Computability and Categoricity of Ultrahomogeneous ... - People
Ultrahomogeneous Structures
Linear Orders
Computability and Categoricity of Ultrahomogeneous Structures Francis Adams
January 17, 2014
Francis Adams Computability and Categoricity of Ultrahomogeneous Structures
Other Examples
Ultrahomogeneous Structures
Linear Orders
Other Examples
Ultrahomogeneous Structures
Definition A structure is ultrahomogeneous if any isomorphism between finitely generated substructures extends to an automorphism of the whole structure.
Francis Adams Computability and Categoricity of Ultrahomogeneous Structures
Ultrahomogeneous Structures
Linear Orders
Other Examples
Countable Ultrahomogeneous Structures
Example Q, the countable dense linear order. The countable atomless boolean algebra. The Rado graph. Equivalence Structures: Every class has the same size. Injection Structures: No ω-orbits.
Francis Adams Computability and Categoricity of Ultrahomogeneous Structures
Ultrahomogeneous Structures
Linear Orders
Other Examples
Effective Categoricity Definition A computable structure A is ∆0α categorical if every computable structure isomorphic to A is ∆0α isomorphic to A. In particular, a computable structure A is computably categorical if every computable structure isomorphic to A is computably isomorphic to A.
Francis Adams Computability and Categoricity of Ultrahomogeneous Structures
Ultrahomogeneous Structures
Linear Orders
Theorem 1
Theorem Every computable ultrahomogeneous structure is ∆02 -categorical.
Francis Adams Computability and Categoricity of Ultrahomogeneous Structures
Other Examples
Ultrahomogeneous Structures
Linear Orders
Other Examples
Proof of Theorem 1
Lemma Checking if two finitely generated substructures are isomorphic is Π01 . We’ll use this lemma along with a back and forth argument, building S partial isomorphisms θn at each stage and letting θ = θn .
Francis Adams Computability and Categoricity of Ultrahomogeneous Structures
Ultrahomogeneous Structures
Linear Orders
Other Examples
Proof of Theorem 1
Suppose we have defined θ2n−1 for {a0 , . . . a2n−1 } with θ2n−1 (ai ) = bi . Choose the least a2n ∈ A \ {a0 , . . . a2n−1 }. There exists a b ∈ B such that ha0 , . . . a2n i ∼ = hb0 , . . . , b2n−1 , bi where the isomorphism can be chosen to extend θ2n−1 . Search for this b using a Π01 -oracle to check if ha0 , . . . a2n i ∼ = hb0 , . . . , b2n−1 , bi. After finding one, call it b2n , S and define θ2n (a2n ) = b2n . Then θ = θn is a ∆02 isomorphism.
Francis Adams Computability and Categoricity of Ultrahomogeneous Structures
Ultrahomogeneous Structures
Linear Orders
Other Examples
Corollary Any relational, or more generally, any locally finite, computable ultrahomogeneous structure is computably categorical. The converse is false; a single Z-orbit is computably categorical.
Francis Adams Computability and Categoricity of Ultrahomogeneous Structures
Ultrahomogeneous Structures
Linear Orders
Other Examples
Weakly Ultrahomogeneous Structures
Definition A structure A is weakly ultrahomogeneous if there exists a finite set {a1 , a2 , . . . , an } ⊆ A such that for all tuples ~x , ~y from A with h~a, ~x i ∼ = h~a, ~y i where each ai is fixed, this isomorphism of substructures extends to an automorphism of A. Call such a set {a1 , a2 , . . . , an } an exceptional set of A.
Francis Adams Computability and Categoricity of Ultrahomogeneous Structures
Ultrahomogeneous Structures
Linear Orders
Other Examples
Weakly Ultrahomogeneous Structures Alternatively, A is weakly ultrahomogeneous if there is a finite set a1 , . . . , an of elements from A such that (A, a1 , . . . , an ) is ultrahomogeneous in the extended language with constants for the ai .
