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

ACUOS: A System for Order-Sorted Modular ACU Generalization

[ES] La generalización, también denominada anti-unificación, es la operación dual de la unificación. Dados dos términos t y t' , un generalizador es un término t'' del cual t y t' son instancias de sustitución. El concepto dual del unificador más general (mgu) es el de generalizador menos general (lgg). En esta tesina extendemos el conocido algoritmo de generalización sin tipos a, primero, una configuración order-sorted con sorts, subsorts y polimorfismo de subtipado; en segundo lugar, la extendemos para soportar generalización módulo teorías ecuacionales, donde los símbolos de función pueden obedecer cualquier combinación de axiomas de asociatividad, conmutatividad e identidad (incluyendo el conjunto vacío de dichos axiomas); y, en tercer lugar, a la combinación de ambos, que resulta en un algoritmo modular de generalización order-sorted ecuacional. A diferencia de las configuraciones sin tipos, en nuestro marco teórico en general el lgg no es único, lo que se debe tanto al tipado como a los axiomas ecuacionales. En su lugar, existe un conjunto finito y mínimo de lggs, tales que cualquier otra generalización tiene a alguno de ellos como instancia. Nuestros algoritmos de generalización se expresan mediante reglas de inferencia para las cuales damos demostraciones de corrección. Ello abre la puerta a nuevas aplicaciones en campos como la evaluación parcial, la síntesis de programas, la minería de datos y la demostración de teoremas para sistemas de razonamiento ecuacional y lenguajes tipados basados en reglas tales como ASD+SDF, Elan, OBJ, CafeOBJ y Maude. Esta tesis también describe una herramienta para el cómputo automatizado de los generalizadores de un conjunto dado de estructuras en un lenguaje tipado módulo un conjunto de axiomas dado. Al soportar la combinación modular de atributos ecuacionales de asociatividad, conmutatividad y existencia de elemento neutro (ACU) para símbolos de función arbitrarios, la generalización ACU modular aporta suficiente poder expresivo a la generalización ordinaria para razonar sobre estructuras de datos tipadas tales como listas, conjuntos y multiconjuntos. La técnica ha sido implementada con generalidad y eficiencia en el sistema ACUOS y puede ser fácilmente integrada con software de terceros.[EN] Generalization, also called anti-uni cation, is the dual of uni cation. Given terms t and t 0 , a generalization is a term t 00 of which t and t 0 are substitution instances. The dual of a most general uni er (mgu) is that of least general generalization (lgg). In this thesis, we extend the known untyped generalization algorithm to, rst, an order-sorted typed setting with sorts, subsorts, and subtype polymorphism; second, we extend it to work modulo equational theories, where function symbols can obey any combination of associativity, commutativity, and identity axioms (includ- ing the empty set of such axioms); and third, to the combination of both, which results in a modular, order-sorted equational generalization algo- rithm. Unlike the untyped case, there is in general no single lgg in our framework, due to order-sortedness or to the equational axioms. Instead, there is a nite, minimal set of lggs, so that any other generalization has at least one of them as an instance. Our generalization algorithms are expressed by means of inference systems for which we give proofs of cor- rectness. This opens up new applications to partial evaluation, program synthesis, data mining, and theorem proving for typed equational rea- soning systems and typed rule-based languages such as ASF+SDF, Elan, OBJ, Cafe-OBJ, and Maude. This thesis also describes a tool for automatically computing the gen- eralizers of a given set of structures in a typed language modulo a set of axioms. By supporting the modular combination of associative, com- mutative and unity (ACU) equational attributes for arbitrary function symbols, modular ACU generalization adds enough expressive power to ordinary generalization to reason about typed data structures such as lists, sets and multisets. The ACU generalization technique has been generally and e ciently implemented in the ACUOS system and can be easily integrated with third-party software.Espert Real, J. (2012). ACUOS: A System for Order-Sorted Modular ACU Generalization. http://hdl.handle.net/10251/1921

E-Generalization Using Grammars

We extend the notion of anti-unification to cover equational theories and present a method based on regular tree grammars to compute a finite representation of E-generalization sets. We present a framework to combine Inductive Logic Programming and E-generalization that includes an extension of Plotkin's lgg theorem to the equational case. We demonstrate the potential power of E-generalization by three example applications: computation of suggestions for auxiliary lemmas in equational inductive proofs, computation of construction laws for given term sequences, and learning of screen editor command sequences.Comment: 49 pages, 16 figures, author address given in header is meanwhile outdated, full version of an article in the "Artificial Intelligence Journal", appeared as technical report in 2003. An open-source C implementation and some examples are found at the Ancillary file

