Ordinal arithmetic: Algorithms and mechanization

Journal of Automated Reasoning

Volumn 34, Issue 4, 2006, Pages 387-423

  Manolios, Panagiotis ^a   Vroon, Daron ^a  

^a Georgia Institute of Technology   (United States)

Author keywords

ACL2; Automated reasoning; Ordinal arithmetic; Ordinal notations; Ordinals; Termination; Theorem proving

Indexed keywords

ALGORITHMS; INTERACTIVE COMPUTER SYSTEMS; MATHEMATICAL TRANSFORMATIONS; NUMBER THEORY; SET THEORY; THEOREM PROVING;

ACL2; AUTOMATED REASONING; ORDINAL ARITHMETIC; ORDINAL NOTATIONS; TERMINATION;

DIGITAL ARITHMETIC;

EID: 33745936124     PISSN: 01687433     EISSN: 15730670     Source Type: Journal    
DOI: 10.1007/s10817-005-9023-9     Document Type: Article

References (65)

1. Apt, K. R. and Olderog, E. R.: Verification of Sequential and Concurrent Programs, Springer, Berlin Heidelberg New York, 1991.
2. Baader, F. and Nipkow, T.: Term Rewriting and All That, Cambridge University Press, Cambridge, UK, 1998.
3. Bancerek, G.: The reflection theorem, J. Formaliz. Math. 2 (1990). See URL http://megrez.mizar.org/mirror/JFM/Vol2/zfrefle.html.
4. Belinfante, J. G.: Computer proofs in Gödel's class theory with equational definitions for composite and cross, J. Autom. Reason. 22(3) (1999), 311-339.
5. Belinfante, J. G. F.: On computer-assisted proofs in ordinal number theory, J. Autom. Reason. 22(3) (1999), 341-378.
6. Belinfante, J. G. F.: Reasoning about iteration in Gödel's class theory, in F. Baader (ed.), Automated Deduction-CADE-19, Proceedings of the 19th International Conference on Automated Deduction, volume 2741 of LNAI, Springer-Verlag, 2003, pp. 228-242.
7. Bertot, Y. and Castéran, P.: Interactive Theorem Proving and Program Development, Coq'Art: The calculus of inductive constructions. Texts in Theoretical Computer Science. Springer, May 2004.
8. Boyer, R. S. and Moore, J S.: A Computational Logic Handbook, 2nd edition, Academic Press, New York, 1997.
9. Brock, B., Kaufmann, M. and Moore, J S.: ACL2 theorems about commercial microprocessors, in M. Srivas and A. Camilleri (eds.), Formal Methods in Computer-Aided Design (FMCAD'96), Springer, 1996, pp. 275-293.
10. Cantor, G.: Beiträge zur Begründung der transfiniten Mengenlehre, Math. Ann, xlvi (1895), 481-512.
11. Cantor, G.: Beiträge zur Bgründung der transfiniten Mengenlehre, Math. Ann. xlix (1897), 207-246.
12. Cantor, G.: Contributions to the Founding of the Theory of Transfinite Numbers. Dover, 1952. Translated by Philip E. B. Jourdain.
13. Church, A. and Kleene, S. C.: Formal definitions in the theory of ordinal numbers, Fundam. Math. 28 (1937), 11-21.
14. Dennis, L. A. and Smaill, A.: Ordinal arithmetic: A case study for rippling in a higher order domain, in R. Boulton and P. Jackson (eds.), Theorem Proving in Higher Order Logics: 14th International Conference, TPHOLs 2001, volume 2152 of LNCS, Springer, 2001, pp. 185-200.
15. Dershowitz, N. and Okada, M.: Proof-theoritic techniques for term rewriting theory, in 3rd IEEE Symp. on Logic in Computer Science, 1988, pp. 104-111.
16. Dershowitz, N. and Reingold, E. M.: Ordinal arithmetic with list structures, in Logical Foundations of Computer Science, 1992, pp. 117-126.
17. Devlin, K.: The Joy of Sets: Fundamentals of Contemporary Set Theory, 2nd edition, Springer, New York, 1992.
18. Doner, J.: Definability in the extended arithmetic of ordinal numbers, Diss. Math. 96 (1972).
19. Doner, J. and Tarski, A.: An extended arithmetic of ordinal numbers, Fundam. Math. 65(1969), 95-127.
20. Gallier, J. H.: What's so special about Kruskal's theorem and the ordinal TV? A survey of some results in proof theory, Ann. Pure Appl. Logic, 53 (1991) 199-260.
