COMPLEJIDAD COMPUTACIONAL DEL AUTOMATA DE MAYORÍA CON SIGNOS

Transcription

1 COMPLEJIDAD COMPUTACIONAL DEL AUTOMATA DE MAYORÍA CON SIGNOS Pedro MONTEALEGRE Univ. Orléans, INSA Cenre Val de Loire, LIFO EA 40, Orléans, France Desafíos maemáicos e informáicos para la consrucción y análisis de redes de regulación biológica CONCEPCIÓN DE ABRIL, 016

2 MAYORÍA BIDIMENSIONAL CON SIGNOS x { 1, 1} n N[v] ={v, v n,v s,v e,v w } (vecindad von Neuman) A =(a u,v ) u,vz, a u,v { 1, 1} (F A (x)) v = 1 P if un(v) a u,vx u > 0, 1 oherwise

3 COMPLEJIDAD COMPUTACIONAL A =(a u,v ) u,vz, a u,v { 1, 1} Predicion( ) A Inpu: A periodic configuraion x and a sie v. Quesion: Does here exiss T>0 such ha (FA T (x)) v =1 Capacidad de simulación de circuios Circuios monóonos planos Circuios planos Circuios monóonos planos reuilizables Coa inferior de la complejidad NC 1 Hard PHard (TuringUniversal) PSPACEHard (InrínsecameneUniversal) 3

4 (a) A wire is consanly blinking (lef op) and ransmis a signal by being consumed (lef boom). A urn (cener lef). An obsrucor prevens a signal (cener). A muliplier (cener righ). These gadges can sraighforwardly be composed o simulae a wiring oolki W. Plus a diode (righ). (b) OR (lef), AND (cener) and FUSIBLE (righ, wih a disinguished cell) gaes. These gaes have a paricular feaure: he four sides are undi ereniaed inpus/oupus. For example, he AND gaes wais for any wo signals o arrive, and hen sends a signal o he wo remaining sides. One can easily ransform hem ino classical gaes using diodes. Mayoría esá en P y es NC 1 Hard 4

5 CASO NO UNIFORME (MOTIVACIÓN?) Exise una mariz simérica (i.e ), al que Predicion( A F A a u,v = a v,u ) es PCompleo (TuringUniversal) (b) AND gae, OR gae and ( wes)and(norh) gadges. Inpus are from norh and wes, and oupus o eas. A F A Exise una mariz para la cual Predicion( ) es PSPACECompleo (InrínsecameneUniversal) 5

6 SI ES UN AUTOMATA A! W A =(C, N, S, E, O) { 1, 1} 5 a b a b W A =(1, 1, 1, 1, 1) 6

7 ( 1, 1, 1, 1, 1) (1, 1, 1, 1, 1) ( 1, 1, 1, 1, 1) (1, 1, 1, 1, 1) ( 1, 1, 1, 1, 1) (1, 1, 1, 1, 1) ( 1, 1, 1, 1, 1) (1, 1, 1, 1, 1) ( 1, 1, 1, 1, 1) F F3 F3 F4 F4 F5 F5 F6 F6 Symmeric Symmeric Symmeric Anisymmeric Anisymmeric Asymmeric Asymmeric Asymmeric Asymmeric 4 SI ES UN AUTOMATA Hay 3 reglas, que se reducen a 1 salvo roación. six Symmeric usmca second line. Figure 7 The six Symmeric usmca are called, from lef o righ in he firs line F1, F, F3 and F 1, F, F 3 in he second line. 4 On he complexiy of planar hreshold dynamics On he complexiy of planar hreshold dynamics Figure 9 The four Asymmeric usmca are called F5, F6 (in he boom line). is equivalen o F1 (resp. F4, resp. F5 ): hey can simulae circu are called, from lef o righ in he firs line F1, F, F3 and gaes he same Figure 8 The Anisymmeric usmca are under called from lef omode. righ: F4 and F 4. Siméricas Anisiméricas Asiméricas Proof. Le : { 1, 1}Z æ { 1, 1}Z bedefined as (x) = x I Lemma. Le F be a symmeric (resp. anisymmeric, usmcahen F 1}Z and for every i œ {1,, 3, 4,resp. 5, 6},asymmeric) any configuraion x œ { 1, NC1Hard, en P,? (en PSPACE) PHard, en PSPACE, Fi = F i. Indeed, noice ha if W is he sign marix of rule Fi Ciclos de largo a lo más Puede ener ciclos Puede ener ciclos of rule F i,9hen W. This implies for every Figure The W four=asymmeric usmca ha are called F5, Fv6 œ (inzhe ÿ ÿ boom line). de largo n superpoly wuv xu = ( wuv )xu Fi (x) = uœn (v) uœn (v) is equivalen o F1 (resp. F4, resp. F5 ): hey can simulae circu and inducively, gaes under he same mode. q q Z Z w (F (x)) = [( 1) = (Fi x (x uv u 1} iæ { 1, (v) 1} uœn (v) wuv as Proof.F Le uœn : { 1, be defined (x) F be a symmeric (resp. anisymmeric, resp. asymmeric) usmcahen q 1 7 Z) (F = ( 1) ( w for every i œ {1,, 3, 4, 5, 6}, any configuraion xuœn œ (v) { 1, 1}uv and Anisymmeric usmca are called from lef o righ: F4 and F 4.

8 Y SI NO ES MAYORÍA Las equivalencias enre casos uniformes siméricos, anisiméricos y asiméricos son válidas. Caso ANDNOT : (F A (x)) v = 1 P if un(v) a u,vx u =5 1 oherwise Exise una mariz Predicion( F A A anisimérica (no uniforme) para la cual ) es PSPACECompleo (InrínsecameneUniversal) 8