Introduction Examples Trees
Computable Ultrahomogeneous Structures
Francis Adams
September 14, 2016
Computable Ultrahomogeneous Structures
Introduction Examples Trees
Computable Model Theory
All structures will be countable structures over a languagewith finitely many non-logical symbols.
DefinitionA structure A is computable if its universe, functions, andrelations are computable.
This is equivalent to saying the atomic diagram of A iscomputable, where the atomic diagram is the set of allsentences true about A which are atomic, or are the negationof an atomic sentence.
Computable Ultrahomogeneous Structures
Introduction Examples Trees
Computable Model Theory
All structures will be countable structures over a languagewith finitely many non-logical symbols.
DefinitionA structure A is computable if its universe, functions, andrelations are computable.
This is equivalent to saying the atomic diagram of A iscomputable, where the atomic diagram is the set of allsentences true about A which are atomic, or are the negationof an atomic sentence.
Computable Ultrahomogeneous Structures
Introduction Examples Trees
Effective Categoricity
I A computable structure A is computably categorical if forevery computable structure B which is isomorphic to A,there is an isomorphism A → B which is computable.
I A computable structure A is relatively computablycategorical if for every structure B which is isomorphic toA, there is an isomorphism A → B which is computablerelative to B.
We can similarly define when a structure is (relatively) ∆02
categorical. Recall that a ∆02 function is a function whose
graph is Σ02, or equivalently is computable from 0′.
Computable Ultrahomogeneous Structures
Introduction Examples Trees
Effective Categoricity
I A computable structure A is computably categorical if forevery computable structure B which is isomorphic to A,there is an isomorphism A → B which is computable.
I A computable structure A is relatively computablycategorical if for every structure B which is isomorphic toA, there is an isomorphism A → B which is computablerelative to B.
We can similarly define when a structure is (relatively) ∆02
categorical. Recall that a ∆02 function is a function whose
graph is Σ02, or equivalently is computable from 0′.
Computable Ultrahomogeneous Structures
Introduction Examples Trees
Effective Categoricity
I A computable structure A is computably categorical if forevery computable structure B which is isomorphic to A,there is an isomorphism A → B which is computable.
I A computable structure A is relatively computablycategorical if for every structure B which is isomorphic toA, there is an isomorphism A → B which is computablerelative to B.
We can similarly define when a structure is (relatively) ∆02
categorical. Recall that a ∆02 function is a function whose
graph is Σ02, or equivalently is computable from 0′.
Computable Ultrahomogeneous Structures
Introduction Examples Trees
Ultrahomogeneous Structures
DefinitionA structure A is ultrahomogeneous if every partialisomorphism of finitely generated substructures ψ : 〈~x〉 → 〈~y〉extends to an automorphism of A.
Computable Ultrahomogeneous Structures
Introduction Examples Trees
Examples
The canonical example of an ultrahomogeneous structure is(Q, <).
Other examples include the random graph and the countableatomless boolean algebra.
Computable Ultrahomogeneous Structures
Introduction Examples Trees
Examples
The canonical example of an ultrahomogeneous structure is(Q, <).Other examples include the random graph and the countableatomless boolean algebra.
Computable Ultrahomogeneous Structures
Introduction Examples Trees
Weakly Ultrahomogeneous Structures
DefinitionA structure A is weakly ultrahomogeneous if there exists afinite set {a1, a2, . . . , an} ⊆ A such that for all tuples ~x , ~y fromA with 〈~a, ~x〉 ∼= 〈~a, ~y〉 where each ai is fixed, this isomorphismof substructures extends to an automorphism of A. Call sucha set {a1, a2, . . . , an} an exceptional set of A.
Computable Ultrahomogeneous Structures
Introduction Examples Trees
Examples
The linear order consisting of a copy of Q, followed by a finitelinear order of length 5, followed by another copy of Q isweakly ultrahomogeneous. The 5 elements in the middle forman exceptional set.
Q a1 a2 a3 a4 a5 Q
Computable Ultrahomogeneous Structures
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TheoremEvery computable weakly ultrahomogeneous structure isrelatively ∆0
2-categorical.
Proof.(Sketch) Use a back-and-forth argument. At each stage youwill have to check if two finitely generated substructures areisomorphic, which is Π0
1.
CorollaryAny locally finite computable weakly ultrahomogeneousstructure is relatively computably categorical. In particular thisholds for relational structures.
Computable Ultrahomogeneous Structures
Introduction Examples Trees
TheoremEvery computable weakly ultrahomogeneous structure isrelatively ∆0
2-categorical.
Proof.(Sketch) Use a back-and-forth argument. At each stage youwill have to check if two finitely generated substructures areisomorphic, which is Π0
1.
