We study the complexity of the isomorphism relation on classes of computable structures. We use the notion of FF-reducibility introduced in [9] to show completeness of the isomorphism relation on many familiar classes in the context of all ${\mathrm{\Sigma }}_{1}^{1}$ equivalence relations on hyperarithmetical subsets of ω.

We exploit properties of certain directed graphs, obtained from the families of sets with special effective enumeration properties, to generalize several results in computable model theory to higher levels of the hyperarithmetical hierarchy. Families of sets with such enumeration features were previously built by Selivanov, Goncharov, and Wehner. For a computable successor ordinal α, we transform a countable directed graph into a structure such that has a isomorphic copy if and only if has a computable isomorphic copy.A computable structure is (...) categorical if for all computable isomorphic copies of , there is an isomorphism from onto , which is . We prove that for every computable successor ordinal α, there is a computable, categorical structure, which is not relatively categorical. This generalizes the result of Goncharov that there is a computable, computably categorical structure, which is not relatively computably categorical.An additional relation R on the domain of a computable structure is intrinsically on if in all computable isomorphic copies of , the image of R is . We prove that for every computable successor ordinal α, there is an intrinsically relation on a computable structure, which not relatively intrinsically . This generalizes the result of Manasse that there is an intrinsically computably enumerable relation on a computable structure, which is not relatively intrinsically computably enumerable.The dimension of a structure is the number of computable isomorphic copies, up to isomorphisms. We prove that for every computable successor ordinal α and every n≥1, there is a computable structure with dimension n. This generalizes the result of Goncharov that there is a structure of computable dimension n for every n≥1.Finally, we prove that for every computable successor ordinal α, there is a countable structure with isomorphic copies in just the Turing degrees of sets X such that relative to X is not . In particular, for every finite n, there is a structure with isomorphic copies in exactly the non- Turing degrees. This generalizes the result obtained by Wehner, and independently by Slaman, that there is a structure with isomorphic copies in exactly the nonzero Turing degrees. (shrink)

We give Scott sentences for certain computable groups, and we use index set calculations as a way of checking that our Scott sentences are as simple as possible. We consider finitely generated groups and torsion-free abelian groups of finite rank. For both kinds of groups, the computable ones all have computable \ Scott sentences. Sometimes we can do better. In fact, the computable finitely generated groups that we have studied all have Scott sentences that are “computable d-\” sentence and a (...) computable \ sentence). In [9], this was shown for the finitely generated free groups. Here we show it for all finitely generated abelian groups, and for the infinite dihedral group. Among the computable torsion-free abelian groups of finite rank, we focus on those of rank 1. These are exactly the additive subgroups of \. We show that for some of these groups, the computable \ Scott sentence is best possible, while for others, there is a computable d-\ Scott sentence. (shrink)

Classification is an important goal in many branches of mathematics. The idea is to describe the members of some class of mathematical objects, up to isomorphism or other important equivalence, in terms of relatively simple invariants. Where this is impossible, it is useful to have concrete results saying so. In model theory and descriptive set theory, there is a large body of work showing that certain classes of mathematical structures admit classification while others do not. In the present paper, we (...) describe some recent work on classification in computable structure theory.Section 1 gives some background from model theory and descriptive set theory. From model theory, we give sample structure and non-structure theorems for classes that include structures of arbitrary cardinality. We also describe the notion of Scott rank, which is useful in the more restricted setting of countable structures. From descriptive set theory, we describe the basic Polish space of structures for a fixed countable language with fixed countable universe. We give sample structure and non-structure theorems based on the complexity of the isomorphism relation, and on Borel embeddings.Section 2 gives some background on computable structures. We describe three approaches to classification for these structures. The approaches are all equivalent. However, one approach, which involves calculating the complexity of the isomorphism relation, has turned out to be more productive than the others. Section 3 describes results on the isomorphism relation for a number of mathematically interesting classes—various kinds of groups and fields. In Section 4, we consider a setting similar to that in descriptive set theory. We describe an effective analogue of Borel embedding which allows us to make distinctions even among classes of finite structures. Section 5 gives results on computable structures of high Scott rank. Some of these results make use of computable embeddings. Finally, in Section 6, we mention some open problems and possible directions for future work. (shrink)

