## Degree spectra and computable dimension in algebraic structures (2002)

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Venue: | Annals of Pure and Applied Logic 115 (2002 |

Citations: | 31 - 6 self |

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@INPROCEEDINGS{Hirschfeldt02degreespectra,

author = {Denis R. Hirschfeldt and Richard A. Shore},

title = {Degree spectra and computable dimension in algebraic structures},

booktitle = {Annals of Pure and Applied Logic 115 (2002},

year = {2002},

pages = {71--113}

}

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### Abstract

\Lambda \Lambda

### Citations

538 | Model theory
- Hodges
- 1993
(Show Context)
Citation Context ...ematics program: computable model theory and computable algebra. We assume the reader is familiar with basic concepts of computability theory, model theory, and algebra; standard references are [40], =-=[21]-=-, and [28], respectively. One of the main concerns of computable model theory is the study of computabilitytheoretic properties of countable structures. We will always assume we are working with compu... |

111 |
An introduction to the theory of groups
- Rotman
- 1994
(Show Context)
Citation Context ...y pair of elements x; y 2 jGj. (A group is 1-step nilpotent if and only if it is Abelian. For the more general denition of n-step nilpotent groups, see any standard textbook on group theory, such as [=-=-=-38].) Let R = (jRj; +; ; 0; 1) be a countably innite ring with unit of characteristic p > 2. The deg(R)-computably presentable group GR is dened to be the set of all triples (a; b; c), a; b; c 2 jRj, ... |

72 |
Soare, Recursively Enumerable Sets and Degrees
- I
- 1987
(Show Context)
Citation Context ...e mathematics program: computable model theory and computable algebra. We assume the reader is familiar with basic concepts of computability theory, model theory, and algebra; standard references are =-=[40]-=-, [21], and [28], respectively. One of the main concerns of computable model theory is the study of computabilitytheoretic properties of countable structures. We will always assume we are working with... |

51 |
A Borel reducibility theory for classes of countable structures
- Friedman, Stanley
- 1989
(Show Context)
Citation Context ...vious. One example is Mekler [31]. In descriptive set theory, the study of Borel reducibilities and Borel completeness has received much attention in recent years, for example in Friedman and Stanley =-=[7]-=-, Hjorth and Kechris [20], and Camerlo and Gao [3]. A recent survey is Kechris [22]. Another example of the use of codings to show that certain phenomena that can occur in general already occur in wha... |

38 |
Degrees coded in jumps of orderings
- Knight
- 1986
(Show Context)
Citation Context ...e thought of as a presentation of itself. 1.17. Denition. The degree spectrum of a countable structure A, which is denoted by DgSp(A), is the set of degrees of presentations of A. As shown by Knight [=-=26-=-], in all nontrivial cases, the degree spectrum of a countable structure is closed upwards. 1.18. Denition. A countable structure A is trivial if for somesnite set S of elements of jAj, every permutat... |

28 |
Borel equivalence relations and classifications of countable models
- Hjorth, Kechris
- 1996
(Show Context)
Citation Context ...kler [31]. In descriptive set theory, the study of Borel reducibilities and Borel completeness has received much attention in recent years, for example in Friedman and Stanley [7], Hjorth and Kechris =-=[20]-=-, and Camerlo and Gao [3]. A recent survey is Kechris [22]. Another example of the use of codings to show that certain phenomena that can occur in general already occur in what would seem to be a much... |

27 |
Nerode, Effective content of field theory
- Metakides, A
- 1979
(Show Context)
Citation Context ...∆ 0 2-categorical structures. The result for algebraically closed and real closed fields is implied by the results in [34]; the result for algebraically closed fields was also independently proved i=-=n [32]. -=-The result for Abelian groups appears in [10], that for linear orderings independently in [13] and [37], and that for ∆ 0 2-categorical structures in [11]. The result for Boolean algebras appears in... |

