Computability-Theoretic Categoricity and Scott Families ·

Computability-Theoretic Categoricity andScott Families

Ekaterina Fokinajoint work with Valentina Harizanov and Daniel Turetsky

Kurt Gödel Research Center for Mathematical LogicUniversity of Vienna

Logic ColloquiumAugust 3–8, 2015

Complexity of isomorphisms

Goal for this talk:We study the complexity of isomorphisms between acomputable structure and its countable/computable copies.

DefinitionA computable structureM is computably categorical(autostable) if, for every computable structure A isomorphic toM, there exists a computable isomorphism fromM onto A.

Theorem (Fröhlich and Shepherdson, 1956)There exist examples of computable fields, extensions of therationals, of both finite and infinite transcendence degrees,which are not computably categorical.

Theorem (Mal’cev, 1962)There exist a computable group which is not computablycategorical.

Computably categorical structures• A vector space is c.c. iff it has a finite basis.• Goncharov and Dzgoev (1980); Remmel (1981): A linear

order is c.c. iff it has only finitely many successor pairs.• Goncharov and Dzgoev; Remmel; LaRoche (1977): A

computable Boolean algebra is c.c. if it has finitely manyatoms.

• Goncharov (1980); Smith (1981): abelian p-groups.• Goncharov, Lempp and Solomon (2003): ordered abelian

groups and ordered Archimedean groups.• Lempp, McCoy, R. Miller and Solomon (2005): trees of

finite hight.• Calvert, Cenzer, Harizanov (2009), and Morozov:

equivalence structures.• Cenzer, Harizanov, and Remmel (2014): injection

structures.• R. Miller, Shlapentokh (2015): algebraic fields with a

splitting algorithm.

Relative computable categoricity

If we consider arbitrary countable copies of a computablestructure:

DefinitionA computable structureM is relatively computably categorical iffor every A isomorphic toM, there is an isomorphism fromMto A, which is computable relative to the atomic diagram of A.Relative computable categoricity implies computablecategoricity.

Theorem (Goncharov, 1977)Computable categoricity of a computable structure does notimply its relative computable categoricity.

Computably categorical not relativelyc.c.

• Hirschfeldt, Khoussainov, Shore, and Slinko: directedgraphs, undirected graphs, partial orders, lattices, rings(with zero-divisors), integral domains of arbitrarycharacteristic, commutative semigroups, and 2-stepnilpotent groups

• R. Miller, Park, Poonen, Schoutens, and A. Shlapentokh:fields.

• Hirschfeldt, Kramer, R. Miller, and Shlapentokhcharacterized relative computable categoricity forcomputable algebraic fields.

Theorem (Hirschfeldt, Kramer, R. Miller, and Shlapentokh,2015)There is a computably categorical algebraic field, which is notrelatively computably categorical.

When computably categorical isrelatively c.c.

For linear orders, Boolean algebras, trees of finite height,abelian p-group, equivalence structures, injection structures,algebraic fields with a splitting algorithm, computablecategoricity implies relative computable categoricity.

d-computable categoricity

DefinitionA computable structureM is d-computably categorical if, forevery computable structure A isomorphic toM, there exists ad-computable isomorphism fromM onto A.In the case when d = 0(n−1), n ≥ 1, we also say thatM is∆0

n-categorical. Thus, computably categorical is the same as0-computably categorical or ∆0

1-categorical.

ExampleThe structures of Frohlich-Shepherdson and Malcev are0′-categorical (∆0

2-categorical).

∆0n-categoricity in familiar classes

• Ash (1986): a complete description of higher levelscategoricity for well-orders.

• McCoy (2003): characterized, under certain restrictions,∆0

2-categorical linear orders and Boolean algebras.• Barker (1995): for every computable ordinal α, there are

∆02α+2-categorical but not ∆0

2α+1-categorical abelianp-groups.

• Lempp, McCoy, R. Miller, and Solomon (2005): for everyn ≥ 1, there is a computable tree of finite height, which is∆0

n+1-categorical but not ∆0n-categorical.

Relative ∆0α-categoricity

Again, consider arbitrary countable copies of a computablestructure:

DefinitionA computable structureM is relatively ∆0

α-categorical if forevery A isomorphic toM, there is an isomorphism fromM toA, which is ∆0

α relative to the atomic diagram of A.Clearly, a relatively ∆0

α-categorical structure is ∆0α-categorical.

Scott families

• Scott families come from Scott isomorphism theorem,which says that for a countable structure A, there is anLω1ω-sentence the countable models of which are exactlythe isomorphic copies of A.

• A Scott family for a structure A is a countable family Φ ofLω1ω-formulas with finitely many fixed parameters from Asuch that:(i) Each finite tuple in A satisfies some ψ ∈ Φ;(ii) If a, b are tuples in A, of the same length, satisfying thesame formula in Φ, then there is an automorphism of A,which maps a to b.

