Search

We adapt the classical notion of learning from text to computable structure theory. Our main result is a model-theoretic characterization of the learnability from text for classes of structures. We show that a family of structures is learnable from text if and only if the structures can be distinguished in terms of their theories restricted to positive infinitary $\Sigma_2$ sentences.

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 effective 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 $K$ be positively computably embeddable in $K'$ by $\Phi$, then for every $\Pi^p_\alpha$ formula $\xi$ in the vocabulary of $K'$ there is a $\Pi^p_\alpha$ formula $\xi^\star$ in the vocabulary of $K$ such that for all $A \in K$, $A \models \xi^\star$ if and only if $\Phi(A) \models \xi$. We use this to obtain new results on the possibility of positive computable embeddings into the class of linear orderings., Comment: 17 pages

We investigate the complexity of embeddings between bi-embeddable structures. In analogy with categoricity spectra, we define the bi-embeddable categoricity spectrum of a structure $\mathcal A$ as the family of Turing degrees that compute embeddings between any computable bi-embeddable copies of $\mathcal A$; the degree of bi-embeddable categoricity of $\mathcal A$ is the least degree in this spectrum (if it exists). We extend many known results about categoricity spectra to the case of bi-embeddability. In particular, we exhibit structures without degree of bi-embeddable categoricity, and we show that every degree d.c.e. above $\mathbf{0}^{(\alpha)}$ for $\alpha$ a computable successor ordinal and $\mathbf{0}^{(\lambda)}$ for $\lambda$ a computable limit ordinal is a degree of bi-embeddable categoricity. We also give examples of families of degrees that are not bi-embeddable categoricity spectra., Comment: 18 pages

We combine computable structure theory and algorithmic learning theory to study learning of families of algebraic structures. Our main result is a model-theoretic characterization of the class $\mathbf{InfEx}_{\cong}$, consisting of the structures whose isomorphism types can be learned in the limit. We show that a family of structures $\mathfrak{K}$ is $\mathbf{InfEx}_{\cong}$-learnable if and only if the structures from $\mathfrak{K}$ can be distinguished in terms of their $\Sigma^{\mathrm{inf}}_2$-theories. We apply this characterization to familiar cases and we show the following: there is an infinite learnable family of distributive lattices; no pair of Boolean algebras is learnable; no infinite family of linear orders is learnable., Comment: 28 pages, 1 figure, forthcoming in Information and Computation

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 InfEx-learning (explanatory learning from informant). Our main contribution is a complete characterization of which families of equivalence structures are 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 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., Comment: 22 pages, forthcoming in Proceedings of Machine Learning Research

We study degree spectra of structures with respect to the bi-embeddability relation. The bi-embeddability spectrum of a structure is the family of Turing degrees of its bi-embeddable copies. To facilitate our study we introduce the notions of bi-embeddable triviality and basis of a spectrum. Using bi-embeddable triviality we show that several known families of degrees are bi-embeddability spectra of structures. We then characterize the bi-embeddability spectra of linear orderings and study bases of bi-embeddability spectra of strongly locally finite graphs., Comment: 15 pages

Computable reducibility is a well-established notion that allows to compare the complexity of various equivalence relations over the natural numbers. We generalize computable reducibility by introducing degree spectra of reducibility and bi-reducibility. These spectra provide a natural way of measuring the complexity of reductions between equivalence relations. We prove that any upward closed collection of Turing degrees with a countable basis can be realised as a reducibility spectrum or as a bi-reducibility spectrum. We show also that there is a reducibility spectrum of computably enumerable equivalence relations with no countable basis and a reducibility spectrum of computably enumerable equivalence relations which is downward dense, thus has no basis., Comment: 17 pages

We study the algorithmic complexity of embeddings between bi-embeddable equivalence structures. We define the notions of computable bi-embeddable categoricity, (relative) $\Delta^0_\alpha$ bi-embeddable categoricity, and degrees of bi-embeddable categoricity. These notions mirror the classical notions used to study the complexity of isomorphisms between structures. We show that the notions of $\Delta^0_\alpha$ bi-embeddable categoricity and relative $\Delta^0_\alpha$ bi-embeddable categoricity coincide for equivalence structures for $\alpha=1,2,3$. We also prove that computable equivalence structures have degree of bi-embeddable categoricity $\mathbf{0},\mathbf{0}'$, or $\mathbf{0}''$. We obtain results on index sets of computable equivalence structure with respect to bi-embeddability., Comment: 18 pages

The study of Borel equivalence relations under Borel reducibility has developed into an important area of descriptive set theory. The dichotomies of Silver and Harrington-Kechris-Louveau show that with respect to Borel reducibility, any Borel equivalence relation strictly above equality on $\omega$ is above equality on ${\cal P}(\omega)$, the power set of $\omega$, and any Borel equivalence relation strictly above equality on the reals is above equality modulo finite on ${\cal P}(\omega)$. In this article we examine the effective content of these and related results by studying effectively Borel equivalence relations under effectively Borel reducibility. The resulting structure is complex, even for equivalence relations with finitely many equivalence classes. However use of Kleene's $O$ as a parameter is sufficient to restore the picture from the noneffective setting. A key lemma is the existence of two effectively Borel sets of reals, neither of which contains the range of the other under any effectively Borel function; the proof of this result applies Barwise compactness to a deep theorem of Harrington establishing for any recursive ordinal $\alpha$ the existence of $\Pi^0_1$ singletons whose $\alpha$-jumps are Turing incomparable.