In this paper we answer the following well-known open question in computable model theory. Does there exist a computable not ‮א‬₀-categorical saturated structure with a unique computable isomorphism type? Our answer is affirmative and uses a construction based on Kolmogorov complexity. With a variation of this construction, we also provide an example of an ‮א‬₁-categorical but not ‮א‬₀-categorical saturated $\Sigma _{1}^{0}$ -structure with a unique computable isomorphism type. In addition, using the construction we give an example of an ‮א‬₁-categorical but (...) not ‮א‬₁-categorical theory whose only non-computable model is the prime one. (shrink)

In this paper we investigate computable models of -categorical theories and Ehrenfeucht theories. For instance, we give an example of an -categorical but not -categorical theory such that all the countable models of except its prime model have computable presentations. We also show that there exists an -categorical but not -categorical theory such that all the countable models of except the saturated model, have computable presentations.

LetEbe a computably enumerable equivalence relation on the setωof natural numbers. We say that the quotient set$\omega /E$realizesa linearly ordered set${\cal L}$if there exists a c.e. relation ⊴ respectingEsuch that the induced structure is isomorphic to${\cal L}$. Thus, one can consider the class of all linearly ordered sets that are realized by$\omega /E$; formally,${\cal K}\left = \left\{ {{\cal L}\,|\,{\rm{the}}\,{\rm{order}}\, - \,{\rm{type}}\,{\cal L}\,{\rm{is}}\,{\rm{realized}}\,{\rm{by}}\,E} \right\}$. In this paper we study the relationship between computability-theoretic properties ofEand algebraic properties of linearly ordered sets realized (...) byE. One can also define the following pre-order$ \le _{lo} $on the class of all c.e. equivalence relations:$E_1 \le _{lo} E_2 $if every linear order realized byE1is also realized byE2. Following the tradition of computability theory, thelo-degrees are the classes of equivalence relations induced by the pre-order$ \le _{lo} $. We study the partially ordered set oflo-degrees. For instance, we construct various chains and anti-chains and show the existence of a maximal element among thelo-degrees. (shrink)

The spectrum of a relation on a computable structure is the set of Turing degrees of the image of R under all isomorphisms between and any other computable structure . The relation is intrinsically computably enumerable if its image under all such isomorphisms is c.e. We prove that any computable partially ordered set is isomorphic to the spectrum of an intrinsically c.e. relation on a computable structure. Moreover, the isomorphism can be constructed in such a way that the image of (...) the minimum element of the partially ordered set is computable. This solves the spectrum problem. The theorem and modifications of its proof produce computably categorical structures whose expansions by finite number of constants are not computably categorical and, indeed, ones whose expansions can have any finite number of computable isomorphism types. They also provide examples of computably categorical structures that remain computably categorical under expansions by constants but have no Scott family. (shrink)

Whenever a structure with a particularly interesting computability-theoretic property is found, it is natural to ask whether similar examples can be found within well-known classes of algebraic structures, such as groups, rings, lattices, and so forth. One way to give positive answers to this question is to adapt the original proof to the new setting. However, this can be an unnecessary duplication of effort, and lacks generality. Another method is to code the original structure into a structure in the given (...) class in a way that is effective enough to preserve the property in which we are interested. In this paper, we show how to transfer a number of computability-theoretic properties from directed graphs to structures in the following classes: symmetric, irreflexive graphs; partial orderings; lattices; rings ; integral domains of arbitrary characteristic; commutative semigroups; and 2-step nilpotent groups. This allows us to show that several theorems about degree spectra of relations on computable structures, nonpreservation of computable categoricity, and degree spectra of structures remain true when we restrict our attention to structures in any of the classes on this list. The codings we present are general enough to be viewed as establishing that the theories mentioned above are computably complete in the sense that, for a wide range of computability-theoretic nonstructure type properties, if there are any examples of structures with such properties then there are such examples that are models of each of these theories. (shrink)

In this paper, the notions of Fα-categorical and Gα-categorical structures are introduced by choosing the isomorphism such that the function itself or its graph sits on the α-th level of the Ershov hierarchy, respectively. Separations obtained by natural graphs which are the disjoint unions of countably many finite graphs. Furthermore, for size-bounded graphs, an easy criterion is given to say when it is computable-categorical and when it is only G2-categorical; in the latter case it is not Fα-categorical for any recursive (...) ordinal α. (shrink)

