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The von Neumann threshold of self-reproducing systems: theory and application

Published online by Cambridge University Press:  14 January 2011

Pierre T. Kabamba*
Department of Aerospace Engineering, The University of Michigan Ann Arbor, MI 48109, USA
Patrick D. Owens
Department of Mechanical Engineering, The University of Michigan Ann Arbor, MI 48109, USA
A. Galip Ulsoy
Department of Mechanical Engineering, The University of Michigan Ann Arbor, MI 48109, USA
*Corresponding author. E-mail:


This paper is devoted to the study of systems of entities that are capable of generating other entities of the same kind and, possibly, self-reproducing. The main technical issue addressed is to quantify the requirements that such entities must meet to be able to produce a progeny that is not degenerative, i.e., that has the same reproductive capability as the progenitor. A novel theory that allows an explicit quantification of these requirements is presented. The notion of generation rank of an entity is introduced, and it is proved that the generation process, in most cases, is degenerative in that it strictly and irreversibly decreases the generation rank from parent to descendent. It is also proved that there exists a threshold of rank such that this degeneracy can be avoided if and only if the entity has a generation rank that meets that threshold – this is the von Neumann rank threshold. On the basis of this threshold, an information threshold is derived, which quantifies the minimum amount of information that must be provided to specify an entity such that its descendents are not degenerative. Furthermore, a complexity threshold is obtained, which quantifies the minimum length of the description of that entity in a given language. As an application, self-assembly for a 2 Degrees of Freedom planar robot is considered, and simulation results are presented. A robot arm capable of picking up and placing the components of another arm, in the presence of errors, is considered to have successfully reproduced if these are placed within an allowable tolerance. The example shows that, due to the kinematics of the robot, errors can grow from one generation to the next, until the reproduction process fails eventually. However, error correction (via error sensing and feedback control) can then be used to prevent such degeneracy. The von Neumann generation rank and information thresholds are computed for this example, and are consistent with the simulation results in predicting degeneracy in the case without error correction, and predicting successful self-reproduction in the case with error correction.

Copyright © Cambridge University Press 2011

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1.von Neumann, J., In: Theory of Self-Reproducing Automata (Burks, A., ed.) (University of Illinois Press, 1966).Google Scholar
2.Owens, P. and Ulsoy, A. G., “Self-Replicating Machines: Preventing Degeneracy,” Proceedings of IMECE 2006: ASME International Mechanical Engineering Congress and Exposition, Chicago, IL (Nov. 5–10, 2006), (see also Control Group Report No. CGR-06–02, The University of Michigan, 2006).Google Scholar
3.Sipper, M., “Fifty years of research on self-replication: An overview,” Artif. Life 4 (3), pp. 237257 (1998).CrossRefGoogle ScholarPubMed
