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Quantum Grammars. Part IV.1: KMS States on Quantum Grammars

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(1)Quantum Grammars. Part IV.1: KMS States on Quantum Grammars Vadim A. Malyshev. To cite this version: Vadim A. Malyshev. Quantum Grammars. Part IV.1: KMS States on Quantum Grammars. [Research Report] RR-3702, INRIA. 1999. �inria-00072966�. HAL Id: inria-00072966 https://hal.inria.fr/inria-00072966 Submitted on 24 May 2006. HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés..

(2) INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE. Quantum Grammars. Part IV.1: KMS states on quantum grammars. Vadim Malyshev. N ˚ 3702 May 31 1999 ` THEME 1. ISSN 0249-6399. apport de recherche.

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(105) Nau. 

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(109) N.  ,

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(114) N,. 

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(121) lq_^§LCsw– \ GIŠ ´H LaLaq£s5 \ H Š \ LIH˜ª Š Š\Vl U‹ LlLCc sp\ €oU“\npH Wor]M L Š$ swXŠ U¦U ŠNoU – e M GIeFH c L¦n H_ª)Š WordM \ e Š l\& ( e ™ s µ Š Wo?²s œ ¥ Š U‹LaXr]s WFŠXp\“U`W L2c ^z\ H LaLI”w X Šewjkc Wgq lLCI swI \¡YZY[\ Y Ie WFr e  Š M/Wg\ cVŠ Y -™ B§œ ™ m # œ T0^'™ miM N{œ (™ m`M Nkœ _ ^ m (™ m _ N{œ —£Š L/U H WgŠll(H rCWgll\ H_Š U e nL c Wo\ ewc W e s_c L®‘npH W c \ Š rlL e nL c e Wo\ ewc ŠŽe¥?\ H_Š Š U &oH npW c \ Š Š r]lŽL & Š U¡e spWo\VX c Wgl\ e eTWoU“e U e ® r Š Wg\VL ŠŽe \ WUVnWorL/”ZŠŽe XpWosw\“Wp–5ªme L L/c \ L¦ŠŽe U“npWor]Š L LCj HlX_\ eoŠ s U¡\ L¦U H M/n_lLCUV\ Š s_LT™›qX_L¦c \ s_Š L q M e LaH spU c s?œ )Lc YpnpWFspU s[Wospq|r Mn LaUVU s™ s[LaWor n sw\ œ – —˜L¡U WolŽl7´?sq \VUUVn?LCr]\ XpM s Wos \ L npWon?L – upD– o e spU Š q_L c RTJV;I X;]Wosq|\ H L¡Œ e ll e  Š s_€/U“X  UV\ Š \VX_\ ŠŽe spU I  IPI X IPI  I X.I  d  I XdI  I. GIH LƒU“X  U“npWor]Lg$ f €oLas_L c Wg\VLCq  ^z eFc qpU¡ Š \ H LYpWor]\`l^ e s_L[U`^_M  e l Š U Š swjkW cVŠ Wosw\]– GIHpŠ U H WoM Š lŽ\ e s Š Woszr]Wos  L Š sw\`L c n c LC\`LCq WFUPW|M Š Y_\VX c L e Œ \ H L/n c L]j ŠŽe XpU+npX c LUVnpWFrL H WoM Š l\ e M Š WFs eB n?f L WFc Wgsp\ q‹eFc\ H Š L2shq &)(Š U“re c s L]\`WL¨´?lWFs n_Š l\`WFL r U`Š WoLCsN\]– B ©Z( Š M ¥k\ Š H lW L¨c Œl^oc LC¥pLd\ Hs L5e s U“cX LC lWoU“np\ Š Woj r]Š LCUV\ $2Š r (‹e s_ LŠ ®‘\ q H Š M LYpLaWFsprU \`ŠŽel^ƒspWg\‘l © e r HpU`^_cVeM La q e Š s_lU€oL c r ewcVc LCU“n e spqpU2\ e \‘ e MWo\`\VL c npW c \ Š r]lŽLaU]– –¡™ me spr e M/M5X_\“Wg\ Š jF L " e r $ U“npWor]L{œ Q LC\ RT V1I X. ]WFspq|r e spU Š q_L c ¦U“X  UV\ Š \VX_\ ŠŽe spU N )cI  IPIhX y )XIPI  I X“u )   q,X8)X   ªP L c L+eF\ c H L c LPW c ( LI\‘ e‹Š swjkW cVŠ gWosw\ e s_GIL]®°H nW c c \ Š rc lL+UVnpe¡WFrŠ LaU $ c`Š X!$ – " e eFc Lc YpWoŠ M/n_lLoX¥ $  Š U¨€oLCspL c Wg\`Laq ^5 qpU IhX.I T'IPI X YZY[YZeFX Ic X YZYZYŽ– L LIW L2Š \‘ sFc jkW Wosw \Y \‘ eF®°c nW \ rlL2U“npWor]LCU $  T $  c / $ X!$  TQ$ /a$  % – " LYpWoM/n_lLo¥ $ U€oLas_L Wg\VLCq ^[ qpU I  X. X.& x–  Ÿs[€oLas_L Wgl Œ eFcY  LC WoYr H LC joLC s Y m \ H L c Li\‘ e ©wŠ ŠswjkW Š cVŠ cWFsw\ƒ Yewy cm e ®‘npW c \m Š rlH L cUVnpWFrc LagU $ em  Š   ¥IcVŠ €FLCs_L c Wom \`Laq  Y ^  c ewŠ c qpU I  Y[YZY I  ¥oWospqN$    – M lW lŽ^¡Œ qpq \ L LW Ld\‘ swjkW WFsF\d™:y _TN{œ°®°nW \ rlL UVnT WorNLa5UX y $ X YZYZY[X \¦  dXeL $ H  W{jo L   Y\w5– ™ \ " eFM c N{W œ c g  7 Š \ f c W m c ®‘^ npW \ c \ Š rly¦L  U“npŠ \ WoH r]LCUVUCX –  GIU`\ HŠ \“XX_U]\ ¥Še \ sHY FUL UV\V WFsp-qpW X c-qi  U“nLa¨rŒ \ ewc c XpM LCWFr e H Œ Y Y Y xzylzxr{.

