A computable structure [Formula: see text] is decidable if, given a formula [Formula: see text] of elementary first-order logic, and a tuple [Formula: see text], we have a decision procedure to decide whether [Formula: see text] holds of [Formula: see text]. We show that there is no reasonable classification of the decidably presentable structures. Formally, we show that the index set of the computable structures with decidable presentations is [Formula: see text]-complete. We also show that for each [Formula: see text] (...) the index set of the computable structures with [Formula: see text]-decidable presentations is [Formula: see text]-complete. (shrink)

Universality has been an important concept in computable structure theory. A class C of structures is universal if, informally, for any structure of any kind there is a structure in C with the same computability-theoretic properties as the given structure. Many classes such as graphs, groups, and fields are known to be universal. This paper is about the class of finitely generated groups. Because finitely generated structures are relatively simple, the class of finitely generated groups has no hope of being (...) universal. We show that finitely generated groups are as universal as possible, given that they are finitely generated: for every finitely generated structure, there is a finitely generated group which has the same computability-theoretic properties. The same is not true for finitely generated fields. We apply the results of this investigation to quasi Scott sentences, and also answer a question of Alvir, Knight, and McCoy. (shrink)

We study the implications of model completeness of a theory for the effectiveness of presentations of models of that theory. It is immediate that for a computable model A\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {A}$$\end{document} of a computably enumerable, model complete theory, the entire elementary diagram E\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E$$\end{document} must be decidable. We prove that indeed a c.e. theory T is model complete if and only if there is a (...) uniform procedure that succeeds in deciding E\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E$$\end{document} from the atomic diagram Δ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varDelta $$\end{document} for all countable models A\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {A}$$\end{document} of T. Moreover, if every presentation of a single isomorphism type A\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {A}$$\end{document} has this property of relative decidability, then there must be a procedure with succeeds uniformly for all presentations of an expansion \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$$$\end{document} by finitely many new constants. (shrink)

Mereological theories are based on the binary relation “being a part of”. The systematic investigations of mereology were initiated by Leśniewski. More recent authors (including Simons, Casati and Varzi, Hovda) formulated a series of first‐order mereological axioms. These axioms give rise to a plenitude of theories, which are of great philosophical interest. The paper considers first‐order mereological theories from the point of view of computable (or effective) algebra. Following the approach of Hirschfeldt, Khoussainov, Shore, and Slinko, we isolate two important (...) computability‐theoretic properties P (namely, degree spectra of structures, and effective dimensions), and consider the following problem: for a given mereological theory T, is it true that its models can realize every possible variant of the property P? If the answer is positive, then we say that the theory T is ‐universal. We obtain the following results about known mereological theories. Any theory T which is weaker than Extensional Closure Mereology (CEM) is ‐universal. A similar fact is true for the theory GM2. On the other hand, any theory stronger that CEM + (C) + (G) is not ‐universal. In particular, General Extensional Mereology is not ‐universal. (shrink)

The paper studies Heyting algebras within the framework of computable structure theory. We prove that the class _K_ containing all Heyting algebras with distinguished atoms and coatoms is complete in the sense of the work of Hirschfeldt et al. (Ann Pure Appl Logic 115(1-3):71-113, 2002). This shows that the class _K_ is rich from the computability-theoretic point of view: for example, every possible degree spectrum can be realized by a countable structure from _K_. In addition, there is no simple syntactic (...) characterization of computably categorical members of _K_ (i.e., structures from _K_ possessing a unique computable copy, up to computable isomorphisms). (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)