4.Freitas, R. A. Jr. and Merkle, Ralph C., Kinematic Self-Replicating Machines, (Landes Bioscience, Georgetown, TX, 2004).Google Scholar
5.Arbib, M. A., “Simple self-reproducing universal automata,” Inf. Control 9, 177189 (1966).CrossRefGoogle Scholar
6.Codd, E. F., Cellular Automata (Academic Press, New York, 1968).Google Scholar
7.Smith, A. R., “Simple Nontrivial Self-Reproducing Machines,” In: Artificial Life II (Langton, C. G., Taylor, C., Farmer, J. D. and Rasmussen, S., eds.) (Vol. X of SFI Studies in the Sciences of Complexity) (Addison-Wesley, Redwood City, CA, 1992) pp. 709725.Google Scholar
8.Burks, A. W., ed., In: Essays on Cellular Automata (University of Illinois Press, Urbana, IL, 1970).Google Scholar
9.Case, J., “Periodiciy in generations of automata,” Math. Syst. Theory 8, 1532 (1974).CrossRefGoogle Scholar
10.Case, J., “Recursion Theorems and Automata which Construct,” Proceedings Of the 1974 Conference on Biologically Motivated Automata Theory, IEEE, New York (1974).Google Scholar
11.Devore, J. and Hightower, R., “The Devore Variation of the Codd Self-Replicating Computer,” In: Artificial Life III (Santa Fe, NM, 1992).Google Scholar
12.Vitanyi, P. M. B., “Sexually reproducing cellular automata,” Math. Biosci. 18, 2354 (1973).CrossRefGoogle Scholar
13.Langton, C. G., “Self-reproduction in cellular automata,” Phys. D 10, 135144 (1984).CrossRefGoogle Scholar
14.Byl, J., “Self-reproduction in small cellular automata,” Phys. D 34, 295299 (1989).CrossRefGoogle Scholar
15.Reggia, J. A., Armentrout, S. L., Chou, H.-H., and Peng, Y., “Simple ystems that exhibit self-directed replication,” Science 259, 12821287 (1993).CrossRefGoogle Scholar
16.Sipper, M., “Non-Uniform Cellular Automata: Evolution in Rule Space and Formation of Complex Structures,” In: Artificial Life IV (Brooks, Rodney A. and Maes, P., eds.) (The MIT Press, Cambridge, MA, 1994) pp. 394399.Google Scholar
17.Ibanez, J., Anabitarte, D., Aspeitia, I., Barrera, O., Barrutieta, A., Blanco, H. and Echarte, F., “Self-Inspection-Based Reproduction in Cellular Automata,” In: ECAL 1995: 3rd European Conference on Artificial Life, Vol. 929 of Lecture Notes in Computer Science (Moran, F., Moreno, A., Merelo, J. J., and Chacon, P., eds.) (Springer -Verlag, Heidelberg, Germany, 1995) pp. 564576.Google Scholar
18.Sayama, H., “Toward the Realization of an Evolving Ecosystem on Cellular Automata,” In: Proceedings of the Fourth International Symposium on Artificial Life and Robotics (AROB4th '99) (Sugisaka, M. and Tanaka, H., eds.) (Beppu, Oita, Japan, 1999) pp. 254257.Google Scholar
19.Tempesti, G., “A New Self-Reproducing Cellular Automaton Capable of Construction and Computation,” In: ECAL '95: Third European Conference on Artificial Life, Vol. 929 of Lecture Notes in Computer Science (Moran, F., Moreno, A., Merelo, J. J., and Chacon, P., eds.) (Springer-Verlag, Heidelberg, Germany, 1995) pp. 555563.Google Scholar
20.Perrier, J.-Y., Sipper, M. and Zahnd, J., “Toward a viable, self-reproducing universal computer,” Phys. D 97, 335352 (1996).CrossRefGoogle Scholar
21.Lohn, J. D. and Reggia, J. A., “Automatic discovery of self-replicating structures in cellular automata,” IEEE Trans. Evol. Comput. 1 (3), 165178 (Sep. 1997).CrossRefGoogle Scholar
22.Chou, H.-H. and Reggia, J. A., “Emergence of self-replicating structures in a cellular automata space,” Phys. D 110 (3–4), 252276 (1997).CrossRefGoogle Scholar
