On a Contribution of Membrane Computing to a Cultural Synthesis of Computer Science, Mathematics, and Biological Sciences

On a Con ibu ion o Memb ane Compu ing o a Cul u al Syn hesis o Compu e Science, Ma hema ics, and Biological Sciences Adam Ob uÃlowicz Ins i u e o Ma hema ics, Polish Academy o Sciences, ´ Sniadeckich 8, P.O.Box 21, 00-956 Wa saw, Poland e-mail: [email p o ec ed] .pl Summa y. Some opic con ibu ion o memb ane compu ing o a cul u al syn hesis o compu e science, ma hema ics and biological sciences is p esen ed. 1 In oduc ion A e a mo e han decade o he esea ches in he a ea o memb ane compu ing, c . [19], ini ia ed by Gheo ghe P˘aun i is wo h o p opose a discussion abou a con ibu ion o hese esea ches o a cul u al syn hesis o compu e science, ma hema ics, and biological sciences. An aim o he p esen pape is o ini ia e he discussion ocusing on hese i s as- pec s which a e amilia o he au ho whose esea ch a ea comp ises ma hema ical ounda ions o compu e science and ounda ions o ma hema ics i sel . The heme o he discussion was inspi ed by [11] and [2], whe e he goal o [11] is o p opose a amewo k ha should allow a heal hy and posi i e ole o specu- la ions in ma hema ics al hough adi ional ma hema ical no ms discou age spec- ula ion whe eas i is he ab ic o heo e ical physics. Thus a cul u al syn hesis o ma hema ics and heo e ical physics is unde s ood in [11] mo e o less explici ly as a mu ual inspi a ion be ween he discussed a eas. We poin ou he e ha specula ion in ma hema ics can be ein o ced by com- pu e expe imen s, c . p o oca i e a icle [10], which could ex end he discussion in [11] o compu e science wi h a ega d o he ela ions be ween ma hema ical hinking and algo i hmic hinking, c . [12]. The esponses in [2] con ain a ious opinions abou a ole o specula ion in ma hema ics om a de ense o an impo ance o p oo s, see S. Mac Lane’s esponse con inued in [14], o an accep ance o specula ion as possessing equal igh s o p oo s. The esponse due o R. Thom in [2] sugges s also a cul u al syn hesis o ma hema ics and biology. 288 A. Ob uÃlowicz We p opose a looking o wa d app oach o a cul u al syn hesis o he a eas men ioned in he i le o he p esen pape , whe e a cul u al syn hesis is also unde s ood as mu ual inspi a ion. In he nex sec ion we ou line some opic con ibu ion o memb ane compu ing o his cul u al syn hesis. 2 Topic con ibu ion We poin ou he ollowing opics as a con ibu ion o memb ane compu ing o he discussed syn hesis: A) biologically inspi ed sel -assembly ( andomized, c . [15] and [16]) P sys ems (wi h memb ane di ision and c ea ion) o sol ing NP comple e p oblems, c . [17] and [21], which ex end algo i hmic hinking by a new idea o a ( an- domized) algo i hm o a cons uc ion o a sel -assembly dis ibu ed sys em ealizing massi ely pa allel compu a ion, B) spiking neu al P sys ems, c . [18], wi h lea ning p oblem solu ion, c . [9], which a e a s ep owa ds digi aliza ion1o neu al ne wo ks wi hin ma hema ical neu- oscience, c . e.g. [3], C) ac al cons uc s gene a ed by P sys ems, c . [8], wi h P sys ems o ob aining homology g oups, c . [5], and a possibili y o expe imen wi h hem ( o p edic some ma hema ical esul s like in [10] o be p o ed wi h ma hema ical p ecision like in [14]) ia P-lingua p og amming en i onmen designed in Se ille, c . [4] and [20]. Conce ning A) one can ask o a ela ion o andomized P sys ems in [15] and andomized Gandy–P˘aun–Rozenbe g machines [16] wi h p obabilis ic compu ing de ices discussed in [1]. Re e ences 1. A o a, S., Ba ak, B., Compu a ional Complexi y: A Mode n App oach, Camb idge Uni . P ess, Camb idge, 2009. 