Strategic Issues, Problems and Challenges in Inductive Theorem Proving

Abstract(Automated) Inductive Theorem Proving (ITP) is a challenging field in automated reasoning and theorem proving. Typically, (Automated) Theorem Proving (TP) refers to methods, techniques and tools for automatically proving general (most often first-order) theorems. Nowadays, the field of TP has reached a certain degree of maturity and powerful TP systems are widely available and used. The situation with ITP is strikingly different, in the sense that proving inductive theorems in an essentially automatic way still is a very challenging task, even for the most advanced existing ITP systems. Both in general TP and in ITP, strategies for guiding the proof search process are of fundamental importance, in automated as well as in interactive or mixed settings. In the paper we will analyze and discuss the most important strategic and proof search issues in ITP, compare ITP with TP, and argue why ITP is in a sense much more challenging. More generally, we will systematically isolate, investigate and classify the main problems and challenges in ITP w.r.t. automation, on different levels and from different points of views. Finally, based on this analysis we will present some theses about the state of the art in the field, possible criteria for what could be considered as substantial progress, and promising lines of research for the future, towards (more) automated ITP

Analytical learning and term-rewriting systems

Analytical learning is a set of machine learning techniques for revising the representation of a theory based on a small set of examples of that theory. When the representation of the theory is correct and complete but perhaps inefficient, an important objective of such analysis is to improve the computational efficiency of the representation. Several algorithms with this purpose have been suggested, most of which are closely tied to a first order logical language and are variants of goal regression, such as the familiar explanation based generalization (EBG) procedure. But because predicate calculus is a poor representation for some domains, these learning algorithms are extended to apply to other computational models. It is shown that the goal regression technique applies to a large family of programming languages, all based on a kind of term rewriting system. Included in this family are three language families of importance to artificial intelligence: logic programming, such as Prolog; lambda calculus, such as LISP; and combinatorial based languages, such as FP. A new analytical learning algorithm, AL-2, is exhibited that learns from success but is otherwise quite different from EBG. These results suggest that term rewriting systems are a good framework for analytical learning research in general, and that further research should be directed toward developing new techniques

Metatheorems about convertibility in typed lambda calculi : applications to CPS transform and "free theorems"

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 1997.Includes bibliographical references (p. 98-96).by Jakov Kuc̆an.Ph.D

A Modular Order-sorted Equational Generalization Algorithm

Generalization, also called anti-unification, is the dual of unification. Given terms t and t , a generalizer is a term t of which t and t are substitution instances. The dual of a most general unifier (mgu) is that of least general generalizer (lgg). In this work, we extend the known untyped generalization algorithm to, first, an order-sorted typed setting with sorts, subsorts, and subtype polymorphism; second, we extend it to work modulo equational theories, where function symbols can obey any combination of associativity, commutativity, and identity axioms (including the empty set of such axioms); and third, to the combination of both, which results in a modular, order-sorted equational generalization algorithm. Unlike the untyped case, there is in general no single lgg in our framework, due to order-sortedness or to the equational axioms. Instead, there is a finite, minimal and complete set of lggs, so that any other generalizer has at least one of them as an instance. Our generalization algorithms are expressed by means of inference systems for which we give proofs of correctness. This opens up new applications to partial evaluation, program synthesis, and theorem proving for typed equational reasoning systems and typed rulebased languages such as ASF+SDF, Elan, OBJ, Cafe-OBJ, and Maude. © 2014 Elsevier Inc. All rights reserved. 1.M. Alpuente, S. Escobar, and J. Espert have been partially supported by the EU (FEDER) and the Spanish MEC/MICINN under grant TIN 2010-21062-C02-02, and by Generalitat Valenciana PROMETEO2011/052. J. Meseguer has been supported by NSF Grants CNS 09-04749, and CCF 09-05584.Alpuente Frasnedo, M.; Escobar Román, S.; Espert Real, J.; Meseguer, J. (2014). A Modular Order-sorted Equational Generalization Algorithm. Information and Computation. 235:98-136. https://doi.org/10.1016/j.ic.2014.01.006S9813623