• A formally Σ0α Scott family is a Σ0

α Scott family consisting ofcomputable Σα formulas. In particular, it follows that aformally c.e. Scott family is a c.e. Scott family consisting offinitary existential formulas.

Syntactic characterization of relativecategoricity

Theorem (Goncharov (1975); Ash, Knight, Manasse, andSlaman (1989); Chisholm, 1990)The following are equivalent for a computable structure A.

1 The structure A is relatively ∆0α-categorical.

2 The structure A has a formally Σ0α Scott family Φ.

3 The structure A has a formally c.e. Scott family Φ ofcomputable Σα formulas.

Relatively ∆0n-categorical structures

• McCoy (2003): A computable linear order is relatively∆0

2-categorical if and only if it is a sum of finitely manyintervals, each of type m, ω, ω∗,Z, or n · η, so that eachinterval of type n · η has a supremum and infimum.

• McCoy: A computable Boolean algebra is relatively∆0

2-categorical if and only if it can be expressed as a finitedirect sum c1 ∨ · · · ∨ cn, where each ci is either atomless,an atom, or a 1-atom.

• Calvert, Cenzer, Harizanov, Morozov (2006): A computableequivalence structure is relatively ∆0

2-categorical if andonly if it either has finitely many infinite equivalenceclasses, or there is an upper bound on the size its finiteequivalence classes.

More examples

• Cenzer, Harizanov, Remmel (2014): A computableinjection structure is relatively ∆0

2-categorical if and only ifit has finitely many orbits of type ω, or finitely many orbitsof type Z .

• Calvert, Cenzer, Harizanov, and Morozov (2009): Acomputable abelian p-group G is relatively ∆0

2-categoricaliff G is reduced and λ(G) ≤ ω , or G is isomorphic to⊕αZ(p∞)⊕ H, where α ≤ ω and H has finite period.

• Every computable injection structure is relatively∆0

3-categorical.• Every computable equivalence structure is relatively

∆03-categorical.

∆0α-categorical not relatively

∆0α-categorical

Theorem (Goncharov)Computable categoricity of a computable structure does notimply its relative computable categoricity.

Theorem (Goncharov, Harizanov, Knight, McCoy,R. Miller, and Solomon (2005); Chisholm, Fokina,Goncharov, Harizanov, Knight, and Quinn, 2009)For every computable ordinal α, there is a ∆0

α-categorical butnot relatively ∆0

α-categorical structure.The same holds for directed graphs, undirected graphs, partialorders, fields, etc.

∆02-categoricity

Theorem (Kach and Turetsky, 2009)There exists a ∆0

2-categorical equivalence structure, which isnot relatively ∆0

2-categorical.On the other hand,• Cenzer, Harizanov, and Remmel (2014): Every

∆02-categorical injection structure is relatively

∆02-categorical.

• Bazhenov (2014): Every ∆02-categorical Boolean algebra is

relatively ∆02-categorical.

Theorem (Goncharov, 1975)A 2-decidable structure is computably categorical if and only if itis relatively computably categorical.Kudinov (1996): an example of 1-decidable and computablycategorical structure, which is not relatively computablycategorical.

Theorem (Ash, 1987)For every computable ordinal α, under certain decidabilityconditions on A, if A is ∆0

α-categorical, then A is relatively∆0α-categorical.

Categoricity and relative categoricity

Theorem (Downey, Kach, Lempp, and Turetsky, 2013)Any 1-decidable computably categorical structure is relatively∆0

2-categorical.

Theorem (Downey, Kach, Lempp, Lewis, Montalbán, andTuretsky, 2015)For every computable ordinal α, there is a computablycategorical structure that is not relatively ∆0

α-categorical.

Fraïssé limitsDefinitionA countable structure is a Fraïssé limit of a class of finitelygenerated structures K if it is ultrahomogeneous, and hasage K.

• The age of a structureM is the class of all finitely generatedstructures that can be embedded inM.

• Fraïssé: a finite or countable class K of finitely generatedstructures is the age of a finite or a countable structure iff K hasthe hereditary property and the joint embedding property.

• A class K has the hereditary property if whenever C ∈ K and Sis a finitely generated substructure of C, then S is isomorphic tosome structure in K.

• A class K has the joint embedding property if for every B, C ∈ Kthere is D ∈ K such that B and C embed into D.

• A structure U is ultrahomogeneous if every isomorphismbetween finitely generated substructures of U extends to anautomorphism of U .

Categoricity of Fraïssé limits I

TheoremLet A be a computable structure which is a Fraïssé limit. ThenA is relatively ∆0

2-categorical.