In this paper we investigate computable isomorphisms of Boolean algebras with operators (BAOs). We prove that there are examples of polymodal Boolean algebras with finitely many computable isomorphism types. We provide an example of a polymodal BAO such that it has exactly one computable isomorphism type but whose expansions by a constant have more than one computable isomorphism type. We also prove a general result showing that BAOs are complete with respect to the degree spectra of structures, computable dimensions, expansions (...) by constants, and the degree spectra of relations. (shrink)

We investigate dependence of recursively enumerable graphs on the equality relation given by a specific r.e. equivalence relation on ω. In particular we compare r.e. equivalence relations in terms of graphs they permit to represent. This defines partially ordered sets that depend on classes of graphs under consideration. We investigate some algebraic properties of these partially ordered sets. For instance, we show that some of these partial ordered sets possess atoms, minimal and maximal elements. We also fully describe the isomorphism (...) types of some of these partial orders. (shrink)

We build an א₁-categorical but not א₀-categorical theory whose only computably presentable model is the saturated one. As a tool, we introduce a notion related to limitwise monotonic functions.

In this paper we introduce effectiveness into model theory of intuitionistic logic. The main result shows that any computable theory T of intuitionistic predicate logic has a Kripke model with decidable forcing such that for any sentence φ, φ is forced in the model if and only if φ is intuitionistically deducible from T.

We construct a computable ℵ0-categorical structure whose first order theory is computably equivalent to the true first order theory of arithmetic.

We define a new type of two player game occurring on a tree. The tree may have no root and may have arbitrary degrees of nodes. These games extend the class of games considered by Gurevich-Harrington in [5]. We prove that in the game one of the players has a winning strategy which depends on finite bounded information about the past part of a play and on future of each play that is isomorphism types of tree nodes. This result extends (...) further the Gurevich-Harrington determinacy theorem from [5]. (shrink)

The aim of this paper is to shed light on our understanding of large scale properties of infinite strings. We say that one string $\alpha $ has weaker large scale geometry than that of $\beta $ if there is color preserving bi-Lipschitz map from $\alpha $ into $\beta $ with small distortion. This definition allows us to define a partially ordered set of large scale geometries on the classes of all infinite strings. This partial order compares large scale geometries of (...) infinite strings. As such, it presents an algebraic tool for classification of global patterns. We study properties of this partial order. We prove, for instance, that this partial order has a greatest element and also possess infinite chains and antichains. We also investigate the sets of large scale geometries of strings accepted by finite state machines such as Büchi automata. We provide an algorithm that describes large scale geometries of strings accepted by Büchi automata. This connects the work with the complexity theory. We also prove that the quasi-isometry problem is a $\Sigma _2^0$ -complete set, thus providing a bridge with computability theory. Finally, we build algebraic structures that are invariants of large scale geometries. We invoke asymptotic cones, a key concept in geometric group theory, defined via model-theoretic notion of ultra-product. Partly, we study asymptotic cones of algorithmically random strings, thus connecting the topic with algorithmic randomness. (shrink)

In this paper, we initiate the study of Ehrenfeucht–Fraïssé games for some standard finite structures. Examples of such standard structures are equivalence relations, trees, unary relation structures, Boolean algebras, and some of their natural expansions. The paper concerns the following question that we call the Ehrenfeucht–Fraïssé problem. Given nω as a parameter, and two relational structures and from one of the classes of structures mentioned above, how efficient is it to decide if Duplicator wins the n-round EF game ? We (...) provide algorithms for solving the Ehrenfeucht–Fraïssé problem for the mentioned classes of structures. The running times of all the algorithms are bounded by constants. We obtain the values of these constants as functions of n. (shrink)

In this paper we show that for any set X ⊆ ω there exists a structure [MATHEMATICAL SCRIPT CAPITAL A] that has no presentation computable in X such that [MATHEMATICAL SCRIPT CAPITAL A]2 has a computable presentation. We also show that there exists a structure [MATHEMATICAL SCRIPT CAPITAL A] with infinitely many computable isomorphism types such that [MATHEMATICAL SCRIPT CAPITAL A]2 has exactly one computable isomorphism type.