23.Chou, H.-H. and Reggia, J. A., “Problem solving during artificial selection of self-replicating loops,” Phys. D 115 (3–4), 293312 (1998).CrossRefGoogle Scholar
24.Morita, K. and Imai, K., “Self-reproduction in a reversible cellular space,” Theor. Comput. Sci. 168, 337366 (1996).CrossRefGoogle Scholar
25.Stauffer, A. and Sipper, M., “On the relationship between cellular automata and L-systems: The self-replication case,” Phys. D 116 (1–2), 7180 (1998).CrossRefGoogle Scholar
26.Signorini, J., “How a SIMD Machine Can Implement a Complex Cellular Automaton? A Case Study: von Neumann's 29-State Cellular Automaton,” Supercomputing '89: Proceedings of the ACM/IEEE Conference (1989) pp. 175–186.Google Scholar
27.Pesavento, U., “An Implementation of von Neumann's self-reproducing machine,” Artif. Life J. 2 (4), 337–254 (1995) (The MIT Press, Cambridge, MA).CrossRefGoogle ScholarPubMed
28.Bratley, P. and Millo, J., “Computer reactions: Self-reproducing programs,” Softw. Pract. Exp. 2, 297400 (1972).Google Scholar
29.Burger, J., Brill, D. and Machi, F., “Self-reproducing programs,” Byte 5, 7274 (1980).Google Scholar
30.Morris, H. C., “Typogenetics: A Logic for Artificial Life,” In: Artificial Life (Langton, C.G., ed.) (Vol. VI of SFI Studies in the Sciences of Complexity) (Addison-Wesley, Reading, MA, 1989) pp. 369395.Google Scholar
31.Varetto, L., “Typogenetics: An artificial genetic system,” J. Theor. Biol. 160, 185205 (1993).CrossRefGoogle ScholarPubMed
32.Koza, J. R., “Artificial Life: Spontaneous Emergence of Self-Replicating and Evolutionary Self-Improving Computer Programs,” In: Artificial Life III (Langton, C.G., ed.) (Vol. XVII of SFI Studies in the Sciences of Complexity) Addison-Wesley, Reading, MA, 1994) pp. 225262.Google Scholar
33.Suzuki, H., “A Necessary Condition for self-reproduction in the semar core,” FUZZ-IEEE '99: 1999 IEEE Int. Fuzzy Syst. Conf. Proc. 1 (1), 123128, (1999).Google Scholar
34.Dewdney, A. K., “Core war,” Sci. Am. 250 (5), 1519 (May 1984).Google Scholar
35.Ray, T., “An Approach to the Synthesis of Life,” In: Artificial Life II (Langton, C.G., Taylor, C., Farmer, J.D., and Rasmussen, S., eds.) (Vol. X of SFI Studies in the Sciences of Complexity) Addison-Wesley, Redwood City, CA, 1992) pp. 371408.Google Scholar
36.Taylor, T. and Hallam, J., “Studying Evolution with Self-Replicating Computer Programs,” In: Proceedings of the Fourth European Conference on Artificial Life (ECAL97) (Husbands, P. and Harvey, I., eds.) (MIT Press/Bradford Books, Cambridge, MA, 1997).Google Scholar
37.Smith, A., Turney, P. and Ewaschuk, R., “Self-replicating machines in continuous space with virtual physics,” Artif. Life 9, 2140 (2003).CrossRefGoogle ScholarPubMed
38.Hutton, T. J., “Evolvable self-replicating molecules in an artificial chemistry,” Artif. Life 8 (4), 341356 (2002).CrossRefGoogle Scholar
39.Stevens, W. M., “NODES: An Environment for Simulating Kinematic Self-Replicating Machines,” Proceedings of the Ninth International Conference on the Simulation and Synthesis of Living Systems (ALIFE9) (2004), pp. 39–44.Google Scholar
40.Rubenstein, M., Krivokon, M. and Shen, W.-M., “Robotic enzyme-based autonomous self-replication,” 2004 IEEE/RSJ Int. Conf. Intell. Robots Syst. (IROS) 3, 26612666 (2004).CrossRefGoogle Scholar