CorollaryAny locally finite computable weakly ultrahomogeneousstructure is relatively computably categorical. In particular thisholds for relational structures.
Computable Ultrahomogeneous Structures
Introduction Examples Trees
Weakly Ultrahomogeneous Structures
For various classes of structures:
I Classify the weakly ultrahomogeneous structures.
I Compare weak ultrahomogeneity with the notions ofeffective categoricity.
Computable Ultrahomogeneous Structures
Introduction Examples Trees
Linear Orders
Theorem (Remmel 1981)If A is a computable linear order, then A is computablycategorical iff A is relatively computably categorical iff A hasfinitely many successivities.
TheoremA countable linear order A is weakly ultrahomogeneous iff Ahas finitely many successivities.
So being computably categorical, relatively computablycategorical, and weakly ultrahomogeneous all coincide.
Computable Ultrahomogeneous Structures
Introduction Examples Trees
Exceptional Sets
TheoremLet A be a weakly ultrahomogeneous linear order. Theminimal exceptional sets are those sets of successivities Swhich are minimal with respect to the following:
i) A \ S doesn’t contain two consecutive successivities.
ii) S contains all elements of the successivity chainsimmediately before or after a copy of Q.
Computable Ultrahomogeneous Structures
Introduction Examples Trees
Minimal Exceptional Sets
The following sets aren’t exceptional:
Q a1 a2 a3 a4 a5 Q
Q a1 a2 a3 a4 a5 Q
Computable Ultrahomogeneous Structures
Introduction Examples Trees
Minimal Exceptional Sets
Including all successivities yields an exceptional set:
Q a1 a2 a3 a4 a5 QBut it isn’t minimal.
Computable Ultrahomogeneous Structures
Introduction Examples Trees
Minimal Exceptional Sets
This set is a minimal exceptional set:
Q a1 a2 a3 a4 a5 Q
Computable Ultrahomogeneous Structures
Introduction Examples Trees
Minimal Exceptional Sets
So is this set:
Q a1 a2 a3 a4 a5 QSo minimal exceptional sets aren’t unique, and don’t evenhave to be isomorphic.
Computable Ultrahomogeneous Structures
Introduction Examples Trees
Equivalence Structures
DefinitionAn equivalence structure is A = (A,E ) where E is anequivalence relation on A.
An equivalence structure A is ultrahomogeneous iff allequivalence classes are the same size.Proof idea: A partial isomorphism induces a permutation ofthe classes, and within each class.
Computable Ultrahomogeneous Structures
Introduction Examples Trees
Equivalence Structures
DefinitionAn equivalence structure is A = (A,E ) where E is anequivalence relation on A.
An equivalence structure A is ultrahomogeneous iff allequivalence classes are the same size.Proof idea: A partial isomorphism induces a permutation ofthe classes, and within each class.
Computable Ultrahomogeneous Structures
Introduction Examples Trees
Classification
TheoremAn equivalence structure A is weakly ultrahomogeneous iff allbut finitely many equivalence classes are the same size. In thiscase, a minimal exceptional set contains exactly one elementfrom each of the exceptional equivalence classes.
Computable Ultrahomogeneous Structures
Introduction Examples Trees
Classification
Proof.[⇒] Suppose that A has infinitely classes of different sizes andlet {a1, . . . , an} be a finite subset. Find elements x , y fromclasses of different sizes so neither is related to any ai .
[⇐] Let {a1, . . . , an} contain exactly one element from each ofthe exceptional classes. Then suppose 〈~a, ~x〉 ∼= 〈~a, ~y〉 withxi → yi . Then either each xi , yi are in an equivalence classwith some ak , or if not they are in classes of the same size. Ineither case the isomorphism extends.
The claim about exceptional sets follows.
Computable Ultrahomogeneous Structures
Introduction Examples Trees
Classification
Proof.[⇒] Suppose that A has infinitely classes of different sizes andlet {a1, . . . , an} be a finite subset. Find elements x , y fromclasses of different sizes so neither is related to any ai .[⇐] Let {a1, . . . , an} contain exactly one element from each ofthe exceptional classes. Then suppose 〈~a, ~x〉 ∼= 〈~a, ~y〉 withxi → yi . Then either each xi , yi are in an equivalence classwith some ak , or if not they are in classes of the same size. Ineither case the isomorphism extends.
The claim about exceptional sets follows.
Computable Ultrahomogeneous Structures
Introduction Examples Trees
Classification
Corollary (Cenzer et al. 2005)A computable equivalence structure is weaklyultrahomogeneous iff it is relatively computably categorical iffit is computably categorical.
