article

Social processes, program verification and all that

Authors:

Andrea Asperti,

Herman Geuvers,

Raja NatarajanAuthors Info & Claims

Mathematical Structures in Computer Science, Volume 19, Issue 5

Pages 877 - 896

https://doi.org/10.1017/S0960129509990041

Published: 01 October 2009 Publication History

Abstract

In a controversial paper (De Millo et al. 1979) at the end of the 1970's, R. A. De Millo, R. J. Lipton and A. J. Perlis argued against formal verifications of programs, mostly motivating their position by an analogy with proofs in mathematics, and, in particular, with the impracticality of a strictly formalist approach to this discipline. The recent, impressive achievements in the field of interactive theorem proving provide an interesting ground for a critical revisiting of their theses. We believe that the social nature of proof and program development is uncontroversial and ineluctable, but formal verification is not antithetical to it. Formal verification should strive not only to cope with, but to ease and enhance the collaborative, organic nature of this process, eventually helping us to master the growing complexity of scientific knowledge.

References

[1]

Alkassar, E., Bogan, S. and Paul, W. J. (2009) Proving the correctness of client/server software. Sadhana 34 (1) 145-191.

[2]

Altenkirch, T., McBride, C. and McKinna, J. (2005) Why dependent types matter. (Available at http://sneezy.cs.nott.ac.uk/epigram/.)

[3]

Asperti, A., Padovani, L., Sacerdoti Coen, C. and Schena, I. (2000) Content-centric logical environments. Short Presentation at the Fifteenth IEEE Symposium on Logic in Computer Science.

[4]

Asperti, A. and Ricciotti, W. (2009) About the formalization of some results by Chebyshev in number theory. In: Proc. of TYPES'08. Springer-Verlag Lecture Notes in Computer Science 5497 19-31.

Digital Library

[5]

Avigad, J., Donnelly, K., Gray, D. and Raff, P. (2007) A formally verified proof of the prime number theorem. ACM Trans. Comput. Log. 9 (1).

Digital Library

[6]

Barbanera, F. and Berardi, S. (1996) A symmetric lambda calculus for classical program extraction. Information and Computation 125 (2) 103-117.

Digital Library

[7]

Boolos, G. (1984) Don't eliminate cut. Journal of Philosophical Logic 13 373-378.

[8]

Bos, J. V. D. (1979) Letter to the editor. Communications of the ACM 22 623.

[9]

Bourbaki, N. (1968) Theory of Sets, Elements of mathematics, Addison Wesley.

[10]

Constable, R. L., Allen, S. F., Bromley, H. M., Cleaveland, W. R., Cremer, J. F., Harper, R. W., Howe, D. J., Knoblock, T. B., Mendler, N. P., Panangaden, P., Sasaki, J. T. and Smith, S. F. (1986) Implementing Mathematics with the Nuprl Development System, Prentice-Hall.

Digital Library

[11]

Coquand, T. (2008) Draft of the Formath Project.

[12]

Corbineau, P., Geuvers, H., Kaliszyk, C., McKinna, J. and Wiedijk, F. (2008) A real semantic web for mathematics deserves a real semantics. In: Lange, C., Schaffert, S., Skaf-Molli, H. and Völkel, M. (eds.) SemWiki. CEUR Workshop Proceedings 360.

[13]

Corbineau, P. and Kaliszyk, C. (2007) Cooperative repositories for formal proofs - a wiki-based solution. In: Kauers, M., Kerber, M., Miner, R. and Windsteiger, W. (eds.) Towards Mechanized Mathematical Assistants. Springer-Verlag Lecture Notes in Computer Science 4573 221- 234.

Digital Library

[14]

De Millo, R. A., Lipton, R. J. and Perlis, A. J. (1979) Social processes and proofs of theorems and programs. Commun. ACM 22 (5) 271-280.

Digital Library

[15]

Dewar, M. (2000) Special issue on OpenMath. ACM SIGSAM Bulletin 34.

