modular construction of finite and complete prefixes of petri net … · 2016. 12. 27. · modular...

24
Modular construction of finite and complete prefixes of Petri net unfoldings Agnes Madalinski, Eric Fabre To cite this version: Agnes Madalinski, Eric Fabre. Modular construction of finite and complete prefixes of Petri net unfoldings. [Research Report] RR-6412, INRIA. 2007. <inria-00204548v3> HAL Id: inria-00204548 https://hal.inria.fr/inria-00204548v3 Submitted on 15 Jan 2008 HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished 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 destin´ ee au d´ epˆ ot et ` a la diffusion de documents scientifiques de niveau recherche, publi´ es ou non, ´ emanant des ´ etablissements d’enseignement et de recherche fran¸cais ou ´ etrangers, des laboratoires publics ou priv´ es. brought to you by CORE View metadata, citation and similar papers at core.ac.uk provided by HAL-Rennes 1

Upload: others

Post on 18-Jul-2021

6 views

Category:

Documents


0 download

TRANSCRIPT

Page 1: Modular construction of finite and complete prefixes of Petri net … · 2016. 12. 27. · Modular construction of nite and complete pre xes of Petri net unfoldings Agnes Madalinski,

Modular construction of finite and complete prefixes of

Petri net unfoldings

Agnes Madalinski, Eric Fabre

To cite this version:

Agnes Madalinski, Eric Fabre. Modular construction of finite and complete prefixes of Petrinet unfoldings. [Research Report] RR-6412, INRIA. 2007. <inria-00204548v3>

HAL Id: inria-00204548

https://hal.inria.fr/inria-00204548v3

Submitted on 15 Jan 2008

HAL is a multi-disciplinary open accessarchive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come fromteaching and research institutions in France orabroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, estdestinee au depot et a la diffusion de documentsscientifiques de niveau recherche, publies ou non,emanant des etablissements d’enseignement et derecherche francais ou etrangers, des laboratoirespublics ou prives.

brought to you by COREView metadata, citation and similar papers at core.ac.uk

provided by HAL-Rennes 1

Page 2: Modular construction of finite and complete prefixes of Petri net … · 2016. 12. 27. · Modular construction of nite and complete pre xes of Petri net unfoldings Agnes Madalinski,

appor t de r ech er ch e

ISS

N02

49-6

399

ISR

NIN

RIA

/RR

--64

12--

FR

+E

NG

Thème COM

INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE

Modular constructionof finite and complete prefixes

of Petri net unfoldings

Agnes Madalinski, Eric Fabre

N° 6412

January 2008

Page 3: Modular construction of finite and complete prefixes of Petri net … · 2016. 12. 27. · Modular construction of nite and complete pre xes of Petri net unfoldings Agnes Madalinski,
Page 4: Modular construction of finite and complete prefixes of Petri net … · 2016. 12. 27. · Modular construction of nite and complete pre xes of Petri net unfoldings Agnes Madalinski,

Unité de recherche INRIA RennesIRISA, Campus universitaire de Beaulieu, 35042 Rennes Cedex (France)

Téléphone : +33 2 99 84 71 00 — Télécopie : +33 2 99 84 71 71

������������ ������������� ��������������������� ����! ��#"%$&��'���($����)��*��+�

�#��,-�'�����.���'�/�����0��������1#�

2�354�687�9;:=<>:@?BAC4�7EDFAHGJI�KLABMON�:=P�KL6

Q)RTSLUWV#X+Y[Z]\_^a`cbedfS�UWVEb+gh8UiU#jlkTmngLopkqdHbrtsfhpuvVd�b+wxmybvdHsfmnz=X�h8U

{|op}T}=h8sed|~cV�sfVEgHRlV�s�gHRTV�k=�J���l���O\��qopkqj@opsH`��p���8��\�����}lo���VLb

�������L���T����� Q)RTmyb>}lo�}=V�s�g�h�klbfmy~cV�s�b�~TmnbedfsHmnzTjcdfVE~Wbf`cbvdHV�U�b��8~cV��lkTVL~�o�b�o�g�h��n��VEg�dHm�h8kihp��gh8Ui}=h8kTV�kqdHbmnkqdfV�s�o�gdfmnkT��dfRTsHh�jT�8Rimnk8dHV�sf��o8gVEb���X�h�UW}=h�kTVLk8d�b���mnk8dHV�sf� o�g�VLb�opk@~W~cmnbedfsHmnzTjcdfVE~�be`cbedfV�U�b�o�sfV+UihT~cV��nVL~o8b#r�VdfsHm'klVdHbL�¢¡£dimyb�¤�VL���'¥akTh0¤¦dfRlopd�dHRTV�jTkc��h8�n~Tm�kT�§h��¨bfjlg�R©o§~cmybvdHsfmnzTjcdHVL~©bf`abedfVLU(� o�g�dHh�sHmybeVEb��mnkªdfRTV�bfV�klbfVWdHRlo0d#m�d�g�o�k¢z=V«V¬c}TsHVLbHbfVL~¢o�b�dHRTV�gh8UW}=h8bfm­dHm�h8k¢hp�)jTkT��h��y~cm�kl�8b�hp�)m�dHb#g�h�UW}=h�kTVLk8dHbL�Q)Rlmnb�� o�g�dHh�sHmnbfVL~W��h8sfU®hp�FdfRTV[jTkc��h8�n~Tm�kT�i}Tsfh0¯amy~cVLb�o#UWh�sHVxgh8UW}lo�g�d'sHV�}TsHVLbfV�kqdHopdfmnh�k�h���dHRTV�be`cbedfVLUdHs�o�uvVEg�dHh�sHm�VEb��co�kl~�U�op¥�VLb)m�d|}=h8bHbemnzT�nVxdHh�o�klop�n`cbeVxdHRTV�be`cbedfV�U°zq`�}lo�sed�b��Q)RTVx}@op}=V�s���hcgjlbfVLb�h8k�dHRTV[~TV�sHm�¯po0dfmnh�k«hp��o��lkTm�dfV�o�kl~�gh�UW}T�nVdHVx}TsHV�T¬O±C²JX�r�³�m�k�dfRTV[jTkc��h8�n~a´

mnkT�µhp�+o&~TmnbedfsHm�zljcdfVE~ªbe`cbedfVLU���^a}=VEgm��@g�o����n`���h�kTV�¤�h�jl�n~¢�nm�¥�VWdfhO~cmnsHVLg�dH��`¶h8zcdHo�m�[email protected]&²JX�r·mnk� o8g�dHh�sHmnbfVL~µ��h�sHU&��¤+m­dHRTh�jTd�gh8UW}TjcdfmnkT���ls�bvd�o�²JX�r;hp�tdfRTVW���nh�zlo���~cmnbedfsHmnzTjcdfVE~§bf`cbvdHV�U]opk@~µdHRTV�k� o8g�dHh�sHmnbfmnkT��m�dL�xQ)RTV�gh8klbvdHsfj@g�dfmnh�kµh���bfjlgHR¶o�¸vUWhc~cjl�no�s|}TsHV�T¬q¹�myb|z@o�bfVL~&h8kO~cVLsfmn¯am�kT��bfjTUWU�opsHm�VEbh���gh8UW}=h�kTVLk8d[z=VLRlo�¯qmnh�jlsHb¨¤�� sE� dE��dHRTV�mns�m�kqdfV�sf� o�g�VLbL�=dfR@o0d�opsHVigh�UWU#jTklmngLo0dfVE~µdfh�dfRlVikTV�mn��Raz=h�jTsf´mnkT��gh�UW}=h8kTV�kqdHbL��Q)RlV��yo0dfdfVLs|mnk8dfVL��s�o0dHV[dHRTV�Uºdfh�dfRlV�mns|�nhcgLop�Fz=V�Rlo�¯am�h8jTs�b��lopkl~&}TsfVL}lopsHV�mnk8dfVLse� o8gVbfjTUWU�opsHm�VEb���h8sWdfRTVµklV¬ad�gh�UW}=h8kTV�k8d�b��»Q)RTmybW�8��h8zlop�n��`.dHo�¥�VLbidfRlV&�¼h8sfU½hp��oªUiVEbfbHop�8V�}@o�bHbemnkT�o��n��h�sHm�dfRTU&�a¤+RTVLsfV[dHRTV����nh�zlo���bf`abedfVLU/myb)klV�¯�V�s|g�h�klbfmy~cV�sHVL~��¾µ¿qÀ5ÁHÂ#à �0Ä��8� rJVdHsfm|kTVdE��jTkc��h8�n~Tm�kT�@�t�lkTm�dfVOgh�UW}T�nVdHV�}lsfV��T¬���~cmybvdHsfmnzTjcdHVL~�bf`abedfVLU&�t}TjT�n�nzlo�g�¥5�gLo0dHV���h8sf`WdfRTVLh�sH`

Page 5: Modular construction of finite and complete prefixes of Petri net … · 2016. 12. 27. · Modular construction of nite and complete pre xes of Petri net unfoldings Agnes Madalinski,

� ������������ �������� " ����������J����($����)��*«�+�/������/ '�#"%$&��'���/���(���)$�����1��+�

��� ��� � � X�V&}@op}TmnV�s�gh�k@bemy~cS�sHVµ~cVLb«bf`cbvdHS�UWVLb«~cmnbedfsHmnzTj��LbL�)~ ���lkTmyb«gh8UWUiVO~cVEb«gh8���nVLgdfmnh�klb~cV�g�h�UW}=h8bHopkqdHb)sHV��nm��Eb)}lops¨~cVLb|m�kqdfV�sf� o�g�VLbL�'X�h�UW}=h8bHopkqdHbL�cm�kqdfVLse� o8gVLb+Vdxbe`cbedfSLUiVEb+~TmnbedfsHm�zlj��Lb|bfh�k8dUWhc~ �L��myb��Eb�}lops�~cVLbFs��EbeVEopjc¬[~cV>r�VdHsfmB��¡ ��VLbed�zTm�VLk�g�h�kTkaj��qjTV>�nV�~ �L}T�nmno���V�~�� jlk[dHV��8bf`cbvdHS�UWV�~cmybvdHsfmnzTj��uvh�jlm­d)~�� jTkTVx}TsHh�}TsHm��d��¨~cV|� o8g�dHh�sHmnbHo0dHm�h8k��Jmn�=}=V�jcd'VLk«V��=V�d)b�� V¬c}TsHmnUiVLs�gh8UWUiV¨�yo#gh8UW}=h8bfm­dHm�h8k«~cVLb~ �L}T��myop�8VLb'~cVLb)~cm�����sHV�kqdHb)gh8UW}=h8bHopk8d�b�~cj�be`cbedfSLUiV8��X�V�dedHV[sfVL}Ts��EbeVLk8dHopdfmnh�k�� o�gdfh�sHmyb��LV¨~Tj�~ �L}T�nmno���VVLbed�V�kW����k���s�op�a}T�njlb�g�h�UW}lo�gdfV��qjTV+��V|~ �L}T��myop�8V)���nh�zlo��C�pVd�}=VLsfUWVd�V�kWh�jTdfsHV+~�� o�klop�n`cbeVLs���V|bf`cbvdHS�UWV}lo�s)UWh8sHgVEopjc¬��

!� h8zauvVEg�dHm­��~cV¢gV¶}lo�}TmnV�s�VLbed��yo©gh8klbvdHsfj@g�dfmnh�k-~cV¶}Ts���l¬aVEb«�lkTmyb�g�h�UW}T�nVdHb¢±�rt²JX)³�~lopklb��nVLb~ �L}T��myop�8VLbt~cVxbe`cbedfSLUiVEbt~cmybvdHsfmnzTj��Eb��Jrt��jlb�}Ts��Lgmyb���UWV�k8dE�8h8k«gHRlV�s�gHRTV#"�h�zcdHV�kTmns�~cmnsfVEg�dHV�UWV�kqd'~cV|dfV��yb}Ts���T¬cVEbxbfh�jlb¨��h8sfUWV#� o�gdfh8sfmyb���V��5V�k$�L¯am­d�opk8d�~cVWgh8klbvdHsHjTm�sHV�~�� opz=h�s�~µjTk§}Ts���T¬cVig�h�UW}T�nVd�~Topklb[��V~ �L}T��myop�8V�~cj be`cbedfSLUiV��8��h8zlop�>}=h8jTs�VLklbfjTm­dHV��nVW� o8g�dfh8sfmybfV�sE��X�VdedHV�gh�k@bvdHsfjlgdfmnh�kª~cV«}Ts����T¬cVLb�beh8jlb��h�sHUWV«UWhc~cjT�yopmnsfV�beV«��h�kl~cV�bfjTs#�yoµkThpdHm�h8k ~cV«s��LbfjTU%��~cj©gh8Ui}=h8sedHV�UWV�kqd�~�� jlk©gh�UW}=hqbfo�k8d�}lo�ss�op}T}=h8sed&"Obfh�k»±CbeVEbH³�m�kqdfVLse� o8gV8±CbH³�§X�VLb#s��EbejlU'�Eb#beh8k8d�g�h�UWU#jTkTm(�qj��Lb�opjc¬.gh8Ui}=hqbfo�k8d�b�¯�h�mybfm�klbL��qjTm+��VEbimnk8dHS���sHV�kqd)"§�nV�jTsW}lsfh8}TsfV�~���}T�nmno���V�}=h�jTs�g�h�klbedfsHjTmnsfV�~cVEbis��LbfjTU%�Lbi¯�V�s�b��nVLb�gh8Ui}=hqbfo�k8d�bbfjTm�¯popkqdHbL�8Vd'o�m�klbfm@~TV¨bfjTm�dfV8�� �V+dHh�jcd'}TsfVLkl~W�yo���h8sfUWV¨~*� jTk�o����8h�sHm�dfRTUWV+"�g�m�s�gjl�nopdfmnh�k�~TV|UWVEbfbHop�8VLbV�kqdfsHV�gh8UW}=h8bHopk8d�b��ch�,��nV�bf`abedfSLUWV����nh�zlo��5k*� VLbed�ueo�U�opmyb)UWo�kTmn}TjT�����- à �L� Á �/. � �©� s��EbeVLo�j©~cV�r�VdfsHmB�t~���}T�nmno���V8�>}Ts���l¬aV��lkTm+g�h�UW}T�nVdE�tbf`cbvdHS�UWV�~cmybedfsHm�zTj�����}lsfhc~TjTm­d�lzTs����cdfR��Lh�sHm�V�~cVEb+gLo0d��L��h8sfmnVLb

Page 6: Modular construction of finite and complete prefixes of Petri net … · 2016. 12. 27. · Modular construction of nite and complete pre xes of Petri net unfoldings Agnes Madalinski,

������������ ���������������������������! "

2t1t

0a

1a

1b

0b 0c

1c

3t

4t

7t5t 6t

Component Component

Interface

A B

#%$�&N

4t4t

0c 0c

0a

1t 2t

3t3t

1a

1c 0a 0a 1c

0c

1a

...#%'�&

UnfA

5t 6t

1b

0a 1c

0b

4t

1b

1c 0a

1b

1c 0a

4t4t

7t

0c

3t

1c

...

#(�&UnfB

²Jmn��jTsHVW���>wxmybvdHsfmnzTjcdHVL~&bf`abedfVLU°o�kl~�dfRlV�jTkc��h��y~cmnkT�8b+hp�Jm�dHb|gh8UW}=h�kTVLk8dHb

) * �������W���� t����#�

r�VdHsfmTkTV�dHb|±�r,+[³���h8sfU ox¤+my~cV��n`#jlbeVE~�UWhc~cV��lg�yo�bHb���h�s�opklo��n`abfmnkT�[gh8klgjlsfsHV�kqd>opkl~i~cmybvdHsfmnzTjcdHVL~Wbe`cbe´dHV�U�bL�.-|UWh8kT�#dfRTV�~cVE~cmyg�o0dHVL~�dHhah��ybL�abeh8UiVxsHVLg�V�k8d)o�}T}TsHh8o8gHRTVEb�Rlo�¯�Vxz=V�V�k�zlo�bfVL~«h�k«dHRTV�beh�´£gLop�n��VE~r�VdHsfm5kTVd�b�jTkc��h��y~cmnkT�8bL�>Q)RTVxjTkc��h��y~cmnkT�#dfVEgHRTklm��qjTV0/ "T�=����1�}Tsfh0¯amn~TVLbto�k«V�2«g�m�VLk8d'sHV�}TsHVLbfV�kqdHopdfmnh�k�hp�o��n��sHjTklb¨h��to«r3+�mnkOdfRTV#dHsfjlV�g�h�klgjlsfsHV�klg�`¢±�h8sx}lo�sedHmno���h8sH~TV�s�³+bfV�U�opkqdfmyg�b �54�¬aVEgjcdHm�h8klb[opsHV#g�h�kc´bfmy~cV�sHVL~«o8bt}lo�sedHmno��n��`ih8sH~cVLsfVE~�bfVdHb�h��FVL¯�V�kqdHb�s�o0dHRTV�stdfR@opk�bfV �qjTV�k@gVLbL�q¤+RTmygHR�sHVLbfjT��dHb�m�k�m�UW}=h�sfdHo�k8dUWVLUih8sf`µbHoE¯amnkT�8b|��h�s�o����8h�sHm­dHRTU�b¨o�klop�n`cbemnkT��z@VLRlo�¯qmnh�jlsHb¨hp��dHRTmnb[kTV�dL��Zµh8bed[hp��dfRlVLbfViop�n��h8sfm�dfRlUWb~Th�klhpd�sHV��n`�h8k�dHRTV+��jT���ljlkc��h��y~cmnkT��hp��o�r,+��8RTh�¤'V�¯�V�sE�pzTjcdts�o0dHRTV�s>h8kWo[�lkTm�dfV¨opk@~ig�h�UW}T�nVdfV+}lsfV��T¬±C²JX�r�³�h���m­d6/87c�l�8�p�@�9"�1B�>Q)RTVx~cV�lkTm�dfmnh�k«h��l¸eg�h�UW}T�nVdfVLkTVLbHb ¹|~cVL}=V�kl~Tbth8kWdfRTV¨}TsHh�}=VLsedv`#h���m�kqdfVLsfVEbvdE�zljcdxm�dxmyb¨��V�klV�sHo��n��`�z@o�bfVL~&h�k&dHRTV�� o8g�d¨dfRlopd[op�n��sfVEo�gHR@opzT�nV�U�opsH¥am�kT�qb+hp�>dfRTV#klVd[o�sfV�sfVL}TsfVEbeVLk8dfVE~mnk�dfRTmyb+}TsHV�T¬��c¤+RTmyg�R�myb+bfj�2«gmnV�kqd+��h�s|bfV�¯�VLsHo��5bed�opkl~To�sH~�}Tjlsf}=hqbeVEb'mnk�UWhc~cVL�­´ gHRlVLg�¥qmnkT�:/n�L�T�F�E��1£�Q)RTmyb�}lo�}=V�s«��hTgjlbfVLb�h�k»}lopsfdfmygjT�yo�s«r,+(g�op�n�nVL~ ¸f~cmybvdHsfmnzTjcdHVL~»bf`cbvdHV�U�b�� ¹ª~TV�lkTVE~ za` o8bfbfV�Ui´

zl��mnkT�§gh8UW}=h�kTVLk8dHb�± bfjTzc´£kTVd�b�³�zq`ªmnk8dfVLse� o8gVLb�± bfRlo�sfVE~.bejTzc´£kTV�dHb�³��;-°g�V�k8dHsHo��t}TsHh�}=VLsedv`§hp�+dHRTVLbfV~TmnbedfsHmnzTjcdfVE~¶be`cbedfV�U�b¨myb¨dfRlopdxdfRlV�mnsxjlkc��h��y~cmnkT��gLopkOz=V#��jTsfdfRlV�s+¸eg�h�UW}TsHVLbHbeVE~q¹�<9�=©o«� o�g�dHh�sHmnbHo0dHm�h8k}lsfh8}=V�sfd `>/����F�T��?a�>�L�@1£�W^a}=VEgm��@g�o����n`8�5dHRTVWjTkc��h8�n~cmnkT�&hp�'o�~TmnbedfsHmnzTjcdfVE~§bf`cbedfV�U�g�o�k§z=ViV�¬c}TsfVEbfbfVL~o8b|dfRlVigh8Ui}=hqbem�dfmnh�kOhp��dfRTV�jTkT��h��y~cm�kl�8bxhp�tm­d�b[gh�UW}=h8kTV�k8d�bL�x¡ k8dfVLsfVEbvdHm�kl���n`��@dHRTVigh8���nVLgdfmnh�kOhp��dfRTV�nhcgLop�ajTkc��h8�n~Tm�kT�qb�UWo�`�z=V)UWh�sHV)gh�UW}lo8g�dJdHRlopk�dHRTV)jTkT�¼h8�y~cm�kT��hp�ldHRTV)�8��h8zlop�cbf`abedfVLU&��o�bJmn�n��jlbedHsHopdfVL~mnkª²Jm��8jTsHV����WQ)RTV�bf`cbvdHV�U]m�kª²Jmn��jTsHV��8± o8³[g�h�klbfmnbed�bxh��tdv¤�hµgh�UW}=h8kTV�k8d�b��

