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.
Preview
Unable to display preview. Download preview PDF.
References
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.
Abramsky, S. & Vickers, S. (1993), ‘Quantales, observational logic and process semantics', Mathematical Structures in Computer Science 3, 161–227.
Bird, R. & de Moor, O. (1997), Algebra of Programming, Prentice Hall.
Cockett, J. R. B. & Seely, R. A. G. (1997), ‘Weakly distributive categories', Journal of Pure and Applied Algebra 114(2), 133–173.
Davey, B. A. & Priestley, H. A. (1990), Introduction to Lattices and Order, Cambridge University Press.
Došen, K. & Schroeder-Heister, P., eds (1993), Substructural Logics, Oxford Science Publications.
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.
Dunn, J. M. (1993), Partial gaggles applied to logics with restricted structural rules, in Došen & Schroeder-Heister (1993), pp. 63–108.
Hesselink, W. J. (1990), ‘Axioms and models of linear logic', Formal Aspects of Computing 2, 139–166.
Hoare, C. A. R. & He, J. (1987), “The weakest prespecification', Information Processing Letters 24, 127–132.
Jones, C. B. (1990), Systematic Software Development Using VDM, second edn, Prentice Hall.
Kleene, S. C. (1971), Introduction to Metamathematics, sixth reprint edn, North Holland.
Lamport, L. (1994), ‘How to write a proof', American Mathematical Monthly 102(7), 600–608.
Martin, A. P., Gardiner, P. & Woodcock, J. C. P. (1997), ‘A tactic calculus — abridged version', Formal Aspects of Computing 8(4), 479–489.
Ono, H. (1993), Semantics of substructural logics, in Došen & Schroeder-Heister (1993), pp. 259–291.
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.
Rosenthal, K. I. (1990), Quantales and their Application, Longman Scientific & Technical.
Simons, M. (1997a), The Presentation of Formal Proofs, GMD-Bericht Nr. 278, Oldenbourg Verlag.
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.
Simons, M. & Weber, M. (1996), ‘An approach to literate and structured formal developments', Formal Aspects of Computing 8(1), 86–107.
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.
Troelstra, A. S. (1992), Lectures on Linear Logic, number 29 in ‘CSLI Lecture Notes', CSLI.
Vickers, S. (1989), Topology via Logic, Cambridge University Press.
Weber, M. (1993), ‘Definition and basic properties of the Deva meta-calculus', Formal Aspects of Computing 5, 391–431.
Weber, M., Simons, M. & Lafontaine, C. (1993), The Generic Development Language Deva: Presentation and Case Studies, LNCS 738, Springer Verlag.
Yetter, D. (1990), ‘Quantales and (non-commutative) linear logic', The Journal of Symbolic Logic 55, 41–64.
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 1997 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Simons, M., Sintzoff, M. (1997). Algebraic composition and refinement of proofs. In: Johnson, M. (eds) Algebraic Methodology and Software Technology. AMAST 1997. Lecture Notes in Computer Science, vol 1349. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0000492
Download citation
DOI: https://doi.org/10.1007/BFb0000492
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-63888-9
Online ISBN: 978-3-540-69661-2
eBook Packages: Springer Book Archive