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Complete Categorical Deduction for Satisfaction as Injectivity

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Algebra, Meaning, and Computation

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 4060))

Abstract

Birkhoff (quasi-)variety categorical axiomatizability results have fascinated many scientists by their elegance, simplicity and generality. The key factor leading to their generality is that equations, conditional or not, can be regarded as special morphisms or arrows in a special category, where their satisfaction becomes injectivity, a simple and abstract categorical concept. A natural and challenging next step is to investigate complete deduction within the same general and elegant framework. We present a categorical deduction system for equations as arrows and show that, under appropriate finiteness requirements, it is complete for satisfaction as injectivity. A straightforward instantiation of our results yields complete deduction for several equational logics, in which conditional equations can be derived as well at no additional cost, as opposed to the typical method using the theorems of constants and of deduction. At our knowledge, this is a new result in equational logics.

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References

  1. Adámek, J., Rosický, J.: Locally Presentable and Accessible Categories. Cambridge University Press, Cambridge (1994), London Math. Society. Lecture Note Series 189

    Book  MATH  Google Scholar 

  2. Andréka, H., Németi, I.: A general axiomatizability theorem formulated in terms of cone-injective subcategories. In: Csakany, B., Fried, E., Schmidt, E. (eds.) Universal Algebra. Colloquia Mathematics Societas János Bolyai, vol. 29, pp. 13–35. North-Holland, Amsterdam (1981)

    Google Scholar 

  3. Andréka, H., Németi, I.: Generalization of the concept of variety and quasivariety to partial algebras through category theory. In: ISSAC 1985 and EUROCAL 1985, vol. 204, Polish Scientific Publishers, Warsaw (1983)

    Google Scholar 

  4. Banaschewski, B., Herrlich, H.: Subcategories defined by implications. Houston Journal Mathematics 2, 149–171 (1976)

    MATH  MathSciNet  Google Scholar 

  5. Bergstra, J., Tucker, J.: Characterization of computable data types by means of a finite equational specification method. In: de Bakker, J., van Leeuwen, J. (eds.) Data Base Techniques for Pictorial Application. LNCS, vol. 81, pp. 76–90. Springer, Heidelberg (1980)

    Google Scholar 

  6. Bergstra, J., Tucker, J.V.: Equational specifications, complete term rewriting systems, and computable and semicomputable algebras. Journal of the Associationfor Computing Machinery 42(6), 1194–1230 (1995)

    MATH  MathSciNet  Google Scholar 

  7. Birkhoff, G.: On the structure of abstract algebras. In: Proceedings of the Cambridge Philosophical Society, vol. 31, pp. 433–454 (1935)

    Google Scholar 

  8. Birkhoff, G., Lipson, J.: Heterogenous algebras. Journal of Combinatorial Theory 8, 115–133 (1970)

    Article  MATH  MathSciNet  Google Scholar 

  9. Borovanský, P., Cîrstea, H., Dubois, H., Kirchner, C., Kirchner, H., Moreau, P.-E., Ringeissen, C., Vittek, M.: Elan. User manual, http://www.loria.fr/equipes/protheo/SOFTWARES/ELAN

  10. Bouhoula, A., Jouannaud, J.-P., Meseguer, J.: Specification and proof in membership equational logic. Theoretical Computer Science 236, 35–132 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  11. Broy, M., Wirsing, M., Pepper, P.: On the algebraic definition of programming languages. ACM Trans. on Prog. Lang. and Systems 9(1), 54–99 (1987)

    Article  MATH  Google Scholar 

  12. Clavel, M., Eker, S., Lincoln, P., Meseguer, J.: Principles of Maude. In: Meseguer, J. (ed.) Proceedings, First International Workshop on Rewriting Logic and its Applications. Electronic Notes in Theoretical Computer Science, vol. 4, Elsevier Science, Amsterdam (1996)

    Google Scholar 

  13. Căzănescu, V.: Local equational logic. In: Ésik, Z. (ed.) FCT 1993. LNCS, vol. 710, pp. 162–170. Springer, Heidelberg (1993)

    Google Scholar 

  14. Diaconescu, R.: Category-based Semantics for Equational and Constraint Logic Programming. PhD thesis, University of Oxford (1994)

    Google Scholar 

  15. Diaconescu, R., Futatsugi, K.: CafeOBJ Report: The Language, Proof Techniques, and Methodologies for Object-Oriented Algebraic Specification. AMAST Series in Computing, vol. 6. World Scientific, Singapore (1998)

    MATH  Google Scholar 

  16. Freyd, P., Kelly, G.: Categories of continuous functors. Journal of Pure and Applied Algebra 2, 169–191 (1972)

    Article  MATH  MathSciNet  Google Scholar 

  17. Goguen, J., Malcolm, G.: Alg. Semantics of Imperative Programs. MIT Press, Cambridge (1996)

    Google Scholar 

  18. Goguen, J., Malcolm, G.: A hidden agenda. Theoretical Computer Science 245(1), 55–101 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  19. Goguen, J., Meseguer, J.: Completeness of many-sorted equational logic. Houston Journal of Mathematics 11(3), 307–334 (1985), Preliminary versions have appeared in: SIGPLAN Notices, July 1981, Vol. 16(7), pp. 24–37; SRI Report CSL-135, May 1982; and Report CSLI-84-15, Stanford (September 1985)