Computable Ultrahomogeneous Structures
Introduction Examples Trees
Injection Structures
DefinitionAn injection structure is A = (A, f ) where f is an injectivefunction on A.
An injection structure is partitioned into orbits of three types:finite cycles, ω-orbits, and Z-orbits.An injection structure is ultrahomogeneous iff it has noω-orbits.
Computable Ultrahomogeneous Structures
Introduction Examples Trees
Classification
TheoremAn injection structure is weakly ultrahomogeneous iff it hasfinitely many ω-orbits. In this case, a minimal exceptional setcontains exactly one member from each ω-orbit.
Proof.[⇒] Let S ⊆ A be finite and look in an ω-orbit containing noelement of S .[⇐] Name an element from each ω-orbit. This will fix each ofthese orbits.
Computable Ultrahomogeneous Structures
Introduction Examples Trees
Classification
TheoremAn injection structure is weakly ultrahomogeneous iff it hasfinitely many ω-orbits. In this case, a minimal exceptional setcontains exactly one member from each ω-orbit.
Proof.[⇒] Let S ⊆ A be finite and look in an ω-orbit containing noelement of S .[⇐] Name an element from each ω-orbit. This will fix each ofthese orbits.
Computable Ultrahomogeneous Structures
Introduction Examples Trees
Effective Categoricity
Theorem (Cenzer et. al. 2014)
I A computable injection structure is computablycategorical iff it is relatively computably categorical iff ithas finitely many infinite orbits
I Such a structure is (relatively) ∆02-categorical iff it has
finitely many ω-orbits or finitely many Z-orbits.
So for computable injection structures, computablecategoricity ⇒ weak ultrahomogeneity ⇒ ∆0
2-categoricity.Neither implication can be reversed.
Computable Ultrahomogeneous Structures
Introduction Examples Trees
Trees
Consider rooted trees, viewed as subsets of ω<ω closed underinitial segments with the empty string λ as a root.These can be realized as:
I A partial order (T , <) where < well-orders thepredecessors of each node.
I (T , f ) where f is a predecessor function: f (t _ n) = tand f (λ) = λ.
In both formulations we assume the root is named.
Computable Ultrahomogeneous Structures
Introduction Examples Trees
Trees
For x ∈ T , let T [x ] be the tree of extensions of x in T .
Let Tn = {x ∈ T : ht(x) = n}.
We define a rank on elements of T . For x ∈ T :
I rkT (x) = 0 if x is a dead-end (leaf node)
I rkT (x) = sup{rkT (y) + 1 : y ∈ T [x ]}
Computable Ultrahomogeneous Structures
Introduction Examples Trees
Trees as Partial Orders
A tree (T , <) is ultrahomogeneous iff T has rank ≤ 1.
Computable Ultrahomogeneous Structures
Introduction Examples Trees
Trees as Partial Orders
Proposition(T , <) is weakly ultrahomogeneous iff only finitely manyelements have rank ≥ 1. In particular, weaklyultrahomogeneous trees have finite height.
Proof.[⇒] For S ⊆ T finite, find a not in the downward closure of Swith successor b.[⇐] Let S ⊆ T be the set of elements with successors.
Computable Ultrahomogeneous Structures
Introduction Examples Trees
Effective Categoricity of Trees
Theorem (Miller)No tree of infinite height is computably categorical.
Theorem (Lempp et al)A computable tree (T , <) is computably categorical iff it isrelatively computably categorical iff it is of finite type.
Computable Ultrahomogeneous Structures
Introduction Examples Trees
Effective Categoricity of Trees
Theorem (Miller)No tree of infinite height is computably categorical.
Theorem (Lempp et al)A computable tree (T , <) is computably categorical iff it isrelatively computably categorical iff it is of finite type.
Computable Ultrahomogeneous Structures
Introduction Examples Trees
Trees with Predecessor
With a predecessor function we are able to determine theheight of any node, so have more (weakly) ultrahomogeneoustrees.
Proposition(T , f ) is ultrahomogeneous if for all n, all a, b ∈ Tn have thesame number of successors.
Equivalently there is a branching function f : ω → ω + 1 suchthat every a ∈ Tn has f (n) many successors.
Computable Ultrahomogeneous Structures
Introduction Examples Trees
Trees with Predecessor
With a predecessor function we are able to determine theheight of any node, so have more (weakly) ultrahomogeneoustrees.
Proposition(T , f ) is ultrahomogeneous if for all n, all a, b ∈ Tn have thesame number of successors.
Equivalently there is a branching function f : ω → ω + 1 suchthat every a ∈ Tn has f (n) many successors.
Computable Ultrahomogeneous Structures
Introduction Examples Trees
Trees with Predecessor
With a predecessor function we are able to determine theheight of any node, so have more (weakly) ultrahomogeneoustrees.
