791 research outputs found
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
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
ACUOS: A System for Modular ACU Generalization with Subtyping and Inheritance
The final publication is available at Springer via http://dx.doi.org/10.1007/978-3-319-11558-0_40Computing 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.This work has been partially supported by the EU (FEDER) and the Spanish MINECO under grants TIN 2010-21062-C02-02 and TIN 2013-45732-C4-1-P, by Generalitat Valenciana PROMETEO2011/052, and by NSF Grant CNS 13-10109. J. Espert has also been supported by the Spanish FPU grant FPU12/06223.Alpuente Frasnedo, M.; Escobar Román, S.; Espert Real, J.; Meseguer, J. (2014). ACUOS: A System for Modular ACU Generalization with Subtyping and Inheritance. En Logics in Artificial Intelligence. Springer. 573-581. https://doi.org/10.1007/978-3-319-11558-0_40S573581Alpuente, M., Escobar, S., Espert, J., Meseguer, J.: A Modular Order-sorted Equational Generalization Algorithm. Information and Computation 235, 98–136 (2014)Alpuente, M., Escobar, S., Meseguer, J., Ojeda, P.: A Modular Equational Generalization Algorithm. In: Hanus, M. (ed.) LOPSTR 2008. LNCS, vol. 5438, pp. 24–39. Springer, Heidelberg (2009)Alpuente, M., Escobar, S., Meseguer, J., Ojeda, P.: Order–Sorted Generalization. ENTCS 246, 27–38 (2009)Alpuente, M., Espert, J., Escobar, S., Meseguer, J.: ACUOS: A System for Modular ACU Generalization with Subtyping and Inheritance. Tech. rep., DSIC-UPV (2013), http://www.dsic.upv.es/users/elp/papers.htmlArmengol, E.: Usages of Generalization in Case-Based Reasoning. In: Weber, R.O., Richter, M.M. (eds.) ICCBR 2007. LNCS (LNAI), vol. 4626, pp. 31–45. Springer, Heidelberg (2007)Clavel, M., Durán, F., Eker, S., Lincoln, P., Martí-Oliet, N., Meseguer, J., Talcott, C. (eds.): All About Maude - A High-Performance Logical Framework. LNCS, vol. 4350. Springer, Heidelberg (2007)Clavel, M., Durán, F., Eker, S., Lincoln, P., Martí-Oliet, N., Meseguer, J., Talcott, C.L.: Reflection, metalevel computation, and strategies. In: All About Maude [6], pp. 419–458Gentner, D.: Structure-Mapping: A Theoretical Framework for Analogy*. Cognitive Science 7(2), 155–170 (1983)Krumnack, U., Schwering, A., Gust, H., Kühnberger, K.-U.: Restricted higher order anti unification for analogy making. In: Orgun, M.A., Thornton, J. (eds.) AI 2007. LNCS (LNAI), vol. 4830, pp. 273–282. Springer, Heidelberg (2007)Kutsia, T., Levy, J., Villaret, M.: Anti-Unification for Unranked Terms and Hedges. Journal of Automated Reasoning 520, 155–190 (2014)Meseguer, J.: Conditioned rewriting logic as a united model of concurrency. Theor. Comput. Sci. 96(1), 73–155 (1992)Muggleton, S.: Inductive Logic Programming: Issues, Results and the Challenge of Learning Language in Logic. Artif. Intell. 114(1-2), 283–296 (1999)Ontañón, S., Plaza, E.: Similarity measures over refinement graphs. Machine Learning 87(1), 57–92 (2012)Plotkin, G.: A note on inductive generalization. In: Machine Intelligence, vol. 5, pp. 153–163. Edinburgh University Press (1970)Pottier, L.: Generalisation de termes en theorie equationelle: Cas associatif-commutatif. Tech. Rep. INRIA 1056, Norwegian Computing Center (1989)Schmid, U., Hofmann, M., Bader, F., Häberle, T., Schneider, T.: Incident Mining using Structural Prototypes. In: García-Pedrajas, N., Herrera, F., Fyfe, C., Benítez, J.M., Ali, M. (eds.) IEA/AIE 2010, Part II. LNCS, vol. 6097, pp. 327–336. Springer, Heidelberg (2010
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
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
Interpretable Graph Networks Formulate Universal Algebra Conjectures
The rise of Artificial Intelligence (AI) recently empowered researchers to
investigate hard mathematical problems which eluded traditional approaches for
decades. Yet, the use of AI in Universal Algebra (UA) -- one of the fields
laying the foundations of modern mathematics -- is still completely unexplored.
This work proposes the first use of AI to investigate UA's conjectures with an
equivalent equational and topological characterization. While topological
representations would enable the analysis of such properties using graph neural
networks, the limited transparency and brittle explainability of these models
hinder their straightforward use to empirically validate existing conjectures
or to formulate new ones. To bridge these gaps, we propose a general algorithm
generating AI-ready datasets based on UA's conjectures, and introduce a novel
neural layer to build fully interpretable graph networks. The results of our
experiments demonstrate that interpretable graph networks: (i) enhance
interpretability without sacrificing task accuracy, (ii) strongly generalize
when predicting universal algebra's properties, (iii) generate simple
explanations that empirically validate existing conjectures, and (iv) identify
subgraphs suggesting the formulation of novel conjectures
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