We consider the feature-generation task wherein we are given a database with entities labeled as positive and negative examples, and the goal is to find feature queries that allow for a linear separation between the two sets of examples. We focus on conjunctive feature queries, and explore two fundamental problems: (a) deciding whether separating feature queries exist (separability), and (b) generating such queries when they exist. In the approximate versions of these problems, we allow a predefined fraction of the examples to be misclassified. To restrict the complexity of the generated classifiers, we explore various ways of regularizing (i.e., imposing simplicity constraints on) them by limiting their dimension, the number of joins in feature queries, and their generalized hypertree width (ghw). Among other results, we show that the separability problem is tractable in the case of bounded ghw; yet, the generation problem is intractable, simply because the feature queries might be too large. So, we explore a third problem: classifying new entities without necessarily generating the feature queries. Interestingly, in the case of bounded ghw we can efficiently classify without ever explicitly generating the feature queries.

In this work we study anti-unification for unranked terms and hedges, permitting context and hedge variables. Hedges are sequences of unranked terms. The anti-unification problem of two hedges s˜ and q˜ is concerned with finding their generalization, a hedge g˜ such that both s˜ and q˜ are substitution instances of g˜. Second-order power is gained by using context variables to generalize vertical differences at the input hedges. Hedge variables are used to generalize horizontal differences. An anti-unification algorithm is presented, which computes a generalization of input hedges and records all the differences. The algorithm is parametric by a skeleton computation function. For instance, we can compute a generalization of a skeleton which represents a constrained longest common subforest, or an agreement subhedge/subtree of the input hedges. The computation of the generalization is done in quadratic time.

We present a rule-based Huet’s style anti-unification algorithm for simply typed lambda-terms, which computes a least general higher-order pattern generalization. For a pair of arbitrary terms of the same type, such a generalization always exists and is unique modulo $$\alpha $$ α -equivalence and variable renaming. With a minor modification, the algorithm works for untyped lambda-terms as well. The time complexity of both algorithms is linear.

We study nominal anti-unification, which is concerned with computing least general generalizations for given terms-in-context. In general, the problem does not have a least general solution, but if the set of atoms permitted in generalizations is finite, then there exists a least general generalization which is unique modulo variable renaming and alpha-equivalence. We present an algorithm that computes it. The algorithm relies on a subalgorithm that constructively decides equivariance between two terms-in-context. We prove soundness and completeness properties of both algorithms and analyze their complexity. Nominal anti-unification can be applied to problems where generalization of first-order terms is needed (inductive learning, clone detection, etc.), but bindings are involved.

We present a rule-based Huet's style anti-unification algorithm for simply-typed lambda-terms in eta-long beta-normal form, which computes a least general higher-order pattern generalization. For a pair of arbitrary terms of the same type, such a generalization always exists and is unique modulo alpha-equivalence and variable renaming. The algorithm computes it in cubic time within linear space. It has been implemented and the code is freely available.