Makkai [10] produced an arithmetical structure of Scott rank $\omega _{1}^{\mathit{CK}}$. In [9]. Makkai's example is made computable. Here we show that there are computable trees of Scott rank $\omega _{1}^{\mathit{CK}}$. We introduce a notion of "rank homogeneity". In rank homogeneous trees, orbits of tuples can be understood relatively easily. By using these trees, we avoid the need to pass to the more complicated "group trees" of [10] and [9]. Using the same kind of trees, we obtain one of rank (...) $\omega _{1}^{\mathit{CK}}$ that is "strongly computably approximable". We also develop some technology that may yield further results of this kind. (shrink)

Ash and Nerode [2] gave natural definability conditions under which a relation is intrinsically r. e. Here we generalize this to arbitrary levels in Ershov's hierarchy of Δmath image sets, giving conditions under which a relation is intrinsically α-r. e.

In [8], the third author defined a reducibility$\le _w^{\rm{*}}$that lets us compare the computing power of structures of any cardinality. In [6], the first two authors showed that the ordered field of reals${\cal R}$lies strictly above certain related structures. In the present paper, we show that$\left \equiv _w^{\rm{*}}{\cal R}$. More generally, for the weak-looking structure${\cal R}$ℚconsisting of the real numbers with just the ordering and constants naming the rationals, allo-minimal expansions of${\cal R}$ℚare equivalent to${\cal R}$. Using this, we show that (...) for any analytic functionf,$\left \equiv _w^{\rm{*}}{\cal R}$. $is noto-minimal.). (shrink)

A set X is prime bounding if for every complete atomic decidable (CAD) theory T there is a prime model U of T decidable in X. It is easy to see that $X = 0\prime$ is prime bounding. Denisov claimed that every $X <_{T} 0\prime$ is not prime bounding, but we discovered this to be incorrect. Here we give the correct characterization that the prime bounding sets $X \leq_{T} 0\prime$ are exactly the sets which are not $low_2$ . Recall that (...) X is $low_2$ if $X\prime\prime$ $\leq_{T} 0\prime$ . To prove that a $low_2$ set X is not prime bounding we use a $0\prime$ -computable listing of the array of sets { Y : Y $\leq_{T}$ X } to build a CAD theory T which diagonalizes against all potential X-decidable prime models U of T. To prove that any $non-low_{2}$ ; X is indeed prime bounding, we fix a function f $\leq_T$ X that is not dominated by a certain $0\prime$ -computable function that picks out generators of principal types. Given a CAD theory T. we use f to eventually find, for every formula $\varphi (\bar{x})$ consistent with T, a principal type which contains it, and hence to build an X-decidable prime model of T. We prove the prime bounding property equivalent to several other combinatorial properties, including some related to the limitwise monotonic functions which have been introduced elsewhere in computable model theory. (shrink)

§0. Introduction. In [16], Barwise described his graduate study at Stanford. He told of his interactions with Kreisel and Scott, and said how he chose Feferman as his advisor. He began working on admissible fragments of infinitary logic after reading and giving seminar talks on two Ph.D. theses which had recently been completed: that of Lopez-Escobar, at Berkeley, on infinitary logic [46], and that of Platek [58], at Stanford, on admissible sets.Barwise's work on infinitary logic and admissible sets is described (...) in his thesis [4], the book [13], and papers [5]—[16]. We do not try to give a systematic review of these papers. Instead, our goal is to give a coherent introduction to infinitary logic and admissible sets. We describe results of Barwise, of course, because he did so much. In addition, we mention some more recent work, to indicate the current importance of Barwise's ideas. Many of the central results are stated without proof, but occasionally we sketch a proof, to indicate how the ideas fit together.Chapters 1 and 2 describe infinitary logic and admissible sets at the time Barwise began his work, circa 1965. From Chapter 3 on, we survey the developments that took place after Barwise appeared on the scene.§1. Background on infinitary logic. In this chapter, we describe the situation in infinitary logic at the time that Barwise began his work. We need some terminology. By a vocabulary, we mean a set L of constant symbols, and relation and operation symbols with finitely many argument places. As usual, by an L-structureM, we mean a universe set M with an interpretation for each symbol of L. (shrink)