    MATH  MathSciNet  Google Scholar 

  20. Goguen, J., Meseguer, J.: Order-sorted algebra I: Equational deduction for multiple inheritance, overloading, exceptions and partial operations. Theoretical Computer Science 105(2), 217–273 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  21. Goguen, J., Winkler, T., Meseguer, J., Futatsugi, K., Jouannaud, J.-P.: Introducing OBJ. In: Goguen, J., Malcolm, G. (eds.) Software Engineering with OBJ: algebraic specification in action, pp. 3–167. Kluwer, Dordrecht (2000)

    Google Scholar 

  22. Grothendieck, A.: Sur quelques points d’algébre homologique. Tôhoku Mathematical Journal 2, 119–221 (1957)

    MathSciNet  Google Scholar 

  23. Herrlich, H., Strecker, G.: Category Theory. Allyn and Bacon (1973)

    Google Scholar 

  24. Isbell, J.R.: Subobjects, adequacy, completeness and categories of algebras. Rozprawy Matematyczne 36, 1–33 (1964)

    MathSciNet  Google Scholar 

  25. Lambek, J.: Completions of Categories. Lecture Notes in Mathematics, vol. 24. Springer, Heidelberg (1966)

    MATH  Google Scholar 

  26. Lane, S.M.: Categories for the Working Mathematician. Springer, Heidelberg (1971)

    MATH  Google Scholar 

  27. Meseguer, J.: Membership algebra as a logical framework for equational specification. In: Parisi-Presicce, F. (ed.) WADT 1997. LNCS, vol. 1376, Springer, Heidelberg (1998)

    Google Scholar 

  28. Mitchell, B.: Theory of categories. Academic Press, New York (1965)

    MATH  Google Scholar 

  29. Németi, I.: On notions of factorization systems and their applications to coneinjective subcategories. Periodica Mathematica Hungarica 13(3), 229–335 (1982)

    Article  MATH  MathSciNet  Google Scholar 

  30. Németi, I., Sain, I.: Cone-implicational subcategories and some Birkhoff-type theorems. In: Csakany, B., Fried, E., Schmidt, E. (eds.) Universal Algebra. Colloquia Mathematics Societas János Bolyai, pp. 535–578. North-Holland, Amsterdam (1981)

    Google Scholar 

  31. Padawitz, P., Wirsing, M.: Completeness of many-sorted equational logic revisited. Bulletin of the European Association for Theoretical Computer Science 24, 88–94 (1984)

    Google Scholar 

  32. Reichel, H.: Initial Computability, Algebraic Specifications, and Partial Algebras. Oxford University Press, Oxford (1987)

    MATH  Google Scholar 

  33. Roşu, G.: A Birkhoff-like axiomatizability result for hidden algebra and coalgebra. In: Jacobs, B., Moss, L., Reichel, H., Rutten, J. (eds.) Proceedings of the First Workshop on Coalgebraic Methods in Computer Science (CMCS 1998). Electronic Notes in Theoretical Computer Science, vol. 11, pp. 179–196. Elsevier Science, Amsterdam (1998)

    Google Scholar 

  34. Roşu, G.: Hidden Logic. PhD thesis, University of California at San Diego (2000), http://ase.arc.nasa.gov/grosu/phd-thesis.ps

  35. Roşu, G.: Complete categorical equational deduction. In: Fribourg, L. (ed.) CSL 2001 and EACSL 2001. LNCS, vol. 2142, pp. 528–538. Springer, Heidelberg (2001)

    Chapter  Google Scholar 

  36. Roşu, G.: Equational axiomatizability for coalgebra. Theoretical Computer Science 260(1-2), 229–247 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  37. Roşu, G.: Axiomatizability in inclusive equational logics. Mathematical Structures in Computer Science, http://ase.arc.nasa.gov/grosu/iel.ps (to appear)

  38. Roşu, G., Goguen, J.: On equational Craig interpolation. Journal of Universal Computer Science 6(1), 194–200 (2000)

    MATH  MathSciNet  Google Scholar 

  39. Rodenburg, P.H.: A simple algebraic proof of the equational interpolation theorem. Algebra Universalis 28, 48–51 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  40. Smolka, G., Nutt, W., Goguen, J., Meseguer, J.: Order-sorted equational computation. In: Nivat, M., Aït-Kaci, H. (eds.) Resolution of Equations in Algebraic Structures Rewriting Techniques, vol. 2, pp. 299–367. Academic, New York (1989)

    Google Scholar 

  41. Wand, M.: First-order identities as a defining language. Acta Informatica 14, 337–357 (1980)

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Computer Science, University of Illinois at Urbana-Champaign, USA

    Grigore Roşu

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  1. Grigore Roşu
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Editors and Affiliations

  1. Japan Advanced Institute of Science and Technology (JAIST),  

    Kokichi Futatsugi

  2. LIX, Projet INRIA TypiCal, École Polytechnique and CNRS, 91400, Palaiseau, France

    Jean-Pierre Jouannaud

  3. CS Dept., University of Illinois at Urbana-Champaign, USA

    José Meseguer

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Roşu, G. (2006). Complete Categorical Deduction for Satisfaction as Injectivity. In: Futatsugi, K., Jouannaud, JP., Meseguer, J. (eds) Algebra, Meaning, and Computation. Lecture Notes in Computer Science, vol 4060. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11780274_9

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  • DOI: https://doi.org/10.1007/11780274_9

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-35462-8

  • Online ISBN: 978-3-540-35464-2

  • eBook Packages: Computer ScienceComputer Science (R0)

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