Proposition(T , f ) is ultrahomogeneous if for all n, all a, b ∈ Tn have thesame number of successors.
Equivalently there is a branching function f : ω → ω + 1 suchthat every a ∈ Tn has f (n) many successors.
Computable Ultrahomogeneous Structures
Introduction Examples Trees
Trees with Predecessor
Proposition(T , f ) is ultrahomogeneous if for all n, all a, b ∈ Tn have thesame number of successors.
Proof.[⇒] If not, let n be the least level with such a, b. Then thereis an automorphism of Tn sending a to b which can’t extend.[⇐] Given an isomorphism of subtrees ϕ : S1 → S2, recursivelydefine permutations of Tn respecting ϕ, yielding anautomorphism of T .
Computable Ultrahomogeneous Structures
Introduction Examples Trees
Trees with Predecessor
Given a tree T , a subtree S ⊆ T , and x ∈ T we can definethe subtree TS [x ] consisting of the node x , its predecessors,together with all nodes in T extending successors of x whichare not in S .
Proposition(T , f ) is weakly ultrahomogeneous iff there is a a finitesubtree S of T such that, for every x ∈ S , the tree TS [x ] isultrahomogeneous.
Computable Ultrahomogeneous Structures
Introduction Examples Trees
Trees with Predecessor
PropositionA tree of height ≤ 2 is weakly ultrahomogeneous if and only ifall but finitely many nodes of height 1 have an equal numberof successors.
PropositionA tree of height 3 is weakly ultrahomogeneous if and only ifthe following conditions hold:
(a) for each node x of height 1, all but finitely manysuccessors of x have an equal number of successors;
(b) there are fixed h and k in ω ∪ {ω} such that all butfinitely many nodes of height 1 have exactly h successorsand each of those successors has exactly k successors.
Computable Ultrahomogeneous Structures
Introduction Examples Trees
n-Equivalence Structures
DefinitionFor n < ω, an n-equivalence structure is a structureA = (A,E1, . . . ,En) where each Ei is an equivalence relationon A. An n-equivalence structure is nested if for i < j ≤ n wehave xEjy → xEiy , i.e. Ej ⊆ Ei as subsets of A× A. Thus theEi classes are partitioned by Ej .
Computable Ultrahomogeneous Structures
Introduction Examples Trees
n-Equivalence Structures
DefinitionFor any n-equivalence structure A = (A,E1, . . . ,En), letE0 = A× A, let En+1 be equality, and define the tree TA asfollows. The universe of TA is the set{[a]i : a ∈ A, i = 1, . . . , n} and the partial ordering isinclusion. This means that for each a and i ≤ n, [a]i is thepredecessor of [a]i+1.
Computable Ultrahomogeneous Structures
Introduction Examples Trees
n-Equivalence Structures
Theorem (Marshall 2015)Let A be a computable n-equivalence structure and TA itscorresponding tree of finite height. Then the following areequivalent:
I A is computably categorical.
I A is relatively computably categorical.
I (TA,≺) is computably categorical.
I (TA,≺) is relatively computably categorical.
I (TA,≺) is of finite type.
Computable Ultrahomogeneous Structures
Introduction Examples Trees
n-Equivalence Structures
TheoremLet A = (A,E1, . . . ,En) be a nested n-equivalence structureand let E0 = A× A and En+1 be equality. Then the followingare equivalent.
1. A is ultrahomogeneous.
2. For each i ≤ n there exists ki such that every Ei class ispartitioned into ki many Ei+1 classes.
3. TA is ultrahomogeneous in the predecessor representation.
Computable Ultrahomogeneous Structures
Introduction Examples Trees
n-Equivalence Structures
CorollaryIf A = (A,E1, . . . ,En) is a nested n-equivalence structure suchthat all equivalence classes are finite, then A isultrahomogeneous if and only if each (A,Ei) isultrahomogeneous.
This restriction is necessary as seen by the 2-equivalencestructure where E1 is two infinite classes and E2 partitions oneE1 class into 3 classes and the other into 5 classes.
Computable Ultrahomogeneous Structures
Introduction Examples Trees
n-Equivalence Structures
TheoremLet A = (A,E1, . . . ,En) be a nested n-equivalence structureand let E0 = A× A and En+1 be equality. Then TA is weaklyultrahomogeneous in the predecessor representation if and onlyif A is weakly ultrahomogeneous.
Computable Ultrahomogeneous Structures
Introduction Examples Trees
To come:
I Boolean algebras
I Abelian p-groups
I Transfinite trees
I and more...
Computable Ultrahomogeneous Structures