arXiv subjects

Vittorio Cipriani

Publications and source records attributed to Vittorio Cipriani.

9 recordsLinked to original sources

arXiv2026

The tolerant learning hierarchy for c.e. sets

Learning from positive data models a scenario in which a learner receives longer and longer initial segments of an enumeration of an unknown set and repeatedly outputs a conjecture about it. Classical identification in the limit requires the learner to eventually stabilise to a hypothesis naming the target exactly, while anomalous learning relaxes this requirement by allowing finitely many errors in the final hypothesis. We develop a natural generalisation for computably enumerable (c.e.) sets in which correctness is determined by an arbitrary equivalence relation on their indices that respects equality of the sets they enumerate. We call the resulting framework tolerant learning. Exact learning and finite-anomaly learning arise as special cases. By comparing tolerance relations according to the families they make learnable, we obtain a new degree structure, the tolerant learning hierarchy. We extend the classical methods of locking sequences and tell-tales to this setting and derive a combinatorial characterisation of those tolerance relations that are universal (that is, under which every family of nonempty computably enumerable sets is learnable). We then show that the hierarchy is a bounded lower semilattice but not an upper semilattice, is atomless and coatomless, and contains chains and antichains of size continuum. We also analyse natural benchmark relations, including c.e.-index analogues of standard equivalence relations from descriptive set theory, and establish substantial incomparability among their learning powers. Finally, we separate non-effective from computable tolerant learning by exhibiting a universal relation that is not computably universal. These results show that semantic tolerance gives rise to a genuine degree theory at the interface of algorithmic learning and the theory of equivalence relations.

math.LO↗

arXiv2025

Dichotomy results for classes of countable graphs

We study classes of countable graphs where every member does not contain a given finite graph as an induced subgraph -- denoted by $\mathsf{Free}(\mathcal{G})$ for a given finite graph $\mathcal{G}$. Our main results establish a structural dichotomy for such classes: If $\mathcal{G}$ is not an induced subgraph of $\mathcal{P}_4$, then $\mathsf{Free}(\mathcal{G})$ is on top under effective bi-interpretability, implying that the members of $\mathsf{Free}(\mathcal{G})$ exhibit the full range of structural and computational behaviors. In contrast, if $\mathcal{G}$ is an induced subgraph of $\mathcal{P}_4$, then $\mathsf{Free}(\mathcal{G})$ is structurally simple, as witnessed by the fact that every member satisfies the computable embeddability condition. This dichotomy is mirrored in the finite setting when one considers combinatorial and complexity-theoretic properties. Specifically, it is known that $\mathsf{Free}(\mathcal{G})^{fin}$ is complete for graph isomorphism and not a well-quasi-order under embeddability whenever $\mathcal{G}$ is not an induced subgraph of $\mathcal{P}_4$, while in all other cases $\mathsf{Free}(\mathcal{G})^{fin}$ forms a well-quasi-order and the isomorphism problem for $\mathsf{Free}(\mathcal{G})^{fin}$ is solvable in polynomial time.

math.LO↗

arXiv2025

Uniformity in learning structures

The standard framework for studying learning problems on algebraic structures assumes that the structures in the target family are pairwise nonisomorphic. Under this assumption, the most widely investigated learning criterion--Ex-learning--becomes inherently equivalent to the well-known paradigm of Bc-learning. This paper explores what happens when the nonisomorphism requirement is removed and analyzes the extent to which these two learning criteria remain uniformly equivalent.

math.LO↗

arXiv2025

On statistical learning of graphs

We study PAC and online learnability of hypothesis classes formed by copies of a countably infinite graph G, where each copy is induced by permuting G's vertices. This corresponds to learning a graph's labeling, knowing its structure and label set. We consider classes where permutations move only finitely many vertices. Our main result shows that PAC learnability of all such finite-support copies implies online learnability of the full isomorphism type of G, and is equivalent to the condition of automorphic triviality. We also characterize graphs where copies induced by swapping two vertices are not learnable, using a relaxation of the extension property of the infinite random graph. Finally, we show that, for all G and k>2, learnability for k-vertex permutations is equivalent to that for 2-vertex permutations, yielding a four-class partition of infinite graphs, whose complexity we also determine using tools coming from both descriptive set theory and computability theory.