41.Penrose, L. S., “Self-reproducing machines,” Sci. Am. 200 (6), 105114 (Jun 1959).CrossRefGoogle Scholar
42.Mange, D., Sanchez, E., Stauffer, A., Tempesti, G., Marchal, P. and Piguet, C., “Embryonics: A new methodology for designing field-programmable gate arrays with self-repairing and self-replicating properties,” IEEE Trans. VLSI Syst. 6 (3), 387399 (Sep 1998).CrossRefGoogle Scholar
43.Beuchat, J.-L. and Haenni, J.-O., “von Neumann's 29-state cellular automata: A hardware implementation,” IEEE Trans. Educ. 43 (3), 300308 (Aug. 2000).CrossRefGoogle Scholar
44.Funes, P. and Polluck, J., “Evolutionary body building: Adaptive physical designs for robots,” Artif. Life 4, 337357 (1998).CrossRefGoogle ScholarPubMed
45.Lipson, H. and Polluck, J., “Automatic design and manufacture of robotic lifeforms,” Nature 406 (6799), 974978 (2000).CrossRefGoogle ScholarPubMed
46.Zykov, V., Mytilnaios, E., Adams, B. and Lipson, H., “Self-reproducing machines: A set of modular robot cubes accomplish a feat fundamental to biological systems,” Nature 435, 163164 (May 12, 2005).CrossRefGoogle Scholar
47.Ichikawa, Y. and Okido, F., “One-dimensional self-reproducing robot,” Proc. IECON'91: 1991 Int. Conf. Ind. Electr., Control Instrum. 2, 963966 (1991).Google Scholar
48.Chirikjian, G. S., Zhou, Y. and Suthakorn, J., “Self-replicating robots for lunar development,” IEEE/ASME Trans. Mechatronics 7 (4), 462472 (Dec. 2002).CrossRefGoogle Scholar
49.Beletski, Y., “Self-reproduction of modular turing machines,” Arch. Autom.Telemech. 16 (2), 161170 (1971).Google Scholar
50.Seife, C., “In clone wars, quantum computers need not apply,” Science 300, 884 (May 9, 2003).CrossRefGoogle Scholar
51.Marchal, B., “Amoeba, Planaria, and Dreaming Machines,” Toward a Practice of Autonomous Systems: Proceedings of the First Conference on Artificial Life (1992) pp. 429–440.Google Scholar
52.Freitas, R. A. Jr., “A self-reproducing interstellar probe,” J. Br. Interplanet. Soc. 33, 251264 (Jul. 1980).Google Scholar
53.Freitas, R. A. Jr., “Report on the NASA/ASEE summer study on advanced automation for space missions,” J. Br. Interplanet. Soc. 34, 407408 (Sep. 1981).Google Scholar
54.Byler, E. and Olsen, R., “Automation and Robotic Technologies for Commercial Space Factories,” EASCON87: Proceedings of the 20th Annual Electronics and Aerospace Systems Conference-Technology for Space Leadership (1987) pp. 93–99.Google Scholar
55.Angelo, J. A. Jr., “Role of Advanced Robotic Systems in Interstellar Exploration,” Proceedings of Space Congress: Yesterday's Vision in Tomorrow's Reality (1993) p. 10.Google Scholar
56.Luk'yanov, V. and Leskov, L. V., “Terraforming of Planets – Prospects and Problems,” Lectures Devoted to the Development of the Scientific Heritage and Ideas of K.E. Tsiolkovsky, 28th Meeting of the Section of K. E. Tsiolkovsky and Problems of Space Manufacturing, Kaluga, Russia (Sep. 14–17, 1993, 1994) pp. 1418.Google Scholar
57.Chadeev, V. M., “Allowance for robot self-replication in industrial automation,” Avtom. Telemekh. 61 (10), 183189 (Oct. 2000).Google Scholar
58.Lackner, K. S. and Wendt, C. H., “Exponential growth of large self-reproducing machine systems,” Math. Comput. Model. 21 (10), 5581 (May 1995).CrossRefGoogle Scholar
59.Storrs Hall, J., “Architectural considerations for self-replicating manufacturing systems,” Nanotechnology 10 (3), 323330 (Sep. 1999).CrossRefGoogle Scholar