Francis Adams Computability and Categoricity of Ultrahomogeneous Structures
Ultrahomogeneous Structures
Linear Orders
Other Examples
Weakly Ultrahomogeneous Structures Alternatively, A is weakly ultrahomogeneous if there is a finite set a1 , . . . , an of elements from A such that (A, a1 , . . . , an ) is ultrahomogeneous in the extended language with constants for the ai . Theorem Every computable weakly ultrahomogeneous structure is ∆02 -categorical. Corollary Every locally finite computable weakly ultrahomogeneous structure is computably categorical. Francis Adams Computability and Categoricity of Ultrahomogeneous Structures
Ultrahomogeneous Structures
Linear Orders
Other Examples
Classification of Linear Orders
Theorem For a countable linear order A, the following are equivalent: 1 A is weakly ultrahomogeneous. 2 A has finitely many successivities. 3 A = L0 + Q + L1 + Q + . . . + Q + Ln where the Li are finite chains, L0 , Ln are possibly empty and |Li | ≥ 2 for 1 ≤ i ≤ n − 1.
Francis Adams Computability and Categoricity of Ultrahomogeneous Structures
Ultrahomogeneous Structures
Linear Orders
Other Examples
Effective Categoricity
Theorem (Remmel 1981) A computable linear ordering is computably categorical iff it has finitely many successivities. Corollary A computable linear order is weakly ultrahomogeneous iff it is computably categorical.
Francis Adams Computability and Categoricity of Ultrahomogeneous Structures
Ultrahomogeneous Structures
Linear Orders
Other Examples
Minimal Exceptional Sets
Q
a1
a2
a3
Francis Adams Computability and Categoricity of Ultrahomogeneous Structures
a4
a5
Q
Ultrahomogeneous Structures
Linear Orders
Other Examples
Minimal Exceptional Sets
The following sets aren’t exceptional:
Q
a1
a2
a3
a4
a5
Q
Q
a1
a2
a3
a4
a5
Q
Francis Adams Computability and Categoricity of Ultrahomogeneous Structures
Ultrahomogeneous Structures
Linear Orders
Other Examples
Minimal Exceptional Sets
Including all successivities yields an exceptional set:
Q
a1
a2
a3
Francis Adams Computability and Categoricity of Ultrahomogeneous Structures
a4
a5
Q
Ultrahomogeneous Structures
Linear Orders
Other Examples
Minimal Exceptional Sets
Including all successivities yields an exceptional set:
Q
a1
a2
a3
But it isn’t minimal.
Francis Adams Computability and Categoricity of Ultrahomogeneous Structures
a4
a5
Q
Ultrahomogeneous Structures
Linear Orders
Other Examples
Minimal Exceptional Sets
This set is a minimal exceptional set:
Q
a1
a2
a3
Francis Adams Computability and Categoricity of Ultrahomogeneous Structures
a4
a5
Q
Ultrahomogeneous Structures
Linear Orders
Other Examples
Minimal Exceptional Sets
So is this set:
Q
a1
a2
a3
Francis Adams Computability and Categoricity of Ultrahomogeneous Structures
a4
a5
Q
Ultrahomogeneous Structures
Linear Orders
Other Examples
Minimal Exceptional Sets
So is this set:
Q
a1
a2
a3
a4
a5
Q
So minimal exceptional sets aren’t unique, and don’t even have to be isomorphic.
Francis Adams Computability and Categoricity of Ultrahomogeneous Structures
Ultrahomogeneous Structures
Linear Orders
Other Examples
Minimal Exceptional Sets
Theorem Let A = L0 + Q + L1 + Q + . . . + Q + Ln be a countable weakly ultrahomogeneous linear order. The S minimal exceptional sets are those subsets S of i≤n Li such that i) A \ S doesn’t contain two consecutive successivities. ii) S contains each last element of L0 , . . . Ln−1 and each first element of L1 , . . . Ln .
Francis Adams Computability and Categoricity of Ultrahomogeneous Structures
Ultrahomogeneous Structures
Linear Orders
Other Examples
Definability Recall D(S) = {x ∈ A : x is definable from S}. Observation If x is definable from S and σ is an automorphism of A fixing S, then σ also fixes x.