cs.LG↗

arXiv2025

On the learning power of Friedman-Stanley jumps

Recently, a surprising connection between algorithmic learning of algebraic structures and descriptive set theory has emerged. Following this line of research, we define the learning power of an equivalence relation $E$ on a topological space as the class of isomorphism relations with countably many equivalence classes that are continuously reducible to $E$. In this paper, we describe the learning power of the finite Friedman-Stanley jumps of $=_{\mathbb{N}}$ and $=_{\mathbb{N}^\mathbb{N}}$, proving that these equivalence relations learn the families of countable structures that are pairwise distinguished by suitable infinitary sentences. Our proof techniques introduce new ideas for assessing the continuous complexity of Borel equivalence relations.

math.LO↗

arXiv2024

Classifying different criteria for learning algebraic structures

In the last years there has been a growing interest in the study of learning problems associated with algebraic structures. The framework we use models the scenario in which a learner is given larger and larger fragments of a structure from a given target family and is required to output an hypothesis about the structure's isomorphism type. So far researchers focused on $\mathbf{Ex}$-learning, in which the learner is asked to eventually stabilize to the correct hypothesis, and on restrictions where the learner is allowed to change the hypothesis a fixed number of times. Yet, other learning paradigms coming from classical algorithmic learning theory remained unexplored. We study the "learning power" of such criteria, comparing them via descriptive-set-theoretic tools thanks to the novel notion of $E$-learnability. The main outcome of this paper is that such criteria admit natural syntactic characterizations in terms of infinitary formulas analogous to the one given for $\mathbf{Ex}$-learning in [6]. Such characterizations give a powerful method to understand whether a family of structure is learnable with respect to the desired criterion.

math.LO↗

arXiv2024

The Weihrauch lattice at the level of $\boldsymbolΠ_1^1\mathsf{-CA}_0$: the Cantor-Bendixson theorem

This paper continues the program connecting reverse mathematics and computable analysis via the framework of Weihrauch reducibility. In particular, we consider problems related to perfect subsets of Polish spaces, studying the perfect set theorem, the Cantor-Bendixson theorem and various problems arising from them. In the framework of reverse mathematics these theorems are equivalent respectively to $\mathsf{ATR}_0$ and $\boldsymbolΠ_1^1\mathsf{-CA}_0$, the two strongest subsystems of second order arithmetic among the so-called big five. As far as we know, this is the first systematic study of problems at the level of $\boldsymbolΠ_1^1\mathsf{-CA}_0$ in the Weihrauch lattice. We show that the strength of some of the problems we study depends on the topological properties of the Polish space under consideration, while others have the same strength once the space is rich enough.

math.LO↗

arXiv2024

Embeddability of graphs and Weihrauch degrees

We study the complexity of the following related computational tasks concerning a fixed countable graph G: 1. Does a countable graph H provided as input have a(n induced) subgraph isomorphic to G? 2. Given a countable graph H that has a(n induced) subgraph isomorphic to G, find such a subgraph. The framework for our investigations is given by effective Wadge reducibility and by Weihrauch reducibility. Our work follows on "Reverse mathematics and Weihrauch analysis motivated by finite complexity theory" (Computability, 2021) by BeMent, Hirst and Wallace, and we answer several of their open questions.

math.LO↗

arXiv2021

Learning algebraic structures with the help of Borel equivalence relations

We study algorithmic learning of algebraic structures. In our framework, a learner receives larger and larger pieces of an arbitrary copy of a computable structure and, at each stage, is required to output a conjecture about the isomorphism type of such a structure. The learning is successful if the conjectures eventually stabilize to a correct guess. We prove that a family of structures is learnable if and only if its learning domain is continuously reducible to the relation $E_0$ of eventual agreement on reals. This motivates a novel research program, that is, using descriptive set theoretic tools to calibrate the (learning) complexity of nonlearnable families. Here, we focus on the learning power of well-known benchmark Borel equivalence relations (i.e., $E_1$, $E_2$, $E_3$, $Z_0$, and $E_{set}$).

math.LO↗