Ωmega: Computer Supported Mathematics

  • Jörg Siekmann
  • Christoph Benzmüller
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3238)


The year 2004 marks the fiftieth birthday of the first computer generated proof of a mathematical theorem: “the sum of two even numbers is again an even number” (with Martin Davis’ implementation of Presburger Arithmetic in 1954).

While Martin Davis and later the research community of automated deduction used machine oriented calculi to find the proof for a theorem by automatic means, the Automath project of N.G. de Bruijn – more modest in its aims with respect to automation – showed in the late 1960s and early 70s that a complete mathematical textbook could be coded and proof-checked by a computer.

Classical theorem proving procedures of today are based on ingenious search techniques to find a proof for a given theorem in very large search spaces – often in the range of several billion clauses. But in spite of many successful attempts to prove even open mathematical problems automatically, their use in everyday mathematical practice is still limited.

The shift from search based methods to more abstract planning techniques however opened up a new paradigm for mathematical reasoning on a computer and several systems of the new kind now employ a mix of interactive, search based as well as proof planning techniques.

The Ωmega system is at the core of several related and well-integrated research projects of the Ωmega research group, whose aim is to develop system support for the working mathematician, in particular it supports proof development at a human oriented level of abstraction. It is a modular system with a central proof data structure and several supplementary subsystems including automated deduction and computer algebra systems. Ωmega has many characteristics in common with systems like NuPrL [ACE + 00], CoQ [Coq03], Hol [GM93], Pvs [ORR + 96], and Isabelle [Pau94,NPW02]. However, it differs from these systems with respect to its focus on proof planning and in that respect it is more similar to the proof planning systems Clam and λClam at Edinburgh [RSG98,BvHHS90].


Mathematical Knowledge Computer Algebra System Control Rule Proof Assistant Open Node 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [ABB00]
    Andrews, P.B., Bishop, M., Brown, C.E.: System description: TPS: A theorem proving system for type theory. In: Conference on Automated Deduction, pp. 164–169 (2000)Google Scholar
  2. [ABF+03]
    Autexier, S., Benzmüller, C., Fiedler, A., Horacek, H., Bao Vo, Q.: Assertion-level proof representation with under-specification. Electronic in Theoretical Computer Science 93, 5–23 (2003)CrossRefGoogle Scholar
  3. [ABI+96]
    Andrews, P.B., Bishop, M., Issar, S., Nesmith, D., Pfenning, F., Xi, H.: TPS: A theorem proving system for classical type theory. Journal of Automated Reasoning 16(3), 321–353 (1996)zbMATHCrossRefMathSciNetGoogle Scholar
  4. [ACE+00]
    Allen, S., Constable, R., Eaton, R., Kreitz, C., Lorigo, L.: The Nuprl open logical environment. In: McAllester [McA00]Google Scholar
  5. [AH02]
    Autexier, S., Hutter, D.: Maintenance of formal software development by stratified verification. In: Baaz, M., Voronkov, A. (eds.) LPAR 2002. LNCS (LNAI), vol. 2514, Springer, Heidelberg (2002)CrossRefGoogle Scholar
  6. [AHMS02]