Ao�kl~B�Fmnk8dfVLsHo8g�dHm�kT�

dHRTsHh�jT�8R¶o�k¶mnk8dfVLse� o8gV8��Q)RTVLsfV�V�¬cmnbedm = 2

}=hqbfbfmnzTm��nm�dfmnVLb¨dHh�}TsfhT~cjlgVt4m�k

UnfA±C²�mn�8jTsfV���± z@³e³

o�kl~n = 3

}=hqbfbfm�zlm��nm­dHmnVLb�dfh©}TsfhT~cjlgVt4mnk

UnfB±C²�mn�8jTsfV �8±CgL³e³�|RTV�k@gV

m · n = 6~cm��=VLsfVLk8d

g�h�U#zTmnklopdfmnh�klb)mnk�dHRTV�jTkc��h��y~cmnkT��hp��dfRlV����nh�zlo���be`cbedfVLU��Q)RTV+� o�gdfh8sfmybfopdfmnh�ki}TsHh�}=VLsedv`#hp��jTkc��h��y~cmnkT�8b�¤)o�b��ls�bed�UWV�kqdfmnh�kTVE~Wzq`BA m�klbf¥�V��@m�k�/n�L��1B�qzTjcd�h�kT�n`

��h8jTkl~im�dHb>�lsHbedtop}T}T�nmyg�o0dHm�h8klb>opsHh�jTk@~i�����C7T�0m�k�~cmyo���kTh8bfmybJ}TsHh�zT�nV�U�b5/ D�1B�E-»g�nh8bfV��n`�sfVL�nopdfVE~�}Tsfh8zT�nV�Ug�h�klbfmybvd�bxmnkªgh�UW}TjcdHm�kl��dfRlViUWmnkTmnUWo���� o�gdfh�sHmybeVE~µ��h�sHU hp�'o�k§jTkc��h��y~cmnkT�l��opkl~¶¤'o8b�gh�k@bemy~cV�sHVL~¶opddHRTV�bHopUWV#}=V�sfmnhc~F/ �T�G?a����1£�x^a}=VEgm��@g�o����n`��@dfRTVLsfV#��VLkTV�s�op�n�n`�V�¬amybedxbfV�¯�VLsHo���� o�gdfh8sfmybfVL~���h8sfU�b|��h�s¨dfRTVjlkc��h��y~cmnkT��hp�'o�~TmnbedfsHmnzTjcdfVE~§bf`abedfVLU&�#Q)RTV�¸e�no�sf�8VLbvde¹�h8kTV�Rlo8bxdfRTVi��jT����g�h�UW}=h�klV�k8d�jTkc��h8�n~Tm�kT�qb�o�b� o8g�dHh�s�b��IH'jcdxhp�tgh8jTsHbfV�klhpd[op�n��sfjTk@b|h���o��8m�¯�VLkOgh8Ui}=h8kTV�kqd¨¤+mn���JsfVLUWo�mnk&}=hqbfbfmnzT��V�mnkµdfRlV#���nh�zlo��bf`cbedfV�U&�+^a`aUWUWVdfsHmyg�op�n�n`��TdfRlVx¸fbeU�op�n�nVLbedv¹�h8kTV�myb|h�zTdHopmnkTVE~�zq`&beVL��VEg�dHm�kT�«sfjlklb|hp��VEo�g�Rµgh8UW}=h�kTVLk8ddHRlo0d�~ch�}@opsfdfmygmn}lo0dHV�dHh�o�sfjTk¶h���dfRTVi���nh�z@op�Jbf`cbvdHV�U&�J4!�qjTmn¯0op�nV�kqdf�n`��5dfRlVLbfV�UWm�klm�U�op�J� o8g�dfh8sHb[g�opkz=V�h�zcd�opmnkTVL~ zq`.sHVLbedfsHmngdfmnkT� ± o8g�dHjlop�n��`.}TsfhpuvVEg�dfmnkT�a³�dHRTV����nh�zlo���jTkc��h��y~cmnkT�§h8k VEo�g�R©g�h�UW}=h�klV�k8dE�ZOhc~cjT�yops�op�n��h8sfm�dfRlUWb5R@oE¯�V>z=VLV�k�}Tsfh8}=h8bfVL~xmnk[dHRTVto�z@h0¯�V>g�h�kqdfsHm�zTjTdfmnh�klb5dHh|g�h�UW}TjcdHV>dfRTVEbeV�UWm�kTmnU�op�

K�KML�NPORQ@SUT

Page 7: Modular construction of finite and complete prefixes of Petri net … · 2016. 12. 27. · Modular construction of nite and complete pre xes of Petri net unfoldings Agnes Madalinski,

� �� �C�� � � � ����� �� ���

� o�gdfh8sHbL��.����P��� ��R����� � ����� � �R ����P�����9������������;� ����P� � ����R�� ��@ � �����WQ)RTmyb�opzTmn�nm­dv`µdfhµopklo���`cbfV���nh�z@op��}TsHh�}=VLsedHm�VEbFhp�lo¨be`cbedfVLU;zq`��nhcg�o��8g�h�UW}TjcdHopdfmnh�k@bFh8k�bfU�op�n�8h�zcuvVLgdHb�mnb�hp�lgh8jTsHbfVtopk�V�¬qdHsfVLUWV��n`~cVEbemnsHo�zT�nVx��VLopdfjTsHV[dfh«o�~T~csHVLbHb'�yopsH��V[UWhc~cjT�yops|be`cbedfVLUWbL�Q)RlV¶h�zcuvVLgdfmn¯�V¶hp��dfRlmnb�}lop}=VLs�myb«dHh ��jTsedHRTV�s�V¬c}T�nh�sHV&dHRTmyb�my~cVLol�¨opkl~»bfRTh�¤�dHRlo0d�dfRlV§bHopUWV

}TsHm�k@gmn}T��VEb�g�opk z=V¶op}l}T��mnVL~�dHh©gh�UW}TjTdfVµ� o8g�dfh8sfmybfVL~ ��h�sHUWb«hp�I�,�P�� � �� ��� ����� ��� ��� ���������� h��jTkc��h8�n~Tm�kT�qb�� A m�dfR.��h8kT��dHV�sHU]h8zauvVEg�dHm�¯�VidHRTV«~cV�sHmn¯0o0dHm�h8k§hp��dfhah8�nb[��h�s�UWhc~cjl�no�s�UWhc~cVL��gHRlVLg�¥qmnkT�@�Q)RTV|}TsHh�zl��VLU o�Uih8jTk8d�b�dHh�zTjTmn�n~cmnkT�#²JX�r hp�Fgh8UW}=h�kTV�kqdHbL��mnk«bfjlgHR«o�¤'oE`#dHRlo0d�dHRTV�mns�gh8UW}=h8bfm­dHm�h8kmyb�g�h�UW}T�nVdfV���h�s#dHRTV��8��h8zlop��bf`cbvdHV�U&�;A m­dHR dHRTV�gh8klbvdHsHo�mnk8d�dHRlo0d�dHRTVLbfV���hcgLop��}TsHV�T¬cVLb#U#jlbediz=Vgh8UW}TjcdfVE~�m�k#o|UihT~cjT�yops�U�opkTkTVLsL��¤+m�dfRTh8jcd�sHV��V�sHV�klg�V>dfh|dfRTV����nh�zlo��8be`cbedfV�U&��-xb�o|bedfs�opmn��Rqde��h8sf¤'opsH~op}l}T��myg�opdfmnh�kF�@dHRTmybx¤�h8jT�y~µV�k@opzT�nV#h8kTV�dHh�sHV�}l�no8gV�o�gh8UW}=h�kTV�kqdxmnk§o�bf`cbvdHV�U opk@~Og�RTVLg�¥�dfR@o0dxdHRTV���nh�z@op��bf`cbvdHV�UºbvdHm��n�Fz=V�Rlo�¯�VLb)¤�VL���B�T¤+m­dHRTh�jTd+R@oE¯amnkT��dHh�g�RTVLg�¥«dfRTV�¤+RTh8��V�bf`cbvdHV�Uºop�8o�m�k��Q)RlV«��m�dfVLsHopdfjTsHV«opz=h8jcd�jlkc��h��y~cmnkT�8b�Rlo8b�o8~T~csHVLbHbeVE~§}TsHh�zT�nV�U�b�dHRlo0d�U�oE`§�nhah�¥¶sHV��yo0dHVL~§dHh&dHRTmyb

�qjTVLbedfmnh�k���²Th8s�V¬TopUW}T�nV���o�dHVLgHRlkTm��qjTV«dfh¶gh8klbvdHsfjlgd#o��@kTm­dHV�g�h�UW}T�nVdfV«}TsHV�T¬ ±C²JX�r�³�hp�|oµbf`akc´g�RTsfh8kTh�j@b�}TsfhT~cjlg�d�hp�+�yopz=V��nVL~¢dHsHo�klbfm­dHm�h8k.bf`cbvdHV�U�b�¤'o�b�}lsfVEbeVLk8dfVE~ªm�k / ��1B�OQ)Rlmnb�dfVEgHRlkTm��qjTV�jlbfVLbdfRlV[}TsHhc~cjlgd)bedfsHjlg�dHjTsHVxhp��dfRTV�be`cbedfVLU¦dfhWbemnUW}T�nm­��`�dfRlV�gh8klbedfsHjlg�dHm�h8k«h��FdHRTV�²JX�r��czTjTd'dHRTmyb'}TsHV�T¬myb#kThpdigh8UW}TjcdfVE~.m�k.� o8g�dfh8sfmybfVL~§��h8sfU&��H'`ªgh�kqdfs�o�bedL��/ �@1'}TsHh�}=h8bfVL~ªoOUWhc~cjT�yops�gh8klbedfsHjlg�dHm�h8kªh��gh8UW}T��V�dfV+}TsHV�T¬cVEb��pdHo�¥qmnkT��m�kqdfh�o�gLgh8jTk8d>dfRTV¨be`aUWUWVdHsfmnVLb>hp��gh8Ui}=h8kTV�kqdHbL� �¨h�¤�VL¯�V�sE�0dfRlV|bf`abedHV�U�bgh8klbfmn~cVLsfVE~¢dfRTVLsfV�opsHV«sHVLbvdHsHmngdfVL~ªdfh§kTh�kc´£sfVLV�kqdfs�opk8d�bf`aklg�RTsfh8kTmybfopdfmnh�klb��JmB� V��§h�kTV�dfs�opklbfm­dHmnh�k.h��¨ogh8UW}=h�kTV�kqd�gLopk«bf`aklgHRTsHh�klmnbfV+h�kl��`i¤+m­dHR«h�kTV|dfs�opklbfm­dHmnh�kWmnk�opkTh�dfRTVLstg�h�UW}=h�klV�k8dE��Q)RTmybtsfVEbvdHsfmyg�dHm�h8k¥am��n�yb�dHRTV)g�h�UW}TsHVLbHbemnh�k��8o�m�k#h��T� o�gdfh8sfmybeVE~���h�sHU�b��FdfRTV+gh��n�nVLg�dHmnh�k#hp�T� o8g�dHh�s�b�h��TdHRTV'�8��h8zlop�qjTkc��h��y~cmnkT�Rlo8b'dHRTV�bHopUWV�bem"!�V�o�b)dfRTV����nh�z@op��jTkc��h8�n~cmnkT��m�d�beV����v�t¡ kOo�~T~cm�dfmnh�k��cdfRlV�~cVLsfmn¯0opdfmnh�k�hp����hcgLop��}lsfV��T¬cVLbjlbfVLb����nh�zlo��cmnkc��h8sfU�o0dHm�h8kF��¡ kWdHRTmyb�}lo�}=V�sE�pdfRTV¨h�zcuvVLgdfmn¯�V|mnb>dHh�jlbfV|h�kT�n`���hTg�op�Tmnkc��h8sfU�o0dHm�h8kF��mB� V8��dHRTVgh8UW}=h�kTV�kqd'UWhc~cVL�=}T�njlb'mnkc��h�sHU�o0dHm�h8k�gh8UWU#jTkTmyg�o0dHVL~«za`�kTV�mn��Raz=h�jTs�b��qmnk�h8sH~TV�stdHhi~cV�dfV�sHUWm�klV¨dHRTV}TsHV�T¬§dfhOsHVdHo�m�k¢��h8s�VLo8gHR.g�h�UW}=h�klV�k8dE��Q)RTV«mnkc��h�sHU�o0dfmnh8k§dHsHo�klbeUWm�dedHVL~ªzq`¢oµg�h�UW}=h�klV�k8d�dfhOm­d�bkTVLm��8Rqz=h�jTs)¤+mn���5dHo�¥�VxdfRTV[��h�sHU®h��Jo�¸ebfjTUWU�opsH`WkTVdE� ¹[m�ka¯�h��n¯am�kl��h�kT�n`�z=VLRloE¯amnh�jTs�b�hp��dfRlV[mnkqdfV�sf� o�g�VdfR@o0d|sfVL�nopdfVEb�dfRlV�U&�Q)RlVµbfjTzauvVEg�d�o8~T~csHVLbHbeVE~ RlV�sfVµg�h�U#zTmnkTVLb�UWo�kq` dfVLg�RTkTmyg�o��|o8be}=VEg�dHbL�'beh8UWV�h��[¤+RTmng�R o�sfV&kThpd

~cmnsfVEg�dH��`�sfVL�nopdfVL~�dfh«dfRTV#}lsfh8zT��VLU�opk@~µg�opkµz=V�gh8klbfmn~cVLsfVE~&o�b�¸em�kl~TV�}=V�kl~TV�k8d�~cm2�g�jT�­dHmnVLbL� ¹ªQ)RlV�sHV´��h�sHV��@dHh�}lsfVEbeVLsf¯�V�dHRTVig�yopsHm�d `�h��>dHRTmnbx�lsHbed�o0dedHV�UW}cdE�=dHRTVibeV�dedHm�kl��mybx¯�h��njTk8d�opsHm��n`&bemnUW}T�nm­�lVE~¶o0d[m­d�bU�o0¬cmnU#jTU&��Q)RTV�opdedHV�k8dHm�h8k�myb)sHVLbedfsHmyg�dfVE~«dHhWjTkc��h8�n~cmnkT�qb)h���bfop��V�r,+��@o�kl~�dfRTV�~TmnbedfsHm�zljcdfVE~�bf`cbvdHV�Umyb�sHVL~cj@gVL~#dHh[dv¤�h�g�h�UW}=h�kTVLk8d�b�sfVL�nopdfVE~#za`�opkimnk8dfVLse� o8gV�±�dfRTV|V�kl~ihp�@dHRTV+}lo�}@VLs�bejl����VEbvd�bJRTh0¤ dHRTmyb}lo�sHo8~cm��8U gLopkWdHRTV�k«z=V¨V�¬qdHV�kl~TVL~l³���Q)RTV¨m�kqdfVLse� o8gV|myb�o�bHbejTUWVE~idfh#z=V¨o#bemnUW}T�nV¨o�jcdfh8U�o0dHh�kWs�o0dHRTV�sdfR@opk o§dHsfjTVµr,+#��opk@~©dHRTVµ}TsHV�T¬�g�h�klbedfsHjlg�dHm�h8k�o8bfbfjTUWVLbWU�o�sf¥amnkT�ªg�h�UW}T�nVdfVLkTVLbHb��)o�kl~�bfmnUi}T�nVo�~TV �qjlo0dHV[h8sH~cVLsHbL�Q)RlV�}lop}=VLsimybih�sH�8o�kTmnbfVL~ o�b���h��n��h0¤|bL�©^cVLg�dHm�h8k �OsHVLgLop�n�nb#dfRlV�zlo8bemyg«dHRTV�h8sfV�dfmyg�op�)zlo�g�¥a��sHh�jTkl~

gh8klgVLsfkTmnkT�µr,+xb��JdHRTV�mns#jTkc��h8�n~Tm�kT�qb#opk@~§dHRTV�~cV�sHm�¯po0dHm�h8k¢h��+oOg�o�kTh�kTmyg�o��>�lkTm�dfV�gh8UW}T��V�dfV«}lsfV�l¬5�^aVEg�dfmnh�k "¶~cV�lklVLbi~cmybvdHsfmnzTjcdHVL~©be`cbedfVLUWbiopkl~.sHVLgLop�n�nb�dfRTV�o8bfbfhcgmyo0dHVL~§� o8g�dHh�sHmnbHo0dHm�h8kª}TsHh�}=VLsedv`§h8kdfRlV�mnsxjTkc��h8�n~cmnkT�@�[^aVLgdfmnh�kO��g�h�k8d�opmnklb|dfRTVigh8k8dfsHmnzTjcdfmnh�kOh���dHRTmyb¨}lo�}=V�s��'m�d[}TsHVLbeVLk8d�b+dHRTV�UWhc~cjT�yops~cVLsfmn¯0opdfmnh�k hp�xo¶�lkTm�dfV&o�kl~©gh8UW}T��V�dfV�}TsfV��T¬ m�k©� o�gdfh�sHmybeVE~ª��h�sHU��>��hTgjlbfm�kl�¶�lsHbedih8k dfRTV&bfmnUi}T�nVg�o8beV�hp��dv¤�h�gh8UW}=h�kTV�kqdHbL��opkl~µdHRTV�k§}TsHh�}=hqbemnkT��dHRTVi��VLkTV�s�op�nm#!Eo0dHm�h�k&dHh�UWh�sHV�gh8UW}T��V�¬µkTV�d ¤�h8sf¥cbhp��g�h�UW}=h�klV�k8d�b��

$ % �'���/����!�����0��������1��

¡ k dHRTmnb�bfVLgdfmnh�k»zlo�bfmng¶~cV�@kTm­dHm�h8klb�gh�k@gV�sHkTmnkT� rJVdHsHmxklVdHb�opk@~ klVd�jTkc��h��y~cmnkT�8b�opsHVO}TsHVLbeVLk8dHVL~��Q)RTVEbeV�o�sfV�UWo�m�kl��`�o�~To�}cdfVE~���sfh8U /87c���8�!1£�

