Computer supported mathematics with Ωmega

Journal of Applied Logic

Volumn 4, Issue 4, 2006, Pages 533-559

  Siekmann, Jörg ^a   Benzmüller, Christoph ^a   Autexier, Serge ^a  

^a SAARLAND UNIVERSITY   (Germany)

Author keywords

Artificial intelligence; Computer supported mathematics; Interactive and automated theorem proving; Omega; Proof assistant systems; Proof planning

Indexed keywords

ABSTRACTING; ALGEBRA; ARTIFICIAL INTELLIGENCE; COMPUTATIONAL METHODS; DATA STRUCTURES; FINITE AUTOMATA; PROBLEM SOLVING; SEARCH ENGINES; THEOREM PROVING;

COMPUTER SUPPORTED MATHEMATICS; INTERACTIVE AND AUTOMATED THEOREM PROVING; OMEGA; PROOF ASSISTANT SYSTEMS; PROOF PLANNING TECHNIQUES;

FORMAL LOGIC;

EID: 33750429904     PISSN: 15708683     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.jal.2005.10.008     Document Type: Article

References (101)

1. S. Allen, R. Constable, R. Eaton, C. Kreitz, L. Lorigo, The Nuprl open logical environment, in: [59]
2. Andrews P.B., Bishop M., Issar S., Nesmith D., Pfenning F., and Xi H. TPS: A theorem proving system for classical type theory. J. Automat. Reason. 16 3 (1996) 321-353
3. S. Autexier, Hierarchical contextual reasoning, PhD thesis, Computer Science Department, Saarland University, Saarbrücken, Germany, 2003
4. Autexier S. The CoRe calculus. In: Nieuwenhuis R. (Ed). Proceedings of the 20th Conference on Automated Deduction (CADE-20), Tallinn, Estonia. Lecture Notes in Artificial Intelligence vol. 3632 (2005), Springer, Berlin 84-98
5. Autexier S., Benzmüller C., Fiedler A., Horacek H., and Bao Vo Q. Assertion-level proof representation with under-specification. Electronic Notes in Theoret. Comput. Sci. 93 (2003) 5-23
6. Autexier S., and Hutter D. Maintenance of formal software development by stratified verification. In: Baaz M., and Voronkov A. (Eds). Proceedings of LPAR'02, Tbilisi, Georgia. Lecture Notes in Computer Science (2002), Springer, Berlin
7. Autexier S., Hutter D., Mossakowski T., and Schairer A. The development graph manager MAYA. In: Kirchner H., and Ringeissen C. (Eds). Proceedings 9th International Conference on Algebraic Methodology and Software Technology (AMAST'02). Lecture Notes in Computer Science vol. 2422 (2002), Springer, Berlin
8. Autexier S., and Mossakowski T. Integrating HOL-CASL into the development graph manager MAYA. In: Armando A. (Ed). Proceedings of FROCOS'02. Lecture Notes in Artificial Intelligence vol. 2309 (2002), Springer, Berlin 2-17
9. Bartle R., and Sherbert D. Introduction to Real Analysis. second ed. (1982), Wiley, New York
10. P. Baumgartner, U. Furbach, PROTEIN, a PROver with a theory INterface, in: [27], pp. 769-773
11. C. Benzmüller, Equality and extensionality in higher-order theorem proving, PhD thesis, Department of Computer Science, Saarland University, Saarbrücken, Germany, 1999
12. Benzmüller C., Bishop M., and Sorge V. Integrating TPS and Ωmega. J. Universal Comput. Sci. 5 (1999) 188-207
13. C. Benzmüller, A. Fiedler, M. Gabsdil, H. Horacek, I. Kruijff-Korbayova, M. Pinkal, J. Siekmann, D. Tsovaltzi, B. Quoc Vo, M. Wolska, Tutorial dialogs on mathematical proofs, In: Proceedings of IJCAI-03 Workshop on Knowledge Representation and Automated Reasoning for E-Learning Systems, Acapulco, Mexico, 2003, pp. 12-22
14. C. Benzmüller, A. Fiedler, A. Meier, M. Pollet, Irrationality of sqrt(2)-a case study in Ωmega, Seki-Report SR-02-03, Department of Computer Science, Saarland University, Saarbrücken, Germany, 2002
15. Benzmüller C., Jamnik M., Kerber M., and Sorge V. Experiments with an agent-oriented reasoning system. In: Baader F., Brewka G., and Eiter T. (Eds). KI 2001: Advances in Artificial Intelligence, Proceedings of Joint German/Austrian Conference on AI, Vienna, Austria. Lecture Notes in Artificial Intelligence vol. 2174 (2001), Springer, Berlin 409-424