(122) Nkv. 

(123)     . \`Lae spU ewc n c cVe Š qpXpr\“UŠ H WFH U \ H Lmc r eFc“c LCU“n e spq Š  s_€)M X_l\ Š n_l Š r Š \ Š LCUCH – cGIH_Š U Le YpWo$ M/n_lL+U“Xp npn eFc \VU \ H H LPspWFM eL e& s_s_L]®°Ln®‘npW Wc \ Š \ rlrL¡lLU“& npWFs¦rLC\ UC– L+´ U`\¨LYpWoM/n_lLo¥ Lar]WFXpU`LPdL+€FL]\ L LPW " r U“npWor]L 3¨Q \ LCUVLP\‘ GIH_Š U+U H_e +UW c`Š r H UV\ c Xpr]\VX c L e Œ(\ H L Š sw\ cVe qXprLaqƒª)WoM Š l\ e s Š WospUC– D0.  N < > Np:=<  35 : NS:$ONO7 _> 53NOG >_A;:w? . —˜c Lz€ Š jFcL H L c L˜c W joe L c ^ Š U Hpc eFc \ e joPeL c j Š L] H e Œ e \ eH L c npWFnL c c UƒH  H L c L c ewL]c lWg\VLCq e q_^_spc`WFŠ M Š r]U e Œ5q Š U“r cc L]\`Š L UV\ Xpr]\VX HLCU|dL L˜r Š spU e q_L LaqK– e c \`L˜\ cWo\TM e UV\npŠ Won?c L U W{ŠjoL˜M L£€oL Š McVL]e \  rzWFspq WolŽ€FL W r WFUVn?LCr]\\ WFsTWFspWgl^Z\ r s_LF–( Ÿs X npWon?L 2L‹r spU q_L Laq|M/W s_l^ƒWosWgl^w\ r¡n lŽLaMUC–. |¸ ¸ †  %' %±„C %›ˆ ¸ ?¼   % ¼ S)L]\VL c M Š s Š U`\ Š r2LCj e lX_\ ŠŽe s e Œ? eFc qpU Š U e s_L e Œ?\ H L+M/W Š s U“X  f Lar\VU e Œ \ H L¡c r e $ M/enpXp\`L c Š U“r Š LCsp«‹r]Loe – Ÿsƒe r ewe c`M/e npX_\VL c UVr Š La$FspŠ frL¡M/W c%c $ LCq/€ c meWFn H U2WoH spq[c \ HŠ L Šc q_LCc \`L c M Š s Š UV\ Š Š rPLCe j e lX_\ ŠŽe s c 2L Š L s +HpszŠ U H sprL Š lM c € H jw® mc U“nLaspU c npWoe n?L  :–  Š \ e L L H U¡WƒlW c%$ €FL‹´L]lc q szH r M/npX_\VL U“r LCsc rLoŠ¥I Š Š r UVe \VXq ŠŽLae U[€ e WFn € Š WoM/M/W Š U® l r]Wgl)q_e ^_spWoMc r]HU Š Œ‹\ L£M/W LaGIq H € WFn Š UC– c  Ÿs H˜ ygx,Š  qpL]\`L M s U`\ r¦LCj lX_\ s Œ+rlWoU“U r]Wol¨U“n s UV^_U`\VLCM/U s’€ Won U UqpL´?s_LaqK– L WoU r/€ WFn U ´YZLaH q  ewe c \“W $ Š Las cc Wospq e ŠMlŽ^j Š W c WFspq e M"Š€ c H WFn eH \ H L eFc`c e ^¦ n cV e Š r]Š LCqpX c LF ¥w\ H Wg \ Š U²Œ ewc ´YZLaq[UVL]\ e Œ'jFL c \ Š rLCU LaWor spq U+q W{+s sq_LCn?LCspqpLCsw\`l^ƒ \ U ML‹n W l \‘^|x NF–  ‡_† 2 ¼  #† ¼ ˆ ¸ „a„ ¸ „ QPWFspq e M € c WoM/M/W c U7dL c L²r e spU Š q_L c LCqmLCW c l Š L cŠ s‹r e M/npX_\VL c U“r Š LCsrL²r e sw\`LY_\C¥ WFU)hzW c%$ge j n cVe rLCU“UVLCUC– IX_\‹”wXpLCUV\ Še spU c L]lWg\`Laqi\ e \ H L/\ H L c M e q_^_spWFM Š rl Š M Š \‹WFnpnLaW c