25 | Computably categorical structures and expansions by constants
- Cholak, Goncharov, et al.
- 1999
(Show Context)
Citation Context ...n any expansion of A bysnitely many constants remains computably categorical. However, preservation of categoricity does not hold in general, as was shown by Cholak, Goncharov, Khoussainov, and Shore =-=[4]-=-. 1.13. Theorem (Cholak, Goncharov, Khoussainov, and Shore). For each k > 0 there exist a computably categorical structure A and an a 2 jAj such that (A; a) has computable dimension k. In fact, as sho... |

25 | Computable isomorphisms, degree spectra of relations and Scott families,” Annals of Pure and Applied Logic 93
- Khoussainov, Shore
- 1998
(Show Context)
Citation Context ... singletons or innite. Harizanov [16] was thesrst to give an example of an intrinsically 0 2 relation with a two-element degree spectrum that includes 0. This was improved by Khoussainov and Shore [2=-=3]-=- as follows. 1.8. Theorem (Khoussainov and Shore). For each n > 0 there exists an intrinsically c.e. relation U on the domain of a computable structure A of computable dimension n such that DgSp A (U)... |

25 |
Recursively categorical linear orderings
- Remmel
- 1981
(Show Context)
Citation Context ...the results in [34]; the result for algebraically closedselds was also independently proved in [32]. The result for Abelian groups appears in [10], that for linear orderings independently in [13] and =-=[37-=-], and that for 0 2 -categorical structures in [11]. The result for Boolean algebras appears in full in [12], though it is implicit in earlier work of Goncharov and, independently, in [29]. In most c... |

21 |
The possible Turing degree of the non-zero member in a two element degree spectrum, Ann
- Harizanov
- 1993
(Show Context)
Citation Context ...the set of degrees of the images of U in all computable presentations of A. It is easy to give examples of relations on computable structures whose degree spectra are singletons or innite. Harizanov [=-=16-=-] was thesrst to give an example of an intrinsically 0 2 relation with a two-element degree spectrum that includes 0. This was improved by Khoussainov and Shore [23] as follows. 1.8. Theorem (Khoussa... |

20 | Relative to any nonrecursive set
- Slaman
(Show Context)
Citation Context ...rings of characteristic 0. Thus we will restrict our attention to nontrivial structures. Any set that is computable in every nonzero degree is in fact computable, but as shown independently by Slaman =-=[39]-=- and Wehner [41], the analogous fact is not true of structures. 1.20. Theorem (Slaman; Wehner). There is a structure A that has presentations of every degree except 0. (In other words, DgSp(A) = D f0g... |

19 |
Problem of the number of non-self-equivalent constructivizations
- Goncharov
- 1980
(Show Context)
Citation Context ...ion early in the development of computable model theory was whether there exist computable structures withsnite computable dimension greater than 1. This question was answered positively by Goncharov =-=[9]-=-. 1.4. Theorem (Goncharov). For each n > 0 there is a computable structure with computable dimension n. Further investigation led to examples of computable structures withsnite computable dimension gr... |

18 |
Intrinsically recursive relations
- Ash, Nerode
- 1981
(Show Context)
Citation Context ... domain of M, that is, relations that are not the interpretation in M of relation symbols in the 4 language of M. The study of relations on computable structures began with the work of Ash and Nerode =-=-=-[2], who were concerned with relations that maintain some degree of eectiveness in dierent computable presentations of a structure. 1.6. Denition. Let U be a relation on the domain of a computable str... |

18 |
The quantity of non-autoequivalent constructivizations
- Goncharov
(Show Context)
Citation Context ...ures, and even all members of many familiar classes of structures, have computable dimension 1 or !. For example, Nurtazin [34] showed that all decidable structures fall into this category. Goncharov =-=[8]-=- later extended this result to 1-decidable structures, and there have been several other well-known algebraic classes of structures for which similar results have been proved. 1.3. Theorem (Goncharov;... |

18 | New directions in descriptive set theory
- Kechris
- 1999
(Show Context)
Citation Context ...educibilities and Borel completeness has received much attention in recent years, for example in Friedman and Stanley [7], Hjorth and Kechris [20], and Camerlo and Gao [3]. A recent survey is Kechris =-=[22]-=-. Another example of the use of codings to show that certain phenomena that can occur in general already occur in what would seem to be a much more restricted setting is the work of Peretyat'kin [35] ... |