Francis Adams Computability and Categoricity of Ultrahomogeneous Structures
Ultrahomogeneous Structures
Linear Orders
Other Examples
Definability Recall D(S) = {x ∈ A : x is definable from S}. Observation If x is definable from S and σ is an automorphism of A fixing S, then σ also fixes x. Proposition Let A be a countable weakly ultrahomogeneous linear order and let M = {a1 < . . . < an } be a minimal exceptional set. Then D(M) is the set of successivities of A.
Francis Adams Computability and Categoricity of Ultrahomogeneous Structures
Ultrahomogeneous Structures
Linear Orders
Other Examples
Definability
Q
a1
a2
a3
a4
Francis Adams Computability and Categoricity of Ultrahomogeneous Structures
a5
Q
a6
a7
Ultrahomogeneous Structures
Linear Orders
Other Examples
Definability
Q
a1
a2
a3
a4
Francis Adams Computability and Categoricity of Ultrahomogeneous Structures
a5
Q
a6
a7
Ultrahomogeneous Structures
Linear Orders
Other Examples
Definability
Q
a1
a2
a3
a4
Francis Adams Computability and Categoricity of Ultrahomogeneous Structures
a5
Q
a6
a7
Ultrahomogeneous Structures
Linear Orders
Other Examples
Definability
Q
a1
a2
a3
a4
Francis Adams Computability and Categoricity of Ultrahomogeneous Structures
a5
Q
a6
a7
Ultrahomogeneous Structures
Linear Orders
Other Examples
Classification of Equivalence Structures Theorem For a countable equivalence structure A, the following are equivalent: 1 A is weakly ultrahomogeneous. 2 All but finitely many equivalence classes of A are of the same size. In this case, a minimal exceptional set is contains exactly one element from each of the exceptional equivalence classes.
Francis Adams Computability and Categoricity of Ultrahomogeneous Structures
Ultrahomogeneous Structures
Linear Orders
Other Examples
Effective Categoricity
By a result of Cenzer, Harizanov, Calvert, Morzov (2005) we have Corollary A computable equivalence structure is weakly ultrahomogeneous iff it is computably categorical
Francis Adams Computability and Categoricity of Ultrahomogeneous Structures
Ultrahomogeneous Structures
Linear Orders
Other Examples
Definability
Proposition Let A = (A, E ) be a countable weakly ultrahomogeneous equivalence structure and let M = {a1 , . . . , an } a minimal exceptional set. Then x ∈ A is definable from M iff x is in an exceptional class of size 2.
Francis Adams Computability and Categoricity of Ultrahomogeneous Structures
Ultrahomogeneous Structures
Linear Orders
Other Examples
Classification of Injection Structures
Proposition An injection structure A is weakly ultrahomogeneous iff it has finitely many ω-orbits. In this case, a minimal exceptional set contains exactly one member from each ω-orbit.
Francis Adams Computability and Categoricity of Ultrahomogeneous Structures
Ultrahomogeneous Structures
Linear Orders
Other Examples
Effective Categoricity By a result of Cenzer, Harizanov, and Remmel A computable injection structure is computably categorical iff it has finitely many infinite orbits Such a structure is ∆02 -categorical iff it has finitely many ω-orbits or finitely many Z-orbits. So for computable injection structures, computable categoricity implies weak ultrahomogeneity which implies ∆02 -categoricity. Neither implication can be reversed as witnessed by computable injection structures consisting of only infinitely Z-orbits, and of only infinitely many ω-orbits, respectively. Francis Adams Computability and Categoricity of Ultrahomogeneous Structures
Ultrahomogeneous Structures
Future Work
Francis Adams Computability and Categoricity of Ultrahomogeneous Structures
Linear Orders
Other Examples
Ultrahomogeneous Structures
Linear Orders
Other Examples
Future Work
The program is: Classify all weakly ultrahomogeneous structures for some class of structures (graphs, BAs, posets, etc.)
Francis Adams Computability and Categoricity of Ultrahomogeneous Structures
Ultrahomogeneous Structures
Linear Orders
Other Examples
Future Work
The program is: Classify all weakly ultrahomogeneous structures for some class of structures (graphs, BAs, posets, etc.) Determine effective categoricity of such structures.
Francis Adams Computability and Categoricity of Ultrahomogeneous Structures
Ultrahomogeneous Structures
Thank you.