    Autexier, S., Hutter, D., Mossakowski, T., Schairer, A.: The development graph manager MAYA. In: Kirchner, H., Ringeissen, C. (eds.) AMAST 2002. LNCS, vol. 2422, p. 495. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  7. [AM02]
    Autexier, S., Mossakowski, T.: Integrating HOL-CASL into the development graph manager MAYA. In: Armando, A. (ed.) FroCos 2002. LNCS (LNAI), vol. 2309, pp. 2–17. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  8. [Aut03]
    Autexier, S.: Hierarchical Contextual Reasoning. PhD thesis, Computer Science Department, Saarland University, Saarbrücken, Germany (2003) (forthcoming)Google Scholar
  9. [BBS99]
    Benzmüller, C., Bishop, M., Sorge, V.: Integrating TPS and Ωmega. Journal of Universal Computer Science 5, 188–207 (1999)Google Scholar
  10. [Ben99]
    Benzmüller, C.: Equality and Extensionality in Higher-Order Theorem Proving. PhD thesis, Department of Computer Science, Saarland University, Saarbrücken, Germany (1999)Google Scholar
  11. [BF94]
    Baumgartner, P., Furbach, U.: PROTEIN, a PROver with a Theory INterface. In: Bundy [Bun94], pp. 769–773Google Scholar
  12. [BFG+03]
    Benzmüller, C., Fiedler, A., Gabsdil, M., Horacek, H., Kruijff-Korbayova, I., Pinkal, M., Siekmann, J., Tsovaltzi, D., Quoc Vo, B., Wolska, M.: Tutorial dialogs on mathematical proofs. In: Proceedings of IJCAI-03 Workshop on Knowledge Representation and Automated Reasoning for E-Learning Systems, Acapulco, Mexico, pp. 12–22 (2003)Google Scholar
  13. [BFMP02]
    Benzmüller, C., Fiedler, A., Meier, A., Pollet, M.: Irrationality of \(\sqrt{2}\) – a case study in Ωmega. Seki-Report SR-02-03, Department of Computer Science, Saarland University, Saarbrücken, Germany (2002)Google Scholar
  14. [BGH03]
    Buchberger, B., Gonnet, G., Hazewinkel, M.: Special issue on mathematical knowledge management. Annals of Mathematics and Artificial Intelligence 38(1-3), 3–232 (2003)MathSciNetGoogle Scholar
  15. [BK98]
    Benzmüller, C., Kohlhase, M.: LEO – a higher-order theorem prover. In: Kirchner and Kirchner [KK98]Google Scholar
  16. [Ble90]
    Bledsoe, W.: Challenge problems in elementary calculus. Journal of Automated Reasoning 6, 341–359 (1990)zbMATHCrossRefGoogle Scholar
  17. [BMS04]
    Benzmüller, C., Meier, A., Sorge, V.: Bridging theorem proving and mathematical knowledge retrieval. In: Hutter and Stephan [HS04] (to appear)Google Scholar
  18. [Bou68]
    Bourbaki, N.: Theory of sets. In: Elements of Mathematics, vol. 1, Addison-Wesley, Reading (1968)Google Scholar
  19. [BS82]
    Bartle, R., Sherbert, D.: Introduction to Real Analysis, 2nd edn. Wiley, Chichester (1982)zbMATHGoogle Scholar
  20. [BS98]
    Benzmüller, C., Sorge, V.: A blackboard architecture for guiding interactive proofs. In: Giunchiglia [Giu98]Google Scholar
  21. [BS99]
    Benzmüller, C., Sorge, V.: Critical agents supporting interactive theorem proving. In: Barahona, P., Alferes, J.J. (eds.) EPIA 1999. LNCS (LNAI), vol. 1695, pp. 208–221. Springer, Heidelberg (1999)CrossRefGoogle Scholar
  22. [BS00]
    Benzmüller, C., Sorge, V.: Ωants – An open approach at combining Interactive and Automated Theorem Proving. In: Kerber and Kohlhase[KK00]Google Scholar
  23. [Bun88]
    Bundy, A.: The use of explicit plans to guide inductive proofs. In: Lusk and Overbeek [LO88], pp. 111–120Google Scholar
  24. [Bun91]
