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

Term-Graph Anti-Unification

We study anti-unification for possibly cyclic, unranked term-graphs and develop an algorithm, which computes a minimal complete set of generalizations for them. For bisimilar graphs the algorithm computes the join in the lattice generated by a functional bisimulation. These results generalize anti-unification for ranked and unranked terms to the corresponding term-graphs, and solve also anti-unification problems for rational terms and dags. Our results open a way to widen anti-unification based code clone detection techniques from a tree representation to a graph representation of the code

One or Nothing: Anti-unification over the Simply-Typed Lambda Calculus

Generalization techniques have many applications, such as template construction, argument generalization, and indexing. Modern interactive provers can exploit advancement in generalization methods over expressive-type theories to further develop proof generalization techniques and other transformations. So far, investigations concerned with anti-unification (AU) over lambda terms and similar type theories have focused on developing algorithms for well-studied variants. These variants forbid the nesting of generalization variables, restrict the structure of their arguments, and are unitary. Extending these methods to more expressive variants is important to applications. We consider the case of nested generalization variables and show that the AU problem is nullary (using capture-avoiding substitutions), even when the arguments to free variables are severely restricted.Comment: 12 pages, submitted to Journal (a revised version of the previous submission with more precise proofs

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 of Unordered Goals

Anti-unification in logic programming refers to the process of capturing common syntactic structure among given goals, computing a single new goal that is more general called a generalization of the given goals. Finding an arbitrary common generalization for two goals is trivial, but looking for those common generalizations that are either as large as possible (called largest common generalizations) or as specific as possible (called most specific generalizations) is a non-trivial optimization problem, in particular when goals are considered to be unordered sets of atoms. In this work we provide an in-depth study of the problem by defining two different generalization relations. We formulate a characterization of what constitutes a most specific generalization in both settings. While these generalizations can be computed in polynomial time, we show that when the number of variables in the generalization needs to be minimized, the problem becomes NP-hard. We subsequently revisit an abstraction of the largest common generalization when anti-unification is based on injective variable renamings, and prove that it can be computed in polynomially bounded time

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

Higher-Order Pattern Anti-Unification in Linear 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 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.This research has been partially supported by the Austrian Science Fund (FWF) project SToUT (P 24087-N18), the Upper Austrian Government strategic program “Innovatives OÖ 2010plus”, the MINECO projects RASO (TIN2015-71799-C2-1-P) and HeLo (TIN2012-33042), the MINECO/FEDER UE project LoCoS (TIN2015-66293-R) and the UdG project MPCUdG2016/055.Peer Reviewe