18 | Effective Model Theory: The number of models and their complexity
- Khoussainov, Shore
- 1999
(Show Context)
Citation Context ...les of the kinds of results we would like to show remain true when we restrict our attention to certain classes of algebraic structures. For more thorough treatments of these and related results, see =-=[24-=-] or the articles in [6]. One way to understand the dierences between noncomputably isomorphic computable presentations of a structure M is to compare (from a computability-theoretic point of view) th... |

18 |
Recursive categoricity and persistence
- Millar
- 1986
(Show Context)
Citation Context ...gSp A (U) = fdeg(A i ) : i 2 !g. A related issue is the question of what happens to the computable dimension of a computably categorical structure when it is expanded bysnitely many constants. Millar =-=[33]-=- showed that, with a relatively small additional amount of decidability, computable categoricity is preserved under expansion bysnitely many constants. 1.12. Theorem (Millar). If A is computably categ... |

15 |
Countable Boolean Algebras and Decidability
- Goncharov
- 1997
(Show Context)
Citation Context ...lt for Abelian groups appears in [10], that for linear orderings independently in [13] and [37], and that for 0 2 -categorical structures in [11]. The result for Boolean algebras appears in full in [=-=12]-=-, though it is implicit in earlier work of Goncharov and, independently, in [29]. In most cases, these results were proved via structure theorems, that is, theorems that connect computability-theoreti... |

15 |
Enumerations, countable structures, and Turing degrees
- Wehner
- 1998
(Show Context)
Citation Context ...eristic 0. Thus we will restrict our attention to nontrivial structures. Any set that is computable in every nonzero degree is in fact computable, but as shown independently by Slaman [39] and Wehner =-=[41]-=-, the analogous fact is not true of structures. 1.20. Theorem (Slaman; Wehner). There is a structure A that has presentations of every degree except 0. (In other words, DgSp(A) = D f0g, where D is the... |

12 |
Recursive labeling systems and stability of recursive structures in hyperarithmetical degrees
- Ash
- 1986
(Show Context)
Citation Context ...esentations of a structure. We say that a computable structure is strictly 0 -categorical if it is 0 -categorical but not 0s-categorical for anys. The following result was proved by Ash in [1]. 1.15. Theorem (Ash). For each computable limit ordinals(includings= 0) and each n 2 !, there is a strictly 0 +2n -categorical well-ordering. The work of computable model theory is not restricted t... |

12 |
Autostability of models, Algebra and Logic 19
- Goncharov, Dzgoev
- 1980
(Show Context)
Citation Context ...plied by the results in [34]; the result for algebraically closedselds was also independently proved in [32]. The result for Abelian groups appears in [10], that for linear orderings independently in =-=[13-=-] and [37], and that for 0 2 -categorical structures in [11]. The result for Boolean algebras appears in full in [12], though it is implicit in earlier work of Goncharov and, independently, in [29]. ... |

11 | Degree spectra of relations on computable structures
- Hirschfeldt
- 2000
(Show Context)
Citation Context ...in result do not involvesnite computable dimension, and thus could in principle still hold for some of the classes mentioned in Theorem 1.3. For instance, in the case of linear orderings, Hirschfeldt =-=[17]-=- has shown that Theorem 1.8 does not hold, but whether Theorem 1.20 holds is still an open question (see [5] for a discussion of this question). The rest of this paper is dedicated to the proof of The... |

10 | The completeness of the isomorphism relation of countable Boolean algebras
- Camerlo, Gao
(Show Context)
Citation Context ...set theory, the study of Borel reducibilities and Borel completeness has received much attention in recent years, for example in Friedman and Stanley [7], Hjorth and Kechris [20], and Camerlo and Gao =-=[3]-=-. A recent survey is Kechris [22]. Another example of the use of codings to show that certain phenomena that can occur in general already occur in what would seem to be a much more restricted setting ... |