&('*) +-,/. 021435,/.768µ¿ �L�:9 - kTVd«mnb«o �qjlo�~TsfjT}l��V

N = (P, T,→, P 0)bejlg�R dfRlopd

Popkl~

TopsHV&~cmnb uvh�mnk8d�beV�dHbWh��

� �%��R�! �opkl~;�����P !����=��� ��qsHVLbf}=VLgdfmn¯�V��n`��→⊆ (P × T )∪ (T × P )

mnb'o�< ��� ��� ����=���l�Topkl~P 0 : P →

=?> K@=�A

Page 8: Modular construction of finite and complete prefixes of Petri net … · 2016. 12. 27. · Modular construction of nite and complete pre xes of Petri net unfoldings Agnes Madalinski,

������������ ���������������������������! 7

N = {0, 1, 2, ...}mybxo«U#jT��dfmybfVd[h8k

PsHV�}TsHVLbfV�k8dHm�kl��dfRTV ��� ����=�� � �� � �����&h��>dfRTV�klVdL��²Th8s[o«kThc~cV

x ∈ P ∪ T�am�dHb � ��� � � •x

myb'~TV�lkTVE~�zq` •x = {y | (y, x) ∈→}o�kl~«m�dHb � �� � � � x• myb)~cV�lklVL~�zq`

x• = {y | (x, y) ∈→}�>¡ k�dfRTmyb)¤�h8sf¥WdfRlV�kTVd�b+opsHVx�nm�UWm�dfVE~�dfh: �U����klVdHbL�cmB� V�����h�s)V�¯8V�sf`�sHVLo8gHRlo�zT�nV

U�o�sf¥amnkT�Mopkl~�VL¯�V�sH`�}T�yo8gV

p ∈ P�M (p) ⊆ {0, 1}

�- �no�z=V��nVL~�kTV�d

N =(

P, T,→, P 0, λ,Λ) mnb�o[kTVd�V�¬adfV�k@~cVL~i¤+m­dHR�o[�no�z=V��TbfVd

Λo�kl~io[�yopz=V��n�nm�kT�

��jlklg�dHm�h8kλ : T → Λ

h�k�dHsHo�klbem�dHm�h8klb��

-�à � ����� � ��9 - UWh�sH}TRTmybfUφ : A → B

z=Vdv¤�V�VLk kTVd�bx =

(

Px, Tx,→x, P0x

) /n��7�1£�'¤+RTVLsfVx ∈ {A,B}

��mnb�o�}lopmns (φP , φT

) ¤+m�dfRφPo�sHV��yo0dfmnh�k§h�k¢}T�yo�g�VLb�o�kl~

φTo�}lo�sedHmyop�J��jTklgdfmnh�kªh�k

dHs�opklbfm­dHm�h8klb��E�ªQ)RTV�mnkTm�dfmyop�'U�o�sf¥amnkT�Omyb�}TsHVLbfV�sH¯�VL~ªza`φo�b#��h��n��h0¤|b��

P 0B = φ

(

P 0A

) opk@~∀pB ∈

P 0B ,∃!pA ∈ P

0A : pAφpB

�>¡£�φmnb|~cV�lkTVE~�h8k

pA ∈ PA

�cdfRlV�k&m­d|myb+op�ybfhW~cV�lklVL~�h�k�z=hpdHR •pA

o�kl~p•A φ

}TsHVLbfV�sH¯�VLb�dfRTV�V�kq¯amnsfh8kTUWV�k8d�hp�|VLo8gHR.dfs�opklbfm�dfmnh�k*�tB = φ (tA)

mnUi}l��mnVLb#dfRlopd�sfVEbvdHsfmyg�dHmnh�klbφop : •tB →

•tAopkl~

φop : t•B → t•AopsHVxz=h�dfR�dfh�dHop�5��jTklgdfmnh�klbL�c¤+RTVLsfV

φop~cV�kTh�dfVEb�dfRlV�sfVL¯�V�s�beVE~

sHV��yo0dHmnh�k�h�k�}T�yo8gVLbL�.+|h�dfmygV�dfRlopd+kTV�d|UWh�sH}TRTmybeU�b)}TsHVLbfV�sf¯8VxsHjTklbL�²Th8s��yopz=V��nVL~§kTVd�b

x =(

Px, Tx,→x, P0x , λx,Λx

) dHRTV�~cV��lkTm�dfmnh�k¢h��'o�UWh8sf}TRlmnbfUφ : A → Bmyb|sfVLm�kT��h�s�gVL~�zq`�V¬adHsHoWsfV��qjTm�sHV�UWVLk8dHb �

ΛB ⊆ ΛA

±�dfRTV#�no�z=V���bfVd¨myb+sHVL~TjlgVL~&zq`φ³��Dom (φT ) =

λ−1A (ΛB) ⊆ TA

±φmyb�~cV�lkTVE~�V�¬co8g�dH��`xh�k�dfs�opklbfm­dHm�h8klb�Rlo�¯am�kT�|o|bfRlopsHVL~[�no�z=V��¼³���o�kl~�m­�

φT (tA) = tBdHRTV�kλA (tA) = λB (tB)

±��yopz=V��F}TsHVLbfV�sH¯am�kT�a³��

&�' & 0��� ���� 1� ���� 0���� , 6 6 , 6Q�¤�h�klhc~cVLbxh���o�kTV�d

N�yopkl~

y′�FopsHV#mnk � �!�2� �������� ��� <�� � C��~cV�kTh�dfVL~¶zq`

y#y′��m��tdfRTV�sHV�V�¬cmnbed

~Tmnbedfmnklg�d|dHsHo�klbfm­dHm�h8klbt, t′ ∈ T

bfjlg�R�dfRlopd •t ∩ •t′ 6= ∅�lo�kl~

(t, y)opkl~

(t′, y′)o�sfV�m�k�dHRTV�sfV��lV¬cmn¯�V

dHs�opklbfm­dHm�¯�V�g�nh8bfjTsHV[hp��dHRTV��lh0¤-sHV��yo0dHm�h8k→�l~cVLkThpdHVL~�zq`

¹�

�«���/����� ¿�� � ¿��J¿ �L�:9 -|k � �������!��� � �R� � � �mybio¶kTVdO = (C,E,→, C0)

��¤+RTVLsfVCmybio§bfVdWhp�

�R��� ���� �=���P �± }T�no8gVEbH³�Emyb¨dfRTVibfVd[hp�6� <@� �2� �±¼dHsHo�klbem�dfmnh8klbH³|opk@~

C0 = c ∈ C : •c = ∅mnb¨dfRlV�bfVd

h��Jm�kTm�dfmyo��Fgh8kl~cm�dfmnh�klb+bHo0dHmnbe��`am�kT�idfRTV���h��n�nh�¤+mnkT���J��h�s+VL¯�V�sH`c ∈ C�| •c |≤ 1

��h�s+V�¯�VLsf`y ∈ C ∪E

�¬ (y#y)

opkl~#dfRTVLsfV+opsHVt�@kTm­dHV��n`�U�o�kq`y′bej@gHR�dfRlopd

y′ ≺ y��¤+RTV�sHV

≺~cV�kTh�dfVEb�dHRTV � ��� �� ��� ����=���l�

dHRTV�dHsHo�klbem�dHm�¯�V�mnsfsHV �lV�¬cm�¯�V�g���hqbejlsfV�h��→�)Q�¤�h«kThc~cVLb¨opsHV��R��� �����!��� �2B�@~cV�kTh�dfVL~

y ‖ y′�@m­�>kTVLm­dHRTV�s

y#y′klh�s

y ¹ y′kTh�s

y′ ¹ y�

! �0� � � �"� �"# � � à � ¿ �L� ¿ ��9 - � �U�� ����������� ��� � �� R |hp��o[kTV�dtbf`cbvdHV�UNmyb�o[}@opmns

β = (O, f)�8¤+RTV�sHV

UWh8sf}TRlmnbfUf : O → N

myb�o¨dfh�dHo��8��jlklg�dHm�h8k�h8kO��o��nbfh[g�op�n�nVL~#o�������������#hp�

Omnk8dfhN�JQ)RlmnbJ�¼h8�y~cm�kT�

gLopk«z=V[beV�VLk�o�b�o��yopz=V��nmnkT���¼jlklg�dHm�h8k�h�k«V�¯�VLk8dHb'opk@~«gh�k@~cm­dHm�h8klbthp�O�azq`W¤+RTmygHR�gh8kc�l��jlsHopdfmnh�klb�hp�

OsHV�}TsHVLbfV�kqdHb)sfjlklb+hp�

N��¡£d¨mnb)��jTsfdfRTVLs|sHV �qjTmnsfVE~«dHRlo0d

βbfopdfmybv��`�oW}lops�bemnUWh�ka`«g�h�kl~cm�dfmnh�k��>��h�s¨op�n�

e1, e2 ∈ E�=m­� •e1 = •e2

o�kl~f(e1) = f(e2)

dfRTVLke1 = e2

�|Q�h�~cV�lkTV��lklm­dHV�gh8Ui}T�nVdHV�}TsHV�l¬aVEb��m�d¨¤+m��n��z=V�jlbfV��jl�Fdfh«g�h�klbfmn~cVLs|oµ±�¯amnsfdfjlo��y³'mnkTm�dfmyop��V�¯�VLk8d

⊥mnkβ�l¤+Rlmng�R&Rlo8b|opk&V�UW}cdv`�}TsHVLbfVdL�

C0o8b)}=h8bede´ beV�d+o�kl~�kThW�yopz=VL��±�mB� V��>klhWm�U�o���V[za`f³�

- zTs�opklg�RTmnkT�ª}TsHhcg�VLbHbµ±=H'r�³β′ hp� N mnb�o � �������»h���oªzTs�opklg�RTmnkT�ª}TsHhcg�VLbHb β �+~TV�kTh�dfVL~ zq`

β′ v β�qm­�O′ mybto�g�o�jlbfo����n`�g���hqbeVE~WbfjTzc´£kTVdth�� O gh�kqdHo�m�kTmnkT�#op�n�lm�klm­dHmno��@g�h�kl~Tm­dHm�h8klbtopkl~�bfjlgHR�dfRlopd �

∀e ∈ E, e ∈ O′ mnUi}T�nmnVLb e• ⊆ O′ o�kl~ f ′ mnbtdfRTVxsHVLbedfsHmyg�dfmnh�k�h�� f dHh C ′ ∪E′ �>²lh�s�VEo�g�R�kTV�d)be`cbedfVLUNdHRTV�sHV�V¬cmnbedHb+o�jTklm��qjTV�± jT}�dHhimybeh8UWh�sH}TRTmybeU«³tUWop¬cm�U�o��>±�¤�� sL� d

v³'zTsHo�klgHRlm�kT�i}TsHhcg�VLbHb'~cV�kTh�dfVE~

za`(Unf(N ), fN )

��h�sx��h8s�beRlh�sfd(UnfN , fN )

�FgLop�n��VE~µdfRTV:����������������Oh��N��¡ k¶dfRTV�beV �qjTVL�C��¤+RTVLk

��h8�n~Tm�kT�qb+opsHV[jTk@opU#zTmn��jTh8jlb'dfRTVL`�¤+m��n�Fz=V�h�UWm­dfdfVE~��lbfhβ¤+mn���Fz=V�mn~TV�k8dHm­�lVE~�¤+m�dfR

O�

$ mn¯�V�k¶o�gh�kT�l��jTs�o0dHm�h8kκhp��o«zTsHo�klg�RTm�kl�«}TsHhcgVEbfb

βh��N�εmybxgLop�n��VE~Oo� !� % � h��

κm�kβm��

dHRTV�sHV�V�¬amybedHb+bfh�UWV�gh8kc�l��jTs�o0dHmnh�kκ′ A κ

bfjlgHR�dfRlopdε = κ′ \ κ

�&('*)�+-,/.10�2431. ( . 576 8

φ.1,:9-+;. 5/5/< L . L ,/5/< $>= )>?

φP)�+

φT @

K�KML�NPORQ@SUT

Page 9: Modular construction of finite and complete prefixes of Petri net … · 2016. 12. 27. · Modular construction of nite and complete pre xes of Petri net unfoldings Agnes Madalinski,

� �� �C�� � � � ����� �� ���

� à ����# ���0�c� � à � �.� � Ä°�����L�:9 - �R���9� ���������=���»hp��o¢zTsHo�klg�RTm�kl�¢}TsfhTgVLbHbβmyb�o§�lkTm�dfVObfVd�h��

V�¯�VLk8d�bκ ⊆ E

bfjlgHR�dfR@o0d���h�s�op�n�e, e′ ∈ κ

�¬(e#e′)

opkl~�����h�s«V�¯�VLsf`e ∈ κ

�e′ ≺ e

mnUW}T�nm�VEbe′ ∈ κ

mnk o�~l~cm­dHm�h8k�m­d«mybWsHV �qjTmnsfVE~.dHRlo0d⊥∈ κ

� ²lh�s�V�¯�VLsf` V�¯�VLk8de ∈ E

��dfRTVOgh8kc�l��jlsHopdfmnh�k[e]

���= {e′|e′ ¹ e}

myb)gLop�n��VE~«dfRTV � @ !� � �R���9� � ���U���=�����[h��e�lopkl~

〈e〉���= [e] \ {e}

~cVLkThpdHVLb�dHRTV�bfVd+hp��R��� ��7� ���R� � �R�� ���! ��|Q)RTV#beVd¨hp��op�n���lkTm�dfV�±�sHVLbf}F�tz@o�bfmngE³)gh8kc�l��jTs�o0dHm�h8klb)hp�>oWzTs�opklg�RTmnkT�W}lsfhcg�VLbHbβmnb|~cV�klhpdfVE~�za`

κβfin

± sfVEbe}F�κ

βbas

³��lo�kl~�dfRlV�bfjT}=VLsHbHgsHm�}Tdβmnb|~csHh�}T}=VE~�¤+RlV�k

β = UnfN�

- � � � �� �myb)o�bfVd)hp�Jgh8kl~cm�dfmnh�k C ′ bejlg�R�dfRlopd'��h�s+op�n��~cmybvdHm�k@g�d c, c′ ∈ C ′ � c ‖ c′ �Topkl~�o-��� tmyb)oU�o0¬cmnUWo��=g�hp´ beV�dt��h�s�dfRTV�beV�d'm�k@g�njlbemnh�k��� �Vdκz=V�o#g�h�kc�l�8jTsHopdfmnh�kWdHRTV�k

Cut(κ)���=(

C0 ∪ κ•)

\ •κmyb�oOgjTd��jTsfdfRTVLsfUWh�sHV���dHRTV�bfVd�h��+}T�yo�g�VLb

f (Cut(κ))mnb�oOsfVLo8gHR@opzT�nV�U�opsH¥am�kl�µhp�

N�>¤+RTmng�R myb

~cVLkThpdHVL~¶za`Mark(κ)

� - UWo�sf¥amnkT�Mhp�Nmyb ��� � ���! � �2 � �¶mnk

βm���dfRTVLsfVWmnb�o�gh8kc�l�8jTsHopdfmnh�k

κhp�βbfjlg�R&dfRlopd

M = Mark(κ)��4t¯�V�sH`�UWo�sf¥amnkT��sfVL}TsfVEbeVLk8dHVL~�m�k

βmnb¨sfVEo�g�RTVL~�m�k

N�=opkl~&V�¯�VLsf`

sHVLo�g�Rlo�zT��V[U�opsH¥am�kl�ihp�Nmnb+sfVL}TsHVLbeVLk8dHVL~�mnk

UnfN�

&('� �1 �1 . , �� �� � �� ���� ,/. , � 07,�����, 6-|k«opzlbedfs�o�gd>}lops�opUWV�dfsHmng'UihT~cV��TRlo8b>z=V�V�k�m�kqdfsHhc~cjlgVE~�mnk /n�L�T�T�8�!1TdHh�gh8}=V)¤+m�dfR«~cm��5V�sHV�kqdt¯0opsHmyopk8d�bhp�@dfsHjTklg�opdfmnkT�[jTkc��h8�n~Tm�kT�qb��J¡£d>jlbfVLbJ}@ops�opUWVdHV�sHb�¤+RTmyg�Ri~TVdfVLsfUWmnkTV'dfRTV)mnkc��h8sfU�o0dHm�h8k#mnk8dHV�kl~cVE~#dHh[z=V}TsHVLbfV�sH¯�VL~imnk�dfRTVxg�h�UW}T�nVdHV|}lsfV�l¬&± m�k«dfRTV[bvd�opkl~To�sH~�g�o�bfV���dHRTmnb�mnb�dHRTVxbfVd�hp�FsfVEo�g�RlopzT�nV+U�opsH¥am�kl�8b�³opk@~�be}=VEgm���`�dHRTV�gmns�gjTU�bedHopk@gVLb)jTkl~cVLs+¤+RTmyg�R&o�k�V�¯�V�kqd|gLopk�z=V�~cVLbfm��8klo0dHVL~�o�b+oWgjcdf´Bh��OVL¯�V�kqdL���¿���� � � � à �����·gjcdfdfmnkT�«gh�kqdfV�¬qd������

UnfN� ���!� � ��

Θ =(

≈,C, {κe}e∈E

) � � �P�������� �≈

� ��F� � ����<@����� � �0��� �� �=���>���κfin

�! �C

� �R����%� � ��.o�~cV��8j@o0dfV ����� � � � � 0 � �!� � �,��� � � ������� � �R� � ��� �=��.���U�C��� ���κfin

�U��� � �����⊂

�� ��� �

κ′ ⊂ κ′′����� � �=�!

