Order-Sorted Unification with Regular Expression Sorts

We extend first-order order-sorted unification by permitting regular expression sorts for variables and in the domains of function symbols. The set of basic sorts is finite. The obtained signature corresponds to a finite bottom-up hedge automaton. The unification problem in such a theory generalizes some known unification problems. Its unification type is infinitary. We give a complete unification procedure and prove decidability

Unital Anti-Unification: Type and Algorithms

Unital equational theories are defined by axioms that assert the existence of the unit element for some function symbols. We study anti-unification (AU) in unital theories and address the problems of establishing generalization type and designing anti-unification algorithms. First, we prove that when the term signature contains at least two unital functions, anti-unification is of the nullary type by showing that there exists an AU problem, which does not have a minimal complete set of generalizations. Next, we consider two special cases: the linear variant and the fragment with only one unital symbol, and design AU algorithms for them. The algorithms are terminating, sound, complete, and return tree grammars from which the set of generalizations can be constructed. Anti-unification for both special cases is finitary. Further, the algorithm for the one-unital fragment is extended to the unrestricted case. It terminates and returns a tree grammar which produces an infinite set of generalizations. At the end, we discuss how the nullary type of unital anti-unification might affect the anti-unification problem in some combined theories, and list some open questions

Higher-Order Equational Pattern Anti-Unification

We consider anti-unification for simply typed lambda terms in associative, commutative, and associative-commutative theories and develop a sound and complete algorithm which takes two lambda terms and computes their generalizations in the form of higher-order patterns. The problem is finitary: the minimal complete set of generalizations contains finitely many elements. We define the notion of optimal solution and investigate special fragments of the problem for which the optimal solution can be computed in linear or polynomial time

Anti-unification and Generalization: A Survey

Anti-unification (AU), also known as generalization, is a fundamental operation used for inductive inference and is the dual operation to unification, an operation at the foundation of theorem proving. Interest in AU from the AI and related communities is growing, but without a systematic study of the concept, nor surveys of existing work, investigations7 often resort to developing application-specific methods that may be covered by existing approaches. We provide the first survey of AU research and its applications, together with a general framework for categorizing existing and future developments.Comment: Accepted at IJCAI 2023 - Survey Trac

A Generic Framework for Higher-Order Generalizations

We consider a generic framework for anti-unification of simply typed lambda terms. It helps to compute generalizations which contain maximally common top part of the input expressions, without nesting generalization variables. The rules of the corresponding anti-unification algorithm are formulated, and their soundness and termination are proved. The algorithm depends on a parameter which decides how to choose terms under generalization variables. Changing the particular values of the parameter, we obtained four new unitary variants of higher-order anti-unification and also showed how the already known pattern generalization fits into the schema

A Variant of Higher-Order Anti-Unification

We present a rule-based Huet's style anti-unification algorithm for simply-typed lambda-terms in η-long -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 α-equivalence and variable renaming. The algorithm computes it in cubic time within linear space. It has been implemented and the code is freely available. © Alexander Baumgartner, Temur Kutsia, Jordi Levy, and Mateu Villaret; licensed under Creative Commons License CC-BY 24th International Conference on Rewriting Techniques and Applications (RTA'13).This research has been partially supported by the projects HeLo (TIN2012-33042) and TASSAT (TIN2010-20967-C04-01), by the Austrian Science Fund (FWF) with the project SToUT (P 24087-N18) and by the Generalitat de Catalunya with the grant AGAUR 2009-SGR-1434.Peer Reviewe

Constraint Solving over Multiple Similarity Relations

Similarity relations are reflexive, symmetric, and transitive fuzzy relations. They help to make approximate inferences, replacing the notion of equality. Similarity-based unification has been quite intensively investigated, as a core computational method for approximate reasoning and declarative programming. In this paper we consider solving constraints over several similarity relations, instead of a single one. Multiple similarities pose challenges to constraint solving, since we can not rely on the transitivity property anymore. Existing methods for unification with fuzzy proximity relations (reflexive, symmetric, non-transitive relations) do not provide a solution that would adequately reflect particularities of dealing with multiple similarities. To address this problem, we develop a constraint solving algorithm for multiple similarity relations, prove its termination, soundness, and completeness properties, and discuss applications