60.Liang, R., “Some alternative reproductive strategies in artificial molecular machines,” J. Theor. Biol. 54, 6384 (1975).CrossRefGoogle Scholar
61.Rebek, J. Jr., “Synthetic self-replicating molecules,” Sci. Am. 271 (1), 4855 (Jul 1994).CrossRefGoogle Scholar
62.Reed, J., Toombs, R. and Barricelli, N. A., “Simulation of biological evolution and machine learning: Selection of self-reproducing numeric patterns by data processing machines, effects of hereditary control, mutation type and crossing,” J. Theor. Biol. 17, 319342 (1967).CrossRefGoogle ScholarPubMed
63.Sharov, A. A., “Self-reproducing systems: Structure, niche relations and evolution,” BioSystems 25, 237249 (1991).CrossRefGoogle ScholarPubMed
64.Adams, B. and Lipson, H., “A Universal Framework for Self-Replication,” In: ECAL 2003, LNAI 2801 (Banzhaf, W., ed.) (Springer-Verlag, Berlin, Heidelberg, Germany, 2003) pp. 19.Google Scholar
65.Gacs, P., “Reliable Cellular Automata with Self-Organization,” Proceedings of the 38th IEEE Symposium on the Foundation of Computer Science (1997) pp. 90–99.Google Scholar
66.Harao, M. and Noguchi, S., “Fault tolerant cellular automata,” J. Comput. Syst. Sci. 11, 171185 (1975).CrossRefGoogle Scholar
67.Bunzli, D. C. and Capcarrere, M. S., “Fault-Tolerant Structures: Towards Robust Self-Replication in a Probabilistic Environment,” In: ECAL 2001, LNAI 2159 (Kelemen, J. and Sosik, P., eds.) (Springer-Verlag, Berlin, Heidelberg, Germany 2001) pp. 9099.Google Scholar
68.Righetti, L., Shokur, S. and Capcarrere, M. S., “Evolution of Fault-Tolerant Self-Replicating Structures,” In: ECAL 2003, LNAI 2801 (Banzhaf, W., ed.) (Springer-Verlag, Berlin, Heidelberg, Germany, 2003) pp. 278288.Google Scholar
69.Shannon, C., “A mathematical theory of communication,” Bell Syst. Tech. J. 27, 379423, 623–656 (Oct. 1948).CrossRefGoogle Scholar
70.Menezes, A. and Kabamba, P., “A Combined Seed-Identification and Generation Analysis Algorithm for Self-Reproducing Systems”, Proceedings of the 2007 American Control Conference, New York, NY (Jun. 2007).Google Scholar
71.DeVor, R. E., Chand, T.-H. and Sutherland, J. W., Statistical Quality Design and Control (MacMillan, New York, 1992) pp. 264267.Google Scholar
73.Sayama, H., “Construction theory, self-replication, and the halting problem,” Complexity 13 (5), 1622 (2008).CrossRefGoogle Scholar
74.Turing, A. M., “On computable numbers, with an application to the entscheidungsproblem,” Proc. London Math. Soc. Ser. 2 (42), 230265 (1936).Google Scholar
75.Mantripragada, R. and Whitney, D. E., “Modeling and controlling variation propagation in mechanical assemblies using state transition models,” IEEE Trans. Robot. Autom. 15 (1), 124140 (Feb. 1999).CrossRefGoogle Scholar
76.Shi, J., Stream of Variation Modeling and Analysis for Multistage Manufacturing Processes (CRC Press, Florida, 2007).Google Scholar
77.Lee, K., Moses, M. and Chirikjian, G. S., “Robotic self-replication in structured environments: Physical demonstrations and complexity measures,” Int. J. Robot. Res. 27 (3–4), 387401 (Mar. 2008).CrossRefGoogle Scholar
78.Wang, Y. and Chirikjian, G. S., “Nonparametric second-order theory of error propagation on motion groups.” Int. J. Robot. Res. 27 (11–12), 12581273 (Nov. 2008)CrossRefGoogle ScholarPubMed