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We study notions of generic and coarse computability in the context of computable structure theory. Our notions are stratified by the $\Sigma_\beta$ hierarchy. We focus on linear orderings. We show that at the $\Sigma_1$ level all linear orderings have both generically and coarsely computable copies. This behavior changes abruptly at higher levels; we show that at the $\Sigma_{\alpha+2}$ level for any $\alpha\in\omega_1^{ck}$ the set of linear orderings with generically or coarsely computable copies is $\mathbf{\Sigma}_1^1$-complete and therefore maximally complicated. This development is new even in the general analysis of generic and coarse computability of countable structures. In the process of proving these results we introduce new tools for understanding generically and coarsely computable structures. We are able to give a purely structural statement that is equivalent to having a generically computable copy and show that every relational structure with only finitely many relations has coarsely and generically computable copies at the lowest level of the hierarchy., Comment: 35 pages

Normal numbers were introduced by Borel. Normality is certainly a weak notion of randomness; for instance, there are computable numbers which are absolutely normal. In the present paper, we introduce a relativization of normality to a fixed representation system. When we require normality with respect to large sets of such systems, we find variants of normality that imply randomness notions much stronger than absolute normality. The primary classes of numbers investigated in this paper are the supernormal numbers and the highly normal numbers, which we will define. These are relativizations of normality which are robust to all reasonable changes of representation. Among other results, we give a proof that the highly normal numbers are exactly those of computable dimension 1, which we think gives a more natural characterization than was previously known of this interesting class.

We introduce a notion of algorithmic randomness for algebraic fields. We prove the existence of a continuum of algebraic extensions of $\mathbb{Q}$ that are random according to our definition. We show that there are noncomputable algebraic fields which are not random. We also partially characterize the index set, relative to an oracle, of the set of random algebraic fields computable relative to that oracle. In order to carry out this investigation of randomness for fields, we develop computability in the context of infinite Galois theory (where the relevant Galois groups are uncountable), including definitions of computable and computably enumerable Galois groups and computability of Haar measure on the Galois groups.

We introduce several highness notions on degrees related to the problem of computing isomorphisms between structures, provided that isomorphisms exist. We consider variants along axes of uniformity, inclusion of negative information, and several other problems related to computing isomorphisms. These other problems include Scott analysis (in the form of back-and-forth relations), jump hierarchies, and computing descending sequences in linear orders.

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 $(u,v)$ 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 Montalb\'an. 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., Comment: Published online by the *Journal of Symbolic Logic*, 23 December 2021. Print version to appear subsequently

We define notions of generically and coarsely computable relations and structures and functions between structures. We investigate the existence and uniqueness of equivalence structures in the context of these definitions

We compute that the index set of PAC-learnable concept classes is $m$-complete $\Sigma^0_3$ within the set of indices for all concept classes of a reasonable form. All concept classes considered are computable enumerations of computable $\Pi^0_1$ classes, in a sense made precise here. This family of concept classes is sufficient to cover all standard examples, and also has the property that PAC learnability is equivalent to finite VC dimension.

A tuple (or subgroup) in a group is said to degenerate to another if the latter is an endomorphic image of the former. In a countable reduced abelian group, it is shown that if tuples (or finite subgroups) degenerate to each other, then they lie in the same automorphism orbit. The proof is based on techniques that were developed by Kaplansky and Mackey in order to give an elegant proof of Ulm's theorem. Similar results hold for reduced countably generated torsion modules over principal ideal domains. It is shown that the depth and the description of atoms of the resulting poset of orbits of tuples depend only on the Ulm invariants of the module in question (and not on the underlying ring). A complete description of the poset of orbits of elements in terms of the Ulm invariants of the module is given. The relationship between this description of orbits and a very different-looking one obtained by Dutta and Prasad for torsion modules of bounded order is explained., Comment: 13 pages, 1 figure

We apply the techniques of computable model theory to the distance function of a graph. This task leads us to adapt the definitions of several truth-table reducibilities so that they apply to functions as well as to sets, and we prove assorted theorems about the new reducibilities and about functions which have nonincreasing computable approximations. Finally, we show that the spectrum of the distance function can consist of an arbitrary single btt-degree which is approximable from above, or of all such btt-degrees at once, or of the bT-degrees of exactly those functions approximable from above in at most n steps., Comment: submitted for publication 9 November 2011

