# Order-sorted equational generalization algorithm revisited

Ontology type: schema:ScholarlyArticle

### Article Info

DATE

2021-09-25

AUTHORS ABSTRACT

Generalization, also called anti-unification, is the dual of unification. A generalizer of two terms t and t′\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$t^{\prime }$\end{document} is a term t′′\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$t^{\prime \prime }$\end{document} of which t and t′\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$t^{\prime }$\end{document} are substitution instances. The dual of most general equational unifiers is that of least general equational generalizers, i.e., most specific anti-instances modulo equations. In a previous work, we extended the classical untyped generalization algorithm to: (1) an order-sorted typed setting with sorts, subsorts, and subtype polymorphism; (2) 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 (3) the combination of both, which results in a modular, order-sorted equational generalization algorithm. However, Cerna and Kutsia showed that our algorithm is generally incomplete for the case of identity axioms and a counterexample was given. Furthermore, they proved that, in theories with two identity elements or more, generalization with identity axioms is generally nullary, yet it is finitary for both the linear and one-unital fragments, i.e., either solutions with repeated variables are disregarded or the considered theories are restricted to having just one function symbol with an identity or unit element. In this work, we show how we can easily extend our original inference system to cope with the non-linear fragment and identify a more general class than one–unit theories where generalization with identity axioms is finitary. More... »

PAGES

1-24

### References to SciGraph publications

• 2007-01-01. Usages of Generalization in Case-Based Reasoning in CASE-BASED REASONING RESEARCH AND DEVELOPMENT
• 2020-02-07. Efficient Safety Enforcement for Maude Programs via Program Specialization in the ÁTAME System in MATHEMATICS IN COMPUTER SCIENCE
• 1998. Membership algebra as a logical framework for equational specification in RECENT TRENDS IN ALGEBRAIC DEVELOPMENT TECHNIQUES
• 2019-05-06. : A High-Performance System for Modular ACU Generalization with Subtyping and Inheritance in LOGICS IN ARTIFICIAL INTELLIGENCE
• 2009. Termination Modulo Combinations of Equational Theories in FRONTIERS OF COMBINING SYSTEMS
• 2014. ACUOS: A System for Modular ACU Generalization with Subtyping and Inheritance in LOGICS IN ARTIFICIAL INTELLIGENCE
• 2009. A Modular Equational Generalization Algorithm in LOGIC-BASED PROGRAM SYNTHESIS AND TRANSFORMATION
• 2011-12-17. Similarity measures over refinement graphs in MACHINE LEARNING

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DOI

http://dx.doi.org/10.1007/s10472-021-09771-1

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