Digital Library

[16]

Dijkstra, E. W. (1986) On a cultural gap. Mathematical Intelligencer 8 (1) 48-52.

[17]

Dold, A. and Vialard, V. (2001) A mechanically verified compiling specification for a lisp compiler. In: Proc. of FSTTCS 2001. Springer-Verlag Lecture Notes in Computer Science 2245 144-155.

Digital Library

[18]

The Economist (2005) Proof and beauty. The Economist, 31st March 2005.

[19]

Fateman, R. (2001) A critique of OpenMath and thoughts on encoding mathematics. (Available at http://www.eecs.berkeley.edu/~fateman/papers/openmathcrit.pdf.)

[20]

Feit, W. and Thompson, J. G. (1963) Solvability of groups of odd order. Pacific Journal of Mathematics 13 775-1029.

[21]

Fowler, M. (2000) The New Methodology. (Available at http://www.martinfowler.com/ articles/newMethodology.html.)

[22]

Geuvers, H. (2009) Proof Assistants: history, ideas and future. Sadhana 34 (1) 3-25.

[23]

Girard, J.-Y., Lafont, Y. and Taylor, P. (1989) Proofs and Types, Cambridge Tracts in Theoretical Computer Science, Cambridge University Press.

Digital Library

[24]

Gonthier, G. (2007) The four colour theorem: Engineering of a formal proof. In: Proc. of ASCM 2007. Springer-Verlag Lecture Notes in Computer Science 5081.

[25]

Gonthier, G. (2008) Formal proof - the four color theorem. Notices of the American Mathematical Society 55 1382-1394.

[26]

Gonthier, G., Mahboubi, A., Rideau, L., Tassi, E. and Thery, L. (2007) A modular formalisation of finite group theory. In: The 20th International Conference on Theorem Proving in Higher Order Logics. Springer-Verlag Lecture Notes in Computer Science 4732 86-101.

Digital Library

[27]

Hales, T. C. (2005) A proof of the Kepler conjecture. Ann. Math. 162 1065-1185.

[28]

Hales, T. C. (2007) The Jordan curve theorem, formally and informally. The American Mathematical Monthly 114 882-894.

[29]

Hales, T. C. (2008) Formal proof. Notices of the American Mathematical Society 55 1370-1381.

[30]

Hall, C., Hammond, K., Jones, S. P. and Wadler, P. (1996) Type classes in Haskell. ACM Transactions on Programming Languages and Systems 18 241-256.

Digital Library

[31]

Halmos, P. (1985) I want to be a Mathematician: An Automathography, Springer-Verlag.

[32]

Hamid, N. A., Shao, Z., Trifonov, V., Monnier, S. and Ni, Z. (2003) A syntactic approach to foundational proof-carrying code. J. Autom. Reasoning 31 (3-4) 191-229.

Digital Library

[33]

Hardy, G. H. (1928) Mathematical proof. Mind 38 1-25.

[34]

Harrison, J. (2007) Floating-point verification. J. UCS 13 (5) 629-638.

[35]

Harrison, J. (2008) Formal proof - theory and practice. Notices of the American Mathematical Society 55 1395-1406.

[36]

Hoare, C. A. R. (1969) An axiomatic basis for computer programming. Commun. ACM 12 (10) 576-580.

Digital Library

[37]

Howard, W. A. (1980) The formulae-as-types notion of construction. In: Seldin, J. P. and Hindley, J. R. (eds.) To H.B. Curry: Essays on Combinatory Logic, Lambda Calculus and Formalism, Academic Press 479-490.

[38]

Hutcheon, P. D. (1995) Popper and Kuhn on the evolution of science. Brock Review 4 (1/2) 28-37.

[39]

Klein, G. (2005) Verified Java bytecode verification. Information Technology 47 (2) 107-110.

[40]

Klein, G. (2009) Operating system verification - an overview. Sadhana 34 (1) 27-69.

