A Dynamical Quasi-Boolean System

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1 ULETNUL Uestăţ Petol Gze Ploeşt Vol LX No / - 9 Se Mtetă - otă - Fză l Qs-oole Sste Gel Mose Petole-Gs Uest o Ploest ots etet est 39 Ploest 68 o el: stt Ths e oes the esto o ol theoetl oet: the oet o Qs-oole sste The oet e se oess tht oles o-el les The e s to eess these tes o les s lot oole les The oet eesets the theoetl et the oel o the ole le oess se o the sste theo ewos: Qs-oole sste oole to seo-oole to toto ste o the eos eet ote-ssste sttol sstes eeloe the lst ew es the stes the el o e-le he eele oles se the holst ole etes o the sttol oess The oess o oel el sttos tht e ot oee le les s lt oe les ew oetol ohes ths e thee s esete ew oet el the Qs-oole sste whh ws toe the tho [6] The oet ws se to oel the sttol oess t ts lto e s ot lte to tht The le oess s oele stt wth the esto o tot elto sste The tellet sttol sste toe [6] ses oe elto oe to ze the eets o the ettos o the le oess to ll the eo (the eee etwee the eeee le the ott le) The t o the sste eesets the eeee le wht the stets he to elze te the le (sttol) oess hs ee she The stte (the sste s seqetl) s ee the stets owlee slls The tos tht et the le - otto ols eos owlee teest teh stles le stles lssoo lte ets eotos hoes et - eeset the ettos [6] lel esto o the oet s esete the wo o the eeee [7] The oet ws e Qs-oole sste ste o seo-oole sste ese the eqtos o the sstes e oth seo-oole eqtos oole eqtos Pseo-oole tos e those set tos whh e ee o te o set e e lose le ole shll sel tteto to the se o lt-le olol eesettos ese o the lo wth oole tos these tos wll e lle seo-oole []

2 Gel Mose Sh tos l jo ole the otzto oles otol theo es theo oetos eseh sete thets oess oel He hs ott otto the st o these tos [] [3] [4] [5] olete lst o hs ltos e o t the eeee [8] The oet o Qs-oole sste s eel theoetl oe theeoe t e se oess whh teees wth el oole es The esto o the l Qs-oole Sstes ths hte we ee the oet o Qs-oole sste whh ossts o seo- oole eqtos The ethos o sol oole seo-oole eqtos sstes o eqtos e o otle oo o He e [] The oet ws set o the e tht the ott o the sttol oess ot e eesse s el les t e llte s seo-oole ols [] eto sete oole sl s to o to the ollow shee (F ): : Z { } () t { } t Z Z s the set o tee es t F ele o set oole sl eto tl sl s o the ollow o: : Z M M Z eto 3 tl seqetl sl s : P s te set o stes { } eto 4 Qs-oole seqetl sl s : : P M M el es whee P { } : P M M Z whee s te set o stes s the set o l l eto 5 Let s ose eto ( ) Qs-oole eto wth We lle the el t o eto (ote wth wth ) s the ollow ols: wth eleets lle l l l ( ) ) the oole t o eto (ote

3 l Qs-oole Sste 3 l l ( ) So Qs-oole eto e ese o to the ol: ( ) eto 6 l seqetl Qs-oole sste s o tos ( ) ee o to the ollow ol: whee: ( ( ( ) ) ( ( ) ) ( ( ) )) ( ( ( ) ) ( ( ) ) ( ( ) )) 3 ( ( ( ) ) ( ( ) ) ( ( ) )) ( ( ) ( )) ( ( ) ( )) ( ( ) ( )) ( ) ( ) 3 X 3 s the sttes set U e the ts sls V s the ettos set Y s the t sls set e seo-oole tos e el tos e oole tos 3 e seo-oole tos e el tos e oole tos 3 3 P s te set o seqees e Qs-oole sls Osetos The te o seo-oole eqto s se ese t s oo te the oole le The sls oesse Qs-oole sste e etos wth lot oole es The tsto tos e seo-oole lot oole tos The eso o hoos these otos o tos s tht the el stto ot e oele s ol lot les oole to e eesete s seo-oole to So to sl the oel the tos 3 3 e oe the eqtos o Qs-oole sste e ese s the ollow ol:

4 4 Gel Mose Us the teolto ol o the seo-oole tos the eqtos o the sste eoe: δ γ δ δ δ γ γ γ L L Oseto The eqtos o the sstes e ese s le olols tht ee o the eos sttes ettos ts o the sste The sstes e lsse o to the let o the tos les sste s lle le seqetl Qs-oole ll ts tos e le lest oe to s o-le the sste s lle o-le seqetl Qs-oole sste The esto o Le Qs-oole Sste le Qs-oole sste s ese s the ollow sste o eqtos: oe to sl the ole the ettos e teete s etel eses (ts o the ste sste) So the eqtos o sste eoe: To sl the ottos e se ste o

5 l Qs-oole Sste 5 le seo-oole eqto e wtte s the ollow otto: l l l l z z z z The stte o the sste s ese s the ollow eqto: The ott o the sste s ese the ollow eqto: The ollow ottos e se:

6 6 Gel Mose The eqtos o the stte the ott o the sste eoe: The tes e lt s ollows: Α 443 L Α 443 L Α 443

7 l Qs-oole Sste 7 Β 443 L Β 443 L Β L 443 L L 443 L 443 The stte s eqto eoes: Β Α Α Α Α Β Β Β Α Α Α Α Α Α Β Α Β Β Β Α Α Α The ott s eqto eoes: the sstes theo the oe lw s ee s eeee: G F whee: X s the sttes set U s the t sls set V s the ettos set G F e tes wth tee oeets The oe lws e eesse s seo-oole eqtos sste

8 8 Gel Mose Oseto el eto ( ) e wtte s oole eto s the ollow oto: q q q q q > q The esto o No-Le Qs-oole Sste the esto o o-le Qs-oole sste oe ses e te to ot o to the let o the tos ole stto s ee the se o o-let o the el tos ths stto hs to e ste o to the te o tos sle se s the se o sste wth le el tos ese s ollows: s s ( ( ) ( s ( ) ) ( ( s ( ) ) ( s ( ) ) ( ( s ) ( s ( ) ) ( ( s ( ) ) ( s ( ) ) ( ) s ( ) ( ) ose the tsoto o seo-oole to s s o olols the sste e ese s the ols elow: o z oz o z z o Fo hooeet we osee ll olols h the os: β β The e o these olols s The sol o eesets the e the olol ws ote wth z olsos The oet esete ths e s theoetl oet tht eesets eelzto o the thetl eqtos es the oet o sste Ths oet s sel oess o oel sttos tht ot e eesse s ol el es ele s tht o oel h eho the le oess eeees oos E He PL - Pseo-oole Otzto htt://totese /~oos/pes/-m-h esse o the st M He (ă es) PL e S - Methoes ooleees e ehehe Oetoelle o Ps 97

9 l Qs-oole Sste 9 3 H e P L - ote o the ootot o seo-oole tos Mthetl Methos o Oetos eseh Vol 8 No H e P L e S - Pseo-oole Po Oetos eseh Vol He PL - Pseo oole Po ltos Lete Notes Mthets Vol 9 Se Vel el/heele/new Yo M o s e G - Fol esto o the Sste Theo se e-le tetol Jol o otes otos & otol Vol 8 7 Mose G - ottos to the Moell otoll the Ole sttol Poess Us tl tellee Tehqes otol thess 8 8 ***-Pete L He Lst o ltos htt://totese/lh_ltos esse o the st ezt Sste s-oole Î est tol este ezettă esee oet teoet ol oetl e sste s-oole oetl ote tlzt î oele oă oes e lă lo e-ee ee lă oetl este e e este lo jtol lolo ele ş ooleee oetl eeztă etl teoet î oele oesl e îăţăât ole zt e teo ssteelo

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