κ′ C κ′′�

" �≈

�� �C

���� }TsfVEbeVLsf¯�VE~ zq` �@kTm­dHV¢V�¬adfV�k@bemnh�klb � � ��� � �!���>��<�� ���;� ����>�U� �R������� ����� �=��� κ′ ≈ κ′′

� �� �I����� � <������� R� %F�ε′� �κ′

� ?� � ���0� �C� �� � �P�� � !� % �ε′′

� �κ′′

R�2���-?� �# %$

κ′′ ⊕ ε′′ ≈ κ′ ⊕ ε′� �� �

# �&$ � �κ′′ C κ′

?� � �κ′′ ⊕ ε′′ C κ′ ⊕ ε′

�' �{κe}e∈E

� ����B��� �:� �J !�2�! � = � �κfin

Q)RTVio�~cV �qjlopdfV#h�s�~cV�sxbf}=VLg�m­�lVEb¨¤+RTmygHR§gh�kT�l��jTs�o0dHm�h8klb|o�sfV�}TsHVLbfV�sH¯�VL~&m�kOdfRlV�g�h�UW}T�nVdfV�}TsfV��T¬o����

C´£UWm�klm�U�op��gh�kT�l��jTs�o0dHm�h8klb¨mnk§VLo8gHR§V �qjTmn¯0o���VLk8d�g�yo�bHbxhp�

≈o�sfVi}TsfVEbeVLsf¯�VE~���Q)RTVW�no8bvd�}lo�sHo�UiV�dfVLs

κe

myb¨kTV�VE~cVL~&dHh�bf}=VLg�m­��`�dfRTVibeV�dxhp�tgh8kc�l��jTs�o0dHmnh�klb+jlbfVL~µ�yo0dfVLs|dHh�~TVLgmy~cV#¤+RTV�dfRTVLs[opkOV�¯�VLk8d[g�o�kz=V�~cVEbemn��kTVE~�o�b|oWgjTde´£h �¶V�¯8V�k8dE�tQ)RTV�g�jcdedHm�kT��gh�kqdfV�¬qd

ΘERV ={

≈mar ,Ctot , {κe = κbas}e∈E

}

gh8sfsHVLbf}=h�kl~Tb[dfh&dHRTV���s�opUWV�¤�h8sf¥Omnk / 7@1£��¤+RlV�sHV≈mar

myb�dHRTV«V �qjTmn¯0o���VLklgV�sfVL�nopdfmnh�k¢h8k¢sHVLo8gHR@opzT�nVU�opsH¥am�kT�#hp�

N�amC� V��

κ′ ≈mar κ′′ m�� Mark(κ′) = Mark(κ′′)

�copkl~Ctot

mnb'o�dHhpd�op�5o�~cV��qjlo0dHV|h8sH~TV�sE�Q)RlVObedHopdfmygµgjcdf´Bh��-VL¯�V�k8d�b«o�sfVO~cV�lkTVE~ m�kl~TV�}=V�kl~TV�k8dH��`�h���opk jTkc��h8�n~cmnkT�©op�n��h8sfm�dfRTU&�'¤+RTV�sHV

��VLo8bemnzT�nVWV�¯�VLk8dHb�opsHViV�¯�VLk8d�b�¤+RThqbeV�g�o�jlbfo���}TsHVL~cVEgVLbHbfh�s�b[opsHVikThpd#bvd�o0dHmngWg�jcde´£h �©V�¯�VLk8d�b�opk@~¶dfRqj@bopsHV[mnklg�njl~cVE~�m�k�dfRTV�}lsfV��T¬�~cV�dfV�sHUWm�klVL~�zq`«dfRThqbeV�g�jcde´£h �¶VL¯�V�k8d�b����¿���� � � � à �)(+* � � �� � � ��VEo�bfm�zT�nV�V�¯�VLk8d�b � � � � �� � � ���

fsbleΘ� �� � ?� �: �� �U�»bvd�o0dfmyg�gjTde´£h �

V�¯�VLk8d�b � �C��� �� � �����cutΘ

� ���� �,�0 � = J� � ��<���� = J� �UnfN

�C�=�,� �R�B��� ���2�� ��<@� � � � ��� ��P�.����������.������,��,�

� ���6�;� <@� �2e� ��� � R� � ���� <@� �23� �

〈e〉 ∩ cutΘ = ∅�

-/.10 <�<32 24+;<�,/, .1) L54 6/7�894:736<;/=?><@�A8<BDCE6<; .1,�F4, < = ' 6 ,/<3G <(+ $ 3 $ F 504)�+;, @IH ) 9:<3G <(+(8J04<(+/<"5K0 <"5 <(+;0ML 31) (U$ 3 L�.1, $ 0 ' .ON<F�P)<F4,�84, . L ( <�. 5-+;< ? <(+;,-5 ) $3( ) 0�2 ) L < L 5 8*,/) .15 .1, +/<(243 $R( < =J' 6RQ '@$ , . ( @ L

=?> K@=�A

Page 10: Modular construction of finite and complete prefixes of Petri net … · 2016. 12. 27. · Modular construction of nite and complete pre xes of Petri net unfoldings Agnes Madalinski,

������������ ���������������������������! ?

! ��� � ��<���� e� � ��� ����� � ��� � <���� � � �� � ����R@ !� � �� � �� � � �4��P����� � # �-�R����%� �¢g�h�sHsfV�´

be}=h�k@~cm�kl� $�� ���9� ���������=���κ ∈ κe

R�2��� ��P�κ ⊆ fsbleΘ \ cutΘ

�κ ≈ [e]

� �� �κC [e]

� # �6���<����

e′� �U����� �!��� �:@ |gh8sfsHVLbf}=h�kl~cmnkT��VL¯�V�k8dB� �

κ = [e′]� $

* �P�������� ����������� ��� �R�! R Pref Θ

N

��� ���2�R� �4� � ?� �I �� �U�I� <@� �2� fsbleΘN

� ��R����%� � ��P�xgLopkTh8kTmngLop�q}lsfV��T¬�U�

UnfN�

+¨hpdHV�dfRlopdPref Θ

N

myb�jTkTm(�qjTV��n`.~cV�dfV�sHUWm�klVL~ zq`ªdfRTV&gjcdfdfmnkT�¢g�h�kqdfV¬adΘ� ^aV�¯�VLsHo��t��jTkl~To�UiVLk8dHo��

}lsfh8}=V�sfdfmnVLb�hp�Pref Θ

N

Rlo�¯�V|z=VLV�k«}TsHh�¯�VLkWmnk /n����1B��¡ k«}lo�sedHmng�jT�yopsE�Pref Θ

N

myb�op�n¤'o�`cb�g�h�UW}T�nVdfV¨¤�� sE� dE�cutΘN�lopk@~�m­d|myb)�lkTm­dHV�m��

≈Rlo8b'�lkTm­dHV��n`�U�o�kq`«V��qjTm�¯pop�nV�klg�V�g��no8bfbfVLb)o�kl~

κe ⊇ κbas

� � �����������&�«���+� �������J�|" �

wxmybvdHsfmnzTjcdHVL~«bf`cbvdHV�U�btopsHV+UWhc~cVL��VE~Wzq`Wo�bfVd�hp��mnk8dHV�s�gh�kTklVLg�dHVL~�gh8UW}=h�kTVLk8dHbtm�kqdfV�s�o�gdfmnkT��dfRTsHh�jT�8RbfRlo�sfVE~�bfjTzc´ be`cbedfV�U�bL�lgLop�n��VE~�m�kqdfVLse� o8gVLbL�'Q)RTmyb|UWhc~cjT�yops¨bedfsHjlg�dHjTsHV�mnb¨o��nbfh�gLop�n��VE~&o�¸v� o�gdfh8sfmybfopdfmnh�k}lsfh8}=V�sfd `8¹+h���dfRlV[be`cbedfV�U&�a~cjlV¨dfh�dHRTVxjlbfV¨h��F}TsHhc~cjlgdHbx± Uih8sfV¨}TsHVLg�mnbfV��n`ihp��}TjT�n��zlo8gH¥cb�³�dfhigh8kTkTVLgdg�h�UW}=h�klV�k8d�b�� Q)RTmyb�� o8g�dfh8sfmybHo0dHm�h8k }Tsfh8}=V�sfd `.mnbWm�klRTV�sHm­dHVL~©za`.dfRTV&jTkc��h8�n~cmnkT�ªhp�¨dfRTVµ~cmybedfsHm�zTjTdfVL~bf`cbedfV�U&�T¤+RTmng�R�U�o�¥�VLb)m­d|}=hqbfbfmnzT��V[dfh�}TsHhcgVLbHb)bfjlgHRµbe`cbedfV�U�b)mnk&oiUWhc~TjT�no�s)U�o�kTkTV�s / D�1B�

��'*) � � � � �� 6���6:. ,1 � . .����T���F�:9 -·~cmybvdHsfmnzTjcdHVL~&bf`abedfVLU/myb'��h8sfU�o����n`«V¬c}TsHVLbHbeVE~�o8b+oWbe}=VEgmyop�Fg�o8beV�hp��oi}Tjl���nzlo�g�¥�/ ��1£���¿:��� � � � à ��������

A,B�� �

C� �B���R����R�:� � = � � �P�����

A�� �B����0��� �� �R� � ��P����� ��� � � ����R�

C��������� �2��� ��

φA : A → C�� �

φB : B → C��P������(� ��� �=�� �R��� �� �=���P � �����U�R��<�� � � @ R !� � �

ΛC ⊇ ΛA ∩ ΛB� * � �i}ljT���nzlo8gH¥ � � ?��� �!� � �� � ��� �R�� ��� ��P�(� �����#�R�� � � � ���= � ����

AφA

→ CφB

← B�

� �C��� �� � � � �N = A×C

N B�� � �C�=�,� �R� ����� �:� �� 4� �

P = {(pA, ?) : pA ∈ PA, pA /∈ Dom (φA)}∪ {(?, pB) : pB ∈ PB, pB /∈ Dom (φB)}∪ {(pA, pB) ∈ PA × PB : φA (pA) = φB (pB)}

±v��³

�� � ��� ����P !����=���P ���

T = {(tA, ?) : tA ∈ TA, tA /∈ Dom (φA) , λA (tA) ∈ ΛA \ ΛB}∪ {(?, tB) : tB ∈ TB, tB /∈ Dom (φB) , λB (tB) ∈ ΛB \ ΛA}∪ {(tA, tB) ∈ TA × TB : φA (tA) = φB (tB)}

±C�8³

* �P� < ��� ��� ����=��� �������%���E B�� �R����������� � �:� �,��� ��� ��P�0� ���,�P����=��� �U� ���P�� � �� � ���R�! �

-xbHbehcg�mnopdfVE~Wdfh�dfRTmyb'}TjT�n�nzlo�g�¥«gh8klbvdHsfj@g�dfmnh�k��8dfRTVLsfV[V¬cmybvd+g�opklh�kTmyg�o��@UWh�sH}TRTmybfUWbψA : N → A

o�kl~ψB : N → B

dHRlo0d>UWo�}�V��nV�UWV�kqdHb�hp�NdfhxdHRTV+gh�sHsHVLbf}@h8kl~cm�kl�|VL��VLUWV�k8d�bJmnk

Aopkl~B�psHVLbf}=VLgdfmn¯�V��n`��

o�kl~�dHRlo0d¨bHo0dfmybe��`φA ◦ ψA = φB ◦ ψB

� � �L��� � ���� ¿ Ä-� À ��� ¿ �:9 - kTV�dNmybxbHopmy~�dfh�z=V�o;��� � �!� ��� � � ��@ � ��� m���m�d[g�o�kµz=V#V�¬a}lsfVEbfbfVL~

o8b+oi}ljT���nzlo8gH¥�h���beVL¯�V�s�op�Fgh�UW}=h�klV�k8d�b)¤+RTVLsfV�dfRTV�mnk8dHV�sHUWVL~cmyopsH`«kTVd�b|opsHV�m�kqdfV�sf� o�g�VLbL��Q)RTV����nh�zlo����� � ���� �=���; ���!�2� �������h��>o�~cmybedfsHm�zTjTdfVL~µbf`cbvdHV�U�g�o�k&z=V�sfVL}TsHVLbfV�k8dHVL~&o�b|o���s�op}TRF�T¤+RlV�sfV#opk&VE~c��Vmyb�~csHo�¤+k¶z=Vdv¤�V�VLkOdv¤�h&gh�UW}=h8kTV�k8d�bxm���dHRTV�`ORlo�¯�V�o�gh�UWUWh�k¶mnk8dfVLse� o8gV8��²lh�s�bfm�UW}T�nmygm�d `Om�k¶dfRlmnb}@op}=V�s�o�~TmnbedfsHm�zljcdfVE~¢be`cbedfVLU

Nmyb���mnUWm�dfVL~§dfh�dv¤�hµgh8UW}=h�kTVLk8dHb

Ao�kl~Bopkl~¢opk¢mnk8dfVLse� o8gV

C�

¤+Rlmng�R gLopk.dHRTV�k z=V�V�¬adfV�k@~cVL~ dfh¢o¶��s�op}TRF�ª²Jmn��jTsHV��¶~cVL}TmngdHbN = A ×C

N B¤+m�dfR�o�bHbehTgmyo0dfVE~

UWh8sf}TRlmnbfU�b��

K�KML�NPORQ@SUT

Page 11: Modular construction of finite and complete prefixes of Petri net … · 2016. 12. 27. · Modular construction of nite and complete pre xes of Petri net unfoldings Agnes Madalinski,

� �� �C�� � � � ����� �� ���

1

1 2

a

t

a

t 2

0

3

1

1

5

2

0

4

t

c

t

c

b

t

b

t

b

3t

0

2t

1c

3t

0c

1t

a 0

1a

2t

1c

3t

0c

4t

0b

5t

2b

1b

2t

1c

0c

3t

N

A B

ψA

φA φB

C

ψB

²Jm��8jTsHV����>Q)RTV�}TjT�n��z@o�g�¥N = A×C

N B

�(' & ���� . � 021*6 � .71(� ��� ��>� �/��1 � 6¡£d+¤'o8b)bfRTh�¤+k�mnk /n�L�@1�dHRlo0d+dfRTV[� o8g�dHh�sHmnbfVL~«��h�sHU®h��Jo�klVd|bf`cbvdHV�U/`am�VL�n~Tb+o#� o8g�dHh�sHmnbfVL~«��h�sHU®h���dHRTVjTkc��h8�n~Tm�kT��hp��bfjlg�R&oWbf`abedfVLU&� $ mn¯�V�k

N = A×CN Bh�kTV�h8zcdHo�m�k@b

UnfN = UnfA ×CO UnfB

±�"8³¤+RTVLsfV

UnfN�UnfA

o�kl~UnfBopsHVtdfRTV)jTkT��h��y~cm�kl�8bJh��

N�Aopk@~B��o�kl~

×CO

~cV�kTh�dfVEb�dfRTV)}Tjl���nzlo�g�¥h�k�zTsHo�klg�RTm�kT�x}TsHhcg�VLbHbeVEb���¤+m­dHR

UnfCo8b�m�kqdfVLse� o8gV��JQ)Rajlb��0dfRlV'jTkc��h8�n~cmnkT�[h��TdfRlV'���nh�zlo��qbf`cbvdHV�U g�o�k

z=V¨V¬c}TsHVLbHbeVE~Wo�b�dfRTV¨}ljT���nzlo8gH¥ih��5dfRTVxjTkc��h��y~cmnkT�8bth���m�dHb�g�h�UW}=h�kTVLk8dHbx± o8b�mn���njlbedfs�o0dHVL~Wmnk�²�mn�8jTsfV "8³��

�. .����T� �F��9 Q)RTV�g�h�UW}=h8bfm�dfmnh�k�za`�}TjT�n��zlo8gH¥�h���d ¤�h�zTs�opklg�RTmnkT�i}TsHhcg�VLbHbeVLb'myb+~cVLsfmn¯�VL~�o8b'��h��n��h0¤|b±�sHV��VLs'dfhW²Jmn��jTsHVJ"�³� FV�d

βA = (OA, fA)o�kl~

βB = (OB, fB)z=V[zTs�opklg�RTmnkT��}TsHhcg�VLbHbeVEbthp�

Aopkl~

B�5sfVEbe}=VEg�dfmn¯�VL��`8�¨Q)RTV�Uih8sf}lRTmnbfU

φA ◦ fAsHV��yo0dHVLb+dHRTV�hcg�g�jTsfsHV�klg�V�kTV�d

OAdfh�kTVd

C��beh�zq`�dHRTV

jTkTmn¯�V�s�bHop��}TsHh�}=VLsedv`«h��UnfC�TdHRTV�sHV�V¬cmybvd�b+oWjTklm��qjTV�UWh�sH}TRTmnbfU

φOA : OA → UnfCdHRlo0d¨V�klbfjTsHVLb

φA ◦ fA = fC ◦ φOA�amB� V��JdHRlo0d)U�op¥�VEb�dHRTV�~cmyop�8sHo�U¦gh8UiU#jTdHo0dHm�¯�V8�.H�`«bf`aUiUWVdHsH`��qh8kTV[o��nbfh#Rlo8b'o

UWh�sH}TRTmybeUφOB : OB → UnfC

� $ mn¯�V�k&dHRTVLbfV�UWh�sH}TRTmybfUWb+dfhUnfC�ldfRTV � �����#�R�� � hp�

OAo�kl~OBmnk�dHRTV�g�opdfV��8h�sH`Wh��Jhcg�g�jTsfsHV�klg�VxklVdHb+myb+~TV�sHm�¯�VE~«��sHh�U/dfRTV�h8sH~cmnklo�sf`�}TjT�n�nzlo�g�¥«h��JkTVd�b+zq`

OA ×CO OB = Unf

(

OA ×UnfCN OB

) ± �q³

Zµh8sfVLh�¯�VLsL�am�d|myb+VEo�bfm��n`�g�RTVLg�¥�VL~�dHRlo0d|dfRTV�sHV�V¬cmybvd�b|o���h8�n~cmnkT�f��sHh�U

OA ×CO OB

dfR@o0d|U�o�¥�VLb)m�dxozTs�opklg�RTmnkT�i}lsfhcg�VLbHb'hp�

A×CN B�,+|h�dfmygV[dHRlo0d#± �q³'V�kqdHo�m��yb'dHRTV�V¬cmybedfV�k@gV�hp�>oisfVEgjTs�bfm�¯�V[}TsHhcgVE~cjTsHV

dfh�gh8Ui}ljcdfV¶dfRTV§}ljT���nzlo8gH¥ hp��zTs�opk@gHRTmnkT�©}TsfhTgVLbHbfVLbL�)zlo8beVL~ h8k dfRTV§sHVLg�jTsHbfmn¯�VOg�h�klbedfsHjlg�dHmnh�k h��jTkc��h8�n~Tm�kT�qb+o�kl~�h�k���h8sfU#jT�yopV�±v��� ��³��

� Ã��E¿ ��� � à � �:9 $ mn¯�V�kUnfN = UnfA ×C

O UnfB��beh8UWV�kThc~cVEbxh��

UnfNopsHV��no�z=V��n��VE~Ozq`O}T�yo�gVEb

opk@~�dHsHo�klbem�dHm�h8klb>hp�C��dHRTsHh�jT�8R�opkq`�h��=dHRTV�± gh8UWU#jcdfmnkT�q³>}lopdfRlb�sHV��yo0dHm�kl�

UnfNdfhCmnk«²Jm��8jTsHVI"T�

Q)RTV0���� ���!� � ��=���&h��UnfN

dfh#dfRlVLbfVxkThc~cVEbtmyb'~cVLkThpdHVL~«za`(UnfN )|C

�.H'`�gh8k8dfs�o�bedL�8dfRlV�� �U��� � � � ���hp�

UnfNh�k�z=VLRloE¯amnh�jTs�b'hp�

Cmnb|~cV�@kTVL~&o�b'dHRTV�m�U�o���V�hp�

UnfNm�k

UnfC�

ΠC (UnfN ) = φOA ◦ ψOA (UnfN ) = φOB ◦ ψ

OB (UnfN ) .