LCq e s_l^ Š s  Ng¥?y :– % C‡p† ¹£ ¸ 7¼  ' % ¼ £‡ ½’ˆo‡ „a‡ '„C ]† (ˆg '† ¸ © Xpr H L]j e lXp\ Še s Š U\ H LM/W Š s e  f Lar\ Š s ”ZXpWFsF\“XpM rGIe H M/npX_\ Š spc € Š e sz\ HŠ L¦cr e MnŠ X_\`L c c U“r H Š LCsrL/r e sw\VLY_\)WFspe q  f Š sz”ZXpWFH sw\VXpe M-€ c W{j Š \‘^ Š Šs˜H W|nH H ^_U Š rCWglr e sw\VLŠY_e \]– e s[LP\ H lWgL‹\`U`\VLCL \ e r ŒspWgUll§q_n L e U“UdU UVŠ  n lL)sƒU“€ n Š Wos[n € c WoWFnUdH WU]”Z– XpGIWoH swL¡\VXU“M”wXW c L LCe r]Œ'\]¥Z\ H \ L¡XpU W{jos_LPLmŒ±Xpq_spLaWgrl\ U¨ŠŽe  s|\ qpL´?\ s_L)LaUIW W{jon LIc`e Œ±XpWs r\Š l Š \‘s^ q Š UV\ © cVŠ  Š X_\ Še s e ewsc $ U“n Š sTc€ c WFHn H UC– Š H Š H Š c Š Š e H Š $ Š s\ H Ln jos L c s_\ LCŠ r]\‘LCU¯œK2U¡L c ™›L € Š swWo\ n cVe U+qpXp rLa\ q Š UVs5n n spH UC^_¥ U Š rCWoU l ŒÁ® ^‹sF\VQL]€F– Lb LaU]s ¥pcVl e j UVL s_€ N{v°s|– Ÿ\ s Ln Hl ^_s U Š UCr]¥?UWFs spe q  U \ e H ML c L Le W n?c L2L c M/Wg\ WoeFswc ^ U jkW cc`Š ŠWoswc \VU e Œ \ c H L ”wXWosw\VXpMLCŠ j e clX_\ ŠŽe sK¥c q Š ŠUV¶ rCXpU“ŠeU`Laq Š c s  y Š +p¥7c yFvZ¥?uoxpe ¥Nax_¥',N Z¥you_Š ¥7H y p¥?uN  WoUI2L]c llWFU Šc s LaW HpŠl L H npe Won?L U]¥7e U`LCLS Qv:eF–Pc — q_¶CŠ L €FLCs_c L Wol c Wo\ e sW LŠ q e UVr LC\`L5r M/n_lLY_LCUI \ W[r]WFXpUVWol'UV\ Xpr\“X L  r r X_lqM q_L]l LCsw\ WFs|UV\ Xpr]\VX L sTMWFs Œ lqpU]–  ¼  #  ¸ ¸  ¹ # ¼ „ % % 7¸ „ ¸  % ?† ¼  #(„ GIH_Š U(rCWoUVL Š U Š sw\`L c LaU`\ Š sp€mqpX_L\ ePŠ s_L]j Š \VW  lŽLs eoŠ UVL2r e M Š sp€ Œ cVe M \ H L‹LasFj Šc`e spMLCsw\ e Œ(\ H L”ZXpWFsF\“XpM UV^_U`\VLCMT– © L]Lq Š U“r]XpU“U ŠŽe s e Œ\ H LCUVLn c`e  lLCM/U Š s  uo,u °–. ¼ '%' ¸ ‡_†  ‡_† 2 ¼ #(† ¼ ˆ ¸ „C„ ¸ „ GIH L c LmrCWos  Lm\ c WFspU`Œ eFc MWo\ Še spU e Œ'n cVe  W  Š l Š \‘^MLaWoU“X c LaU e s U“c n Š s¦er e MnpeolŽL]Š YZLaU]¥Fc  H_Š r eFH c r]WossŠŽe e \  L c LCqXpH rLaq/\ e c W¡hze W ce $oe j5n cVe rLCU“UI™›\ H WoH \ Š U²Š \ H L]^/W c LPc s e \¨€ Š joLCs  ^ e WoŒ(sU“qXpr HM q_n ^_spswWoM\I\ Š r]WoU spe U`s_Œ L‹r]M/WoWgsƒ\ ´?spspq U œ¨Š sTWos\‘q| e \ ®°q L]Š ^TMW LCspL‹U Šs e sp\ WolŒ ”ZXp”ZWFXpswWo\VswXp\“M XpM € c MW{j LCŠr \‘^oWo¥_s UVL]r]L Wol§ u sp:–Wg\“X LF%– 3²YpWoM/n_lLCU . . z{z`|#}~€‚ƒ.