2. A iyah, M., e al., Responses o [11], Bull. Ame . Ma h. Soc. 30 (1994), pp. 178–207. 3. Boh e, S. M., Spiking Neu al Ne wo ks, P o esso sch i , Leiden Uni e si y, 2003. 4. D´ıaz-Pe nil, D., P´e ez-Hu ado, I., P´e ez-Jim´enez, M. J., Riscos-N´u˜nez, A., A P- Lingua P og amming En i onmen o Memb ane Compu ing, in: Memb ane Com- pu ing, Lec u e No es in Compu e Science 5391, Sp inge , Be lin, 2009, pp. 187–203. 5. D´ıaz-Pe nil, D., Gu i´e ez-Na anjo, M. A., Real, P., Sanchez-Canales, V., A cellula way o ob ain homology g oups in bina y 2D images, in: Eigh B ains o ming Week on Memb ane Compu ing, Se ille 2010, ed. M. A. Ma inez-del-Amo e al., RGNC Repo 01/2010, Se ille Uni e si y, pp. 89–99. 1in a simila way as cellula au oma a digi alize physics acco ding o E. F edkin, S. Wol- am, c . [6], [7], [13] Syn hesis o Compu e Science, Ma hema ics, and Biology 289 6. F edkin, E., An in oduc ion o digi al philosophy, In e na . J. Theo . Phys. 42 (2003), pp. 189–247. 7. F edkin, E., Digi al mechanics, Phys. D 45 (1990), pp. 254–270. 8. Gu i´e ez-Na anjo, M. A., P´e ez-Jim´enez, M. J., F ac als and P sys ems, in: Fou h B ains o ming Week on Memb ane Compu ing, Se ille 2006, ol. II, ed. C. G aciani e al., Fenix Edi o a, 2006, pp. 65–86. 9. Gu i´e ez-Na anjo, M. A., P´e ez-Jim´enez, M. J., A spiking neu al P sys ems based model o Hebbian lea ning, in: P oceedings o 9 h Wo kshop on Memb ane Com- pu ing, Edinbu gh, July 28 – July 31, 2008, ed. P. F isco e al., Technical Repo HW-MASC-TR-0061, School o Ma hema ical and Compu e Sciences, He io –Wa Uni e si y, Edinbu gh, UK, 2008, pp. 189–207. 10. Ho gan, J., The dea h o p oo , Scien i ic Ame ican 269, no. 4 (Oc obe 1993), pp. 74– 82. 11. Ja e, A., Quinn, F., Theo e ical ma hema ics: Towa ds a cul u al syn hesis o ma h- ema ics and heo e ical physics, Bull. Ame . Ma h. Soc. 29 (1993), pp. 1–13, 12. Knu h, D. E., Algo i hmic hinking and ma hema ical hinking, Ame . Ma h. Mon hly 92 (1985), pp. 170–181. 13. K˚u ka, P., Topological and Symbolic Dynamics, Soci´e ´e Ma h´ema ique de F ance, Pa is, 2003. 14. S. Mac Lane, Despi e physicis s, p oo is essen ial in ma hema ics, Syn hese 111 (1997), pp. 147–154. 15. A. Ob uÃlowicz, P obabilis ic P sys ems, in: Memb ane Compu ing, ed. Gh. P˘aun e al., Lec u e No es in Compu e Science 2597, Sp inge , Be lin, 2003, pp. 377–387. 16. A. Ob uÃlowicz, Randomized Gandy–P˘aun–Rozenbe g machines, in: Memb ane Com- pu ing, Lec u e No es in Compu e Science 6501, Sp inge , Be lin, 2011, pp. 305–324. 17. P˘aun, Gh., P sys ems wi h ac i e memb anes: A acking NP comple e p oblems, Jou - nal o Au oma a, Languages and Combina o ics 6 (2000), pp. 75–90. 18. P˘aun, Gh., P´e ez-Jim´enez, M. J., Spiking neu al P sys ems. Recen esul s, esea ch opics, p esen ed a he 6 h B ains o ming Week on Memb ane Compu ing, Se illa 2008, web page h p://psys ems.disco.unimib.i /download/leidenGR65.pd 19. P˘aun, Gh., Rozenbe g, G., Salomaa, A., The Ox o d Handbook o Memb ane Com- pu ing, Ox o d, 2009. 20. The P-Lingua websi e, h p://www.p-lingua.o g 21. Zand on, C., Fe e i, C., Mau i, G., Sol ing NP comple e p oblems wi h ac i e mem- b anes, in: Uncon en ional Models o Compu a ion, UMC’2K, ed. I. An oniu e al., Be lin, 2001, pp. 289–301.