± 7�³

=?> K@=�A

Page 12: Modular construction of finite and complete prefixes of Petri net … · 2016. 12. 27. · Modular construction of nite and complete pre xes of Petri net unfoldings Agnes Madalinski,

������������ ���������������������������! D

f

fC

fA

fBψOA ψO

B

ψA ψB

φBφAφOA φO

B

C

A B

UnfC

OA OB

ON = OA ×UnfC

O OB A×C

NB

²Jmn��jTsHV6"���X�h�UWU#jcd�o0dHm�¯�Vx~Tmno���s�opU hp��dHRTVx}TjT�n��zlo8gH¥cbth��FzTs�opklg�RTm�kl��}TsHhcgVEbfbfVLbtopkl~�klVdHbL�ao�kl~WdfRlV�mnssHV��yo0dHmnh�k�zq`�o8bfbfhcgmyo0dHVL~«�¼h8�y~cm�kT�qb��

-¨��dfV�sHklopdfmn¯�V��n`��8dfRTmybt}TsHhpuvVLgdfmnh�k«gLopk�z=Vxh�zcd�opmnkTVL~�zq`idHo�¥qmnkT���@sHbed(UnfN )|C

opkl~WdfRlV�k«}=VLse��h8sfUWmnkT�o ��!��� �0����� � mC� V���zq`µUWV�sH��mnkT��mybfh�UWh�sH}TRTmyg#gh8kc�l��jlsHopdfmnh�klb¨m�k

(UnfN )|Cmnk¶h8sH~TV�s¨dHh��8Vd�o�¯pop�nmn~

zlsHo�klgHRlm�kT�#}lsfhcg�VLbHb�h��C��r�sHh0uvVEg�dHm�h8klb�h�k

Aopkl~Bh��

UnfNopsHVx~cV�lklVL~�bemnUWm��yopsH��`�o�b�m­d�b�mnU�op��VEb

za`ψOA

opkl~ψOB

sHVLbf}=VLgdfmn¯�V��n`���r�sHhpuvVLg�dHm�h8klb)klopdfjTs�op�n��`«V�¬qdHV�kl~�dfh«opkq` H'r�h��N�

+|h�dfmygV�dfRlopd¨dfRTV#sHVLbedfsHmyg�dfmnh�k&h��UnfN

dfh�o�bejTz@beVdxh��>kThc~cVEb|U�oE`�V�s�o�bfV�g�opj@bfo���m�dv`�h8sxgh�k��lmngdsHV��yo0dHmnh�klb|z=V�d ¤'V�V�kOdHRTVLbfV#kThT~cVLbL�=¤+RTmygHR¶U�oE`�dfRlV�k§op}T}=VEops[o�bxgh8klgjTsHsHV�k8dxm�k

(UnfN )|C¤+RTVLsfVEo�b

dHRTV�`W¤'V�sHV|klhpd�mnkUnfN�>Q)RTVxbHopUWV¨}TRTV�klh�UWV�kTh8k«hcg�g�jTsHbto8bt¤�V��n�=¤+m­dHR«}lsfhpuvVLgdfmnh�klbL��Q)RTV�sHV��h8sfV¨o

}lsfhpuvVLgdfmnh�kΠC (UnfN )

mnbxbfo�mn~�dfh«z=V � ��� �0� !��R �������©m��¢VL¯�V�sH`�g�h�kc�l�8jTs�o0dfmnh�kκ′mnk

ΠC (UnfN )myb'dfRlV�mnUWo���V[h��JoWgh�kc�@��jTs�o0dHm�h8k«h��κmnk

UnfN�lopkl~�g�o�jlbfo���m�dv`�sHV��yo0dfmnh�k@b�h8k�V�¯�VLk8d�b)hp�

κ′o�sfV[kTh�d

�nhqbvdE��YxzlbfV�sH¯�V�dHRlo0d�}lsfhpuvVLgdfmnh�klb�h�k¢mnk8dHV�sf� o�gVEbCdHRlo0d#o�sfV�opjTdfh�U�opdHoT��mB� V8�«rJV�dfsHm�kTV�dHb�¤+m�dfRlh�jcd

g�h�klg�jTsHsfVLklg`8�ao�sfV�zq`�~cV��lkTm�dfmnh�k&kTh�k�UWmybe�nVLo8~cmnkT�l�Q)RTV«� o�g�d#dfRlopd#dfRTV�bfm�UW}T�nV«}TsHhpuvVLgdfmnh�klb#~cV�@kTVL~ opz=h0¯�V�U�oE`¢z=V«UWmnbf�nVLo�~Tm�kT�Omyb�dfRlV«VEbfbfV�kqdfmyop�

sHVLo8beh8k§¤+Rq`¶dfRTmyb�}lop}=V�s��nm�UWm�dHb�m­d�b�bHgh8}=VidfhObfjlg�R.bemnUW}T��V«m�kqdfV�sf� o�g�VLb�o�b�opjcdHh�U�o0d�oT��²Th�s#bemnUWm��yo�ssHVLo8beh8klbL� /n��1�mnk8dHsfhT~cjlgVE~>��� � �!�� �<9����� � �!�2� ������� @�5¤+RTmn��V:/8?�1�}lsfh8}@hqbeVE~�dfh�¤�h8sf¥�¤+m�dHR �� ��� � �2 �R�� ���� � �������;� ��� �R�! R �� ��|m�k h�s�~cVLsidHh.}=V�sf��h�sHU½UWhc~cjT�yops�gh�UW}TjTdHo0dHm�h8klbL�»¡ kqdfsHhc~cjlg�m�kT�.RTVLsfV&dHRTVLbfVdHVLg�RTkTmyg�o��asHV�@kTV�UWV�kqdHb�¤�h8jT�n~igV�sfd�opmnkT��`�z=V)}=hqbfbfmnzT��V8�0zTjcd�¤�h8jT�n~igh8klbemy~cV�s�opzl��`���hqo�~�~cV�¯�VL��h8}TUWV�k8d�b¤+m�dHR«dfVEgHRlkTmngLop�5~TVdHo�m��ybtdfR@o0d)o�sfV¨kTh�d'sHVLop�n�n`Wo0d�dHRTV[g�V�kqdfV�s'h���dHRTV�bvdHjl~c`8��¡£d'mybtdfRlV�sHV��h�sHVxg�RTh8bfV�k�dHh��hTgjlb�h8k«dfRTV�bf}=VLg�m­�@g[~cm 2«g�jT�­dHm�VEb'sfVL�nopdfVE~Wdfhih�zcd�opmnkTmnkT���@kTm­dHV�gh8Ui}l��V�dfVx}lsfV�l¬aVEb�¤+m­dHR�o#UWhc~cjl�no�so�}T}TsHh8o8gHRF�@opd|dHRTV#V�¬a}=VLklbfV�hp�to��nm�UWm�dfVE~ObfVdedHm�kl�«¤+RlV�sHV�m�kqdfVLse� o8gV�kTV�dHb[opsHV�kTh�dx��V�klV�s�op�FklVdHbxzTjcdbfmnUW}T��V�opjcdHh�U�opdHoT�

- � � � .��. � �. . �J�l���µ� Ã��=¿ � � �"# 9 rtsfhpuvVEg�dfmnh�k@bJop�n�nh�¤ jlb�dfh[zTjTmn�n~�dHRTV��B���P������ � �����#�R�� � �R��<����!�����h��

UnfN�=¤+RTmygHROsfVEbvdHsfmyg�d�b)dHRTV�� o�gdfh8sHb

UnfAopkl~

UnfBdfh�dHRTV#z=V�Rlo�¯amnh�jTs�b+hp��g�h�UW}=h�kTVLk8dHb

A,BdHRlo0d|sHV�U�opmnk��¼VEo�bfm�zl��V�mnk�dfRTV�VLk8dfmnsHV�bf`abedfVLUN��YxkTV�g�o�k�¤+sHm�dfV

UnfN = ΠA (UnfN )×CO ΠB (UnfN )

± �q³o�kl~OVEo�g�R �0��� ��� �� ���� ���

Πx (UnfN )��¤+RTV�sHV

x ∈ {A,B}��myb[o�}TsHV�T¬Ohp�

Unfx�#Q)RTVEbeV�� o�g�dHh�s�b

o�sfV[UWmnkTm�U�o���mnk�dfRTV�beVLklbeV[dfRlopd)d�op¥am�kl�Wopkq`�bedfsHmyg�d+}TsHV�T¬�hp��dfRTVLU°¤�h�jl�n~�}TsHV�¯�V�kqd)sHVLg�h�¯�VLsfmnkT�#dfRTV��jl���

UnfNza`�}TjT�n��zlo8gH¥5�

ZµmnkTm�U�o��5� o�gdfh�s�b)g�o�k�z=V�gh8Ui}ljcdfVE~�m�k&oiUihT~cjT�yops)U�opkTkTVLs)¤+m�dfRTh8jcd|g�h�UW}TjcdHm�kl�UnfN

m�dHbfV����v�Y[kTV�Rlo�b

ΠA (UnfN ) = UnfA ×CO ΠC (UnfB)

± ?�³±Copkl~ bf`aUiUWV�dfsHmngLop�n��`.��h8s

ΠB (UnfN )³ / �T��?�1£�-Q)RTmnb�sHV��yo0dHm�h8k V�¬a}lsfVLbHbfVLb�dHRlo0d«¥akTh0¤+mnkT�¢dfRTVµz=V�´

R@oE¯amnh�jTs�b�h��Bh�k«dfRlVxmnk8dHV�sf��o8gVxkTV�d

Cmyb'bfj�2«gmnV�kqd�dHh

AdHhi~cVdHV�sHUimnkTVx¤+Rlmng�R�hp��m�dHb'sfjlklb�sfVLUWo�m�k

}=hqbfbfmnzT��V[mnk�dHRTV����nh�z@op�5bf`cbvdHV�UN�|± ?p³�RTh8�n~lb��Topkl~«dHRqjlb|op�n��h0¤|b�h8kTVxdfhW}=V�sf��h�sHU/o�UWhc~cjT�yops+gh�Ui´

}ljcdHopdfmnh�k�h��JUWm�kTmnU�op��� o8g�dfh8sHbL�co8b+behch�k�o�b)}TsHhpuvVLgdfmnh�klb)h�kCopsHV[kTh8k�Uimybf��VEo�~cmnkT�l�>¡£d+myb+}TsfVEgmybeVL��`

K�KML�NPORQ@SUT

Page 13: Modular construction of finite and complete prefixes of Petri net … · 2016. 12. 27. · Modular construction of nite and complete pre xes of Petri net unfoldings Agnes Madalinski,

�L� �� �C�� � � � ����� �� ���

dfh�h8zcdHo�m�kWo[bfm�UWmn�no�s�}Tsfh8}=V�sfd `���h8s>o�kq`�mnk8dfVLse� o8gV)kTVdCdfRlopd �� ��� � �2 �R� ������ ����������� ��� �R�� �� J¤�V�sHV

~cVL¯�V��nh�}=VL~�mnk / ?�1B�T¤+RTmn�nV /n��1��nopdfV�s+}TsHh�}=hqbeVE~�dHRTV�op��dfVLsfk@o0dfV[dfVEgHRTklm��qjTV�hp����� � �!��R�<������ ���!�2� ������� ��²lh�s�Uih8sfV�g�h�UW}T�nV¬.~cmybvdHsfmnzTjcdHVL~ bf`cbvdHV�U�b��JdHo�¥am�kT�µdHRTV�bfRlo�}@V�hp�¨oµkTV�d ¤�h8sf¥¢hp�¨gh�UW}=h8kTV�kqdHbL�

UWm�klm�U�op�l� o�gdfh�s�b�g�o�kWz=V|h8zcdHo�m�kTVE~iza`�V¬adHV�kl~cmnkT��dHRTV¨o�z=h�¯�V+}TsHm�k@gmn}T��V|h��5UWhc~cjl�no�stgh�UW}TjTdHo0dHm�h8klbL�¡£d�dfRTVLk�d�op¥�VEb5dHRTV���h�sHU hp�To+UWVLbHbfo���V�}@o�bHbemnkT�|op�n��h8sfm�dfRlU��E¤+RTmng�R�sHjTklbFh8k[dHRTVtm�kqdfV�s�o�gdfmnh�k�bvdHsfjlgdfjTsHVhp�Nopkl~�}Tsfh8��sHVLbfbfm�¯�V��n`|jl}5~To0dHVLbFmnkc��h8sfU�o0dHm�h8k�h�k�mnk8dHV�sf� o�gVEb���Q)RTV�op�n��h8sfm�dfRlU g�o�k�z=Vt}TsHh�¯�VE~[V¬To�gd

��h�s#bf`cbvdHV�U�b���mn¯amnkT�µh�k.o�dfsHV�V��Jo�kl~¢myb�h�kl��`§o�}T}TsHhE¬cmnU�o0dfVWmnkªh�dfRTVLs�gLo�bfVLbL� A¢V�sHV��VLs�dHRTV«sfVEo�~cVLsdfh�/ �T� ?�1F��h�s|~cV�dHo�m��yb��

� ���W��������º '�������� '�� $����)��*«�+�

Q)RTV+h�zcuvVLg�dHm�¯8V)hp�@dHRTmyb>}lop}=V�s>mybJdHh�gh8UW}TjcdfV)�lkTm�dfV|o�kl~igh8UW}T��V�dfV+}TsfV��T¬cVLb+± ²JX�r�³�h��5o�~cmybvdHsfmnzTjcdHVL~bf`abedfVLU

Nmnk � o�gdfh�sHmybeVE~©��h8sfU&� Q�hª��hcg�jlb«h�k dfRTV¶VLbHbeVLk8dHmno��¨~cm2«gjT��dfmnVLb�h��[dHRTmnb�}TsHh�zT�nV�U&�'dHRTV

~cmybfg�jlbHbemnh�k mnbi�ls�bedW�nm�UWm�dfVE~©dfh¢dfRTVOg�o�bfV�hp�¨dv¤�h.gh8Ui}=h8kTV�kqdHbAopkl~

Bmnk8dHV�s�o�gdfmnkT�§dfRTsHh�jT�8R�o

gh8UWUih8kOm�kqdfV�sf� o�g�VC�5¤+RTmygHR¶V�k@bejTsHVLb

N = A ×CN Bo8b¨m­d[¤)o�bxbfV�V�k§opz=h0¯�V��6-�d[dfRlV�V�kl~Ohp�>dHRTmyb

bfVLg�dHm�h8kF�ERTh0¤�V�¯�VLsL��dHRTV��8V�kTVLsHo���mybfo0dHmnh�k¨dHh|Uih8sfV�g�h�UW}T�nV¬W±¼dHsfVLV´ beRlo�}=VL~l³�be`cbedfVLUWbF¤+mn���8z=V�V¬TopUWmnkTVL~��¡£d[myb¨��jTsedHRTV�sHUWh�sHV#o8bfbfjTUWVL~&dHRlo0dxdHRTV�mnk8dfVLse� o8gV

Cmyb[opk¶o�jcdfh8UWopdfh8kF�=kThpd�o«�8V�kTVLsHo���bfop��V#kTV�dL�5mnk

h�s�~cVLs'dfh�o�¯�h�my~«dHRTV�V¬adfs�o#dHVLg�RTkTmyg�op��~cm2«gjT��d `�h���~cVLop�nmnkT�W¤+m­dHR�UWmybf��VEo�~cmnkT�W}TsfhpuvVEg�dfmnh�k@b��-·dfsHmn¯qmyop��beh8��jcdHmnh�k�dfh«h8zcdHo�m�kOopkµ²JX�r»h��

Nmnk&� o�gdfh�sHmybeVE~���h�sHUº¤�h8jT�n~&z=V�dHh«bedHo�sed|��sfh8U�bfo�`

oi���nh�z@op��²JX�r��PrefN

�lo�kl~�dfh�}TsfhpuvVEg�d+m�d|h�k�dHRTV�gh�UW}=h8kTV�kqdHb+o�b'��h8���nh0¤|b��

PrefN v ΠA (PrefN )×CO ΠB (PrefN )

±C�8³�|h0¤�VL¯�V�sE�LdfRTmyb�mnUi}=hqbeVEbFdfh¨¤'h�sH¥¨�ls�bvdJh�k�dfRTV��8��h8zlop�qbe`cbedfV�U&��¤+Rlmng�R#g�h�jT�y~�z=VtV�¬adfsHV�UWV��n`[V¬c}=V�k@bemn¯�Vm���dHRTV�bf`cbedfV�U/mnb)�no�sf�8V��J¡ k�}lo�sedHmng�jT�no�sL�qdfRTmyb)}TsHV�¯�V�kqdHb�dHo�¥am�kT�Wo�~c¯popkqdHop�8V¨hp��dHRTV�gh8Ui}lsfVLbHbfm�h8k«�8o�m�k}TsHh�¯amy~cVL~«zq`W� o�gdfh�sHmybeVE~i��h8sfU�bL�ao�kl~�dfRTmyb'}TsfVL¯�V�kqdHb'o�b�¤�VL���5}TsHhcgVEbfbfm�kl��dfRTV�bf`cbvdHV�U¦za`�}lopsfdHbL��Q)RTVh�zcuvVLgdfmn¯�V�h���dfRTmyb|bfVLg�dHm�h8k�mnb)dfhWzljTm��y~>���������� � �§dfRTV�� o8g�dHh�s�b'hp��oW²JX�r hp�

UnfN�

Q�hµh8zcdHo�m�kªdfRTVLU���m�d#mnb�kTh�d#V�kTh8jT��R¢dfh¶bfm�UW}T�n`§zTjlm��y~.²JX�r hp�'dHRTV�g�h�UW}=h�kTVLk8dHbmnk.��V�klV�s�op�B�

dfRlV�mns+g�h�U#zTmnklo0dHm�h8k�zq`«}TjT�n��z@o�g�¥�¤'h�jT�y~�kTh�d)z=V�gh�UW}T�nVdHVx��h8s'dfRTV��8�nh�zlop��be`cbedfVLU���Q)RqjlbL�adfRTV�my~cVEomybidfhªzTjlm��y~�²JX�r¦h��[g�h�UW}=h�kTVLk8d�b�dHRlo0d«¤+m��n�¨op�ybeh¢}TsHh0¯qmy~cV&bfj�2«gmnV�kqdf�n` sHmygHR�z=V�Rlo�¯am�h8jTs�b�h8k©dHRTVmnk8dfVLse� o8gVim�k§h8sH~cVLs¨dHh�VLklbejTsHV��8��h8zlop��gh8Ui}l��V�dfV�klVLbfbL��Q)RTmyb�~cVLUWo�kl~Tb[hp�'gh8jTsHbfV�o�k§op�8sfVLV�UWV�kqdxh��dfRlV�dv¤�h�gh�UW}=h8kTV�k8d�b�opz=h�jcd�¤+RlopdJz=VLRlo�¯qmnh�jlsHb�dfRTV)m�kqdfV�sf� o�g�V'bfRTh�jl�n~#}TsHh�¯amy~cV��popkl~�bfh[opk�V¬TgHR@opkT�8Vhp�Jmnkc��h8sfU�o0dHm�h8k�z=V�d ¤�VLV�kµgh8Ui}=h8kTV�kqdHbL�

- bedfs�opmn��Rqde��h�sH¤'opsH~�¤)oE`«dHh�m�UW}T�nV�UWV�kqd|dHRTmnb|my~cVLoWmyb)dHh«bvd�opsfd+¤+m�dHR&�nhcg�o����n`�g�h�UW}T�nVdHV�}TsfV��T¬cVLbL�opk@~�dfRTVLk&dHh�dfs�opk@beUWm�d�± opk@~&sHVLgVLm�¯�V�³�dHRTV#z=V�Rlo�¯am�h8jTs�b)sHV �qjTmnsHVL~�h�k&dHRTV#m�kqdfV�sf� o�g�V��@z=V��h8sfV�sfVEgh8U�´}TjcdHm�kl�&��hTg�op�>²JX�r ¤+m­dHR§dfRlVLbfViV�¬adfs�o�sHV �qjTmnsHV�UWV�kqdHbL��Vd�gp���¨h�¤'V�¯�V�sE�5dHRTmnb�UWm��8R8d�sHVLbfjT��d�mnk¢UWo�kq`m�dfV�s�o0dHmnh�klb'z=Vdv¤�V�VLk�dHRTV�gh8UW}@h8kTV�kqdHbL�tX�h�klbfmy~cV�s'dfRTV�V�¬TopUW}T�nV[m�k&²Jm��8jTsHVx�l�c¤+Rlmng�R�~TV�}Tmyg�d�b