There is a Turing computable embedding $\Phi $ of directed graphs $\mathcal {A}$ in undirected graphs. Moreover, there is a fixed tuple of formulas that give a uniform effective interpretation; i.e., for all directed graphs $\mathcal {A}$, these formulas interpret $\mathcal {A}$ in $\Phi $. It follows that $\mathcal {A}$ is Medvedev reducible to $\Phi $ uniformly; i.e., $\mathcal {A}\leq _s\Phi $ with a fixed Turing operator that serves for all $\mathcal {A}$. We observe that there is a graph G (...) that is not Medvedev reducible to any linear ordering. Hence, G is not effectively interpreted in any linear ordering. Similarly, there is a graph that is not interpreted in any linear ordering using computable $\Sigma _2$ formulas. Any graph can be interpreted in a linear ordering using computable $\Sigma _3$ formulas. Friedman and Stanley [4] gave a Turing computable embedding L of directed graphs in linear orderings. We show that there is no fixed tuple of $L_{\omega _1\omega }$ -formulas that, for all G, interpret the input graph G in the output linear ordering $L$. Harrison-Trainor and Montalbán [7] have also shown this, by a quite different proof. (shrink)

We study completely decomposable torsion-free abelian groups of the form $\mathcal{G}_S := \oplus_{n \in S} \mathbb{Q}_{p_n}$ for sets $S \subseteq \omega$. We show that $\mathcal{G}_S$has a decidable copy if and only if S is $\Sigma^0_2$and has a computable copy if and only if S is $\Sigma^0_3$.

We show that for every computable limit ordinal α, there is a computable structure A that is $\Delta _\alpha ^0 $ categorical, but not relatively $\Delta _\alpha ^0 $ categorical (equivalently. it does not have a formally $\Sigma _\alpha ^0 $ Scott family). We also show that for every computable limit ordinal a, there is a computable structure A with an additional relation R that is intrinsically $\Sigma _\alpha ^0 $ on A. but not relatively intrinsically $\Sigma _\alpha ^0 $ (...) on A (equivalently, it is not definable by a computable $\Sigma _\alpha $ formula with finitely many parameters). Earlier results in [7], [10], and [8] establish the same facts for computable successor ordinals α. (shrink)

Let ???? be a computable structure and let R be a new relation on its domain. We establish a necessary and sufficient condition for the existence of a copy ℬ of ???? in which the image of R (¬R, resp.) is simple (immune, resp.) relative to ℬ. We also establish, under certain effectiveness conditions on ???? and R, a necessary and sufficient condition for the existence of a computable copy ℬ of ???? in which the image of R (¬R, resp.) (...) is simple (immune, resp.). (shrink)

We ask when, for a pair of structures $\mathcal {A}_1,\mathcal {A}_2$, there is a uniform effective procedure that, given copies of the two structures, unlabeled, always produces a copy of $\mathcal {A}_1$. We give some conditions guaranteeing that there is such a procedure. The conditions might suggest that for the pair of orderings $\mathcal {A}_1$ of type $\omega _1^{CK}$ and $\mathcal {A}_2$ of Harrison type, there should not be any such procedure, but, in fact, there is one. We construct an (...) example for which there is no such procedure. The construction involves forcing. On the way to constructing our example, we prove a general result on modifying Cohen generics. (shrink)

We improve on and generalize a 1960 result of Maltsev. For a field F, we denote by $H(F)$ the Heisenberg group with entries in F. Maltsev showed that there is a copy of F defined in $H(F)$, using existential formulas with an arbitrary non-commuting pair of elements as parameters. We show that F is interpreted in $H(F)$ using computable $\Sigma _1$ formulas with no parameters. We give two proofs. The first is an existence proof, relying on a result of Harrison-Trainor, (...) Melnikov, R. Miller, and Montalbán. This proof allows the possibility that the elements of F are represented by tuples in $H(F)$ of no fixed arity. The second proof is direct, giving explicit finitary existential formulas that define the interpretation, with elements of F represented by triples in $H(F)$. Looking at what was used to arrive at this parameter-free interpretation of F in $H(F)$, we give general conditions sufficient to eliminate parameters from interpretations. (shrink)