[41]

Lakatos, I. (1976) Proofs and Refutations: The Logic of Mathematical Discovery, Cambridge University Press.

[42]

Lamport, L. (1979) Letter to the editor. Communications of the ACM 22 624.

[43]

Langley, S. P. (1891) Experiments in Aerodynamics, Kessinger Publishing.

[44]

Lecat, M. (1939) Erreurs de mathématiciens: des origines à nos jours, Ancienne Librairie Castaigne, Brussels.

[45]

Lee, J. K. (2002) Philosophical perspectives on proof in mathematics education. Philosophy of Mathematics Education Journal 16.

[46]

Leinenbach, D., Paul, W. J. and Petrova, E. (2005) Towards the formal verification of a C0 compiler: Code generation and implementation correctness. In: Third IEEE International Conference on Software Engineering and Formal Methods (SEFM 2005), IEEE Computer Society 2-12.

Digital Library

[47]

Leroy, X. (2006) Formal certification of a compiler back-end or: programming a compiler with a proof assistant. In: Proc. of the 33rd ACM SIGPLAN-SIGACT Symposium on Principles of Programming Languages, POPL 2006, Charleston, South Carolina, USA 42-54.

Digital Library

[48]

MacKenzie, D. (2005) What in the name of Euclid is going on here? Science 207 (5714) 1402- 1403.

[49]

Maurer, W. D. (1979) Letter to the editor. Communications of the ACM 22 625-629.

[50]

Necula, G. C. and Lee, P. (1996) Proof-carrying code. Technical Report CMU-CS-96-165, Carnegie Mellon University.

[51]

Parigot, M. (1992) Lambda-mu calculus: An algorithmic interpretation of classical natural deduction. In: Proc. of the Logic Programming and Automated Reasoning International Conference LPAR'92. Springer-Verlag Lecture Notes in Computer Science 624 190-201.

Digital Library

[52]

Popper, K. (1948) Logic without assumptions. Aristotelian society proceedings 47 251-292.

[53]

Popper, K. (1963) Conjectures and Refutations. The Growth of Scientific Knowledge, Routledge.

[54]

Prawitz, D. (1965) Natural Deduction: a proof theoretical study, Almqvist and Wiksell.

[55]

Schieber, P. (1987) The wit and wisdom of Grace Hopper. OCLC Newsletter 167.

[56]

Sozeau, M. and Oury, N. (2008) First-class type classes. In: TPHOLs 278-293.

Digital Library

[57]

Strecker, M. (1998) Construction and Deduction in Type Theories, Ph.D. thesis, Universität Ulm.

[58]

Tristan, J.-B. and Leroy, X. (2008) Formal verification of translation validators: a case study on instruction scheduling optimizations. In: Proc. of the 35th ACM SIGPLAN-SIGACT Symposium on Principles of Programming Languages, POPL 2008 17-27.

Digital Library

[59]

Wadler, P. and Blott, S. (1989) How to make ad-hoc polymorphism less ad hoc. In: POPL '89: Proc. of the 16th ACM SIGPLAN-SIGACT symposium on Principles of programming languages, ACM 60-76.

Digital Library

[60]

Wenzel, M. (1997) Type classes and overloading in higher-order logic. In: TPHOLs 307-322.

Digital Library

[61]

Wenzel, M. (1999) Isar - a generic interpretative approach to readable formal proof documents. In: Theorem Proving in Higher Order Logics. Springer-Verlag Lecture Notes in Computer Science 1690 167-184.

Digital Library

[62]

Wiedijk, F. (2001) Estimating the cost of a standard library for a mathematical proof checker. (Available at http://www.cs.ru.nl/~freek/notes/mathstdlib2.pdf.)

[63]

Wiedijk, F. (2007) The Qed manifesto revisited. In: Matuszwski, R. and Zalewska, A. (eds.) From Insight to Proof, Festschrift in Honour of Andrzej Trybulec. Studies in Logic, Grammar and Rhetoric, University of Bia¿pystok 10 (23) 121-133.