PrefAopk@~PrefB�pdv¤�h��nhcg�o��T²JX�r��p¤+m�dfR�o�bfmnUi}l��V+mnk8dfVLse� o8gV)h8kT��`�gh8klbemybedfmnkT�[h��@dfRlV)dfs�opklbfm­dHmnh�k

t1o�kl~#dHRTV

}T�yo�g�Vc0�PrefA

kTV�VE~Tb�h�kTV|hcgLgjTsHsfVLklgV)h��=dHRTV|mnk8dHV�sf��o8gV+dfs�opklbfm�dfmnh�kt1�EH'`igh8k8dfs�o�bedL�

PrefBkTVLVL~Tb

dv¤�hihcgLgjTsHsfVLklgVEbthp�t1dfhWz=V�g�h�UW}T�nVdfV8�>Q)Rqj@b��

PrefARlo8b�dfhiz=V�V¬adfVLkl~cVL~�zq`«h8kTV[hcg�gjlsfsHV�klg�V¨h��

t1 dfRTmyb�kTVL¤©z=V�Rlo�¯am�h8jTsJh8k�dHRTV)mnk8dfVLse� o8gV'Rlo8b�dfh�z@V)}TsHh�}lo��8o0dHVL~�dfh

PrefB�J¡ kio8~T~cm�dfmnh�kF�ph8kTV'kTVLVL~Tb

dfhWg�RTVLg�¥W��h�s'�8��h8zlop�5gh8Ui}T�nVdHV�kTVEbfb'zq`«g�h�klbfmn~TV�sHm�kT�����nh�zlo��@U�opsH¥am�kT�qb��aopk@~�dfRTmyb'UWmn��Rqd)sfV��8jlm�sHV|dHRTVV¬adHV�klbfm�h8k�hp��dHRTV�}TsfV��T¬cVLb'hp��dfRTVEbeV�gh8Ui}=h8kTV�kqdHb'jTkqdfmn�F���nh�zlo��@dfsHjTklgLo0dfmnh�k�}=h8m�kqdHb+opsHVxsHVLo�g�RTVL~F�J¡ kdfRlVWV�kl~��FdfRTmyb�UWm��8R8d�sHVLbfjT��d�m�k¢U�opka`µV�¬cg�Rlo�kT��VEb���Q�h&o�¯�h�my~µdHRTmnb�bem�dfjlopdfmnh�kF��dHRTVWz=V�Rlo�¯am�h8jTs[h��'ogh8UW}=h�kTV�kqd�h�k.m�dHb�m�kqdfVLse� o8gV�mnb�gLop}cdHjTsfVE~ªjTkl~TV�s#dfRlV«��h8sfU(h��|o> R� � ������ � ��B�Jdfs�opklbfUWm�dedfVE~ªmnk¸eh�kTV#beRTh�dL� ¹¢Q)RTV�bejTUWU�opsH`�kTVdxmyb|h8zcdHo�m�kTVE~���sfh8U�opkµV�¬adfV�kl~TVL~µgLopkTh8kTmyg�op��}TsfV��T¬�za`W¸esHV��h��y~cmnkT�0¹m�dHb+sfVEbvdHsfmyg�dHmnh�k�dfhWdfRTV�mnk8dHV�sf��o8gV8�

� > )>5/< 50 $ 5�. L #�� & ) L <�0 $ ,�<�� F $ 31. 576�. ?Pref

N

.1, $ 31+;< $>= 6 N�. G < L . L 2�F4313 '�$R(� ? )�+/0�8 $ ,�. L # O & @� F 5�. L N�< L <(+ $ 3504< 0�. L .10 $ 3 2JF4313 '@$!(�� ( )/G <(+;. L N�)>?Pref

N

.1,-3 $ + N�<(+ @

=?> K@=�A

Page 14: Modular construction of finite and complete prefixes of Petri net … · 2016. 12. 27. · Modular construction of nite and complete pre xes of Petri net unfoldings Agnes Madalinski,

������������ ���������������������������! �8�

a 0

a 1

a 0

t 2

t 1

t 2 e 2

e 3

e 4

c 0

a 0

a 1

t 1e 1

c 0

c 0

#%$�&Pref

A

c 0

c 0

t 1

c 0

t 1

t 1

c 0

c 0

c 0

t 1

#'�& . L 5/<(+ $!( 5/.1) L

t 3

b 1

b 0

b 1 d 1

b 0

b 1

b 0

d 2

b 1

b 1b 1

b 0

b 1

d 1

d 0e 2 t 4

e 4 t 1

t 5

t 3e 1

e 3 t 1

t 3e 5 t 4

e 8

d 0

e 6

t 6t 1e 9

e 7 t 3

e 11

e 10

c 0

c 0 c 0

c 0

#(�&Pref

B

²Jm��8jTsfV����>¡ ���njlbedfs�o0dHm�kl�isHV �qjTmnsfVE~�m�kqdfV�s�o�gdfmnh�k�¯amno�dfRlV�mnk8dHV�sf� o�g�V�z=Vdv¤�V�VLk&g�h�UW}=h�kTVLk8d�b

��'*) � � . ,- ���,�� � �� � 1�� � � � 07,����(, 6Q)RlV#V¬adHV�kl~cVE~OgLopkTh8kTmyg�op��}lsfV��T¬&hp�to�g�h�UW}=h�kTVLk8d

Ah�sBmnbxzTjTmn�­d[¤+m�dHROsHV��qops�~�dfh�m�dHbxmnk8dHV�sf��o8gV8�

^cjlgHR©oµ}TsHV�T¬ªgLop}cdHjTsHVLb�dfRTV�z=V�Rlo�¯amnh�jTs�h��'dHRlo0d�mnk8dHV�sf��o8gV«mnk.sHV��yo0dHm�h8k¢dfhOm�dHb�gh8Ui}=h8kTV�kqdL�&¡£d�mybh8zcdHo�mnkTVL~«zq`�sfVEbvdHsfmyg�dHm�kT��dHRTV[g�jcdedHm�kl�ig�h�k8dHV¬adL�amnk�}lo�sedHmng�jT�no�s�dHRTV�beV�d'hp��gh8kc�l��jlsHopdfmnh�klb�¤+RTmyg�R�opsHVj@beVE~���h�s)dHRTV�gjcdf´Bh��¶g�sfm�dHV�sfmnh8kF���¿:��� � � � à ��� ����

A�R�� �R����� ��� � �2��.��?� ��F���2 ��� ��:� �

C�&* � � � � � �%� �" ���� �B��� � � ����R�

C� ?� �

� � ������ � ���2 ���:ΘC =

(

≈mar ,Ctot , {κe}e∈EA

) � �C�=�,� �R���E��?�∀e ∈ EA

κe =

{

{ [e′] : e′ ∈ EA, ΠC (e′) 6= ∅ }� �

ΠC (e) 6= ∅# %$

{ [e′] : e′ ∈ EA, ΠC ([e]4 [e′]) = ∅ }���� � ���.� � # �&$ ±=Dq³

� �P�����4

� ��P� ���� � � ��!� � � ���� � � �U� � � � �

Q)RlV�sHVLbedfsHmngdfmnh�k�h���dfRTV�g�jcdedHm�kT�«g�h�k8dHV¬adΘC

mnkAUWVLo�klb'dfRlopd �

±Co8³:-|k mnk8dHV�sf� o�gV�V�¯�VLk8dµ±Copk V�¯�VLk8d�g�h�sHsfVEbe}=h�kl~Tm�kT�OdHh¢o¶dfs�opklbfm�dfmnh�k©hp�¨dfRlV�mnk8dfVLse� o8gVE³�gLopk z=V~cVLbfm��8klo0dHVL~�o�b+oWgjcdf´Bh��OV�¯�VLk8d+h8kT��`«m��Jm�dHb|gh�sHsHVLbf}=h�kl~cmnkT��V�¯�V�kqd+mnb|op�ybehWopk�mnk8dfVLse� o8gV�V�¯�V�kqdL�

±�z=³ A RlV�sHVLo�bL�@dfRTVWgh�sHsfVEbe}=h�k@~cm�kl��V�¯�VLk8de′hp�to�}Tsfmn¯0opdfV�gjcdf´Bh��ªV�¯�VLk8d

e±�mB� V8��opkOV�¯�VLk8d[¤+RTmyg�R

~chaVLbWkThpd«g�h�sHsfVLbf}=h�kl~.dfh.o�kq`.dfs�opkqdfm�dfmnh�k�hp�¨dHRTV&mnk8dHV�sf� o�g�VE³�Rlo�bidfhªz=Vµg�RTh8bfV�k�bfjlg�R©dHRlo0ddfRTV�sHV�opsHV[kThWm�kqdfV�sf� o�g�V[VL¯�V�kqdHb�¸ez=Vdv¤�V�VLka¹

eopk@~

e′�cmB� V8��mnk

ΠC ([e]4 [e′])��Q)RTmyb+gh8kl~cm�dfmnh�k

mnb+kTVLg�VLbHbfo�sf`Wdfh«gjcd|mnkc�lkTm�dfV#gHR@opmnklb)hp�J}TsHmn¯0o0dHV[VL¯�V�kqdHb+m�k�dfRTV�jTkT��h��y~cm�kl�l���¿:��� � � � à ���+* �P� ������ ����������� ��� � ��

Pref ΘC

A

��� ���2� �R�;� �-?� � � 6�U��� <���� = fsbleΘC

A

� �� �����R�?� ��V�¬adfV�k@~cVL~&g�o�kTh�kTmyg�o��5}TsHV�T¬ � �

A� �%� �" �6��= ���2 � � ���� �

C�

²Jmn��jTsHVLb 7T± oq³[o�kl~�± gE³�beRlh�¤�dfRTV�V¬adHV�kl~cVE~¢gLopkTh8kTmngLop�>}TsfV��T¬cVLb�hp��dHRTV«gh�UW}=h8kTV�k8d�bAopkl~Bm�k

²Jmn��jTsHV��c�¢¡ k dHRTV�bfV �qjTVL��dfRTV&gjcdf´Bh�� V�¯�V�kqdHbio�sfV�~cs�oE¤+k©o8b�~ch8jTzT�nV�z=hE¬cVLb�m�k �l�8jTsHVLbL��¤+RTVLsfVEo�bmnk§dHRTV�gLo�bfVihp�'V¬adHV�kl~cVE~¢gjcdf´Bh�� V�¯�VLk8d�b[dfRTV�h�jTdfV�s�z=h�¬Omyb�~cs�o�¤+k§¤+m­dHRªo&~To�bfRTVE~¶�nm�kTV8�

Pref ΘC

A

K�KML�NPORQ@SUT

Page 15: Modular construction of finite and complete prefixes of Petri net … · 2016. 12. 27. · Modular construction of nite and complete pre xes of Petri net unfoldings Agnes Madalinski,

�E� �� �C�� � � � ����� �� ���

a 0 c 0

a 0 c 1

3t

2te 1

e 3

c 0

e 2 1t

a 1

a 1

1te 0

# $�&Pref

ΘCA

2t 0y

3t 1y

sA0

(a0, c0)

sA1

(a0, c1)

#%'�&ΠC (A)

c 0

2te 0

b 0

3te 2

c 1

c 0

2te 3

b 2

c 1

3te 6

b 2c 0

e 1 4t

b 1

e 4 5t

b 0

e 5 4t

b 1

#(�&Pref

ΘCB

2t 0y

1y3t

2y2t

3t 3y

sB1

(c1, b0)

sB2

(c0, b2)

sB0

(c0, b0)

sB3

(c1, b2)

# =�&ΠC (B)

²�mn�8jTsfV 7 �.4>¬adfVLkl~cVL~&}TsfV��T¬cVLb+opkl~�dfRTV�mns¨gh8sfsHVLbf}=h�kl~cmnkT�ibfjTUiU�o�sf`«kTV�dHb

gh8m�k@gmy~cVLb|¤+m­dHR&dHRTV#gLopkTh�klmngLop�F}TsHV�T¬&bfm�klg�V�dfRTV#h�kT�n`�gjcdf´Bh��§V�¯�VLk8dxo�kl~&m­d�b¨gh8sfsHVLbf}=h�kl~cmnkT�WV�¯�VLk8dopsHV[mnk8dfVLse� o8gV�V�¯�VLk8d�b�� �|h0¤�V�¯�VLsL�adfRTmyb+mnb+kThpd)dHRTV�g�o�bfVx��h8s

Pref ΘC

B

m�k&²Jmn��jTsHV 7T±CgL³��Q)RTV�V�¬adfV�k@~cVL~}TsHV�T¬�mnb)�no�sf�8V�s�dfRlo�k�dfRlV�bedHo�kl~To�sH~«}TsHV�l¬5�c¤+RTmygHR�¤�h�jl�n~�z=V�h�zcd�opmnkTVL~�za`�beVdfdfmnkT�

e4o�b)oigjTde´£h �

V�¯�VLk8d+bfm�klg�V[⊥] ≈mar [e4]

opkl~[⊥]Ctot [e4]

��¨h�¤'V�¯�V�sE�e4~ThaVLb�kTh�d)g�h�sHsfVEbe}=h�kl~WdfhWopk�V�¬adfV�k@~cVL~

gjTde´£h �¢VL¯�V�k8d�bemnklgV#m�dxmybxo«}TsHm�¯po0dfV�VL¯�V�kqd[opkl~ΠC ([e4]4 [⊥]) = {e2, e3}

�¨Q)Rlmnbxo�}T}T�nm�VEbxop�ybeh�dfhdfRlV�V�¯�VLk8d

e5�>Q)RTV�V�¯�V�kqd

e6mnb+o�k�V�¬qdHV�kl~TVL~�g�jcde´£h �¶bfmnklgV�m­d�beVL�­�>opkl~�m­d�b+g�h�sHsfVEbe}=h�kl~Tm�kT��V�¯�VLk8d

e2opsHV[mnk8dHV�sf� o�gV�VL¯�V�kqdHbL�-¨k¶V�¬adfV�k@~cVL~¶gLopkTh8kTmyg�op��}TsHV�l¬µmnb�gh�UW}T�nVdHVibfm�klg�V�m­d�myb[o�g�opklh�kTmyg�o���}TsfV��T¬ ± bfV�V�rtsfh8}=h8bfm­dHm�h8k

�c� D�mnkF/��E�@1�³��>¡£d|Rlo8b'dfh�z@V�bfRTh�¤+k�dHRlo0d|m�d|mnb)�lkTm�dfV8� � à � à � � � � à � �

Pref ΘC

A

� �,�P�� � �

� ÃFÃ�� 9 H'`«r�sfh8}=h8bfm­dHm�h8k��T���E�#mnk /��E�@1Fm�d+myb)V�klh�jT�8R�dfh�beRlh�¤»dHRlo0d+VEo�g�R�mnkc�lkTm�dfV≺´£g�Rlopmnk�mnk

UnfAg�o�k z=Vµg�jcdL� Q�¤�hªgLo�bfVLbWopsHV&gh8klbemy~cVLsfVE~�� ¡£�xdHRTV�sHV&o�sfV�mnkc�lkTm�dfVL��` UWo�kq` mnk8dHV�sf� o�gV&V�¯�VLk8d�b���mB� V8�V�¯�VLk8d�b¨¤+RTmyg�R¶g�h�sHsfVEbe}=h�kl~�dfh

C�=mnkOdfRTVig�Rlo�m�kµdHRTV�kOdHRTVigHR@opmnk¶g�o�kµz=VigjTd�bemnklgV#dfRTV�kajTU#z=V�s[h��

U�opsH¥am�kT�qb'mnb)�lkTm�dfV�o�kl~�dfRajlb|bfh�UWV�UWo�sf¥amnkT�imyb+o��lk@op�FUWo�sf¥amnkT�ihp��bfV�¯�VLsHo��5m�kqdfVLse� o�g�V�V�¯�VLk8dHbL�Y¨dHRTV�sH¤+mybeV8�8dfRlV�sHVxmyb�h8kT�n`�o#�lkTm�dfV�kqjTU#z=V�s+hp��mnk8dHV�sf� o�gV[V�¯�VLk8dHb'mnk�dfRTV�gHRlo�m�k���^am�k@gVxdfRlV�gHR@opmnk

myb�m�kT�lkTm�dfV¨m�d'gh�kqdHo�m�klbtopk�mnkc�lkTm�dfV|d�opmn�lhp��h8kT��`i}Tsfmn¯0opdfV�± kTh�kT´Bmnk8dHV�sf��o8gV�³JVL¯�V�kqdHbL�>^amnklgV|dfRTV¨kqjlU#z=V�shp��U�o�sf¥amnkT�8b>myb>�@kTm­dHVxbfh�UWV|UWo�sf¥amnkT��mybto��lk@op�@UWo�sf¥amnkT��hp�FbfV�¯�VLsHo��T}TsHm�¯po0dfV|V�¯�VLk8d�b�mnkWdHRTV|dHo�m��=opk@~dfRajlb)dHRTV�gHRlo�m�kµg�opk�z=V�gjcdE�

¤

��' & � � � 0 � �,/.76Q)RTV)bfjTUWUWo�sf`�kTVd>hp�@o[gh8Ui}=h8kTV�kqd�gLop}cdHjTsfVEb�dfRTV)z=V�Rlo�¯am�h8jTs�h��@o[gh�UW}=h8kTV�kqdJ¤�� sL� dL�Jm­d�bJm�kqdfVLse� o8gV��opk@~�m­d|myb+~TV�sHm�¯�VE~«��sHh�U°m­d�b+V¬adfVLkl~cVL~&g�o�kTh�kTmyg�o��5}TsHV�T¬��

=?> K@=�A

Page 16: Modular construction of finite and complete prefixes of Petri net … · 2016. 12. 27. · Modular construction of nite and complete pre xes of Petri net unfoldings Agnes Madalinski,

������������ ���������������������������! ��"

fA

φA φA

fA

C

ΠC(A)

Pref ΘC

A

A

²�mn��jlsfV�����X'opkTh8kTmngLop�5UWh8sf}TRlmnbfU/sHV��yo0dfmnkT�idfRTV�bfjTUWUWo�sf`«kTV�dΠC (A)

dHhidfRlV�m�kqdfV�sf� o�g�VC�

��¿:��� � � � à ��������Pref ΘC

A

�R��?� � � �� � � � �R� � �� ���P� �R���� �������B�U�A� �%� �" �I��� 5��� � � ����R�

C�� � ���

cutΘC

A

� ��?� �I �� E� � � �� � � � �R� � � � � � � <@� �2� � � ��ΠC (A) = (S, Y, s0,→)

�C��� �� ����P�6�� ��� � ���loop((Pref ΘC

A )|C)�.��?�

loop� ��� � � � @

∀e ∈ (cutΘC

A )|C���������

e•�� �

e′•

� �P�����e′

� ?� � � ���!���! � ��� ��� ��� ��<�� �2 � �e�+* �P��� �9� �� ��U�

ΠC (A)� �����U�� � ��� � � ��P��� �9�C�!

�U�(Pref ΘC

A )|C# � � � ?� � ����������� � � ��� �=��� ���; � � �!