21. Gentzen, G.: Die Widerspruchsfreiheit der reinen Zahlentheorie, Math. Ann. 112 (1936), 493-565.
22. English translation in M. E. Szabo (ed.), The Collected Works of Gerhard Gentzen, North-Holland, Amsterdam, 1969, pp. 132-213.
23. Gordon, M. J. C. and Melham, T. F. (eds.): Introduction to HOL: A Theorem Proving Environment for Higher Order Logic, Cambridge University Press, Cambridge, UK, 1993.
24. Greve, D., Wilding, M. and Hardin, D.: High-speed, analyzable simulators, in Kaufmann et al. [25], 2000, pp. 113-135.
25. Greve, D. A.: Symbolic simulation of the JEM1 microprocessor, in Formal Methods in Computer-Aided Design - FMCAD, LNCS. Springer, 1998.
26. Kaufmann, M., Manolios, P. and Moore, J S. (eds.), Computer-Aided Reasoning: ACL2 Case Studies, Kluwer, Boston, MA, June 2000.
27. Kaufmann, M., Manolios, P. and Moore, J S.: Computer-Aided Reasoning: An Approach, Kluwer, Boston, MA, July 2000.
28. Kaufmann, M. and Moore, J. S.: ACL2 homepage. See URL http://www.cs.-utexas.edu/users/moore/acl2.
29. Kunen, K.: Set Theory - An Introduction to Independence Proofs, volume 102 of Studies in Logic and the Foundations of Mathematics, North-Holland, Amsterdam, 1980.
30. Manolios, P.: Correctness of pipelined machines, in W. A. Hunt Jr. and S. D. Johnson (eds.), Formal Methods in Computer-Aided Design - FMCAD 2000, volume 1954 of LNCS, Springer, London, UK, 2000, pp. 161-178.
31. Manolios, P.: Mechanical verification of reactive systems. PhD thesis, University of Texasat Austin, August 2001. See URL http://www.cc.gatech.edu/ ~manolios/publications.html.
32. Manolios, P. and Moore, J S.: Partial functions in ACL2, in M. Kaufmann and J S. Moore (eds.), Proceedings of the ACL2 Workshop 2000. The University of Texas at Austin, Technical Report TR-00-29, November 2000.
33. Manolios, P. and Moore, J S.: Partial functions in ACL2, J., Autom. Reason. 31(2) (2003), 107-127.
34. Manolios, P., Namjoshi, K. and Sumners, R.: Linking theorem proving and model-checking with well-founded bisimulation, in N. Halbwachs and D. Peled (eds.), Computer-Aided Verification-CAV '99, volume 1633 of LNCS, Springer, 1999, pp. 369-379.
35. Manolios, P. and Vroon, D.: Algorithms for ordinal arithmetic, in F. Baader, (ed.), 19th Inter-national Conference on Automated Deduction - CADE-19, volume 2741 of LNAI, Springer, July/August 2003, pp. 243-257.
36. Manolios, P. and Vroon, D.: Ordinal arithmetic in ACL2, in Kaufmann, M. and Moore, J S. (eds.), Fourth International Workshop on the ACL2 Theorem Prover and Its Applications (ACL2-2003), July 2003. See URL http://www.cs.utexas.edu/ users/moore/ac12/workshop-2003/.
37. Manolios, P. and Vroon, D.: Integrating reasoning about ordinal arithmetic into ACL2, in Formal Methods in Computer-Aided Design: 5th International Conference - FMCAD-2004, LNCS, Springer, November 2004.
38. Medina-Bulo, I., Palomo-Lozano, F. and Alonso-Jimenez, J. A.: Implementation in ACL2 of well-founded polynomial orderings, in M. Kaufmann and J S. Moore (eds.), Proceedings of the ACL2 Workshop 2002, 2002.
39. Miller, L. W.: Normal functions and constructive ordinal notations, J. Symbolic Logic 41 (June 1976), 439-459.
40. Moore, J S., Lynch, T. and Kaufmann, M.: A mechanically checked proof of the AMD5K86 floating-point division program, IEEE Trans. Comput. 47(9) (September 1998), 913-926.
41. Morris, F. and Jones, C.: An early program proof by Alan Turing, IEEE Ann, Hist, Comput. 6(2) (April-June 1984), 139-143.
42. Owre, S. Rushby, J. and Shankar, N.: PVS: A prototype verification system, in D. Kapur (ed), 11th International Conference on Automated Deduction (CADE), Lecture Notes in Artificial Intelligence, vol. 607, Springer, June 1992, pp. 748-752.