    Bundy, A.: A science of reasoning. In: Plotkin, G., Lasser, J.-L. (eds.) Computational Logic: Essays in Honor of Alan Robinson, pp. 178–199. MIT Press, Cambridge (1991)Google Scholar
  25. [Bun94]
    Bundy, A. (ed.): CADE 1994. LNCS, vol. 814. Springer, Heidelberg (1994)zbMATHGoogle Scholar
  26. [Bun02]
    Bundy, A.: A critique of proof planning. In: Kakas, A.C., Sadri, F. (eds.) Computational Logic: Logic Programming and Beyond. LNCS (LNAI), vol. 2408, pp. 160–177. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  27. [BvHHS90]
    Bundy, A., van Harmelen, F., Horn, C., Smaill, A.: The Oyster-Clam System. In: Stickel, M.E. (ed.) CADE 1990. LNCS, vol. 449, pp. 647–648. Springer, Heidelberg (1990)Google Scholar
  28. [CBE+92]
    Carbonell, J., Blythe, J., Etzioni, O., Gil, Y., Joseph, R., Kahn, D., Knoblock, C., Minton, S., Pérez, M.A., Reilly, S., Veloso, M., Wang, X.: PRODIGY 4.0: The Manual and Tutorial. CMU Technical Report CMUCS- 92-150, Carnegie Mellon University (June 1992)Google Scholar
  29. [CGG+92]
    Char, B., Geddes, K., Gonnet, G., Leong, B., Monagan, M., Watt, S.: First leaves: a tutorial introduction to Maple V. Springer, Heidelberg (1992)zbMATHGoogle Scholar
  30. [Chu40]
    Church, A.: A Formulation of the Simple Theory of Types. Journal of Symbolic Logic 5, 56–68 (1940)zbMATHCrossRefMathSciNetGoogle Scholar
  31. [Coq03]
    Coq Development Team. The Coq Proof Assistant Reference Manual. INRIA (1999-2003), See
  32. [CS98]
    Cheikhrouhou, L., Siekmann, J.: Planning diagonalization proofs. In: Giunchiglia [Giu98], pp. 167–180Google Scholar
  33. [CS00]
    Cheikhrouhou, L., Sorge, V.: 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 (2000)Google Scholar
  34. [Dav65]
    Davis, M. (ed.): The Undecidable: Basic Papers on undecidable Propositions, unsolvable Problems and Computable Functions. Raven Press Hewlett, New York (1965)Google Scholar
  35. [Dav83]
    Davis, M.: The prehistory and early history of automated deduction. In: Siekmann, J., Wrightson, G. (eds.) Automation of Reasoning, Classical Papers on Computational Logic 1967–1970 of Symbolic Computation, vol. 2, Springer, Heidelberg (1983)Google Scholar
  36. [Dav01]
    Davis, M.: The early history of automated deduction. In: Robinson, A., Voronkov, A. (eds.) Handbook of Automated Reasoning, vol. I, ch. 1, pp. 3–15. Elsevier Science, Amsterdam (2001)CrossRefGoogle Scholar
  37. [dN99]
    de Nivelle, H.: Bliksem 1.10 user manual. Technical report, Max-Planck- Institut für Informatik (1999)Google Scholar
  38. [Dor01]
    The Doris system (2001), available at
  39. [EHN94]
    Erol, K., Hendler, J., Nau, D.: Semantics for hierarchical task network planning. Technical Report CS-TR-3239, UMIACS-TR-94-31, Computer Science Department, University of Maryland (March 1994)Google Scholar
  40. [Fie01a]
    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, pp. 1295–1300. Morgan Kaufmann, San Francisco (2001)Google Scholar
  41. [Fie01b]
    Fiedler, A.: P.rex: An interactive proof explainer. In: Goré et al. [GLN01]Google Scholar
  42. [Fie01c]
    Fiedler, A.: User-adaptive proof explanation. PhD thesis, Department of Computer Science, Saarland University, Saarbrücken, Germany (2001)Google Scholar
  43. [FK00a]
    Franke, A., Kohlhase, M.: System description: MBase, an open mathematical knowledge base. In: McAllester [McA00]Google Scholar
  44. [FK00b]
