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Order-Sorted Equality Enrichments Modulo Axioms

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Gutiérrez, Raúl
Meseguer, José
Rocha, Camilo
Abstract (Spanish)
Los predicados de igualdad y desigualdad incorporados basados ​​en la comparación de formas canónicas en especificaciones algebraicas se usan con frecuencia porque son prácticos y eficientes. Sin embargo, su uso sitúa las especificaciones algebraicas con semántica inicial del álgebra más allá de las herramientas de prueba de teoremas basadas, por ejemplo, en técnicas de inducción explícitas o sin inducción, y de otras herramientas formales para verificar propiedades clave como la confluencia, la terminación y la completitud suficiente. En cambio, tales especificaciones serían susceptibles de análisis formal si se les agregara un predicado de igualdad definido ecuacionalmente que enriqueciera los tipos de datos algebraicos. Además, tener un predicado de igualdad definido ecuacionalmente es muy útil por derecho propio, particularmente en la demostración inductiva de teoremas. ¿Es posible definir efectivamente una transformación teórica E↦E≃ que extienda una especificación algebraica E a una especificación E≃ que tenga un predicado de igualdad definido ecuacionalmente? Este artículo responde afirmativamente a esta pregunta para una amplia clase de especificaciones E condicionales clasificadas por orden que son de orden decreciente, confluentes en el suelo y que terminan operacionalmente los axiomas del módulo B y tienen una subfirma de constructores. Los axiomas B pueden consistir en axiomas de asociatividad, de conmutatividad o de asociatividad-conmutatividad, de modo que los constructores sean módulo B libre. Probamos que la transformación E↦E≃ conserva todas las propiedades de E recién mencionadas. La transformación se ha automatizado en Maude mediante reflexión y se utiliza en varias herramientas formales de Maude.
Abstract (English)
Built-in equality and inequality predicates based on comparison of canonical forms in algebraic specifications are frequently used because they are handy and efficient. However, their use places algebraic specifications with initial algebra semantics beyond the pale of theorem proving tools based, for example, on explicit or inductionless induction techniques, and of other formal tools for checking key properties such as confluence, termination, and sufficient completeness. Such specifications would instead be amenable to formal analysis if an equationally-defined equality predicate enriching the algebraic data types were to be added to them. Furthermore, having an equationally-defined equality predicate is very useful in its own right, particularly in inductive theorem proving. Is it possible to effectively define a theory transformation E↦E≃ that extends an algebraic specification E to a specification E≃ having an equationally-defined equality predicate? This paper answers this question in the affirmative for a broad class of order-sorted conditional specifications E that are sort-decreasing, ground confluent, and operationally terminating modulo axioms B and have a subsignature of constructors. The axioms B can consist of associativity, or commutativity, or associativity-commutativity axioms, so that the constructors are free modulo B. We prove that the transformation E↦E≃ preserves all the just-mentioned properties of E . The transformation has been automated in Maude using reflection and is used in several Maude formal tools.
Extent
20 páginas.
URI: https://repositorio.escuelaing.edu.co/handle/001/1859
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References

Baader, F., Nipkow, T.: Term Rewriting and All That. Cambridge University Press (1998)

Bergstra, J., Tucker, J.: Characterization of Computable Data Types by Means of a Finite Equational Specification Method. In: de Bakker, J.W., van Leeuwen, J. (eds.) ICALP 1980. LNCS, vol. 81, pp. 76–90. Springer, Heidelberg (1980)

Clavel, M., Durán, F., Eker, S., Lincoln, P., Martí-Oliet, N., Meseguer, J., Talcott, C.: All About Maude - A High-Performance Logical Framework. LNCS, vol. 4350. Springer, Heidelberg (2007)

Durán, F., Lucas, S., Marché, C., Meseguer, J., Urbain, X.: Proving Operational Termination of Membership Equational Programs. Higher Order Symbolic Computation 21(1-2), 59–88 (2008)

Durán, F., Lucas, S., Meseguer, J.: Termination Modulo Combinations of Equational Theories. In: Ghilardi, S., Sebastiani, R. (eds.) FroCoS 2009. LNCS, vol. 5749, pp. 246–262. Springer, Heidelberg (2009)

Durán, F., Meseguer, J.: On the Church-Rosser and Coherence Properties of Conditional Order-Sorted Rewrite Theories. Journal of Logic and Algebraic Programming (2011) (to appear)

Goguen, J., Meseguer, J.: Order-Sorted Algebra I: Equational Deduction for Multiple Inheritance, Overloading, Exceptions and Partial Operations. Theoretical Computer Science 105, 217–273 (1992)

Goguen, J.A.: How to Prove Algebraic Inductive Hypotheses Without Induction. In: Bibel, W., Kowalski, R. (eds.) CADE 1980. LNCS, vol. 87, pp. 356–373. Springer, Heidelberg (1980)

Gutiérrez, R., Meseguer, J., Rocha, C.: Order-Sorted Equality Enrichments Modulo Axioms (Extended Version). Tech. rep., University of Illinois at Urbana-Champaing (December 2011), http://hdl.handle.net/2142/28597

Hendrix, J.: Decision Procedures for Equationally Based Reasoning. Ph.D. thesis, Department of Computer Science, University of Illinois at Urbana-Champaign, Urbana, IL, USA (2008)

Hendrix, J., Clavel, M., Meseguer, J.: A Sufficient Completeness Reasoning Tool for Partial Specifications. In: Giesl, J. (ed.) RTA 2005. LNCS, vol. 3467, pp. 165–174. Springer, Heidelberg (2005)

Lucas, S., Marché, C., Meseguer, J.: Operational Termination of Conditional Term Rewriting Systems. Information Processing Letters 95(4), 446–453 (2005)

Meseguer, J.: Membership Algebra as a Logical Framework for Equational Specification. In: Parisi-Presicce, F. (ed.) WADT 1997. LNCS, vol. 1376, pp. 18–61. Springer, Heidelberg (1998)

Meseguer, J., Goguen, J.A.: Initially, Induction and Computability. Algebraic Methods in Semantics (1986)

Musser, D.R.: On Proving Inductive Properties of Abstract Data Types. In: Proc. of the 7th Annual ACM SIGPLAN-SIGACT Symposium on Principles of Programming Languages, POPL 1980, pp. 154–162. ACM Press (1980)

Masaki, N., Kokichi, F.: On Equality Predicates in Algebraic Specification Languages. In: Jones, C.B., Liu, Z., Woodcock, J. (eds.) ICTAC 2007. LNCS, vol. 4711, pp. 381–395. Springer, Heidelberg (2007)

Rocha, C., Meseguer, J.: Theorem Proving Modulo Based on Boolean Equational Procedures. In: Berghammer, R., Möller, B., Struth, G. (eds.) RelMiCS/AKA 2008. LNCS, vol. 4988, pp. 337–351. Springer, Heidelberg (2008)

Rocha, C., Meseguer, J.: Proving Safety Properties of Rewrite Theories. In: Corradini, A., Klin, B., Cîrstea, C. (eds.) CALCO 2011. LNCS, vol. 6859, pp. 314–328. Springer, Heidelberg (2011)