S$ � �� �

s0�R���!���� � ��� �@ � ��P� �0��� ��� ��

�R��� ���� �=��� � �(Pref ΘC

A )|C�5������� ��<@� � � ��P�����6� �C� � �R�� ��� � � ��2����� �2��� ��0

fA : Pref ΘC

A → ΠC (A)�� �

φA : ΠC (A) → C !�2��� ?� �

φA ◦ fA = φA ◦ fA# ��R� �,� � ���%$������ ?� �B ��%� �P��� �

ΠC (A)�

�R�� �� � ?� �ibfjTUWU�opsH`�klVd �U�A� �%� �" �6��� J��� � � ����R�

C�

�!� � ������=��� � ������ ��� �� � � ��� � �� !�=��� �R���%��� � � :� � � <@� � e� �

ΠC (A)� J@ R � � � � �R���.��?� ?� �

���� � �����Mark([e])

� � �R����� ��� � �2A��P� �� � ���9���2� �� 0���

Pref ΘC

A

����� ��� �����! � ���� ���� � ����� � ��� � ���� ��� �R� @ ���R���E��� ?� � � � �

s = e•�

Q)RTV:�%�9� � h8}=V�s�o0dHm�h8kµ��h��y~TbxdHRTVWV¬adfVLkl~cVE~¶}TsHV�T¬OsHVLbedfsHmyg�dfVE~µdfh�dHRTVWm�kqdfVLse� o8gV��FmC� V��#dfRTV�bvd�o0dHVLbsHVLo8gHRlVL~§za`¶mnk8dHV�sf� o�g�V«gjcdf´Bh��©VL¯�V�k8d�b

eopk@~§dfRTVLm�s�gh8sfsHVLbf}=h�kl~cmnkT��V�¯�VLk8d�b

e′opsHVWUWV�sH��VL~F��¤+RTmyg�R

o��n��h0¤|b)dHh«sHV�}=VEo0d|dHRTV[¸fbeV��8jlV�klgV¹���sfh8Ue′dHheo�kOopsHzTm�dfs�opsH`«kajTU#z=V�sxh���dHm�UWVLbL�¨{+VLgLop�n��dfRlopd¨dfRTV

g�jcdedHmnkT��g�h�kqdfV¬adΘC

myb>~cVEbemn��kTVE~#mnkWbfjlg�RWo[¤'o�`�dfRlopd>mnkqdfV�sf� o�g�V+gjcdf´Bh��«V�¯8V�k8d�b�Rlo�¯�V+o�gh�sHsHVLbe}=h�k@~a´mnkT��VL¯�V�k8d|dfRlopd|myb|o��nbfh�opk&mnk8dfVLse� o8gV�V�¯�VLk8dL��+|h�dfV�op�ybfhidfRlopdL�@~cjTV�dfhWdHRTV�sfVEbvdHsfmyg�dHm�h8k�h���m�kqdfV�sf� o�g�VLbdHh�o�jcdfh8UWopdHol�qdHRTV�bejTUWU�opsH`«kTVd�b|opsHV�op�ybfhWopjcdHh�U�o0d�oT�²Jm��8jTsfV 7&~cV�}Tmyg�d�b�dHRTV�bejlUiU�o�sf`¶kTVd�b�h���dHRTV�gh�UW}=h8kTV�kqdHb�m�k ²�mn��jlsfV���dHh��8VdfRlV�s�¤+m�dfRªdfRlV�mns

g�h�sHsfVEbe}=h�k@~cm�kT��V¬adfVLkl~cVL~W}TsfV��T¬cVLbL�JQ)RTV+klVdΠC (A)

mnk�²Jmn��jTsHVI7T± z@³>gh8m�k@gmy~cVLb>¤+m�dfRWdfRTV|mnk8dfVLse� o8gVC�)Q)RTmyb+myb|kThpd|dHRTV#gLo�bfV�¤+m�dfR

ΠC (B)mnkO²Jm��8jTsHV 7c±C~l³�)Q)RTVLsfV#opsHV[dv¤�h�bvd�o0dHVLb+�yopz=VL��VE~�zq`�V�m�dfRTVLs

c0h8sc1��RTh0¤�VL¯�V�sE�0dfRTVL`�o�sfV+o8bfbfhcgmyo0dHVL~#¤+m�dfR«~cm��5V�sHV�kqd�U�o�sf¥amnkT�8b�m�k

B±��8m�¯�VLkimnkWzTs�o�g�¥�Vd�b�kTV¬ad>dHh

dHRTVxbvd�o0dHVLb�³��J¡£d'g�opk�z=V¨bfV�VLkidHRlo0d'bejTUWU�opsH`�kTVd�btg�o�sfsH`#UimnkTmnU�op�@m�kc��h8sfU�opdfmnh�k�o�z=h�jcd�gh8Ui}=h8kTV�kqdHbbfmnklgV[dHRTV�mns¨bvd�o0dHVLb+opsHV[�nmnkT¥�VL~�¤+m­dHR�dHRTV�U�opsH¥am�kT�qb'sfVEo�g�RTVL~�za`�m�kqdfVLse� o�g�V�V�¯�VLk8dHbL�YxzlbfV�sH¯�V�dfRlopdxdfRTVWbvd�o0dHVLb¨h��>dfRlVibejlUiU�o�sf`�kTV�d�opsHV�o�bHbehcg�mnopdfVL~&¤+m�dfR¶jTklm��qjTV�U�opsH¥am�kl�8b¨bfm�k@gV

dHRTV�V¬adHV�kl~cVE~�}TsfV��T¬��E��sfh8U;¤+Rlmng�R�dHRTV'bfjTUWUWo�sf`[klVdJmyb�~cVLsfmn¯�VL~F��mnbJ~cV��lkTVL~�jlbfmnkT�xo �� ���o�~cV��qjlo0dHVh8sH~TV�sE�JQ)RTmyb'UWVLo�klbL�am�k�dHV�sHUWb'h���mnk8dHV�sf� o�gV[V�¯�VLk8dHbL�qdHRlo0d+RloE¯amnkT�#dv¤�him�kqdfV�sf� o�g�VxVL¯�V�kqdHb'¤+m�dfR�V��qjlop��@klop�FU�opsH¥qmnkT�qb�mnUW}T��mnVLb)dHRlo0d|h�kTV�h���dfRTVLU°myb+oWgjTde´£h �OVL¯�V�kqdL�lopkl~�dfRqjlb)dHRTV�`�o�sfV�UiVLsf�8VL~�za`«dfRTV�%�9� � h�}=V�s�o0dHh�sE�w¨jlV�dfhWdfRTV�klopdfjTsHV�hp��dHRTV�bejlUiU�o�sf`«kTVd+dHRTV���h��n�nh�¤+mnkT�WsfVL�nopdfmnh�k�RTh8�n~TbL�

� Ã � Ã � � � � Ã � (ΠC (UnfA) = ΠC

(

UnfΠC (A)

) �

¡ k hpdfRlV�si¤�h�s�~TbL�JdHRTV�bfjTUWU�opsH`¢kTV�dimybio�zT��V�dHh¢~cVLbHgsHm�z=V�op�n�'}=h8bHbemnzT�nV�z=V�Rlo�¯am�h8jTs�b�h��Ah�k dfRTV

mnkqdfV�sf� o�g�V�� � ÃFà � 9 �Vd|jlb|beRTh0¤

(UnfA)|C = UnfΠC (A)

�ls�bedL�¡£dxmyb|¤�VL����¥akTh0¤;dfRlopd|dHRTV#jTkT�¼h8�y~cm�kT��h��

Ag�opkµz=V[¸esfV���h��y~cVL~q¹�h8kOo«g�o�kTh�kTmyg�o��F}TsHV�T¬

Pref ΘC

A

�h8stV��8jlm�¯pop�nV�k8dH��`�dHRlo0d

Pref ΘC

A

myb�bfj�2«gmnV�kqd�dfh#sHVLg�h�¯�VLs>dfRlV|��jT�n�=jlkc��h��y~cmnkT�UnfA�>Q)RTV¨my~cVLo�myb�dHRlo0d

K�KML�NPORQ@SUT

Page 17: Modular construction of finite and complete prefixes of Petri net … · 2016. 12. 27. · Modular construction of nite and complete pre xes of Petri net unfoldings Agnes Madalinski,

��� �� �C�� � � � ����� �� ���

b 0

e 4 t 1e 3 t 1

t 1e 8

c 0

c 0 c 0

c 0

t 1

e 7 t 1

e 12 e 13 t 1

t 3e 1

b 1

b 0

e 2 t 4

d 0

d 1b 1

b 1

d 2

d 0

d 1b 1

t 4

t 5e 6

t 6e 9

e 11

t 3e 5

b 1 b 1

c 0 b 0 b 0

t 3e 10

c 0 c 0 b 0

b 0

b 0

#%$�&Pref

ΘCB

y 3

y 5

s (c , b , d ) 0 0 0 0

s (c , b , d ) 1 0 0 1

s (c , b , d ) 2 0 0 2

y 1y 0

y 2

y 4

#'�&ΠC (B)

s (c , b , d ) 1 0 0 1

y 5

y 3

s (c , b , d ) 2 0 0 2

y 1

s (c , b , d ) 0 0 0 0

#(�& 5/+;.10�< =ΠC (B)

o0�¼dHV�s�opka`igjcdf´Bh���V�¯�VLk8de�8¤+m�dfR«g�h�sHsfVEbe}=h�kl~Tm�kT��V�¯8V�k8d

e′�8h8kTV|gLopk�¸e���njTV¹+o�gh�}a`#h��

(Pref ΘC

A )|e′+�

dfRlV�sfVEbvdHsfmyg�dHm�h8k�hp�Pref ΘC

A

dfh«kThc~cVLb+dfRlopd¨o�sfV�m�k&dHRTV���jcdfjTsHV�h��e′�@~cVLkThpdHVL~&zq`

e′+�'w¨h8m�kl�WdfRlopd

sHV�}=VLopdfVE~c��`[o�kl~[��h8s�op�n�8g�jcde´£h �iV�¯�VLk8d�b�o����nh0¤|b5dfh¨sHVLgh0¯�V�s�dfRlV���jT���qjTkc��h8�n~Tm�kT�@��Q)Rlmnb�myb�o|g�h�klbfV �qjTVLklgVhp�>dfRlV�gh8UW}T��V�dfVLkTVLbHb|}TsHh�}=VLsedv`��|Q)RlV�bfo�UiV#RTh��y~Tb|��h�s

(UnfA)|Co�kl~

(Pref ΘC

A )|Cz=VLg�o�jlbeV#¤+RTV�k

oWgjTde´£h �¶V�¯�V�kqd+myb|opk�mnk8dfVLse� o8gV�VL¯�V�k8dE�Tm­d�b|gh�sHsHVLbe}=h�k@~cm�kl��V�¯�VLk8d|mnb|op�ybeh�opk�mnk8dfVLse� o8gV�V�¯�VLk8dE�Yxz@beVLsf¯�V>dHRlo0d�dfRlV'gh�UW}T�nVdHm�h8k�h�}=V�s�o0dHm�h8k�h��

(Pref ΘC

A )|C�po�b�m�dJ¤'o8b�~TVLbHgsHm�z=VE~�o�z=h�¯�V8��opUWh8jTk8d�b

V¬To8g�df�n`OdHhOg�h�UW}TjcdHm�kT�µdfRlV«jTkc��h��y~cmnkT�Ohp�ΠC (A)

�Jzq`¢~cV��lkTm�dfmnh�k.h��ΠC (A)

�&Q)RTmyb�}Tsfh0¯�VLb�dfRlopd(UnfA)|C

mybtmnbfh�UWh�sH}TRTmyg'dHhUnfΠC (A)

��^ah�dfRTVEbeV|dv¤�h#hcg�gjlsfsHV�klg�V+kTVd�btRloE¯�V|dfRTVxbHopUWV|}TsHhpuvVLgdfmnh�kh�k

C± ¤+RTmng�RµbemnUW}T�n`�opUWh�jlk8dHb)dfhidHsfmnUWUWm�kl�idfRlVLbfV�h8zauvVEg�dHbL�TmB� V8�>UWV�sH��mnkT�imybeh8Uih8sf}lRTmng�gh�kc�@��jTs�o0´

dfmnh�k@bH³�¤

��¿ �c� �9 - bejTUWU�opsH`.kTVdΠC (A)

mybih8zcdHo�m�klVL~©zq` �nhah8}Tm�kl�¶dfRTVµsfVEbvdHsfmyg�dHm�h8k©m�klbedfVEo�~©h��¨dfRlV}TsHhpuvVLg�dHm�h8k&hp�

Pref ΘC

A

dHhWdHRTV�z=VLRloE¯amnh�jlsHb)h��JdHRTV�mnk8dHV�sf��o8gVC±�kTh�dfsHmnUiUWmnkT�«mnb¨o�}T}T�nm�VE~l³��){|VLg�o����

dfR@o0d>dHRTV+dfsHm�UWUWmnkT��¤�h�jl�n~iUWV�sH��V)mybeh8UWh�sH}TRTmyg'g�h�kc�l�8jTs�o0dfmnh8k#h��=dfRTV|sfVEbvdHsfmyg�dHm�h8kF��o�kl~�dfRlopdΠC (A)g�o�sfsHmnVLb¨UWmnkTmnU�op�>m�kT��h�sHUWopdfmnh�k¢opz=h�jTd

A± mC� V���dfRTV�bedHopdfVLb[h��

ΠC (A)o�sfVWo�bHbehcg�mnopdfVL~O¤+m�dfR¢U�opsH¥8´

mnkT�8b+mnkPref ΘC

A

³�)Q)RqjlbL�TdfRlV�dfsHm�UWUWmnkT��¤'h�jT�y~�UWV�sH��V�bvd�o0dHVLb+mnk&dfRTV#sfVEbvdHsfmyg�dHm�h8k�¤+RTmygHRµUWm��8R8d¨z=Vo�bHbfhcgmyo0dfVE~¢¤+m�dHR ~cm��5V�sHV�kqdiU�opsH¥am�kT�qb�mnk

Pref ΘC

A

�¢Q)Rlmnb�myb#mn�n��j@bvdHsHopdfVL~.mnk ²Jm��8jTsHV«�@� �&jlbfmnkT�OdHRTVgh8UW}=h�kTV�kqdxm�k§²Jm��8jTsfV#�@± gE³��[Q)RTV�g�h�kc�l�8jTs�o0dfmnh�k@b

[e12]opkl~

[e13]opsHV�mnbfh�UWh�sH}TRTmyg�m�kOdfRTV�sHVLbedfsHmng´

dfmnh�k ±�¤+m�dfRlh�jcd#gh�k@bemy~cV�sHmnkT��dfRTV���sHV�`�VL~Oh8jcd�kThc~cVEb[mnk¶dHRTVi�l�8jTsHVE³���opk@~OdfRajlbL�F¤�h8jT�y~¶z=VWUWV�sH��VL~op��dfRlh�jT�8ROdHRTV�`¶g�h�sHsfVEbe}=h�kl~µdHh&~cm��=VLsfVLk8d�U�o�sf¥amnkT��mnk¶dHRTVW}TsHV�T¬

Mark([e12 ]) = (c0, b0, d2)opk@~

Mark([e13 ]) = (c0, b0, d1)�>²TsHh�U�dfRTV¨bfjTUWUWo�sf`#klVdtmnk�²Jmn��jTsHV+�l� �c±�z=³�h8kTV|g�o�kWsfVLUih0¯�V'dHRTV|��hch�}lb

gh8k8d�opmnkTm�kl�y0�y2o�kl~

y4sfVEbejT��dfmnkT�im�k&o � dfsHm�UWVL~*�aklVd|beRlh�¤+k�mnk��l� �c±CgL³��t²TsHh�U¦dHRTV�}=h�mnk8d+hp��¯am�VL¤

hp�¨dfRTV&g�h�UW}=h�kTVLk8dA�t¤+RTmng�R sfVEgV�mn¯�VLb

ΠC (B)�¼sHh�U

B��m�dWmnbikTVEgVLbHbHopsH`§dfh§¥akTh0¤/V�� �l�

y1zTjcd

kTh�dy0��h8s#dHRTV�g�h�klbedfsHjlg�dHm�h8k.h��+dHRTV�}TsHV�T¬��F+|h�dfV�dfRlopdWop�n��dfs�opklbfm­dHm�h8klb#m�k

ΠC (B)gh8sfsHVLbf}=h�kl~

dfh¢hTg�gjlsfsHV�klg�VLb#hp�t1���¨o�¯am�kT�§��h�jTkl~�o§g�jcde´£h �»V�¯�VLk8d�g�h�k8d�opmnkTmnkT�

y1dHRTV�sHV�mnkl~cm�sHVLgdf�n`.V�¬amybed�o

gjTde´£h ��V�¯�VLk8d�gh8k8d�opmnkTm�kl�y0bfmnklgV+dfRTV|V�¬cVLgjTdfmnh�kWhp�

y0sfVEo�g�RTVLb�dfRTV¨bHopUWV|bedHopdfV�� $ VLkTV�s�op�n�n`��pdfRTVLsfV

V¬cmybvd+op�ybeh�hpdfRlV�s)gLo�bfVLbt¤+RlV�sfVxmybfh�UWh�sH}TRTmyg¨gh8kc�l��jlsHopdfmnh�klb�g�o�k«z=V[UWV�sH��V8�J¡ k�o8~T~cm�dfmnh�kF�ah8kTV[gh8jT�y~

=?> K@=�A

Page 18: Modular construction of finite and complete prefixes of Petri net … · 2016. 12. 27. · Modular construction of nite and complete pre xes of Petri net unfoldings Agnes Madalinski,

������������ ���������������������������! �@7

}=VLse��h8sfU®o���jTsfdfRlV�s�dfsHmnUiUWmnkT��zq`�V¬c}lopk@~cm�kl��dfRTV[V¬adfVLkl~cVE~«}TsfV��T¬��qRTh0¤�V�¯�VLsL�8dfRTmyb�¤'h�jT�y~�UWVLo�k�dHhdHs�o�~cV´£h �Oz=Vdv¤�V�VLk�dHRTV�¤+my~adfRµo�kl~�dfRlV��nV�kl�pdfR&hp��dHRTV�sfVEbejl�­dHm�kT��bfjTUWUWo�sf`«kTV�dL�

��'� � � � ��(� 0 � �� � �1�� � � ��07,���� � � 6�. 0 � .71(� ��¿:��� � � � à �����5��<���� ��� � �!� � � �R�: ��@ � � �

N = A ×CN B

?� ��g�o�kTh�klmngLop�J}TsHV�T¬µ� o8g�dfh8s #�� � � $�U� �R����� ��� ���

A� �:� ���� � � � �

A = A×CN ΠC (B)

±v�E�8³

PrefΘ∗

A = ΠA

(

Pref Θ∗

A

) ±v�8�E³

� �P�����Θ∗ =

{

≈mar ,⊂, {κe = κbas}e∈E

} � * � � ���� � � �!� � ��� �� �=��� � J�� �R�� � � ��� � �Pref

Θ∗

B

YxzlbfV�sH¯�V�dHRlo0d[m�kµdHRTmybxbfVdf´BjT}µdHRTVibeV��nVLgdfVE~µo8~cV �qjlopdfV�h8sH~TV�s¨mnbxbfmnUi}l��`�dfRTVibfVdxmnklg�njlbfm�h8k⊂ � �¤+Rlmng�R�op�n��h0¤|bFdfRTV)gh�UW}lo�sfmybfh�k�hp�Tz@o�bfmngtg�h�kc�l�8jTs�o0dfmnh�k@b�z=h�dfR#mnk

PrefΘ∗

A

�0mnkPref

Θ∗

B

o�kl~�mnkPref Θ∗

Nmnkµopk&opk@op�nh���h�j@b�U�opkTkTVLsL��Q)RTmnb+}TsHh�}=V�sfdv`�¤+mn���Fz=V�gsHjlgmyop��mnk�dHRTV�}TsHhahp��h��Jr�sfh8}=h8bfm­dHmnh�k:"T�YxzlbfV�sH¯�Vxop�ybeh#dfRlopd