9 |
1965], ‘On a correspondence between rings and groups
- Malcev
(Show Context)
Citation Context ...ilpotent Groups In this section, we prove Theorem 1.22 in the case of 2-step nilpotent groups. Much of the proof consists of verifying the eectiveness of a coding of rings into groups due to Mal'cev [=-=30]-=-. Combined with the results of Section 5, this will enable us to provide 39 analogs of Lemmas 2.6-2.9, which can then be used to establish analogs of Propositions 2.10-2.13. We recall the following de... |

8 |
Groups with a finite number of constructivizations
- Goncharov
- 1981
(Show Context)
Citation Context ...gebraically closed and real closed fields is implied by the results in [34]; the result for algebraically closed fields was also independently proved in [32]. The result for Abelian groups appears in =-=[10], -=-that for linear orderings independently in [13] and [37], and that for ∆ 0 2-categorical structures in [11]. The result for Boolean algebras appears in full in [12], though it is implicit in earlier... |

7 |
1981], ‘Stability of nilpotent groups of class 2 and prime exponent
- Mekler
(Show Context)
Citation Context ... kinds have long been used to transfer model-theoretic properties from classes of structures in which they are easy to determine to other classes in which they are less obvious. One example is Mekler =-=[31]-=-. In descriptive set theory, the study of Borel reducibilities and Borel completeness has received much attention in recent years, for example in Friedman and Stanley [7], Hjorth and Kechris [20], and... |

7 |
Romanovskii; Nilpotent groups of finite algorithmic dimension, Siberian Math Journal 30
- Goncharov, Molokov, et al.
- 1989
(Show Context)
Citation Context ...(implicitly) graphs appear in [9], and the result for lattices is an easy consequence of the results in that paper. The result for 2-step nilpotent groups (which improves a result in [10]) appears in =-=[15]-=-, and that for integral domains in [27]. We now turn to the other computability-theoretic properties of structures with which this paper is concerned, and give examples of the kinds of results we woul... |

6 |
1997], ‘On the spectrum of the degrees of decidable relations
- Goncharov, Khoussainov
(Show Context)
Citation Context ... a computable structure A of computable dimension n such that DgSp A (U) consists of n distinct c.e. degrees, including 0. The n = 2 case of the above theorem is also due to Goncharov and Khoussainov =-=[14]-=-. The following result was also proved in [23]. 1.9. Theorem (Khoussainov and Shore). For each computable partial ordering P there exists an intrinsically c.e. relation U on the domain of a computable... |

6 |
Strong and weak constructivizations and enumerable families, Algebra and Logic 13
- Nurtazin
- 1974
(Show Context)
Citation Context ...ble structures with computable dimension 1 or !. Indeed, most familiar structures, and even all members of many familiar classes of structures, have computable dimension 1 or !. For example, Nurtazin =-=[34]-=- showed that all decidable structures fall into this category. Goncharov [8] later extended this result to 1-decidable structures, and there have been several other well-known algebraic classes of str... |

5 | Questions in computable algebra and combinatorics
- Downey, Remmel
- 1999
(Show Context)
Citation Context ...asses mentioned in Theorem 1.3. For instance, in the case of linear orderings, Hirschfeldt [17] has shown that Theorem 1.8 does not hold, but whether Theorem 1.20 holds is still an open question (see =-=[5]-=- for a discussion of this question). The rest of this paper is dedicated to the proof of Theorem 1.22. Most of the cases are handled by coding graphs with the desired properties into models of the giv... |

5 |
Limit equivalent constructivizations, Mathematical logic and theory of algorithms
- GONCHAROV
- 1982
(Show Context)
Citation Context ...osedselds was also independently proved in [32]. The result for Abelian groups appears in [10], that for linear orderings independently in [13] and [37], and that for 0 2 -categorical structures in [=-=11]-=-. The result for Boolean algebras appears in full in [12], though it is implicit in earlier work of Goncharov and, independently, in [29]. In most cases, these results were proved via structure theore... |