Francis Adams Computability and Categoricity of Ultrahomogeneous Structures
Linear Orders
Other Examples
Linear Orders
Computability and Categoricity of Ultrahomogeneous Structures Francis Adams
January 17, 2014
Francis Adams Computability and Categoricity of Ultrahomogeneous Structures
Other Examples
Ultrahomogeneous Structures
Linear Orders
Other Examples
Ultrahomogeneous Structures
Definition A structure is ultrahomogeneous if any isomorphism between finitely generated substructures extends to an automorphism of the whole structure.
Francis Adams Computability and Categoricity of Ultrahomogeneous Structures
Ultrahomogeneous Structures
Linear Orders
Other Examples
Countable Ultrahomogeneous Structures
Example Q, the countable dense linear order. The countable atomless boolean algebra. The Rado graph. Equivalence Structures: Every class has the same size. Injection Structures: No ω-orbits.
Francis Adams Computability and Categoricity of Ultrahomogeneous Structures
Ultrahomogeneous Structures
Linear Orders
Other Examples
Effective Categoricity Definition A computable structure A is ∆0α categorical if every computable structure isomorphic to A is ∆0α isomorphic to A. In particular, a computable structure A is computably categorical if every computable structure isomorphic to A is computably isomorphic to A.
Francis Adams Computability and Categoricity of Ultrahomogeneous Structures
Ultrahomogeneous Structures
Linear Orders
Theorem 1
Theorem Every computable ultrahomogeneous structure is ∆02 -categorical.
Francis Adams Computability and Categoricity of Ultrahomogeneous Structures
Other Examples
Ultrahomogeneous Structures
Linear Orders
Other Examples
Proof of Theorem 1
Lemma Checking if two finitely generated substructures are isomorphic is Π01 . We’ll use this lemma along with a back and forth argument, building S partial isomorphisms θn at each stage and letting θ = θn .
Francis Adams Computability and Categoricity of Ultrahomogeneous Structures
Ultrahomogeneous Structures
Linear Orders
Other Examples
Proof of Theorem 1
Suppose we have defined θ2n−1 for {a0 , . . . a2n−1 } with θ2n−1 (ai ) = bi . Choose the least a2n ∈ A \ {a0 , . . . a2n−1 }. There exists a b ∈ B such that ha0 , . . . a2n i ∼ = hb0 , . . . , b2n−1 , bi where the isomorphism can be chosen to extend θ2n−1 . Search for this b using a Π01 -oracle to check if ha0 , . . . a2n i ∼ = hb0 , . . . , b2n−1 , bi. After finding one, call it b2n , S and define θ2n (a2n ) = b2n . Then θ = θn is a ∆02 isomorphism.
Francis Adams Computability and Categoricity of Ultrahomogeneous Structures
Ultrahomogeneous Structures
Linear Orders
Other Examples
Corollary Any relational, or more generally, any locally finite, computable ultrahomogeneous structure is computably categorical. The converse is false; a single Z-orbit is computably categorical.
Francis Adams Computability and Categoricity of Ultrahomogeneous Structures
Ultrahomogeneous Structures
Linear Orders
Other Examples
Weakly Ultrahomogeneous Structures
Definition A structure A is weakly ultrahomogeneous if there exists a finite set {a1 , a2 , . . . , an } ⊆ A such that for all tuples ~x , ~y from A with h~a, ~x i ∼ = h~a, ~y i where each ai is fixed, this isomorphism of substructures extends to an automorphism of A. Call such a set {a1 , a2 , . . . , an } an exceptional set of A.
Francis Adams Computability and Categoricity of Ultrahomogeneous Structures
Ultrahomogeneous Structures
Linear Orders
Other Examples
Weakly Ultrahomogeneous Structures Alternatively, A is weakly ultrahomogeneous if there is a finite set a1 , . . . , an of elements from A such that (A, a1 , . . . , an ) is ultrahomogeneous in the extended language with constants for the ai .
