Algebraic composition and refinement of proofs

  • Martin Simons
  • Michel Sintzoff
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1349)

Abstract

We present an algebraic calculus for proof composition and refinement. Fundamentally, proofs are expressed at successive levels of abstraction, with the perhaps unconventional principle that a formula is considered to be its own most abstract proof, which may be refined into increasingly concrete proofs. Consequently, we suggest a new paradigm for expressing proofs, which views theorems and proofs as inhabiting the same semantic domain. This algebraic/model-theoretical view of proofs distinguishes our approach from conventional typetheoretical or sequent-based approaches in which theorems and proofs are different entities. All the logical concepts that make up a formal system — formulas, inference rules, and derivations — are expressible in terms of the calculus itself. Proofs are constructed and structured by means of a composition operator and a consequential rule-forming operator. Their interplay and their relation wrt. the refinement order are expressed as algebraic laws.

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References

  1. Abramsky, S. (1994), Interaction categories and communicating sequential processes, in A. W. Roscoe, ed., ‘A Classical Mind: Essays in Honour of C.A.R. Hoare', Prentice Hall, pp. 1–16.Google Scholar
  2. Abramsky, S. & Vickers, S. (1993), ‘Quantales, observational logic and process semantics', Mathematical Structures in Computer Science 3, 161–227.Google Scholar
  3. Bird, R. & de Moor, O. (1997), Algebra of Programming, Prentice Hall.Google Scholar
  4. Cockett, J. R. B. & Seely, R. A. G. (1997), ‘Weakly distributive categories', Journal of Pure and Applied Algebra 114(2), 133–173.Google Scholar
  5. Davey, B. A. & Priestley, H. A. (1990), Introduction to Lattices and Order, Cambridge University Press.Google Scholar
  6. Došen, K. & Schroeder-Heister, P., eds (1993), Substructural Logics, Oxford Science Publications.Google Scholar
  7. Dunn, J. M. (1990), Gaggle theory: An abstraction of Galois connections and residuation, with applications to negation, implication, and various logical operators, in J. van Eijck, ed., ‘European Workshop on Logics in AI (JELIA'90)', LNCS 478, Springer Verlag.Google Scholar
  8. Dunn, J. M. (1993), Partial gaggles applied to logics with restricted structural rules, in Došen & Schroeder-Heister (1993), pp. 63–108.Google Scholar
  9. Hesselink, W. J. (1990), ‘Axioms and models of linear logic', Formal Aspects of Computing 2, 139–166.Google Scholar
  10. Hoare, C. A. R. & He, J. (1987), “The weakest prespecification', Information Processing Letters 24, 127–132.Google Scholar
  11. Jones, C. B. (1990), Systematic Software Development Using VDM, second edn, Prentice Hall.Google Scholar
  12. Kleene, S. C. (1971), Introduction to Metamathematics, sixth reprint edn, North Holland.Google Scholar
  13. Lamport, L. (1994), ‘How to write a proof', American Mathematical Monthly 102(7), 600–608.Google Scholar
  14. Martin, A. P., Gardiner, P. & Woodcock, J. C. P. (1997), ‘A tactic calculus — abridged version', Formal Aspects of Computing 8(4), 479–489.Google Scholar
  15. Ono, H. (1993), Semantics of substructural logics, in Došen & Schroeder-Heister (1993), pp. 259–291.Google Scholar
  16. Pratt, V. (1995), Chu spaces and their interpretation as concurrent objects, in J. van Leeuwen, ed., ‘Computer Science Today: Recent Trends and Developments', LNCS 1000, Springer Verlag, pp. 392–405.Google Scholar
  17. Rosenthal, K. I. (1990), Quantales and their Application, Longman Scientific & Technical.Google Scholar
  18. Simons, M. (1997a), The Presentation of Formal Proofs, GMD-Bericht Nr. 278, Oldenbourg Verlag.Google Scholar
  19. Simons, M. (1997b), Proof presentation for Isabelle, in E. L. Gunter & A. Felty, eds, “Theorem Proving in Higher Order Logics — 10th International Conference', LNCS 1275, Springer Verlag, pp. 259–274.Google Scholar
  20. Simons, M. & Weber, M. (1996), ‘An approach to literate and structured formal developments', Formal Aspects of Computing 8(1), 86–107.Google Scholar
  21. Sintzoff, M. (1993), Endomorphic typing, in B. Möller, H. A. Partsch & S. A. Schumann, eds, 'Formal Program Development', LNCS 755, Springer Verlag, pp. 305–323.Google Scholar
  22. Troelstra, A. S. (1992), Lectures on Linear Logic, number 29 in ‘CSLI Lecture Notes', CSLI.Google Scholar
  23. Vickers, S. (1989), Topology via Logic, Cambridge University Press.Google Scholar
  24. Weber, M. (1993), ‘Definition and basic properties of the Deva meta-calculus', Formal Aspects of Computing 5, 391–431.Google Scholar
  25. Weber, M., Simons, M. & Lafontaine, C. (1993), The Generic Development Language Deva: Presentation and Case Studies, LNCS 738, Springer Verlag.Google Scholar
  26. Yetter, D. (1990), ‘Quantales and (non-commutative) linear logic', The Journal of Symbolic Logic 55, 41–64.Google Scholar

Copyright information

© Springer-Verlag 1997

Authors and Affiliations

  • Martin Simons
    • 1
    • 2
  • Michel Sintzoff
    • 3
  1. 1.GMD Research Institute for Computer Architecture and Software TechnologyGermany
  2. 2.Forschungsgruppe Softwaretechnik (FR5-6)Technische Universität BerlinBerlinGermany
  3. 3.Dept. of Computing Science and EngineeringUniversité catholique de LouvainLouvain-la-NeuveBelgium

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