5 |
Recursively represented Boolean algebras
- LaRoche
(Show Context)
Citation Context ...n [13] and [37], and that for 0 2 -categorical structures in [11]. The result for Boolean algebras appears in full in [12], though it is implicit in earlier work of Goncharov and, independently, in [=-=29]-=-. In most cases, these results were proved via structure theorems, that is, theorems that connect computability-theoretic properties of structures in the relevant classes to 3 their structural propert... |

4 |
An integral domain with finite algorithmic dimension, personal communication
- Kudinov
- 1977
(Show Context)
Citation Context ...the result for lattices is an easy consequence of the results in that paper. The result for 2-step nilpotent groups (which improves a result in [10]) appears in [15], and that for integral domains in =-=[27]-=-. We now turn to the other computability-theoretic properties of structures with which this paper is concerned, and give examples of the kinds of results we would like to show remain true when we rest... |

3 |
Finitely Axiomatizable Theories
- Peretyat’kin
- 1997
(Show Context)
Citation Context ... [22]. Another example of the use of codings to show that certain phenomena that can occur in general already occur in what would seem to be a much more restricted setting is the work of Peretyat'kin =-=[35-=-] onsnitely axiomatizable theories, which touches on both classical and 9 computable model theory. Probably best-known of all, of course, is the use of highly eective reducibilities in complexity theo... |

3 |
Degree spectra of relations on structures of finite computable dimension
- Hirschfeldt
- 2002
(Show Context)
Citation Context ...n intrinsically c.e. relation U on the domain of a computable structure A such that (DgSp A(U), � T) ∼ = P. If P has a least element then we can pick A and U so that 0 ∈ DgSp A(U). Later, Hirsch=-=feldt [18] a-=-nd Khoussainov and Shore [25] independently obtained the following strengthenings of the previous two theorems. 1.10 Theorem (Hirschfeldt; Khoussainov and Shore). Let a0, . . . , an−1 be c.e. degree... |

2 |
2000], ‘From computable categoricity to computable dimension ω by expansion by a constant
- Hirschfeldt, Khoussainov, et al.
(Show Context)
Citation Context ...ainov, and Shore). For each k > 0 there exist a computably categorical structure A and an a 2 jAj such that (A; a) has computable dimension k. In fact, as shown by Hirschfeldt, Khoussainov, and Shore =-=[19]-=-, not evensniteness of computable dimensionality is always preserved under expansion by a constant. 1.14. Theorem (Hirschfeldt, Khoussainov, and Shore). There are a computably categorical structure A ... |

2 |
Eective content of theory
- Metakides, Nerode
- 1979
(Show Context)
Citation Context ...nd 0 2 -categorical structures. The result for algebraically closed and real closedselds is implied by the results in [34]; the result for algebraically closedselds was also independently proved in [=-=32-=-]. The result for Abelian groups appears in [10], that for linear orderings independently in [13] and [37], and that for 0 2 -categorical structures in [11]. The result for Boolean algebras appears i... |

2 |
The undecidability of some simple theories, unpublished
- Rabin, Scott
(Show Context)
Citation Context ... DgSp G (S) and that if S is intrinsically c.e. then so is U . We now give a relatively simple example of the application of Proposition 4.1 to rings, via a coding based on one due to Rabin and Scott =-=[36-=-]. Let G be a symmetric, irre exive, countably innite graph with edge relation E. We may assume that there exist x; y; z 2 G such that E(x; y), E(x; z), and E(y; z), since the theory of graphs with th... |

1 |
Groups with a number of constructivizations
- Goncharov
- 1981
(Show Context)
Citation Context ...r algebraically closed and real closedselds is implied by the results in [34]; the result for algebraically closedselds was also independently proved in [32]. The result for Abelian groups appears in =-=[10-=-], that for linear orderings independently in [13] and [37], and that for 0 2 -categorical structures in [11]. The result for Boolean algebras appears in full in [12], though it is implicit in earlie... |