    Franke, A., Kohlhase, M.: System description: Mbase, an open mathematical knowledge base. In: McAllester, D. (ed.) CADE 2000. LNCS, vol. 1831, Springer, Heidelberg (2000)CrossRefGoogle Scholar
  45. [Gan99]
    Ganzinger, H. (ed.): CADE 1999. LNCS (LNAI), vol. 1632. Springer, Heidelberg (1999)zbMATHGoogle Scholar
  46. [Geb99]
    H. Gebhard. Beweisplanung für die Beweise der Vollständigkeit verschiedener Resolutionskalküle in Ωmega. Master’s thesis, Department of Computer Science, Saarland University, Saarbrücken, Germany, 1999. Google Scholar
  47. [Giu98]
    Giunchiglia, F. (ed.): AIMSA 1998. LNCS (LNAI), vol. 1480. Springer, Heidelberg (1998)zbMATHGoogle Scholar
  48. [GLN01]
    Goré, R., Leitsch, A., Nipkow, T. (eds.): IJCAR 2001. LNCS (LNAI), vol. 2083. Springer, Heidelberg (2001)zbMATHGoogle Scholar
  49. [GM93]
    Gordon, M., Melham, T.: Introduction to HOL – A theorem proving environment for higher order logic. Cambridge University Press, Cambridge (1993)zbMATHGoogle Scholar
  50. [Had44]
    Hadamard, J.: The Psychology of Invention in the Mathematical Field. Dover Publications, New York; edition 1949, 1944Google Scholar
  51. [HJL99]
    Th. Hillenbrand, A.: Jaeger, and B. Löchner. System description:Waldmeister – improvements in performance and ease of use. In: Ganzinger [Gan99], pp. 232–236Google Scholar
  52. [HS04]
    Hutter, D., Stephan, W. (eds.): Festschrift in Honour of Jörg Siekmann’s 60s Birthday. LNCS (LNAI). Springer, Heidelberg (2004) (to appear)Google Scholar
  53. [Hua94]
    Huang, X.: Reconstructing Proofs at the Assertion Level. In: Bundy [Bun94], pp. 738–752Google Scholar
  54. [Hut00]
    Hutter, D.: Management of change in structured verification. In: Proceedings of Automated Software Engineering, ASE-2000, IEEE, Los Alamitos (2000)Google Scholar
  55. [KF01]
    Kohlhase, M., Franke, A.: MBase: Representing knowledge and context for the integration of mathematical software systems. Journal of Symbolic Computation; Special Issue on the Integration of Computer Algebra and Deduction Systems 32(4), 365–402 (2001)zbMATHMathSciNetGoogle Scholar
  56. [KK98]
    Kirchner, C., Kirchner, H. (eds.): CADE 1998. LNCS (LNAI), vol. 1421. Springer, Heidelberg (1998)zbMATHGoogle Scholar
  57. [KK00]
    Kerber, M., Kohlhase, M. (eds.): 8th Symposium on the Integration of Symbolic Computation and Mechanized Reasoning (Calculemus-2000). AK Peters, Wellesley (2000)Google Scholar
  58. [KR00]
    Kirchner, H., Ringeissen, C. (eds.): FroCos 2000. LNCS, vol. 1794. Springer, Heidelberg (2000)zbMATHGoogle Scholar
  59. [LO88]
    Lusk, E., Overbeek, R. (eds.): CADE 1988. LNCS, vol. 310. Springer, Heidelberg (1988)zbMATHGoogle Scholar
  60. [MAH01]
    Mossakowski, T., Autexier, S., Hutter, D.: Extending development graphs with hiding. In: Hussmann, H. (ed.) FASE 2001. LNCS, vol. 2029, pp. 269–283. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  61. [MB88]
    Manthey, R., Bry, F.: SATCHMO: A theorem prover implemented in Prolog. In: Lusk and Overbeek [LO88], pp. 415–434Google Scholar
  62. [McA00]
    McAllester, D. (ed.): CADE 2000. LNCS, vol. 1831. Springer, Heidelberg (2000)zbMATHGoogle Scholar
  63. [McC94]
    McCune, W.W.: Otter 3.0 reference manual and guide. Technical Report ANL-94-6, Argonne National Laboratory, Argonne, Illinois 60439, USA (1994)Google Scholar
  64. [McC97]
