We study the learnability of classes of computable structures under models that allow finitely many mind changes. Extending classical notions of explanatory learning from informant and from text, we introduce new paradigms where the information source and the convergence requirements are modified. In particular, we define Δ 2 0 -learning, where each atomic fact may be presented with finitely many errors before stabilizing, and c.e.- and d.c.e.-learning , where information is restricted to positive atomic facts that may either never be retracted (c.e.) or be retracted at most once (d.c.e.). We provide syntactic characterizations for these notions: Δ 2 0 -learning coincides with definability by finite existential sentences, and c.e.-learning coincides with TxtEx-learning. For d.c.e.-learning we give partial characterization in restricted languages. Furthermore, for n -learning from informant, where learners are allowed at most n mind changes, we establish a complete syntactic characterization in terms of specific infinitary formulas of bounded depth.

Abstract We prove an effective version of the Lopez-Escobar theorem for continuous domains. Let $Mod(\tau )$ be the set of countable structures with universe $\omega $ in vocabulary $\tau $ topologized by the Scott topology. We show that an invariant set $X\subseteq Mod(\tau )$ is $\Pi ^0_\alpha $ in the Borel hierarchy of this topology if and only if it is definable by a $\Pi ^p_\alpha $ -formula, a positive $\Pi ^0_\alpha $ formula in the infinitary logic $L_{\omega _1\omega }$ . As a corollary of this result we obtain a new pullback theorem for positive computable embeddings: Let $\mathcal {K}$ be positively computably embeddable in $\mathcal {K}'$ by $\Phi $ , then for every $\Pi ^p_\alpha $ formula $\xi $ in the vocabulary of $\mathcal {K}'$ there is a $\Pi ^p_\alpha $ formula $\xi ^{*}$ in the vocabulary of $\mathcal {K}$ such that for all $\mathcal {A}\in \mathcal {K}$ , $\mathcal {A}\models \xi ^{*}$ if and only if $\Phi (\mathcal {A})\models \xi $ . We use this to obtain new results on the possibility of positive computable embeddings into the class of linear orderings.

While most research in Gold-style learning focuses on learning formal languages, we consider the identification of computable structures, specifically equivalence structures. In our core model the learner gets more and more information about which pairs of elements of a structure are related and which are not. The aim of the learner is to find (an effective description of) the isomorphism type of the structure presented in the limit. In accordance with language learning we call this learning criterion $\mathbf{InfEx}$-learning (explanatory learning from informant). Our main contribution is a complete characterization of which families of equivalence structures are $\mathbf{InfEx}$-learnable. This characterization allows us to derive a bound of $\mathbf{0”}$ on the computational complexity required to learn uniformly enumerable families of equivalence structures. We also investigate variants of $\mathbf{InfEx}$-learning, including learning from text (where the only information provided is which elements are related, and not which elements are not related) and finite learning (where the first actual conjecture of the learner has to be correct). Finally, we show how learning families of structures relates to learning classes of languages by mapping learning tasks for structures to equivalent learning tasks for languages.

Abstract Let E be a computably enumerable (c.e.) equivalence relation on the set ω of natural numbers. We say that the quotient set $\omega /E$ (or equivalently, the relation E ) realizes a linearly ordered set ${\cal L}$ if there exists a c.e. relation ⊴ respecting E such that the induced structure ( $\omega /E$ ; ⊴) is isomorphic to ${\cal L}$ . Thus, one can consider the class of all linearly ordered sets that are realized by $\omega /E$ ; formally, ${\cal K}\left( E \right) = \left\{ {{\cal L}\,|\,{\rm{the}}\,{\rm{order}}\, - \,{\rm{type}}\,{\cal L}\,{\rm{is}}\,{\rm{realized}}\,{\rm{by}}\,E} \right\}$ . In this paper we study the relationship between computability-theoretic properties of E and algebraic properties of linearly ordered sets realized by E . 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 by E 1 is also realized by E 2 . Following the tradition of computability theory, the lo -degrees are the classes of equivalence relations induced by the pre-order $ \le _{lo} $ . We study the partially ordered set of lo -degrees. For instance, we construct various chains and anti-chains and show the existence of a maximal element among the lo -degrees.

We continue recent work on computable structure theory in the setting of ω1. We prove the analogue of a result from Fokina et al. (2012 J. Symbolic Logic, 77, 122–132) saying that isomorphism of computable structures lies ‘on top’ among Σ11 equivalence relations on ω. Our equivalence relations are on ω1. In the standard setting, Σ11 sets are characterized in terms of paths through trees. In the setting of ω1, we use a new characterization of Σ11 sets that involves clubs in ω1. Finally, we present some new results about ω1-computable categoricity for fields.

Abstract 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 equivalence relations on hyperarithmetical subsets of ω.

Abstract The first main result isolates some conditions which fail for the class of graphs and hold for the class of Abelianp-groups, the class of Abelian torsion groups, and the special class of “rank-homogeneous” trees. We consider these conditions as a possible definition of what it means for a class of structures to have “Ulm type”. The result says that there can be no Turing computable embedding of a class not of Ulm type into one of Ulm type. We apply this result to show that there is no Turing computable embedding of the class of graphs into the class of “rank-homogeneous” trees. The second main result says that there is a Turing computable embedding of the class of rank-homogeneous trees into the class of torsion-free Abelian groups. The third main result says that there is a “rank-preserving” Turing computable embedding of the class of rank-homogeneous trees into the class of Boolean algebras. Using this result, we show that there is a computable Boolean algebra of Scott rank .

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

Abstract This paper calculates, in a precise way. the complexity of the index sets for three classes of computable structures: the class of structures of Scott rank , the class , of structures of Scott rank , and the class K of all structures of non-computable Scott rank. We show that I(K) is m-complete is m-complete relative to Kleene's and is m-complete relative to .