1 |
Nilpotent groups of algorithmic dimension
- Goncharov, Molokov, et al.
- 1989
(Show Context)
Citation Context ...(implicitly) graphs appear in [9], and the result for lattices is an easy consequence of the results in that paper. The result for 2-step nilpotent groups (which improves a result in [10]) appears in =-=[15]-=-, and that for integral domains in [27]. We now turn to the other computability-theoretic properties of structures with which this paper is concerned, and give examples of the kinds of results we woul... |

1 |
Degree spectra of relations on structures of computable dimension
- Hirschfeldt
(Show Context)
Citation Context ...ntrinsically c.e. relation U on the domain of a computable structure A such that (DgSp A (U); 6 T ) = P. If P has a least element then we can pick A and U so that 0 2 DgSp A (U). Later, Hirschfeldt [=-=18]-=- and Khoussainov and Shore [25] independently obtained the following strengthenings of the previous two theorems. 1.10. Theorem (Hirschfeldt; Khoussainov and Shore). Let a 0 ; : : : ; a n 1 be c.e. de... |

1 |
Borel equivalence relations and classi of countable models
- Hjorth, Kechris
- 1996
(Show Context)
Citation Context ...kler [31]. In descriptive set theory, the study of Borel reducibilities and Borel completeness has received much attention in recent years, for example in Friedman and Stanley [7], Hjorth and Kechris =-=[20]-=-, and Camerlo and Gao [3]. A recent survey is Kechris [22]. Another example of the use of codings to show that certain phenomena that can occur in general already occur in what would seem to be a much... |

1 |
Solution of the Goncharov-Ash problem and the spectrum problem in the theory of computable models
- Khoussainov, Shore
- 2000
(Show Context)
Citation Context ... the domain of a computable structure A such that (DgSp A (U); 6 T ) = P. If P has a least element then we can pick A and U so that 0 2 DgSp A (U). Later, Hirschfeldt [18] and Khoussainov and Shore [=-=25]-=- independently obtained the following strengthenings of the previous two theorems. 1.10. Theorem (Hirschfeldt; Khoussainov and Shore). Let a 0 ; : : : ; a n 1 be c.e. degrees. There exists an intrinsi... |

1 |
An integral domain with algorithmic dimension, personal communication
- Kudinov
(Show Context)
Citation Context ...the result for lattices is an easy consequence of the results in that paper. The result for 2-step nilpotent groups (which improves a result in [10]) appears in [15], and that for integral domains in =-=[27]-=-. We now turn to the other computability-theoretic properties of structures with which this paper is concerned, and give examples of the kinds of results we would like to show remain true when we rest... |

1 |
From computable categoricity to infinite computable dimension by expansion by a constant
- Hirschfeldt, Khoussainov, et al.
(Show Context)
Citation Context ...ainov, and Shore). For each k > 0 there exist a computably categorical structure A and an a ∈ |A| such that (A, a) has computable dimension k. In fact, as shown by Hirschfeldt, Khoussainov, and Shor=-=e [19]-=-, not even finiteness of computable dimensionality is always preserved under expansion by a constant. 1.14 Theorem (Hirschfeldt, Khoussainov, and Shore). There are a computably categorical structure A... |

1 |
3rd ed. (Addison-Wesley Publishing Co
- Lang, Algebra
- 1993
(Show Context)
Citation Context ...ogram: computable model theory and computable algebra. We assume the reader is familiar with basic concepts of computability theory, model theory, and algebra; standard references are [40], [21], and =-=[28]-=-, respectively. One of the main concerns of computable model theory is the study of computabilitytheoretic properties of countable structures. We will always assume we are working with computable lang... |

1 |
The undecidability of some simple theories
- Rabin, Scott
- 1981
(Show Context)
Citation Context ... ` DgSpG(S) and that if S is intrinsically c.e. then so is U . We now give a relatively simple example of the application of Proposition 4.1 to rings, via a coding based on one due to Rabin and Scott =-=[36]-=-. Let G be a symmetric, irreflexive, countably infinite graph with edge relation E. We may assume that there exist x; y; z 2 G such that E(x; y), E(x; z), and E(y; z), since the theory of graphs with ... |