16. C. Benzmüller, M. Kohlhase, LEO-a higher-order theorem prover, in: [54]
17. Benzmüller C., Meier A., and Sorge V. Bridging theorem proving and mathematical knowledge retrieval. In: Hutter D., and Stephan W. (Eds). Festschrift in Honour of Jörg Siekmann's 60s Birthday. Lecture Notes in Artificial Intelligence vol. 2605 (2004), Springer, Berlin
18. C. Benzmüller, V. Sorge, A blackboard architecture for guiding interactive proofs, in: [46]
19. C. Benzmüller, V. Sorge, Ωants-An open approach at combining Interactive and Automated Theorem Proving, in: [53]
20. Benzmüller C., Sorge V., Jamnik M., and Kerber M. Can a higher-order and a first-order theorem prover cooperate?. In: Baader F., and Voronkov A. (Eds). Proceedings of the 11th International Conference on Logic for Programming Artificial Intelligence and Reasoning (LPAR). Lecture Notes in Artificial Intelligence vol. 3452 (2005), Springer, Berlin 415-431
21. Benzmüller C.E., and Vo Q.B. Mathematical domain reasoning tasks in natural language tutorial dialog on proofs. In: Veloso M., and Kambhampati S. (Eds). Proceedings of the Twentieth National Conference on Artificial Intelligence (AAAI-05), Pittsburgh, PA (2005), AAAI Press/MIT Press 516-522
22. Benzmüller C., and Sorge V. Critical agents supporting interactive theorem proving. In: Borahona P., and Alferes J.J. (Eds). Proceedings of the 9th Portuguese Conference on Artificial Intelligence (EPIA'99), Evora, Portugal. Lecture Notes in Artificial Intelligence vol. 1695 (1999), Springer, Berlin 208-221
23. Bledsoe W. Challenge problems in elementary calculus. J. Automat. Reason. 6 (1990) 341-359
24. Bourbaki N. Theory of sets. Elements of Mathematics vol. 1 (1968), Addison-Wesley, Reading, MA
25. A. Bundy, The use of explicit plans to guide inductive proofs, in: [57], pp. 111-120
26. Bundy A. A science of reasoning. In: Plotkin G., and Lasser J.-L. (Eds). Computational Logic: Essays in Honor of Alan Robinson (1991), MIT Press, Cambridge, MA 178-199
27. In: Bundy A. (Ed). Proceedings of the 12th Conference on Automated Deduction. Lecture Notes in Artificial Intelligence vol. 814 (1994), Springer, Berlin
28. Bundy A. A critique of proof planning. Computational Logic: Logic Programming and Beyond. Lecture Notes in Computer Science vol. 2408 (2002), Springer, Berlin 160-177
29. Bundy A., van Harmelen F., Horn C., and Smaill A. The Oyster-Clam system. In: Stickel M. (Ed). Proceedings of the 10th Conference on Automated Deduction, Kaiserslautern, Germany. Lecture Notes in Computer Science vol. 449 (1990), Springer, Berlin 647-648
30. Char B., Geddes K., Gonnet G., Leong B., Monagan M., and Watt S. First Leaves: A Tutorial Introduction to Maple V (1992), Springer, Berlin
31. L. Cheikhrouhou, J. Siekmann, Planning diagonalization proofs, in: [46], pp. 167-180
32. L. Cheikhrouhou, V. Sorge, PDS-A three-dimensional data structure for proof plans, In: Proceedings of the International Conference on Artificial and Computational Intelligence for Decision, Control and Automation in Engineering and Industrial Applications (ACIDCA'2000), Monastir, Tunisia, 2000
33. Church A. A formulation of the simple theory of types. J. Symbolic Logic 5 (1940) 56-68
34. Coq Development Team, The Coq proof assistant reference manual, INRIA, 1999-2003, see http://coq.inria.fr/doc/main.html
35. Davis M. The prehistory and early history of automated deduction. In: Siekmann J., and Wrightson G. (Eds). Automation of Reasoning, vol. 2, Classical Papers on Computational Logic 1967-1970 of Symbolic Computation (1983), Springer, Berlin