(124) N.  ,

(125) . *g*P *PI*+  N  – ! – h£Wgl^ZU H LCj7–aQPWFspq e M c WoM/M/W c U]–]QPXpU“U Š Wos‹h£Wg\ H – © X c jFL]^_U]¥ZNa}o} ¥]j–gup¥ me –ky_¥ZNCxZva® NCu?–  y  – ! – h£Wgl^ZU H LCj7– © \ e r H WoUV\ Š r|L]j e lXp\ Še s j Š W˜€ c Won H € c WFM/MW c UC–(  Q+  ! bdc LCn c`Š sw\ me –²uou x¥ h£W c r H Na}o} – © LCLPWglU e S Š U“r c L]\VL+hzWo\ H LaMWo\ Š r]U²WFspq ! npnpl Š r]Wo\ Še spUC¥?Na}o} ¥ Pe –Fu¥FnpnK–Zy_va®Ÿy+Fyp–  u  – ! – h£Wgl^ZU H LCj7– bdcVe  W  Š l Š \‘^iW cVe Xpspqz\ H LS")XWosw\VXpM  c W{j Š \‘^o–  QP  ! bdc LCn cVŠ sw\ Pe –'uw}ou¥ Na}o} –   ! – – «‹e lM e € eFcVe j¥ – ! – )UVn?LCsU $ ^o– s\ H L s e \ ŠŽe s e Œ Wgl€ eFcVŠ \ H MT–pQPXUVU Š WFsThzWg\ H LCM/Wg\ Š rCWgl © X c jFL]^_UC¥KNC}  ¥Zj–KNCu¥ Pe – ¥pnpnK–?uk®Ÿy –   – ! – h£Wgl^ZU H LCj7¥Q– ! –ph Š s_l e UC–  Š p UIQPWosq e M " Š L]lqpUC– « lXw2L c –KNC}F}pNo– c Š e  Š e c ewc !  c © Š Š H Š  + 3 © n – c`Š  s_€FWgL \Vc \`LClŽLl c C¥ lWgS€–K–+NaQ }wvg}p– sU sK– nL Wo\ lŽ€FL WoUWFspq ")XpWosw\VXM \VWo\ U`\ rCWglhzLCr WFs rCU]–  ,v ¡ ¯– S)WFq Š rg¥ « – b¨Š U $

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