago On the back-and-forth relation on Boolean Algebras.

Picture - Levels 1, 2 and 3 bf-relations for 1- and 2-indecomposable bf-types projection a 0 b 0 b 1 c 0 c 2 b 0 c 1 b 1 bf-relations for 3-indecomposable bf-types projection c 0 c 1 c 2 d 0 d 1 d 4 d 2 d 3 Antonio Montalb´ an. U. of Chicago On the back-and-forth relation on Boolean Algebras.

Picture - Level 4 bf-relations for 4-indecomposable bf-types projection d 0 d 1 d 2 d 3 d 4 e 0 e 1 e 3 e 7 e 8 e 2 e 4 e 5 e 6 Antonio Montalb´ an. U. of Chicago On the back-and-forth relation on Boolean Algebras.

Picture - Level 5 bf-relations for 5-indecomposable bf-types e 0 e 1 e 2 e 3 e 4 e 5 e 7 projection f 0 f 1 f 2 f 5 f 16 f 21 f 25 f 3 f 6 f 20 f 4 f 15 f 10 e 6 projection f 24 � � ������ � � � � f 11 f 23 � � � � �������������� ������ � � � � � � � � f 12 f 22 � �������������� � ������ � � � � f 17 f 13 � ������ � ������ � � � � f 7 f 18 f 14 � � � ������ � ������ � � � � � � � � f 8 f 19 � � � ������ � � � f 9 Antonio Montalb´ an. U. of Chicago On the back-and-forth relation on Boolean Algebras.

Quantifier Elimination. Theorem Every infinitary Σ n +1 formula is equivalent to an infinitary Σ 1 formula over the predicates R α for α ∈ BF n . Antonio Montalb´ an. U. of Chicago On the back-and-forth relation on Boolean Algebras.

Quantifier Elimination. α = � α 1 , ..., α m � and ¯ β = � β 1 , ..., β k � ∈ BF <ω Notation: Given ¯ let n � � β ( x ) ⇐ ⇒ ∃ y 1 ˙ ∨ . . . ˙ ∨ y m = x R α 1 ( y 1 ) & ... & R α m ( y m ) & R ¯ α, ¯ � � ∃ z 1 ˙ ∨ . . . ˙ ∨ z k = ¬ x R β 1 ( z 1 ) & ... & R β k ( z k ) where ∃ y 1 ˙ ∨ . . . ˙ ∨ y m = x is short for “there is a partition y 1 , ..., y m of x such that...” Theorem Let B be a BA, and R ⊆ B. TFAE 1 If A ∼ = B and ( A , Q ) ∼ = ( B , R ) then Q is Σ 0 , A n +1 . 2 R can be defined in B by a comp infinitary Σ c n +1 formula. α i , ¯ 3 There is a 0 ( n ) -comp seq { (¯ β i ) } i ∈ ω ⊆ BF <ω such that n x ∈ R ⇐ ⇒ � β i ( x ) i ∈ ω R ¯ α i , ¯ The equivalence between (1) and (2) is due to Ash, Knight, Manasse, Slaman; Chisholm. Antonio Montalb´ an. U. of Chicago On the back-and-forth relation on Boolean Algebras.

n -approximable Boolean Algebras Theorem Let B be a presentation of a Boolean algebra. TFAE. n +1 -diagram of B is Σ 0 1 The Σ c n +1 ; 2 The relations R α ( B ) for α ∈ BF n are computable in 0 ( n ) . Definition If a BA satisfies these conditions, we say it’s n-approximable . Question: Does every n + 1-approximable BA have an n -approximable copy? Antonio Montalb´ an. U. of Chicago On the back-and-forth relation on Boolean Algebras.

Difficulties at level 5 Definition α ∈ BF n is a isomorphism type if whenever T n ( A ) = T n ( B ) = α , A ∼ = B . α ∈ BF n is an exclusive type if whenever T n ( A ) = α and a ∈ A either A ↾ a ≡ n A or A ↾ ( ¬ a ) ≡ n A , but not both. Observation: For n ≤ 4, and α ∈ BF n , α is an exclusive type = ⇒ α is an isomorphism type. This is not true for n = 5. Antonio Montalb´ an. U. of Chicago On the back-and-forth relation on Boolean Algebras.

Picture - Levels 1 and 2 bf-relations for 1- and 2-indecomposable bf-types projection a 0 b 0 b 1 c 0 c 2 b 0 c 1 b 1 1-indecomposable bf-types Name Example R u atom atom b 0 b 1 non-zero infinite 2-indecomposable bf-types Name ( · ) 1 Example R u atom atom c 0 b 0 c 1 b 1 infinite inf-atoms c 2 b 1 atomless atomless Antonio Montalb´ an. U. of Chicago On the back-and-forth relation on Boolean Algebras.

Picture - Level 3 bf-relations for 3-indecomposable bf-types projection c 0 c 1 c 2 d 0 d 1 d 4 d 2 d 3 Name ( · ) 2 Example R u atom atom d 0 c 0 d 1 c 1 1-atom 1-atom atomic & infinite 2-atom, 1-atomless d 2 c 1 d 3 c 1 atominf Int ( ω + η ) d 4 c 2 atomless atomless Antonio Montalb´ an. U. of Chicago On the back-and-forth relation on Boolean Algebras.