We study the relative computational power of structures related to the ordered field of reals, specifically using the notion of generic Muchnik reducibility. We show that any expansion of the reals by a continuous function has no more computing power than the reals, answering a question of Igusa, Knight, and Schweber [7]. On the other hand, we show that there is a certain Borel expansion of the reals that is strictly more powerful than the reals and such that any Borel (...) quotient of the reals reduces to it. (shrink)

We consider the following generalization of the notion of a structure recursive relative to a set X. A relational structure A is said to be a Γ-structure if for each relation symbol R, the interpretation of R in A is ∑math image relative to X, where β = Γ. We show that a certain, fairly obvious, description of classes ∑math image of recursive infinitary formulas has the property that if A is a Γ-structure and S is a further relation on (...) A, then the following are equivalent: For every isomorphism F from A to a Γ-structure, F is ∑math image relative to X, The relation is defined in A by a ∑math image formula with parameters. (shrink)

Let R be a real closed field. An integer part I for R is a discretely ordered subring such that for every \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${r \in R}$$\end{document}, there exists an \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${i \in I}$$\end{document} so that i ≤ r < i + 1. Mourgues and Ressayre (J Symb Logic 58:641–647, 1993) showed that every real closed field has an integer part. The procedure of Mourgues and (...) Ressayre appears to be quite complicated. We would like to know whether there is a simple procedure, yielding an integer part that is \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Delta^0_2(R)}$$\end{document} —limit computable relative to R. We show that there is a maximal Z-ring \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${I \subseteq R}$$\end{document} which is \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Delta^0_2(R)}$$\end{document}. However, this I may not be an integer part for R. By a result of Wilkie (Logic Colloquium ’77), any Z-ring can be extended to an integer part for some real closed field. Using Wilkie’s ideas, we produce a real closed field R with a Z-ring \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${I \subseteq R}$$\end{document} such that I does not extend to an integer part for R. For a computable real closed field, we do not know whether there must be an integer part in the class \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Delta^0_2}$$\end{document}. We know that certain subclasses of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Delta^0_2}$$\end{document} are not sufficient. We show that for each \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${n \in \omega}$$\end{document}, there is a computable real closed field with no n-c.e. integer part. In fact, there is a computable real closed field with no n-c.e. integer part for any n. (shrink)

We prove that for every Scott set S there are S-saturated real closed fields and S-saturated models of Presburger arithmetic.

We study the complexity of infinite chains and antichains in computable partial orderings. We show that there is a computable partial ordering which has an infinite chain but none that is ${\Sigma _{1}^{1}}$ or ${\Pi _{1}^{1}}$ , and also obtain the analogous result for antichains. On the other hand, we show that every computable partial ordering which has an infinite chain must have an infinite chain that is the difference of two ${\Pi _{1}^{1}}$ sets. Our main result is that there (...) is a computably axiomatizable theory K of partial orderings such that K has a computable model with arbitrarily long finite chains but no computable model with an infinite chain. We also prove the corresponding result for antichains. Finally, we prove that if a computable partial ordering ${\mathcal{A}}$ has the feature that for every ${\mathcal{B} \cong \mathcal{A}}$ , there is an infinite chain or antichain that is ${\Delta _{2}^{0}}$ relative to ${\mathcal{B}}$ , then we have uniform dichotomy: either for all copies ${\mathcal{B}}$ of ${\mathcal{A}}$ , there is an infinite chain that is ${\Delta _{2}^{0}}$ relative to ${\mathcal{B}}$ , or for all copies ${\mathcal{B}}$ of ${\mathcal{A}}$ , there is an infinite antichain that is ${\Delta _{2}^{0}}$ relative to ${\mathcal{B}}$. (shrink)