Order-sorted Equational Generalization Algorithm Revisited

[EN] Generalization, also called anti-unification, is the dual of unification. A generalizer of two terms t and t' is a term t '' of which t and t' 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. Alpuente Frasnedo, M.; Escobar Román, S.; Meseguer, J.; Sapiña-Sanchis, J. (2022). Order-sorted Equational Generalization Algorithm Revisited. Annals of Mathematics and Artificial Intelligence. 90(5):499-522. https://doi.org/10.1007/s10472-021-09771-1 Alpuente, M., Escobar, S., Meseguer, J., Ojeda, P.: A modular equational generalization algorithm. In: Hanus, M. (ed.) Logic-based program synthesis and transformation, 18th international symposium, LOPSTR 2008, Valencia, Spain, July 17-18, 2008, Revised Selected Papers, Springer, Lecture Notes in Computer Science, vol. 5438, pp 24–39 (2008), https://doi.org/10.1007/978-3-642-00515-2_3 Alpuente, M., Escobar, S., Meseguer, J., Ojeda, P.: Order-sorted generalization. Electr Notes Theor Comput Sci 246, 27–38 (2009). https://doi.org/10.1016/j.entcs.2009.07.013 Alpuente, M., Escobar, S., Espert, J., Meseguer, J.: ACUOS: A system for modular ACU generalization with subtyping and inheritance. In: Proceedings of the 14th Eur