43. Paulson, L. C.: Set theory for verification: I. From foundations to functions, J. Autom. Reason. 11(3) (1993), 353-389.
44. Paulson, L. C.: Isabelle: A Generic Theorem Prover, Springer LNCS 828, New York, NY, 1994.
45. Paulson, L. C.: Set theory for verification: II. Induction and recursion, J. Autom. Reason. 15(2) (1995), 167-215.
46. Paulson, L. C.: The reflection theorem: A study in meta-theoretic reasoning, in A. Voronkov (ed.), 18th International Conf. on Automated Deduction: CADE-18, number 2392 in LNAI, Springer, 2002, pp. 377-391.
47. Paulson, L. C.: The relative consistency of the axiom of choice mechanized using Isabelle, LMS J. Comput. Math. 6 (2003), 198-248.
48. Paulson, L. C. and Grabczewski, K.: Mechanizing set theory: Cardinal arithmetic and the axiom of choice, J. Autom. Reason. 17 (1996), 291-323.
49. Rogers Jr, H.: Theory of Recursive Functions and Effective Computability, MIT Press, Cambridge, MA, 1987.
50. Rudnicki, P.: An overview of the MIZAR project, in 1992 Workshop on Types for Proofs and Programs, 1992.
51. Ruiz-Reina, J.-L., Alonso, J.-A., Hidalgo, M.-J. and Martin, F.-J.: Multiset relations: A tool for proving termination, in Kaufmann, M. and Moore, J S. (eds.), Proceedings of the ACL2 Workshop 2000, The University of Texas at Austin, Technical Report TR-00-29, November 2000.
52. Russinoff, D. M.: A mechanically checked proof of correctness of the AMD5K86 floating-point square root microcode, Formal Methods in System Design Special Issue on Arithmetic Circuits, 1997.
53. Russinoff, D. M.: A mechanically checked proof of IEEE compliance of a register-transfer-level specification of the AMD-K7 floating-point multiplication, division, and square root instructions, LMS J. Comput. Math. 1 (December 1998), 148-200.
54. Russinoff, D. M.: A mechanically checked proof of correctness of the AMDK5 floating-point square root microcode, Form. Methods Syst. Des. 14 (1999), 75-125.
55. Russinoff, D. M. and Flatau, A.: RTL verification: A floating-point multiplier, Kaufmann, M., Manolios, P. and Moore, J S. (eds.), Computer-Aided Reasoning: ACL2 Case Studies, Kluwer, Boston, MA, 2000, pp. 201-231.
56. Schütte, K.: Proof Theory. Springer, 1977.
57. Translation from German by J. N. Crossley. The book is a completely rewritten version of Beweistheorie, Springer, 1960.
58. Setzer, A.: Ordinal systems, in B. Cooper and J. Truss (eds.), Sets and Proofs, Cambridge University Press, Cambridge, 1999, pp. 301-331.
59. Setzer, A.: Ordinal systems part 2: One inaccessible, in Logic Colloquium '98, volume 13 of ASL Lecture Notes in Logic, 2000, pp. 426-448.
60. Sumners, R.: An incremental stuttering refinement proof of a concurrent program in ACL2, in Kaufmann, M. and Moore, J S. (eds.), Proceedings of the ACL2 Workshop 2000. The University of Texas at Austin, Technical Report TR-00-29, November 2000.
61. Sustik, M.: Proof of Dixon's lemma using the ACL2 theorem prover via an explicit ordinal mapping, in Kaufmann, M. and Moore, J S. (eds.), Fourth International Workshop on the ACL2 Theorem Prover and Its Applications (ACL2-2003), July 2003. See URL http://www.cs.utexas.edu/users/moore/acl2/ workshop-2003/.
62. Troelstra, A. S. and Schwichtenberg, H.: Basic Proof Theory, 2nd edition, Cambridge University Press, Cambridge, 2000.
63. Turing, A. M.: Systems of logic based on ordinals, Proc. Lond. Math. Soc. 45(2) (1939), 161-228. See URL http://www.turingarchive.org/.
64. Turing, A. M.: Checking a large routine, in Report of a Conference on High Speed Automatic Calculating Machines, University Mathematical Laboratory, Cambridge, June 1949, pp. 67-69.
65. Veblen, O.: Continuous increasing functions of finite and transfinite ordinals, Trans. Am. Math. Soc. 9 (1908), 280-292.