Cited By

Emerich JVisser EMurphy-Hill ELopes C(2016)How are programs found? speculating about language ergonomics with Curry-HowardProceedings of the 2016 ACM International Symposium on New Ideas, New Paradigms, and Reflections on Programming and Software10.1145/2986012.2986030(212-223)Online publication date: 20-Oct-2016
https://dl.acm.org/doi/10.1145/2986012.2986030
Lange C(2013)Ontologies and languages for representing mathematical knowledge on the Semantic WebSemantic Web10.5555/2590215.25902174:2(119-158)Online publication date: 1-Apr-2013
https://dl.acm.org/doi/10.5555/2590215.2590217
Asperti A(2013)A Formal Proof of Borodin-Trakhtenbrot’s Gap TheoremCertified Programs and Proofs10.1007/978-3-319-03545-1_11(163-177)Online publication date: 11-Dec-2013
https://dl.acm.org/doi/10.1007/978-3-319-03545-1_11
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Index Terms

Social processes, program verification and all that
1. Mathematics of computing
  1. Continuous mathematics
    1. Calculus
      1. Lambda calculus
2. Theory of computation
  1. Logic
    1. Proof theory

Index terms have been assigned to the content through auto-classification.

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cover image Mathematical Structures in Computer Science

Mathematical Structures in Computer Science Volume 19, Issue 5

October 2009

193 pages

ISSN:0960-1295

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Cambridge University Press

United States

Publication History

Published: 01 October 2009

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Cited By

Emerich JVisser EMurphy-Hill ELopes C(2016)How are programs found? speculating about language ergonomics with Curry-HowardProceedings of the 2016 ACM International Symposium on New Ideas, New Paradigms, and Reflections on Programming and Software10.1145/2986012.2986030(212-223)Online publication date: 20-Oct-2016
https://dl.acm.org/doi/10.1145/2986012.2986030
Lange C(2013)Ontologies and languages for representing mathematical knowledge on the Semantic WebSemantic Web10.5555/2590215.25902174:2(119-158)Online publication date: 1-Apr-2013
https://dl.acm.org/doi/10.5555/2590215.2590217
Asperti A(2013)A Formal Proof of Borodin-Trakhtenbrot’s Gap TheoremCertified Programs and Proofs10.1007/978-3-319-03545-1_11(163-177)Online publication date: 11-Dec-2013
https://dl.acm.org/doi/10.1007/978-3-319-03545-1_11
Asperti A(2012)Proof, message and certificateProceedings of the 11th international conference on Intelligent Computer Mathematics10.1007/978-3-642-31374-5_2(17-31)Online publication date: 8-Jul-2012
https://dl.acm.org/doi/10.1007/978-3-642-31374-5_2
Turner R(2011)SpecificationMinds and Machines10.1007/s11023-011-9239-x21:2(135-152)Online publication date: 1-May-2011
https://dl.acm.org/doi/10.1007/s11023-011-9239-x
Pincas U(2011)Program Verification and Functioning of Operative Computing RevisitedMinds and Machines10.1007/s11023-011-9237-z21:2(337-359)Online publication date: 1-May-2011
https://dl.acm.org/doi/10.1007/s11023-011-9237-z
Asperti ACoen C(2010)Some considerations on the usability of interactive proversProceedings of the 10th ASIC and 9th MKM international conference, and 17th Calculemus conference on Intelligent computer mathematics10.5555/1894483.1894498(147-156)Online publication date: 5-Jul-2010
https://dl.acm.org/doi/10.5555/1894483.1894498
Formisano AOmodeo E(2010)Theory-specific automated reasoningA 25-year perspective on logic programming10.5555/1880413.1880417(37-63)Online publication date: 1-Jan-2010
https://dl.acm.org/doi/10.5555/1880413.1880417

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