Idea of the proof:Because of ultrahomogeneity, we can construct isomorphismsbetween A and an isomorphic structure B using aback-and-forth argument, as long as we can determine, forevery a ∈ A and b ∈ B, whether there is an isomorphism fromthe structure generated by a to the structure generated by bthat maps a to b in order. This can be determined by (B)′, sincethere is such an isomorphism precisely if there is no atomicformula ϕ with A |= ϕ(a) and B 6|= ϕ(b). This is a Π0

1 conditionrelative to A⊕ B ≡T B.Therefore, we can use (B)′ as an oracleto perform the back-and-forth construction of an isomorphism,and so there is a ∆0

2(B) isomorphism.

Categoricity of Fraïssé limits II

Note that if the language of A is finite and relational, then thereare only finitely many atomic formulas ϕ to consider, and theset of such formulas can be effectively determined. Hence, ifthe language is finite and relational, then a Fraïssé limit isnecessarily relatively computably categorical.

TheoremThere is a 1-decidable structure F that is a Fraïssé limit andcomputably categorical, but not relatively computablycategorical. Moreover, the language for such F can be finite orrelational.Idea of the proof: take structures by Kudinov or Downey, Kach,Lempp, Turetsky and make them Fraïssé limits.

Categoricity for Trees

• Lempp, McCoy, R. Miller, and Solomon: characterizedcomputably categorical trees of finite height, and showedthat for these structures, computable categoricity coincideswith relative computable categoricity.

• Lempp, McCoy, R. Miller, and Solomon: for every n ≥ 1,there is a computable tree of finite height, which is∆0

n+1-categorical but not ∆0n-categorical.

• There is no known characterization of ∆02-categoricity or

higher level categoricity for trees of finite height.

TheoremThere is a computable ∆0

2-categorical tree of finite height,which is not relatively ∆0

2-categorical.

root

c0 c1

· · ·

ce

ae be me

ne

0 1

· · ·

e

· · ·

Ri : X i with parameters pi is not a Scott family for T .

If ϕ(x) is a computable Σ2 formula and a ∈ T , then T |= ϕ(a) ifand only if Ts |= ϕ(a) for co-finitely many stages s.

ce

ae be me

ne

0 1

· · ·

e

Qj : If Mj∼= T , then there is a 0′-computable isomorphism

between Mj and T .

Trees of infinite height

R. Miller established that no computable tree of infinite height iscomputably categorical.


2-categorical tree of infinite height,which is not relatively ∆0

2-categorical.

Abelian p-groups• A group G is called a p-group if for all g ∈ G, the order of g

is a power of p.• Barker: for every computable ordinal α, there is a

∆02α+2-categorical but not ∆0

2α+1-categorical abelianp-group.

• Z(pn) is the cyclic group of order pn.• Z(p∞) is the quasicyclic (Prüfer) abelian p-group, the direct

limit of the sequence Z(pn).• Goncharov; Smith: computably categorical abelian

p-groups are those that can be written as:(i)

⊕lZ(p∞)⊕ F for l ≤ ω and F is a finite group; or

(ii)⊕nZ(p∞)⊕ H ⊕

⊕ωZ(pk ), where n, k ∈ ω and H is a

finite group.• For these groups, computable categoricity and relative

computable categoricity coincide.

Categoricity for abelian p-groupsCalvert, Cenzer, Harizanov, and Morozov (2009): a computableabelian p-group G is relatively ∆0

2-categorical if and only if:(i) G is isomorphic to

⊕lZ(p∞)⊕ H, where l ≤ ω and H has

finite period; or(ii) All elements in G are of finite height (equivalently, G isreduced with λ(G) ≤ ω).


2-categorical abelian p-group, which isnot relatively ∆0

2-categorical.

Idea of the proof:Define

G =⊕ω

Z(p∞)⊕⊕k∈A

Z(pk ),

where A is a specific ∆02 set.

Torsion free groupsDefinitionA homogenous completely decomposable abelian group is agroup of the form

⊕i∈κ

H, where H is a subgroup of the additive

group of the rationals, (Q,+).

• A HCDAP is computably categorical iff κ is finite.• For P a set of primes, define Q(P) to be the subgroup of

(Q,+) generated by { 1pk : p ∈ P ∧ k ∈ ω}.

Theorem (Downey and Melnikov, 2013)A computable homogenous completely decomposable abeliangroup of infinite rank is ∆0

2-categorical if and only if it isisomorphic to

⊕ω

Q(P), where P is c.e. and the set (Primes− P)

is semi-low.Recall: a set S ⊆ ω is semi-low if the set HS = {e : We ∩S 6= ∅}is computable from ∅′.

Categoricity for HCDAGs

TheoremA computable homogenous completely decomposable abeliangroup of infinite rank is relatively ∆0

2-categorical if and only if itis isomorphic to

⊕ω

Q(P), where P is a computable set of

primes.

Idea of proof:By Downey-Melnikov, P is c.e. Then we prove that P is alsoco-c.e.