    McCune, W.: Solution of the Robbins problem. Journal of Automated Reasoning 19(3), 263–276 (1997)zbMATHCrossRefMathSciNetGoogle Scholar
  65. [Mei00]
    Meier, A.: TRAMP: Transformation of Machine-Found Proofs into Natural Deduction Proofs at the Assertion Level. In: McAllester [McA00]Google Scholar
  66. [Mei03]
    Meier, A.: Proof Planning with Multiple Strategies. PhD thesis, Department of Computer Science, Saarland University, Saarbrücken, Germany (2003)Google Scholar
  67. [Mel96]
    Melis, E.: Island planning and refinement. Seki-Report SR-96-10, Department of Computer Science, Saarland University, Saarbrücken, Germany (1996)Google Scholar
  68. [Mel98a]
    Melis, E.: AI-techniques in proof planning. In: Prade, H. (ed.) Proceedings of the 13th European Conference on Artifical Intelligence, Brighton, UK, pp. 494–498. John Wiley & Sons, Chichester (1998)Google Scholar
  69. [Mel98b]
    Melis, E.: AI-techniques in proof planning. In: European Conference on Artificial Intelligence, Brighton, pp. 494–498. Kluwer, Dordrecht (1998)Google Scholar
  70. [MM00]
    Melis, E., Meier, A.: Proof planning with multiple strategies. In: Palamidessi, C., Moniz Pereira, L., Lloyd, J.W., Dahl, V., Furbach, U., Kerber, M., Lau, K.-K., Sagiv, Y., Stuckey, P.J. (eds.) CL 2000. LNCS (LNAI), vol. 1861, pp. 644–659. Springer, Heidelberg (2000)CrossRefGoogle Scholar
  71. [MMP02]
    Meier, A., Melis, E., Pollet, M.: Towards extending domain representations. Seki Report SR-02-01, Department of Computer Science, Saarland University, Saarbrücken, Germany (2002)Google Scholar
  72. [MPS01]
    Meier, A., Pollet, M., Sorge, V.: Classifying Isomorphic Residue Classes. In: Moreno-Díaz Jr., R., Buchberger, B., Freire, J.-L. (eds.) EUROCAST 2001. LNCS, vol. 2178, pp. 494–508. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  73. [MPS02]
    Meier, A., Pollet, M., Sorge, V.: Comparing Approaches to the Exploration of the Domain of Residue Classes. Journal of Symbolic Computation, Special Issue on the Integration of Automated Reasoning and Computer Algebra Systems 34(4), 287–306 (2002)zbMATHMathSciNetGoogle Scholar
  74. [MS99]
    Melis, E., Siekmann, J.: Knowledge-based proof planning. Artificial Intelligence 115(1), 65–105 (1999)zbMATHCrossRefMathSciNetGoogle Scholar
  75. [MS00]
    Meier, A., Sorge, V.: Exploring properties of residue classes. In: Kerber and Kohlhase [KK00]Google Scholar
  76. [MZM00]
    Melis, E., Zimmer, J., Müller, T.: Integrating constraint solving into proof planning. In: Kirchner and Ringeissen [KR00]Google Scholar
  77. [NPW02]
    Nipkow, T., Paulson, L.C., Wenzel, M.: Isabelle/HOL. LNCS, vol. 2283. Springer, Heidelberg (2002)zbMATHCrossRefGoogle Scholar
  78. [ORR+96]
    Owre, S., Rajan, S., Rushby, J.M., Shankar, N., Srivas, M.: PVS: Combining specification, proof checking, and model checking. In: Alur, R., Henzinger, T.A. (eds.) CAV 1996. LNCS, vol. 1102, pp. 411–414. Springer, Heidelberg (1996)Google Scholar
  79. [Pau94]
    Paulson, L.: Isabelle. LNCS, vol. 828. Springer, Heidelberg (1994)zbMATHGoogle Scholar
  80. [Pol73]
    Polya, G.: How to Solve it. Princeton University Press, Princeton (1973)Google Scholar
  81. [Rob65]
    Robinson, J.A.: A Machine-Oriented Logic Based on the Resolution Principle. J. ACM 12(1), 23–41 (1965)zbMATHCrossRefGoogle Scholar
  82. [RSG98]