Francis Adams Computability and Categoricity of Ultrahomogeneous Structures
Ultrahomogeneous Structures
Linear Orders
Other Examples
Weakly Ultrahomogeneous Structures Alternatively, A is weakly ultrahomogeneous if there is a finite set a1 , . . . , an of elements from A such that (A, a1 , . . . , an ) is ultrahomogeneous in the extended language with constants for the ai . Theorem Every computable weakly ultrahomogeneous structure is ∆02 -categorical. Corollary Every locally finite computable weakly ultrahomogeneous structure is computably categorical. Francis Adams Computability and Categoricity of Ultrahomogeneous Structures
Ultrahomogeneous Structures
Linear Orders
Other Examples
Classification of Linear Orders
Theorem For a countable linear order A, the following are equivalent: 1 A is weakly ultrahomogeneous. 2 A has finitely many successivities. 3 A = L0 + Q + L1 + Q + . . . + Q + Ln where the Li are finite chains, L0 , Ln are possibly empty and |Li | ≥ 2 for 1 ≤ i ≤ n − 1.
Francis Adams Computability and Categoricity of Ultrahomogeneous Structures
Ultrahomogeneous Structures
Linear Orders
Other Examples
Effective Categoricity
Theorem (Remmel 1981) A computable linear ordering is computably categorical iff it has finitely many successivities. Corollary A computable linear order is weakly ultrahomogeneous iff it is computably categorical.
Francis Adams Computability and Categoricity of Ultrahomogeneous Structures
Ultrahomogeneous Structures
Linear Orders
Other Examples
Minimal Exceptional Sets
Q
a1
a2
a3
Francis Adams Computability and Categoricity of Ultrahomogeneous Structures
a4
a5
Q
Ultrahomogeneous Structures
Linear Orders
Other Examples
Minimal Exceptional Sets
The following sets aren’t exceptional:
Q
a1
a2
a3
a4
a5
Q
Q
a1
a2
a3
a4
a5
Q
Francis Adams Computability and Categoricity of Ultrahomogeneous Structures
Ultrahomogeneous Structures
Linear Orders
Other Examples
Minimal Exceptional Sets
Including all successivities yields an exceptional set:
Q
a1
a2
a3
Francis Adams Computability and Categoricity of Ultrahomogeneous Structures
a4
a5
Q
Ultrahomogeneous Structures
Linear Orders
Other Examples
Minimal Exceptional Sets
Including all successivities yields an exceptional set:
Q
a1
a2
a3
But it isn’t minimal.
Francis Adams Computability and Categoricity of Ultrahomogeneous Structures
a4
a5
Q
Ultrahomogeneous Structures
Linear Orders
Other Examples
Minimal Exceptional Sets
This set is a minimal exceptional set:
Q
a1
a2
a3
Francis Adams Computability and Categoricity of Ultrahomogeneous Structures
a4
a5
Q
Ultrahomogeneous Structures
Linear Orders
Other Examples
Minimal Exceptional Sets
So is this set:
Q
a1
a2
a3
Francis Adams Computability and Categoricity of Ultrahomogeneous Structures
a4
a5
Q
Ultrahomogeneous Structures
Linear Orders
Other Examples
Minimal Exceptional Sets
So is this set:
Q
a1
a2
a3
a4
a5
Q
So minimal exceptional sets aren’t unique, and don’t even have to be isomorphic.
Francis Adams Computability and Categoricity of Ultrahomogeneous Structures
Ultrahomogeneous Structures
Linear Orders
Other Examples
Minimal Exceptional Sets
Theorem Let A = L0 + Q + L1 + Q + . . . + Q + Ln be a countable weakly ultrahomogeneous linear order. The S minimal exceptional sets are those subsets S of i≤n Li such that i) A \ S doesn’t contain two consecutive successivities. ii) S contains each last element of L0 , . . . Ln−1 and each first element of L1 , . . . Ln .
Francis Adams Computability and Categoricity of Ultrahomogeneous Structures
Ultrahomogeneous Structures
Linear Orders
Other Examples
Definability Recall D(S) = {x ∈ A : x is definable from S}. Observation If x is definable from S and σ is an automorphism of A fixing S, then σ also fixes x.
Francis Adams Computability and Categoricity of Ultrahomogeneous Structures
Ultrahomogeneous Structures
Linear Orders
Other Examples
Definability Recall D(S) = {x ∈ A : x is definable from S}. Observation If x is definable from S and σ is an automorphism of A fixing S, then σ also fixes x. Proposition Let A be a countable weakly ultrahomogeneous linear order and let M = {a1 < . . . < an } be a minimal exceptional set. Then D(M) is the set of successivities of A.
