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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.

The final publication is available at Springer via http://dx.doi.org/10.1007/978-3-319-11558-0_40, Computing generalizers is relevant in a wide spectrum of automated reasoning areas where analogical reasoning and inductive inference are needed. The ACUOS system computes a complete and minimal set of semantic generalizers (also called “anti-unifiers”) of two structures in a typed language modulo a set of equational axioms. By supporting types and any (modular) combination of associativity (A), commutativity (C), and unity (U) algebraic axioms for function symbols, ACUOS allows reasoning about typed data structures, e.g. lists, trees, and (multi-)sets, and typical hierarchical/structural relations such as is a and part of. This paper discusses the modular ACU generalization tool ACUOS and illustrates its use in a classical artificial intelligence problem.

[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 expre, [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.

[EN] Four commercial diets containing different levels of crude protein and crude lipid (44/25, 43/21, 46/20 and 47/20%) were assayed in duplicated groups in juvenile meagre (Argyrosomus regius) (initial individual weights were 94 g) in an experiment lasting 173 days. The essential amino acid contents (expressed in g/kg of diet basis) in diets 46/20 and 47/20 were higher than in diets 44/25 and 43/21. The HUFAs represented 184 and 207 g/kg in diets 46/20 and 47/20, respectively and 98 and 116 g/kg in diets 44/25 and 43/21, respectively. The fish fed diet 47/20 obtained the best growth and efficiency results, reaching a final individual weight of 393 g, followed by the meagre fed with diet 46/20. Meagre from the 47/20 group retained more of the ingested protein and energy than those fed diets 46/20. Fish fed 44/25 and 43/21 obtained the significantly lowest protein and energy efficiency. The retention of individual amino acids (AAs) in fish fed diets generally decreased in order of diets 46/20, 43/21 and 44/25. The IAA retention of meagre fed diet 47/20 was around 24.8% in phenylalanine and 39.7% in lysine. The results of the current experiment show that the fish fed commercial diet 47/20 obtained the best results in meagre growth, followed by fish fed diet 46/20. Diets 43/21 and 44/25 presented the worst growth and feed efficiency results.

[EN] Web-TLR is a Web verification engine that is based on the well-established Rewriting Logic--Maude/LTLR tandem for Web system specification and model-checking. In Web-TLR, Web applications are expressed as rewrite theories that can be formally verified by using the Maude built-in LTLR model-checker. Whenever a property is refuted, a counterexample trace is delivered that reveals an undesired, erroneous navigation sequence. Unfortunately, the analysis (or even the simple inspection) of such counterexamples may be unfeasible because of the size and complexity of the traces under examination. In this paper, we endow Web-TLR with a new Web debugging facility that supports the efficient manipulation of counterexample traces. This facility is based on a backward trace-slicing technique for rewriting logic theories that allows the pieces of information that we are interested to be traced back through inverse rewrite sequences. The slicing process drastically simplifies the computation trace by dropping useless data that do not influence the final result. By using this facility, the Web engineer can focus on the relevant fragments of the failing application, which greatly reduces the manual debugging effort and also decreases the number of iterative verifications.

Web-TLR is a software tool designed for model-checking Web applications that is based on rewriting logic. Web applications are expressed as rewrite theories that can be formally verified by using the Maude built-in LTLR model-checker. Whenever a property is refuted, it produces a counterexample trace that underlies the failing model checking computation. However, the analysis (or even the simple inspection) of large counterexamples may prove to be unfeasible due to the size and complexity of the traces under examination. This work aims to improve the understandability of the counterexamples generated by Web-TLR by developing an integrated framework for debugging Web applications that integrates a trace-slicing technique for rewriting logic theories that is particularly tailored to Web-TLR. The verification environment is also provided with a user-friendly, graphical Web interface that shields the user from unnecessary information. Trace slicing is a widely used technique for execution trace analysis that is effectively used in program debugging, analysis and comprehension. Our trace slicing technique allows us to systematically trace back rewrite sequences modulo equational axioms (such as associativity and commutativity) by means of an algorithm that dynamically simpli es the traces by detecting control and data dependencies, and dropping useless data that do not infuence the final result. Our methodology is particularly suitable for analyzing complex, textually-large system computations such as those delivered as counter-example traces by Maude model-checkers. The slicing facility implemented in Web-TLR allows the user to select the pieces of information that she is interested into by means of a suitable pattern-matching language supported by wildcards. The selected information is then traced back through inverse rewrite sequences. The slicing process drastically simpli es the computation trace by dropping useless data that do not influence the nal result. By usin