CorollaryThere is a computable homogenous completely decomposableabelian group, which is ∆0

2-categorical but not relatively∆0

2-categorical.

Computable Spectrum and Degree ofCategoricity

LetM be any computable structure.

DefinitionThe categoricity spectrum ofM is the set

CatSpec(M) = {d|M is d-computably categorical}.

A Turing degree d is the degree of categoricity ofM if d is theleast degree in CatSpec(M), if it exists.

Degrees of Categoricity

Theorem (F., Kalimullin, R. Miller, 2010)For n ∈ ω, each degree d-c.e. in and above 0(n) is the degreeof categoricity of a computable structure.

• The result was lifted to hyperarithmetical levels by Csima,Franklin and Shore.

• The result is also true for directed graphs, undirectedgraphs, partial orders, fields, etc...

Degrees of categoricity for equivalencestructures

Easily follows from Calvert, Cenzer, Harizanov, and Morozov(2006):

TheoremThe only degrees of categoricity for relatively ∆0

2-categoricalequivalence structures are 0 and 0′.Barbara Csima and Keng Meng Ng recently announced that theonly possible degrees of categoricity for equivalence structurescan be 0,0′,0′′.

Degrees of categoricity II

Theorem (Cenzer, Harizanov, and Remmel)The degrees of categorictiy of computable injections structurescan only be 0, 0′ and 0′′.

Theorem (Frolov, 2015)

• The degrees of categoricity of relatively ∆02-categorical

linear orders can only be 0 and 0′.• For every d ≥ 0′′, d is the degree of categoricity of a linear

order.

Theorem (Bazhenov, 2014)The degrees of categoricity of relatively ∆0

2-categorical(equivalently, ∆0

2-categorical) Boolean algebras can only be 0and 0′.

Degrees of categoricity for Booleanalgebras

TheoremThe degrees of categoricity of relatively ∆0

3-categorical Booleanalgebras can only be 0, 0′ and 0′′.Use McCoy’s characterization of relatively ∆0

3-categoricalBoolean algebras:

Theorem (McCoy)A Boolean algebra is relatively ∆0

3-categorical iff it can beexpressed as finite direct sums of algebras that are atoms,atomless, 1-atoms, rank 1 atomic, or isomorphic to the intervalalgebra I(ω + η).

Boolean algebras

• A Boolean algebra B is atomic if for every a ∈ B there is anatom b ≤ a. An equivalence relation ∼ on a Booleanalgebra A is defined by:

a ∼ b iff each of a ∩ b and b ∩ a is ∅ or a union of finitelymany atoms of A.

• A Boolean algebra A is a 1-atom if A/ ∼ is a two-elementalgebra.

• A Boolean algebra A is rank 1 if A/ ∼ is a nontrivialatomless Boolean algebra.

• McCoy: a countable rank 1 atomic Boolean algebra isisomorphic to I(2 · η).

Proof sketchTheorem (McCoy)

• A Boolean algebra is relatively ∆03-categorical iff it can be

expressed as finite direct sums of algebras that are atoms,atomless, 1-atoms, rank 1 atomic, or isomorphic to theinterval algebra I(ω + η).

• Fix a relatively ∆03-categorical Boolean algebra B. Assume

that B has a summand which is either rank 1 atomic orisomorphic to the interval algebra I(ω + η).

• All of the potential summands in the characterization ofrelatively ∆0

3-categorical Boolean algebras havecomputable isomorphic copies in which the set of finiteelements is computable.

• We show that both the rank 1 atomic algebra and I(ω + η)have computable isomorphic copies where the set of finiteelements is Σ0

2-complete.

Proof sketch• Let C be a computable copy of I(ω + η) in which the set of

atoms {ai : i ∈ ω} is computable.• Let ϕ(i , x) be a computable formula such that

i ∈ ∅′′ ⇔ ∃<∞x ϕ(i , x).

• If ϕ(i , s) holds:a2i

b0i

b00i

b000ib001

i

b01i

b010ib011

i

b1i

b10i

b100ib101

i

b11i

b110ib111

i

Degrees of categoricity for p-groups

Recall:

Theorem (Calvert, Cenzer, Harizanov, and Morozov)A computable abelian p-group G is relatively ∆0

2-categorical ifand only if:

• G ∼=⊕

lZ(p∞)⊕H, where l ≤ ω and H has finite period; or

• All elements in G are of finite height.

PropositionThe categoricity degrees of computable relatively∆0

2-categorical abelian p-groups can only be 0 and 0′.

Open questions

• Do ∆02-categoricity and relative ∆0

2-categoricity for linearorders coincide?

• Do ∆11-categoricity and relative ∆1

1-categoricity (forarbitrary structures) coincide?

Thank you!

Computability-Theoretic Categoricity and Scott Families ·

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