36. Davis M. The early history of automated deduction. In: Robinson A., and Voronkov A. (Eds). Handbook of Automated Reasoning vol. I (2001), Elsevier Science, Amsterdam 3-15 Chapter 1
37. In: Davis M. (Ed). The Undecidable: Basic Papers on Undecidable Propositions, Unsolvable Problems and Computable Functions (2004), Dover Publications, New York
38. H. de Nivelle, Bliksem 1.10 user manual, Technical Report, Max-Planck-Institut für Informatik, 1999
39. Fiedler A. Dialog-driven adaptation of explanations of proofs. In: Nebel B. (Ed). Proceedings of the 17th International Joint Conference on Artificial Intelligence (IJCAI), Seattle, WA (2001), Morgan Kaufmann, San Mateo, CA 1295-1300
40. A. Fiedler, P.rex: An interactive proof explainer, in: [48]
41. A. Fiedler, User-adaptive proof explanation, PhD thesis, Department of Computer Science, Saarland University, Saarbrücken, Germany, 2001
42. A. Franke, M. Kohlhase, System description: MBase, an open mathematical knowledge base, in: [59]
43. A. Franke, M. Kohlhase, System description: Mbase, an open mathematical knowledge base, in: [59]
44. In: Ganzinger H. (Ed). Proceedings of the 16th Conference on Automated Deduction. Lecture Notes in Artificial Intelligence vol. 1632 (1999), Springer, Berlin
45. H. Gebhard, Beweisplanung für die Beweise der Vollständigkeit verschiedener Resolutionskalküle in Ωmega, Master's thesis, Department of Computer Science, Saarland University, Saarbrücken, Germany, 1999
46. In: Giunchiglia F. (Ed). Proceedings of 8th International Conference on Artificial Intelligence: Methodology, Systems, Applications (AIMSA'98). Lecture Notes in Artificial Intelligence vol. 1480 (1998), Springer, Berlin
47. Gordon M., and Melham T. Introduction to HOL-A Theorem Proving Environment for Higher Order Logic (1993), Cambridge University Press, Cambridge
48. In: Goré R., Leitsch A., and Nipkow T. (Eds). Automated Reasoning-1st International Joint Conference, IJCAR 2001. Lecture Notes in Artificial Intelligence vol. 2083 (2001), Springer, Berlin
49. Hadamard J. The Psychology of Invention in the Mathematical Field (1949), Dover Publications, New York 1944
50. Th. Hillenbrand, A. Jaeger, B. Löchner, System description: Waldmeister-improvements in performance and ease of use, in: [44], pp. 232-236
51. X. Huang, Reconstructing proofs at the assertion level, in: [27], pp. 738-752
52. Hutter D. Management of change in structured verification. Proceedings of Automated Software Engineering, ASE-2000 (2000), IEEE
53. In: Kerber M., and Kohlhase M. (Eds). 8th Symposium on the Integration of Symbolic Computation and Mechanized Reasoning (Calculemus-2000) (2000), AK Peters
54. In: Kirchner C., and Kirchner H. (Eds). Proceedings of the 15th Conference on Automated Deduction. Lecture Notes in Artificial Intelligence vol. 1421 (1998), Springer, Berlin
55. In: Kirchner H., and Ringeissen C. (Eds). Frontiers of Combining systems: Third International Workshop, FroCoS 2000. Lecture Notes in Artificial Intelligence vol. 1794 (2000), Springer, Berlin
56. Kohlhase M., and Franke A. MBase: Representing knowledge and context for the integration of mathematical software systems. J. Symbolic Comput. 32 4 (2001) 365-402 (special issue on the Integration of Computer Algebra and Deduction Systems)
57. In: Lusk E., and Overbeek R. (Eds). Proceedings of the 9th Conference on Automated Deduction, Argonne, Illinois. Lecture Notes in Computer Science vol. 310 (1988), Springer, Berlin
58. R. Manthey, F. Bry, SATCHMO: A theorem prover implemented in Prolog, in: [57], pp. 415-434
59. In: McAllester D. (Ed). Proceedings of the 17th Conference on Automated Deduction. Lecture Notes in Artificial Intelligence vol. 1831 (2000), Springer, Berlin