In this paper we study the question as to which computable algebras are isomorphic to non-computable \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Pi_{1}^{0}$$\end{document}-algebras. We show that many known algebras such as the standard model of arithmetic, term algebras, fields, vector spaces and torsion-free abelian groups have non-computable\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Pi_{1}^{0}$$\end{document}-presentations. On the other hand, many of this structures fail to have non-computable \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Sigma_{1}^{0}$$\end{document}-presentation.

In this paper we study the question as to which computable algebras are isomorphic to non-computable $\Pi_{1}^{0}$ -algebras. We show that many known algebras such as the standard model of arithmetic, term algebras, fields, vector spaces and torsion-free abelian groups have non-computable $\Pi_{1}^{0}$ -presentations. On the other hand, many of this structures fail to have non-computable $\Sigma_{1}^{0}$ -presentation.

This paper shows how the notion of randomness defines, in a natural way, an algebra. It turns out that the algebra is computably enumerable and finitely generated. The paper investigates algebraic and effective properties of this algebra.

A unary algebra is an algebraic system A = , where ƒ 0 ,…,ƒ n are unary operations on A and n ∈ ω. In the paper we develop the theory of effective unary algebras. We investigate well-known questions of constructive model theory with respect to the class of unary algebras. In the paper we construct unary algebras with a finite number of recursive isomorphism types. We give the notions of program, uniform, and algebraic dimensions of models, and then we (...) investigate these notions on unary algebras. We find connections between algebraic and effective properties of r.e. representable unary algebras. We also deal with finitely generated r.e. unary algebras. We show the connections between trees and unary algebras. Our interests also concern recursive automorphisms groups, r.e. subalgebra and congruence lattices of effective unary algebras. (shrink)

Cholak, Goncharov, Khoussainov, and Shore [1] showed that for each k > 0 there is a computably categorical structure whose expansion by a constant has computable dimension k. We show that the same is true with k replaced by ω. Our proof uses a version of Goncharov's method of left and right operations.

Finite automata have been recently used as alternative, discrete models in theoretical physics, especially in problems related to the dichotomy between endophysical/intrinsic and exophysical/ extrinsic perception . These studies deal with Moore experiments; the main result states that it is impossible to determine the initial state of an automaton, and, consequently, a discrete model of Heisenberg uncertainty has been suggested. For this aim the classical theory of finite automata — which considers automata with initial states — is not adequate, and (...) a new approach is necessary. A study of finite deterministic automata without initial states is exactly the aim of this paper. We will define and investigate the complexity of various types of simulations between automata. Minimal automata will be constructed and proven to be unique up to an isomorphism. We will build our results on an extension of Myhill-Nerode technique; all constructions will make use of “automata responses” to simple experiments only, i.e., no information about the internal machinery will be considered available. (shrink)

The New Evil Demon problem has been hotly debated since the case was introduced in the early 1980’s (e.g. Lehrer and Cohen 1983; Cohen 1984), and there seems to be recent increased interest in the topic. In a forthcoming collection of papers on the New Evil Demon problem (Dutant and Dorsch, forthcoming), at least two of the papers, both by prominent epistemologists, attempt to resist the problem by appealing to the distinction between justification and excuses. My primary aim here is (...) to critically evaluate this new excuse maneuver as a response to the New Evil Demon problem. -/- Their response attempts to give us reason to reject the idea that victims of the New Evil Demon have justification for believing as they do. I shall argue that this approach is ultimately unsuccessful, however much of value can be learned from these attempts. In particular, progress in the debate can be made by following those who advance the excuse maneuver and make explicit the connection between epistemic justification and epistemic norms. By doing so, the questions being debated are clarified, as is the methodology being used to attempt to answer them. (shrink)

There are various equivalent formulations of the Church-Turing thesis. A common one is that every effective computation can be carried out by a Turing machine. The Church-Turing thesis is often misunderstood, particularly in recent writing in the philosophy of mind.