    Richardson, J., Smaill, A., Green, I.: System description: Proof planning in higher-order logic with λClam. In: Kirchner and Kirchner [KK98]Google Scholar
  83. [RV01]
    Riazanov, A., Voronkov, A.: Vampire 1.1 (system description). In: Goré et al. [GLN01]Google Scholar
  84. [S+95]
    Schönert, M., et al.: GAP – Groups, Algorithms, and Programming. Lehrstuhl D für Mathematik, Rheinisch Westfälische Technische Hochschule, Aachen, Germany (1995)Google Scholar
  85. [SBF+02]
    Siekmann, J., Benzmüller, C., Fiedler, A., Meier, A., Pollet, M.: Proof development with OMEGA: Sqrt(2) is irrational. In: Baaz, M., Voronkov, A. (eds.) LPAR 2002. LNCS (LNAI), vol. 2514, pp. 367–387. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  86. [SBF+03]
    Siekmann, J., Benzmüller, C., Fiedler, A., Meier, A., Normann, I., 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), pp. 271–314. Kluwer Academic Publishers, Dordrecht (2003) ISBN 1-4020-1656-5Google Scholar
  87. [SHB+99]
    Siekmann, J., Hess, S., Benzmüller, C., Cheikhrouhou, L., Fiedler, A., Horacek, H., Kohlhase, M., Konrad, K., Meier, A., Melis, E., Pollet, M., Sorge, V.: LOUI: Lovely Ωmega User Interface. Formal Aspects of Computing 11, 326–342 (1999)CrossRefGoogle Scholar
  88. [Sie92]
    Siekmann, J.: Geschichte des automatischen beweisens (history of automated deduction). In: Deduktionssysteme, Automatisierung des Logischen Denkens. R. Oldenbourg Verlag, 2nd edition (1992); Also in English with ElsewoodGoogle Scholar
  89. [Sie04]
    Siekmann, J.: History of computational logic. In: Gabbay, D., Woods, J. (eds.) The Handbook of the History of Logic, vol. I-IX, Elsevier, Amsterdam (2004) (to appear)Google Scholar
  90. [Sor00]
    Sorge, V.: Non-Trivial Computations in Proof Planning. In: Kirchner and Ringeissen [KR00]Google Scholar
  91. [Sor01]
    Sorge, V.: ΩANTS – A Blackboard Architecture for the Integration of Reasoning Techniques into Proof Planning. PhD thesis, Department of Computer Science, Saarland University, Saarbrücken, Germany (2001)Google Scholar
  92. [SSY94]
    Sutcliffe, G., Suttner, C., Yemenis, T.: The TPTP problem library. In: Bundy [Bun94]Google Scholar
  93. [vdH01]
    van der Hoeven, J.: GNU TeXmacs: A free, structured, wysiwyg and technical text editor. In: Actes du congrès Gutenberg, number 39-40 in Actes du congrès Gutenberg, Metz, pp. 39–50 (May 2001)Google Scholar
  94. [Vor02]
    Voronkov, A. (ed.): CADE 2002. LNCS (LNAI), vol. 2392. Springer, Heidelberg (2002)zbMATHGoogle Scholar
  95. [WAB+99]
    Weidenbach, C., Afshordel, B., Brahm, U., Cohrs, C., Engel, T., Keen, E., Theobalt, C., Topic, D.: System description: SPASS version 1.0.0. In: Ganzinger [Gan99], pp. 378–382Google Scholar
  96. [Wel94]
    Weld, D.: An introduction to least commitment planning. AI Magazine 15(4), 27–61 (1994)Google Scholar
  97. [Wie02]
    Wiedijk, F.: The fifteen provers of the world. Unpublished Draft (2002)Google Scholar
  98. [ZK02]
    Zimmer, J., Kohlhase, M.: System description: The Mathweb Software Bus for distributed mathematical reasoning. In: Voronkov [Vor02], pp. 138–142Google Scholar
  99. [ZZ95]
    Zhang, J., 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, pp. 298–303. Morgan Kaufmann, San Mateo (1995)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  • Jörg Siekmann
    • 1
  • Christoph Benzmüller
    • 1
  1. 1.Saarland UniversitySaarbrückenGermany

Personalised recommendations