Computable Ultrahomogeneous Structures · This is equivalent to saying the atomic diagram of Ais ... I A computable structure Ais computably categorical if for ... Computable Ultrahomogeneous

Introduction Examples Trees

Computable Ultrahomogeneous Structures

Francis Adams

September 14, 2016



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 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.



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′.



















Ultrahomogeneous Structures

DefinitionA structure A is ultrahomogeneous if every partialisomorphism of finitely generated substructures ψ : 〈~x〉 → 〈~y〉extends to an automorphism of A.



Examples

The canonical example of an ultrahomogeneous structure is(Q, <).

Other examples include the random graph and the countableatomless boolean algebra.



Examples

The canonical example of an ultrahomogeneous structure is(Q, <).Other examples include the random graph and the countableatomless boolean algebra.



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.



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



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.



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.



Weakly Ultrahomogeneous Structures

For various classes of structures:

I Classify the weakly ultrahomogeneous structures.

I Compare weak ultrahomogeneity with the notions ofeffective categoricity.



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.



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.



Minimal Exceptional Sets

The following sets aren’t exceptional:

Q a1 a2 a3 a4 a5 Q

Q a1 a2 a3 a4 a5 Q




Including all successivities yields an exceptional set:

Q a1 a2 a3 a4 a5 QBut it isn’t minimal.




This set is a minimal exceptional set:

Q a1 a2 a3 a4 a5 Q




So is this set:

Q a1 a2 a3 a4 a5 QSo minimal exceptional sets aren’t unique, and don’t evenhave to be isomorphic.



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.



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.



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.



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.



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.



Classification

Corollary (Cenzer et al. 2005)A computable equivalence structure is weaklyultrahomogeneous iff it is relatively computably categorical iffit is computably categorical.



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.



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.



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.




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.



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.



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 ]}



Trees as Partial Orders

A tree (T , <) is ultrahomogeneous iff T has rank ≤ 1.



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.



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.



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.



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.

















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 .




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.




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.



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 .




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.




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.




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.




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.




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.



To come:

I Boolean algebras

I Abelian p-groups

I Transfinite trees

I and more...


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Computable Ultrahomogeneous Structures · This is equivalent to saying the atomic diagram of Ais ... I A computable structure Ais computably categorical if for ... Computable Ultrahomogeneous