We initiate the computability-theoretic study of ringed spaces and schemes. In particular, we show that any Turing degree may occur as the least degree of an isomorphic copy of a structure of these kinds. We also show that these structures may fail to have a least degree., Comment: This version contains new introduction and corrections of minor errors

Working in the Blum-Shub-Smale model of computation on the real numbers, we answer several questions of Meer and Ziegler. First, we show that, for each natural number d, an oracle for the set of algebraic real numbers of degree at most d is insufficient to allow an oracle BSS-machine to decide membership in the set of algebraic numbers of degree d + 1. We add a number of further results on relative computability of these sets and their unions. Then we show that the halting problem for BSS-computation is not decidable below any countable oracle set, and give a more specific condition, related to the cardinalities of the sets, necessary for relative BSS-computability. Most of our results involve the technique of using as input a tuple of real numbers which is algebraically independent over both the parameters and the oracle of the machine.

We examine the relation of BSS-reducibility on subsets of the real numbers. The question was asked recently (and anonymously) whether it is possible for the halting problem H in BSS-computation to be BSS-reducible to a countable set. Intuitively, it seems that a countable set ought not to contain enough information to decide membership in a reasonably complex (uncountable) set such as H. We confirm this intuition, and prove a more general theorem linking the cardinality of the oracle set to the cardinality, in a local sense, of the set which it computes. We also mention other recent results on BSS-computation and algebraic real numbers.

The main result of this paper, as previously presented to arxiv, was incorrect. See the full text for details and for reference to the remaining results., Comment: This submission has been withdrawn by the author [arXiv admin]

Continuous first-order logic is used to apply model-theoretic analysis to analytic structures (e.g. Hilbert spaces, Banach spaces, probability spaces, etc.). Classical computable model theory is used to examine the algorithmic structure of mathematical objects that can be described in classical first-order logic. The present paper shows that probabilistic computation (sometimes called randomized computation) can play an analogous role for structures described in continuous first-order logic. The main result of this paper is an effective completeness theorem, showing that every decidable continuous first-order theory has a probabilistically decidable model. Later sections give examples of the application of this framework to various classes of structures, and to some problems of computational complexity theory.

This survey paper examines the effective model theory obtained with the BSS model of real number computation. It treats the following topics: computable ordinals, satisfaction of computable infinitary formulas, forcing as a construction technique, effective categoricity, effective topology, and relations with other models for the effective theory of uncountable structures.

The \emph{index set} of a computable structure $\mathcal{A}$ is the set of indices for computable copies of $\mathcal{A}$. We determine the complexity of the index sets of various mathematically interesting structures, including arbitrary finite structures, $\mathbb{Q}$-vector spaces, Archimedean real closed ordered fields, reduced Abelian $p$-groups of length less than $\omega^{2}$, and models of the original Ehrenfeucht theory. The index sets for these structures all turn out to be $m$-complete $\Pi_{n}^{0}$, $d$-$\Sigma_{n}^{0}$, or $\Sigma_{n}^{0}$, for various $n$. In each case, the calculation involves finding an \textquotedblleft optimal\textquotedblright% \ sentence (i.e., one of simplest form) that describes the structure. The form of the sentence (computable $\Pi_{n}$, $d$-$\Sigma_{n}$, or $\Sigma_{n}$) yields a bound on the complexity of the index set. When we show $m$% -completeness of the index set, we know that the sentence is optimal. For some structures, the first sentence that comes to mind is not optimal, and another sentence of simpler form is shown to serve the purpose. For some of the groups, this involves Ramsey theory.

