Abstract
It has been established that Physarum polycephalum slime-mould organisms retain the time-period of a regular pulse of brief stimuli. We ask whether a period can equally well be imparted not via regular pulses, but via more general periodic functions—for example via stimulus intensity varying sinusoidally with time, or even varying with time as a function with unknown period (whence the organisms not merely retain the period, but in a sense compute it); we discuss this theoretically, and also outline, though defer to future work, an experimental investigation. As motivation, we note that the ability to determine a function’s period is computationally highly desirable, not least since from such ability follow methods of integer factorization. Specifically, the phenomena described herein afford a novel (albeit inefficient), non-quantum implementation of Shor’s algorithm; inefficiency aside, this offers interesting, alternative perspectives on approaches to factorization and on the computational uses of Physarum.
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Notes
- 1.
- 2.
The exact form of stimulus considered in [15] is an imposition of “unfavourable [cooler, less humid] conditions”—specifically, a temperature of 23 \(^{\circ }\mathrm {C}\) and a humidity of \(60\,\%\), rather than the usual, favourable 26 \(^{\circ }\mathrm {C}\) and \(90\,\%\). In the experimental set-up of [15], a ‘pulse’ of such conditions lasts for 10 min; for context, the duration of experiments of this sort is typically of the order of hours (rather than, say, minutes or days).
- 3.
The response sought and observed in [15] is a drop in the locomotive speed of the organism.
- 4.
We suppose for convenience that n is odd, composite and not a prime power, which supposition is unproblematic since the condition can be checked efficiently by Turing machine, and, should it fail, a prime factor can be found, again by Turing machine, again efficiently, without ever having to invoke Shor’s algorithm.
- 5.
Contrast this situation with more general periodic sequences, in which a value may be revisited several times during one period: certainly the period of the sequence 1, 2, 1, 3, 1, 2, 1, 3, ... is not 2, despite what the positions of the ‘1’s may lead one to suppose.
- 6.
In an experimental context, this exposure consists of interpreting the training function as mapping time to stimulus intensity. See Sect. 3.2.1 for details.
- 7.
We treat the values of a and n as being fixed and understood, and accordingly leave these values implicit by writing ‘f’ rather than ‘\(f_{a,n}\)’ or similar.
- 8.
In the language of our function f above, this pulse function is given by, say, \(f' :t \mapsto {\left\{ \begin{array}{ll} 1 &{} \text {if } {\left\lfloor t\right\rfloor \equiv 0 \pmod r }\\ 0 &{} \text {otherwise} \end{array}\right. }\), which represents a unit-duration pulse every r units of time. In the experimental set-up of [15], the unit of time (and, hence, the duration of each pulse) is of the order of 10 min, and r is of the order of six (though other periods are also considered).
- 9.
- 10.
Recall that, initially, we focus on Physarum, and defer the consideration of other substrates largely to subsequent work.
References
Adamatzky, A.: Physarum Machines: Computers from Slime Mould. World Scientific Series on Nonlinear Science, Series A, vol. 74 (2010). doi:10.1142/9789814327596
Belousov, B.: A periodic reaction and its mechanism. In: Oscillations and Traveling Waves in Chemical Systems. Wiley (1985). ISBN: 978-0-471-89384-4
Blakey, E.: Factorizing RSA keys, an improved analogue solution. New Gener. Comput. 27(2), 159–176 (2008). doi:10.1007/s00354-008-0059-3
Blakey, E.: A model-independent theory of computational complexity: from patience to precision and beyond. Doctoral thesis, Computer Science, University of Oxford (2010). http://ora.ox.ac.uk/objects/uuid%3A5db40e2c-4a22-470d-9283-3b59b99793dc. Accessed 7 Jul 2013
Blakey, E.: Complexity-style resources in cryptography. Inf. Comput. 226, 3–15 (2013). doi:10.1016/j.ic.2013.03.002
Brent, R.: An improved Monte Carlo factorization algorithm. BIT Numer. Math. 20(2), 176–184 (1980). doi:10.1007/BF01933190
Brent, R.P.: Recent progress and prospects for integer factorisation algorithms. In: Du, D.-Z., Eades, P., Sharma, A.K., Lin, X., Estivill-Castro, V. (eds.) COCOON 2000, LNCS, vol. 1858, pp. 3–22. Springer, Heidelberg (2000)
Gale, E., Adamatzky, A., de Lacy Costello, B.: Are slime moulds living memristors? arXiv:1306.3414 [cs.ET] (2013). http://arxiv.org/abs/1306.3414. Accessed 7 Jul 2013
Krieger, J., Spitzer, S.: Non-traditional, non-volatile memory based on switching and retention phenomena in polymeric thin films. In: Proceedings of Non-Volatile Memory Technology Symposium, pp. 121–124 (2004). doi:10.1109/NVMT.2004.1380823
Lu, C.-Y., Browne, D., Yang, T., Pan, J.-W.: Demonstration of a compiled version of Shor’s quantum factoring algorithm using photonic qubits. Phys. Rev. Lett. 99(25) (2007). doi:10.1103/PhysRevLett. 99.250504
Martín-López, E., Laing, A., Lawson, T., Alvarez, R., Zhou, X.-Q., O’Brien, J.: Experimental realization of Shor’s quantum factoring algorithm using qubit recycling. Nat. Photon. 6(11), 773–776 (2012). doi:10.1038/nphoton.2012.259
Nakagaki, T., Yamada, H., Tóth, Á.: Intelligence: Maze-solving by an amoeboid organism. Nature 407(6803), 470 (2000). doi:10.1038/35035159
Nielsen, M., Chuang, I.: Quantum Computation and Quantum Information. Cambridge University Press (2000). ISBN: 0-521-63503-9
Rivest, R., Shamir, A., Adleman, L.: A method for obtaining digital signatures and public-key cryptosystems. Commun. ACM 21(2), 120–126 (1978). doi:10.1145/359340.359342
Saigusa, T., Tero, A., Nakagaki, T., Kuramoto, Y.: Amoebae anticipate periodic events. Phys. Rev. Lett. 100(1) (2008). doi:10.1103/PhysRevLett.100.018101
Sel’kov, E.: Self-oscillations in glycolysis. Euro. J. Biochem. 4(1), 79–86 (1968). doi:10.1111/j.1432-1033.1968.tb00175.x
Shor, P.: Polynomial time algorithms for prime factorization and discrete logarithms on a quantum computer. SIAM J. Comput. 26(5), 1484–1509 (1997). doi:10.1137/S0097539795293172
Smolin, J., Smith, G., Vargo, A.: Pretending to factor large numbers on a quantum computer. arXiv:1301.7007 [quant-ph], 2013). http://arxiv.org/abs/1301.7007. Accessed 7 Jul 2013
Strukov, D., Snider, G., Stewart, D., Williams, R.: The missing memristor found. Nature 453(7191), 80–83 (2008). doi:10.1038/nature06932
Acknowledgments
We thank Andy Adamatzky for many formative discussions about this work, for sharing his expertise concerning Physarum experimentation and computation, and for his kind invitation to contribute this chapter. We thank the anonymous reviewers of this work for their useful comments. We acknowledge the generous financial support of the European Commission, which funded the author’s position when undertaking this research.
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Blakey, E. (2016). Towards a Non-quantum Implementation of Shor’s Factorization Algorithm. In: Adamatzky, A. (eds) Advances in Physarum Machines. Emergence, Complexity and Computation, vol 21. Springer, Cham. https://doi.org/10.1007/978-3-319-26662-6_23
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