Francis Adams Computability and Categoricity of Ultrahomogeneous Structures
Ultrahomogeneous Structures
Linear Orders
Other Examples
Definability
Q
a1
a2
a3
a4
Francis Adams Computability and Categoricity of Ultrahomogeneous Structures
a5
Q
a6
a7
Ultrahomogeneous Structures
Linear Orders
Other Examples
Definability
Q
a1
a2
a3
a4
Francis Adams Computability and Categoricity of Ultrahomogeneous Structures
a5
Q
a6
a7
Ultrahomogeneous Structures
Linear Orders
Other Examples
Definability
Q
a1
a2
a3
a4
Francis Adams Computability and Categoricity of Ultrahomogeneous Structures
a5
Q
a6
a7
Ultrahomogeneous Structures
Linear Orders
Other Examples
Definability
Q
a1
a2
a3
a4
Francis Adams Computability and Categoricity of Ultrahomogeneous Structures
a5
Q
a6
a7
Ultrahomogeneous Structures
Linear Orders
Other Examples
Classification of Equivalence Structures Theorem For a countable equivalence structure A, the following are equivalent: 1 A is weakly ultrahomogeneous. 2 All but finitely many equivalence classes of A are of the same size. In this case, a minimal exceptional set is contains exactly one element from each of the exceptional equivalence classes.
Francis Adams Computability and Categoricity of Ultrahomogeneous Structures
Ultrahomogeneous Structures
Linear Orders
Other Examples
Effective Categoricity
By a result of Cenzer, Harizanov, Calvert, Morzov (2005) we have Corollary A computable equivalence structure is weakly ultrahomogeneous iff it is computably categorical
Francis Adams Computability and Categoricity of Ultrahomogeneous Structures
Ultrahomogeneous Structures
Linear Orders
Other Examples
Definability
Proposition Let A = (A, E ) be a countable weakly ultrahomogeneous equivalence structure and let M = {a1 , . . . , an } a minimal exceptional set. Then x ∈ A is definable from M iff x is in an exceptional class of size 2.
Francis Adams Computability and Categoricity of Ultrahomogeneous Structures
Ultrahomogeneous Structures
Linear Orders
Other Examples
Classification of Injection Structures
Proposition An injection structure A is weakly ultrahomogeneous iff it has finitely many ω-orbits. In this case, a minimal exceptional set contains exactly one member from each ω-orbit.
Francis Adams Computability and Categoricity of Ultrahomogeneous Structures
Ultrahomogeneous Structures
Linear Orders
Other Examples
Effective Categoricity By a result of Cenzer, Harizanov, and Remmel A computable injection structure is computably categorical iff it has finitely many infinite orbits Such a structure is ∆02 -categorical iff it has finitely many ω-orbits or finitely many Z-orbits. So for computable injection structures, computable categoricity implies weak ultrahomogeneity which implies ∆02 -categoricity. Neither implication can be reversed as witnessed by computable injection structures consisting of only infinitely Z-orbits, and of only infinitely many ω-orbits, respectively. Francis Adams Computability and Categoricity of Ultrahomogeneous Structures
Ultrahomogeneous Structures
Future Work
Francis Adams Computability and Categoricity of Ultrahomogeneous Structures
Linear Orders
Other Examples
Ultrahomogeneous Structures
Linear Orders
Other Examples
Future Work
The program is: Classify all weakly ultrahomogeneous structures for some class of structures (graphs, BAs, posets, etc.)
Francis Adams Computability and Categoricity of Ultrahomogeneous Structures
Ultrahomogeneous Structures
Linear Orders
Other Examples
Future Work
The program is: Classify all weakly ultrahomogeneous structures for some class of structures (graphs, BAs, posets, etc.) Determine effective categoricity of such structures.
Francis Adams Computability and Categoricity of Ultrahomogeneous Structures
Ultrahomogeneous Structures
Thank you.
Francis Adams Computability and Categoricity of Ultrahomogeneous Structures
Linear Orders
Other Examples