60. McCune W. Solution of the Robbins problem. J. Automat. Reason. 19 3 (1997) 263-276
61. W.W. McCune, Otter 3.0 reference manual and guide, Technical Report ANL-94-6, Argonne National Laboratory, Argonne, IL 60439, USA, 1994
62. A. Meier, TRAMP: Transformation of machine-found proofs into natural deduction proofs at the assertion level, in: [59]
63. A. Meier, Proof planning with multiple strategies, PhD thesis, Department of Computer Science, Saarland University, Saarbrücken, Germany, 2004
64. Meier A., and Melis E. Multi: A multi-strategy proof planner (system description). In: Nieuwenhuis R. (Ed). Proceedings of the 20th Conference on Automated Deduction (CADE-20), Tallinn, Estonia. Lecture Notes in Artificial Intelligence vol. 3632 (2005), Springer, Berlin 250-254
65. A. Meier, E. Melis, M. Pollet, Towards extending domain representations, Seki Report SR-02-01, Department of Computer Science, Saarland University, Saarbrücken, Germany, 2002
66. Meier A., Pollet M., and Sorge V. Classifying isomorphic residue classes. In: Moreno-Diaz R., Buchberger B., and Freire L. (Eds). A Selection of Papers from the 8th International Workshop on Computer Aided Systems Theory (EuroCAST 2001). Lecture Notes in Computer Science vol. 2178 (2001), Springer, Berlin 494-508
67. Meier A., Pollet M., and Sorge V. Comparing approaches to the exploration of the domain of residue classes. In: Linton S., and Sebastiani R. (Eds). J. Symbolic Comput. 34 4 (2002) 287-306 (special issue on the Integration of Automated Reasoning and Computer Algebra Systems)
68. A. Meier, V. Sorge, Exploring properties of residue classes, in: [53]
69. A. Meier, E. Melis, J. Siekmann, Proof planning with multiple strategies, Artificial Intelligence, submitted for publication
70. E. Melis, Island planning and refinement, Seki-Report SR-96-10, Department of Computer Science, Saarland University, Saarbrücken, Germany, 1996
71. Melis E. AI-techniques in proof planning. In: Prade H. (Ed). Proceedings of the 13th European Conference on Artificial Intelligence, Brighton, UK (1998), John Wiley & Sons, Chichester, UK 494-498
72. Melis E. AI-techniques in proof planning. European Conference on Artificial Intelligence, Brighton, UK (1998), Kluwer, Dordrecht 494-498
73. Melis E., and Meier A. Proof planning with multiple strategies. In: Loyd J., Dahl V., Furbach U., Kerber M., Lau K., Palamidessi C., Pereira L.M., Sagivand Y., and Stuckey P. (Eds). First International Conference on Computational Logic (CL-2000), London, UK. Lecture Notes in Artificial Intelligence vol. 1861 (2000), Springer, Berlin 644-659
74. Melis E., and Siekmann J. Knowledge-based proof planning. Artificial Intelligence 115 1 (1999) 65-105
75. Melis E., and Siekmann J. Activemath: An Intelligent Tutoring System for Mathematics vol. 3070 (2004), Springer, Berlin pp. 91-101
76. E. Melis, J. Zimmer, T. Müller, Integrating constraint solving into proof planning, in: [55]
77. Mossakowski T., Autexier S., and Hutter D. Extending development graphs with hiding. In: Hussmann H. (Ed). Proceedings of Fundamental Approaches to Software Engineering (FASE 2001), Genova. Lecture Notes in Computer Science vol. 2029 (2001), Springer, Berlin 269-283
78. Nipkow T., Paulson L.C., and Wenzel M. Isabelle/HOL: A Proof Assistant for Higher-Order Logic. Lecture Notes in Computer Science vol. 2283 (2002), Springer, Berlin
79. Owre S., Rajan S., Rushby J.M., Shankar N., and Srivas M. PVS: Combining specification, proof checking, and model checking. In: Alur R., and Henzinger T. (Eds). Computer-Aided Verification, CAV '96, New Brunswick, NJ. Lecture Notes in Computer Science vol. 1102 (1996), Springer, Berlin 411-414