There are familiar examples of computable structures having various computable Scott ranks. There are also familiar structures, such as the Harrison ordering, which have Scott rank $\omega_1^{CK}+1$. Makkai produced a structure of Scott rank $\omega_1^{CK}$, which can be made computable, and simplified so that it is just a tree. In the present paper, we show that there are further computable structures of Scott rank $\omega_1^{CK}$ in the following classes: undirected graphs, fields of any characteristic, and linear orderings. The new examples share with the Harrison ordering, and the tree just mentioned, a strong approximability property., Comment: Advances in Logic (Proceedings of the North Texas Logic Conference, October 8--10, 2004), Contemporary Mathematics 425 (2007), American Mathematical Society, 49--66

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.

We introduce a reducibility on classes of structures, essentially a uniform enumeration reducibility. This reducibility is inspired by the Friedman-Stanley paper on using Borel reductions to compare classes of countable structures. This reducibility is calibrated by comparing several classes of structures. The class of cyclic graphs and the class of finite prime fields are equivalent, and are properly below the class of arbitrary finite graphs. The class of finite graphs and the class of finite linear orders are maximal among all classes of finite structures. We also prove some general characterizations of reducibility to certain classes. Examples of large chains and antichains of classes are constructed.

We compare three notions of effectiveness on uncountable structures. The first notion is that of a $\real$-computable structure, based on a model of computation proposed by Blum, Shub, and Smale, which uses full-precision real arithmetic. The second notion is that of an $F$-parameterizable structure, defined by Morozov and based on Mal'tsev's notion of a constructive structure. The third is $\Sigma$-definability over $HF(\real)$, defined by Ershov as a generalization of the observation that the computably enumerable sets are exactly those $\Sigma_1$-definable in $HF(\mathbb{N})$. We show that every $\real$-computable structure has an $F$-parameterization, but that the expansion of the real field by the exponential function is $F$-parameterizable but not $\real$-computable. We also show that the structures with $\real$-computable copies are exactly the structures with copies $\Sigma$-definable over $HF(\real)$. One consequence of this equivalence is a method of approximating certain $\real$-computable structures by Turing computable structures., Comment: Added a major section comparing real computation to $\Sigma$-definability, plus some additional references. Formerly entitled "$\mathbb{R}$-computation and $F$-parameterizability."

The Turing degree spectrum of a countable structure $\mathcal{A}$ is the set of all Turing degrees of isomorphic copies of $\mathcal{A}$. The Turing degree of the isomorphism type of $\mathcal{A}$, if it exists, is the least Turing degree in its degree spectrum. We show there are countable fields, rings, and torsion-free abelian groups of arbitrary rank, whose isomorphism types have arbitrary Turing degrees. We also show that there are structures in each of these classes whose isomorphism types do not have Turing degrees., Comment: 17 pages

Theories of classification distinguish classes with some good structure theorem from those for which none is possible. Some classes (dense linear orders, for instance) are non-classifiable in general, but are classifiable when we consider only countable members. This paper explores such a notion for classes of computable structures by working out a sequence of examples. We follow recent work by Goncharov and Knight in using the degree of the isomorphism problem for a class to distinguish classifiable classes from non-classifiable. In this paper, we calculate the degree of the isomorphism problem for Abelian $p$-groups of bounded Ulm length. The result is a sequence of classes whose isomorphism problems are cofinal in the hyperarithmetical hierarchy. In the process, new back-and-forth relations on such groups are calculated., Comment: 15 pages

Theories of classification distinguish classes with some good structure theorem from those for which none is possible. Some classes (dense linear orders, for instance) are non-classifiable in general, but are classifiable when we consider only countable members. This paper explores such a notion for classes of computable structures by working out several examples. One motivation is to see whether some classes whose set of countable models is very complex become classifiable when we consider only computable members. We follow recent work by Goncharov and Knight in using the degree of the isomorphism problem for a class to distinguish classifiable classes from non- classifiable. For some classes (undirected graphs, fields of fixed characteristic, and real closed fields) we show that the isomorphism problem is \Sigma^1_1 complete (the maximum possible), and for others it is of relatively low complexity. For instance, for algebraically closed fields, archimedean real closed fields, and vector spaces, we show that the isomorphism problem is \Pi^0_3 complete., Comment: 15 pages, updated references, general revision, previously entitled "The isomorphism problem for familiar classes of computable structures I", long version of published paper