80. Paulson L. Isabelle: A Generic Theorem Prover. Lecture Notes in Computer Science vol. 828 (1994), Springer, Berlin
81. Polya G. How to Solve it (1973), Princeton University Press, Princeton, NJ
82. A. Riazanov, A. Voronkov, Vampire 1.1 (system description), in: [48]
83. J. Richardson, A. Smaill, I. Green, System description: Proof planning in higher-order logic with λClam, in: [54]
84. Robinson J.A. A machine-oriented logic based on the resolution principle. J. ACM 12 1 (1965) 23-41
85. Schönert M., et al. GAP-Groups, Algorithms, Programming. Lehrstuhl D für Mathematik (1995), Rheinisch Westfälische Technische Hochschule, Aachen, Germany
86. Siekmann J. Geschichte des Automatischen Beweisens (History of Automated Deduction). Deduktionssysteme, Automatisierung des Logischen Denkens. second ed. (1992), R. Oldenbourg Verlag also in English with Elsewood
87. Siekmann J. History of computational logic. In: Gabbay D., and Woods J. (Eds). The Handbook of the History of Logic vols. I-IX (2004), Elsevier, Amsterdam
88. Siekmann J., Benzmüller C., Fiedler A., Meier A., Normann I., and Pollet M. Proof development in OMEGA: The irrationality of square root of 2. In: Kamareddine F. (Ed). Thirty Five Years of Automating Mathematics. Kluwer Applied Logic Series vol. 28 (2003), Kluwer Academic Publishers, Dordrecht. 1-4020-1656-5 271-314
89. Siekmann J., Benzmüller C., Fiedler A., Meier A., and Pollet M. Proof development with OMEGA: Sqrt(2) is irrational. In: Baaz M., and Voronkov A. (Eds). Logic for Programming, Artificial Intelligence, and Reasoning, 9th International Conference, LPAR 2002. Lecture Notes in Artificial Intelligence vol. 2514 (2002), Springer, Berlin 367-387
90. Siekmann J., Hess S., Benzmüller C., Cheikhrouhou L., Fiedler A., Horacek H., Kohlhase M., Konrad K., Meier A., Melis E., Pollet M., and Sorge V. LOUI: Lovely ΩmegaU ser Interface. Formal Aspects of Computing 11 (1999) 326-342
91. V. Sorge, Non-trivial computations in proof planning, in: [55]
92. V. Sorge, ΩANTS-a blackboard architecture for the integration of reasoning techniques into proof planning, PhD thesis, Department of Computer Science, Saarland University, Saarbrücken, Germany, 2001
93. G. Sutcliffe, C. Suttner, T. Yemenis, The TPTP problem library, in: [27]
94. J. van der Hoeven, GNU TeXmacs: A free, structured, wysiwyg and technical text editor, in: Actes du congrès Gutenberg, Metz, Actes du congrès Gutenberg, vols. 39-40, May 2001, pp. 39-50
95. C. Weidenbach, B. Afshordel, U. Brahm, C. Cohrs, Th. Engel, E. Keen, C. Theobalt, D. Topic, System description: SPASS version 1.0.0, in: [44], pp. 378-382
96. Weld D. An introduction to least commitment planning. AI Magazine 15 4 (1994) 27-61
97. F. Wiedijk, The Seventeen Provers of the World, in: Lecture Notes in Artificial Intelligence, Springer, Berlin, 2005, in press
98. Zhang J., and Zhang H. SEM: A system for enumerating models. In: Mellish C.S. (Ed). Proceedings of the 14th International Joint Conference on Artificial Intelligence (IJCAI), Montreal, Canada (1995), Morgan Kaufmann, San Mateo, CA 298-303
99. Zimmer J., and Kohlhase M. System description: The Mathweb software bus for distributed mathematical reasoning. In: Voronkov A. (Ed). Proceedings of the 18th International Conference on Automated Deduction. Lecture Notes in Artificial Intelligence vol. 2392 (2002), Springer, Berlin 138-142
100. Zimmer J., and Melis E. Constraint solving for proof planning. J. Automat. Reason. 33 (2004) 51-88
101. Zimmer J. A framework for agent-based brokering of reasoning services. In: Monroy R., Arroyo-Figueroa G., Sucar L.E., and Azuela J.H.S. (Eds). MICAI. Lecture Notes in Computer Science vol. 2972 (2004), Springer, Berlin