PrefΘ∗

A

mnb)o#}TsHV�T¬«hp�UnfA = UnfA ×

CO UnfΠC (B)

opkl~�beh#dHRTV[}TsHh0uvVEg�dHm�h8kh8kAgLopk�z=V�op}T}T�nmnVL~�dHhWm�dL�T¤+RTmyg�R�sHVLbfjT��dHb)mnk&o�dHsfjTV�zTs�opklg�RTmnkT�i}TsHhcgVEbfb'h��

A�

²Th8s'dHRTV�sHjTkTklm�kT�WV¬To�Ui}l��V�hp��dfRlmnb+}lop}=VLsL�adfRTV�g�o�kTh�kTmyg�o��5}TsfV��T¬«� o�gdfh8sHb'h��No�kl~�dfRTVLm�s¨gh�Ui´

}=hqbem�dHm�h8k mnk8dfh oª}lsfV��T¬�hp�UnfN

opsHVO~cV�}Tmyg�dHVL~ m�k-²Jm��8jTsHV;?a��YxzlbfV�sH¯�V&dfRlopd«dHRTV¶��hcgLop�xgjcde´£h �~TVLgmybfm�h8klb'opsHV �qjTm�dfV�gh8klbeVLsf¯po0dfmn¯�V¨o�kl~«U�o�`WzTjlm��y~�zTs�opklg�RTVLb�hp��o�X'²�r dfRlopd+opsHV¨kThpd)kTVEgVLbHbfo�sf`�dHhdHRTV[hpdfRlV�s�� o�g�dHh�sE��²Th8s�V¬To�Ui}l��V8�8dfRTV�gjcdf´Bh��&VL¯�V�k8d

e6hp�

PrefΘ∗

A

mnk�²Jm��8jTsfV ?c± o8³tmnb�kTh�d'sHVLo�g�Rlopzl��Vmnk

PrefΘ∗

A ×CO Pref

Θ∗

B

��Q)RTmybxmyb[~cjTV�dfh�dHRTV�� o�g�d[dfRlopd[mnkl~cVL}=V�kl~cVLk8d�dfsHjTklg�opdfmnh�klb[opsHV#}=V�sf��h�sHUiVE~mnk�dfRTV�~cV��lkTm�dfmnh�k&hp�JdfRTV�dv¤�h�X'²Jr�� -¨b|oWgh8klbeV��qjTV�klg�V[h���dfRTmyb|gh8klbeVLsf¯po0dfmn¯�V[g�jcde´£h �§g�sfm�dfVLsfmnh�kF�czq`z@o�g�¥�}TsHh0uvVEg�dHm�kT��dHRTVxgh�UW}=hqbem�dfmnh�k

PrefΘ∗

A ×CO Pref

Θ∗

B

h8kWdfRTV|dv¤�hiX'²JrPref

Θ∗

A

opk@~Pref

Θ∗

B

h�kTVU�o�`µ�8Vd�bfU�op�n��VLsx� o�gdfh8sHbL��¡ k¢hpdHRTV�s�¤�h8sH~TbL�5dfRTVidv¤�hOX'²Jr ~ch&kThpd�gh�sHsHVLbf}@h�k@~&dfh�dfRTVWUWmnUimnU�op�g�h�¯8V�sfmnkT�ihp��dHRTV�mns|g�h�UW}=h8bfm�dfmnh�kF���¿ .� � � � ?� �3 !� � � ���� � ��

ΠC (B)�C�9�! �� �� ���R���2�R� ?� ��� ��� �<9�=�����R � �

A# ��� ������� � ��� � � ��� �

�U� ��P� ���2 � � ����R�C � � �0�R�����

A$ � ��� ���P@

Pref Θ∗

A v PrefΘ∗

A

Q)Rlmnb�myb�beh©z=VLgLopjlbfVOdfRTVªbejTUWU�opsH`�kTV�d&o�~T~Tb�UWh�sHV¶g�h�klbedfs�opmnklb�dHh g�h�UW}lopsHVOU�opsH¥amnkT�8b«mnk»dfRTVg�h�klbedfsHjlgdfmnh�k§hp�

Pref Θ∗

A

��o�kl~¶dfRajlb�dfhµ~TVdfVEg�d#gjcdf´Bh�� VL¯�V�k8d�bL�WQ)Rlmnb�sfVEbejT��d�dfRajlb�V¬c}TsHVLbHbeVEb¨dHRlo0d¤+RlV�k&gh�UW}=h�klV�k8d�b

Aopkl~B~chikTh�d)�nmnUim�d)dHRTV�z=V�Rlo�¯am�h8jTsHb�h���dfRTV�mnk8dHV�sf� o�g�V

C�adfRlV�X�r�².h���VLo8gHR

g�h�UW}=h�klV�k8d¨¤+mn����z@V#g�h�UW}T�nVdHV��(�¨h�¤�VL¯�V�sE�Tmnkµ��V�klV�s�op�5dHRTmyb¨gh8UW}T��V�dfVLkTVLbHb+myb|kThpd¨kTVEgVEbfbHopsH`�dfh«�8VddHRTV����nh�z@op�>g�h�UW}T�nVdHV�kTVLbHb[��h8s

N�Jo0�¼dfVLs�g�h�UW}=h8bfm­dHm�h8k§hp��dHRTVWdv¤�h¶X'²Jr�� A§V�kTh0¤ beRlh�¤�dfRlopd�dfRTV

dv¤�h�X'²�r»~ch�gh8U#zTm�klV�mnk8dfh«oi���nh�z@op�n��`«g�h�UW}T�nVdHV�}TsHV�T¬�hp�N�

� à � à � � � � à ���

ΠA

(

Pref Θ∗

N

)

, ΠA

(

Pref Θ∗

A×CNB

)

v ΠA

(

Pref Θ∗

A×CNΠC (B)

)

, ΠA

(

Pref Θ∗

A

)

, PrefΘ∗

A

{|VLg�o����nmnkT�¢±C�8³���dHRTmyb[sHVLbfjT��d�V¬c}TsHVLbHbeVEb|dHRlo0d�dfRTVidv¤�hOX'²JrPref

Θ∗

A

o�kl~Pref

Θ∗

B

��h�sHU]o�}TjT�n��z@o�gH¥g�h�¯8V�sfmnkT�xhp��o��lkTm�dfV#gh�UW}T�nVdHV�}TsHV�T¬�hp�

UnfN�

� ÃFà � 9 �Vd�jlb��lsHbed�g�h�UW}lopsHV�dfRlV�}TsfhpuvVEg�dfmnh�k¢h�kAh���dfRTVW��jT�n��jlkc��h��y~cmnkT�8b�hp�

Nopkl~¢hp�

A =A×C

N ΠC (B)�TsHopdfRTVLs'dHRlopk�dfRTVLm�s¨g�o�kTh�kTmyg�o��5}TsHV�T¬cVLbL� $ mn¯�V�k

UnfA×CNΠC (B) = UnfA ×

CO UnfΠC (B)

±v����³� = 5"9 $ ,"2 )�. L 5 < = )<F 5 ' 6 � $ 3 5 <(+�� )<N�31<�+-5K0 $ 5"504< )�+ = <(+

⊂.1, L )>5 $>= <�� F $ 5/< 8 $ L = 50 $ 5"5K0 < 24+;<���2 ' F4.13 = F , . L N

⊂.1, $3(U$ L ) L . (U$ 3 24+;<��J2 # = <(,/24. 5/<

⊂' <(. L N L )>5 $>= <�� F $ 5/< & , . L ( <

⊂.1,-. L ( 3OF = < = . L $ L $>= < � F $ 5/< )�+ = <(+ @� L )>5 L < ( <�,/, $ +;.13 6 0�. L .10 $ 3 8 $ ,-.1313OF4,;5 + $ 5 < = . L 504< <32 $ 0�2431< $R' )/G <

K�KML�N���SUT��RQ��!O�� ���

Page 19: Modular construction of finite and complete prefixes of Petri net … · 2016. 12. 27. · Modular construction of nite and complete pre xes of Petri net unfoldings Agnes Madalinski,

�L� �� �C�� � � � ����� �� ���

a 0

e 0 t1

a 1

3ee 2 t1

a 1

a 0

e 5 t1 e 6

a 1

a 0

e 1

e 4

(c0, sB0)

(c1, sB1)

(c0, sB2)

(c1, sB3)

(c0, sB2)

(t2, yB0)

(t2, yB2)

(t3, yB3)

(t3, yB1)

# =�&Pref

Θ∗

A

e 1 t4e 0

b 1

b 2

e 3 e 4 t5

b 0

b 0

e 2

(c0, sA0)

(c1, sA1)

(c0, sA0)

(c1, sA1)

(t3, yA1)

(t2, yA0)

(t2, yA2)

# < &Pref

Θ∗

B

0a 0b

4t 1e

1b

1t 0e

2t 2e

1c 0a

e 1t

1a

5t

8e 1t 7e 4t

1ba 1

2b 0c

3t 4e

2t 6e 5e

3te 9

1c 0a 0b

2b 0c

1a

0c

3

# ? &Pref

Θ∗

A ×C

O PrefΘ∗

B

²Jm��8jTsHV ? ��X'o�kTh�kTmyg�o��5}lsfV��T¬���o8g�dHh�s�b+opkl~�dfRTVLm�s¨g�h�U#zTmnklo0dHm�h8k�zq`�}TjT�n��zlo8gH¥5�,+|h�dfmygV�dfR@o0d|VL¯�V�k8de6hp�=� o8g�dfh8s|±C~l³�¯0o�kTmybeRTVEb�mnkidfRlV+}Tjl���nzlo�g�¥5��Q�h�}TsHVLbfV�sH¯�V'm�dL�8h8kTV+¤�h�jl�n~ikTVLVL~�dHRTV|}lopsfd>h��

fmnk�~To8beRTVE~

�nm�kTVEb��T¤+RTmygHR&myb)mnkl~cVLVL~�jlbeVL��VEbfb'dHhW��V�d|oWgh8Ui}l��V�dfV[�@kTm­dHV�}TsHV�T¬�hp�UnfN�

h�klV�Rlo�b

ΠA (UnfA) = ΠA

(

UnfA×CNΠC (B)

)

= UnfA ×CO ΠC

(

UnfΠC (B)

)

= UnfA ×CO ΠC (UnfB)

= ΠA

(

UnfA ×CO UnfB

)

= ΠA (UnfN )±e�9"q³

Q)RTV|bfVLg�h�kl~iV �qjlop�nm�d `#j@beVEb+± ?p³���dHRTV+dfRTmnsH~Wh�kTV|~cV�sHmn¯�VLbJ��sHh�U r�sfh8}F�8�c��dfRTV+��h�jTsfdfRWV��qjlop�nm­dv`#jlbfVLb|± ?p³op�qopmnk¶opkl~OdfRTVW�no8bvd[h8kTVig�h�sHsfVEbe}=h�kl~lb+dHh¢±�"8³��#Q)RTmyb�beRlh�¤|b¨dHRlo0d

UnfΠC (B)

gh8k8dHo�mnklbxV¬To�gdf�n`�dHRTVmnkc��h�sHU�o0dfmnh�k opz=h�jcd

BdfRlopd«myb�kTVEgVLbHbHopsH`ªdfh

Amnk h�s�~cV�sWdHh ~cVdHV�sHUimnkTVµ¤+R@o0d�o�sfV&m�dHb«}=h8bHbfm�zT�nV

z=V�R@oE¯amnh�jTs�btm�kN�E4!�qjTmn¯0o���VLk8df�n`8�qdfRTmyb'sHVLbfjT��d'V¬c}TsHVLbHbeVEb�dfR@o0d

Nopkl~ARloE¯�Vxmy~cVLk8dfmyg�o��5sfjTk@bt��sHh�U

dfRlV�}=V�sHbf}=VLgdfmn¯�V[h��A�

H'V���h�sHVtgh�UW}lo�sfmnkT�)dHRTV�}TsfhpuvVEg�dfmnh�k@bFhp� � ���������� [h��adHRTVLbfVtjTkc��h8�n~Tm�kT�qb����nVdJjlb�U�o�¥�V>dv¤�h¨sfVLU�opsH¥abL�Yxz@beVLsf¯�V��lsHbed�dfRlopd�dHRTV�VL¯�V�k8d�b�hp�

UnfΠC (B)

V�m�dfRTVLs�be`aklg�RTsHh�kTmybfV�¤+m�dfR#VL¯�V�kqdHb�h��UnfA

h8s�¯0o�kTmybeRmnk�dfRTV¨}ljT���nzlo8gH¥&±e�E��³��Jmnk«hpdfRlV�st¤�h8sH~lb���dfRTV[bejTUWU�opsH`�kTV�d

ΠC (B)~chaVEb�kThpd�Rlo�¯�V|}TsHmn¯0o0dHV|V�¯�VLk8dHbL�

bfh���h8s+V�¯�V�sH`«V�¯�VLk8deh��

UnfAh�kTV�Rlo8b

ΠA (e) 6= ∅�

^cVLgh8kl~c�n`��5sHVLgLop�n��dHRlo0d[dHRTViV�¯�V�kqdHb[hp�UnfΠC (B)

o�sfV��no�z=V��nVL~Ozq`&dHRTViU�opsH¥qmnkT��hp�BdfRlV�`µ}TsHhp´

~cjlg�VL~���o�b[V¬c}T�yopmnkTVL~Omnk¶dfRlVWgh8klbvdHsHjlg�dfmnh8kOh���bfjTUWUWo�sf`µkTVd�bW± bfV�V�dfRlVi�yo�bed�bfV�k8dHV�klg�Vim�k¢wxV�v�F�8³��

=?> K@=�A

Page 20: Modular construction of finite and complete prefixes of Petri net … · 2016. 12. 27. · Modular construction of nite and complete pre xes of Petri net unfoldings Agnes Madalinski,

������������ ���������������������������! ��?

^ch§m�k©dfRTV�}TjT�n��z@o�g�¥ ±e�E�p³�tbf`aklg�RTsfh8kTmnbfVL~.V�¯�VLk8d�biopsHV«dHRqjlbWo�bHbfhcgmyo0dfVE~¢dfhªoµ��jT�n�)U�o�sf¥amnkT�¶hp�N�

h8zcdHo�mnkTVL~�zq`«UWV�sH��mnkT�#dfRTV�U�opsH¥am�kl�#}lsfhc~TjlgVE~«mnkBdHh�dfRlV[U�o�sf¥amnkT��}TsfhT~cjlgVE~«mnk

A�>Q)RTmyb'Rlh��y~Tb

o8b'¤�V��n�5��h�s+V�¯�V�kqdHb)hp�'±e�E��³tdfRlopd|opsHVx}TsHmn¯0o0dHV¨dHhA�c}TsHh�¯amy~cVL~�h�kTV�sfVL��mnVLb'h8k�dfRTV��yo�bed+bf`aklgHRTsHh�klmnbfVL~

VL¯�V�kqd�dfhO��V�d�dHRTV«U�o�sf¥amnkT�&}TsHhc~cj@gVL~¢h8k¢dHRTVB}lo�sedE�µYx¯�VLsHo����B��VL¯�V�sH`¶V�¯�VLk8d#h��

UnfAo�}T}=VLo�sHb

¤+m�dHR�dHRTV�U�opsH¥qmnkT�im�d|¤�h�jl�n~�R@oE¯�V�}TsHhc~cjlg�VL~�m�kN�

A¢V�klh�¤ }TsHh�¯�V[dfRTV�mnklg�njlbfm�h8k�hp�ΠA

(

Pref Θ∗

N

) m�kqdfhΠA

(

Pref Θ∗

A

) � FV�d

ez=V�o���VEo�bfm�zl��V«VL¯�V�k8d�hp�

UnfN± mC� V��µkThpdibvdHsfmyg�dH��`¢}TsfVEgVL~TVL~ªzq`ªoµg�jcde´£h ��V�¯�VLk8d#h��)dfRTV

gLopkTh8kTmyg�op�T²JX�r'³��qopkl~WbfjlgHRWdfRlopdΠA (e) 6= ∅

��Q)RTV|g�h�kc�l�8jTs�o0dfmnh�kκ = [e]

dHV�sHUimnklopdfVLb>¤+m�dfR�U�opsH¥q´mnkT�

mh��N��H'`¢±v��"�³'dHRTV�sHV�V�¬amybedHb|opd|�nVLo8bvd|h8kTV�gh8kc�l��jlsHopdfmnh�k

κ′ = [e′]h��

UnfA± o�kl~�}=hqbfbfm�zl��`

bfV�¯8V�sHo��¼³�bfjlgHR�dHRlo0dΠA (κ) = ΠA (κ′)

�.H'`�dfRTV¨�ls�bvd�sHV�U�opsH¥iopz=h0¯�V��ΠA (κ′)

mnbtmybfh�UWh�sH}TRTmyg)dfhκ′�

bfhl�q¤+m�dfR�o��nm��8R8d)opzTj@beV¨h��FkThpd�o0dHm�h8klb��qh�kTVxgLopk«¤+sHm­dHVΠA (κ) = κ′

�>ZOh�sHV�h0¯�V�sL��za`�gh8klbvdHsfj@g�dfmnh�k«hp�dHRTV|bfjTUWU�opsH`�kTV�d

ΠC (B)�8h�kTV|g�o�kWgHRlhah8bfV

κ′ = [e′]mnk�bejlg�R�o[¤)oE`�dfRlopd>dfRlV+U�opsH¥am�kl��o�bHbfhcgmyo0dfVE~

dHhe′mnk

UnfAmnb�}TsfVEgmybeVL��`µV �qjlo���dHh

m��o�kl~¶bfmnUWm��yopsH��`&��h�s�op�n��U�o0d�gHRTmnkT��V�¯8V�k8d�b[mnk¶dfRTVWmybeh8UWh�sf´

}lRTmnbfU�V�¬a}lsfVEbfbfVL~�zq`ΠA (κ) = κ′

� A m�dfRµdHRTmyb¨g�RTh8mng�V��@V�¯�V�kqde′myb|kTVEgVLbHbHopsHm��n`«��VLo�bfmnzT��V#m�k

UnfA�

YxdfRTV�sH¤+mybeV8���nVdf ′ 6= e′

z=V�o&g�jcde´£h � VL¯�V�kqd�mnkκ′opk@~¶�nVd

g′z=VWm�dHb�gh�sHsfVEbe}=h�k@~cm�kl��VL¯�V�kqdL��z=hpdHR

z=VLm�kl��o�bHbfhcgmyo0dfVE~�dHh�U�opsH¥amnkT�m′ ��^am�klg�V+dfRTV¨g�jcdedHm�kl�#gh�kqdfV�¬qd Θ∗

m�kUnfA

sHV��nm�VEb>h�k�bfVdtm�klg���j@bemnh�ko8b+o�~cV��qjlo0dHV[h8sH~TV�sL�ch�kTV�R@o�b

g′ ∈ [f ′] @ [e′]�f ′, g′opsHV[sHVLbf}=VLg�dHm�¯8V��n`�sHV��yo0dHVL~�dfh�V�¯�V�kqdHb

f, gmnkκ�

^ch�z=hpdHRfo�kl~

gopsHV+o8bfbfhcgmyo0dHVL~#dfh�UWo�sf¥amnkT�

m′ �qopk@~iUWh�sHV�h0¯�V�s f ∈ [g] @ [e]��^aVdtmnklg���jlbfmnh�kW¤'o8b

o��ybeh�g�RTh8bfV�k�dfh�~cV�lklV�dHRTV#gjTdedfmnkT��gh8k8dHV¬adΘ∗m�k

UnfN�l¤+Rlmng�R&U�op¥8VLb

fo�g�jcde´£h ��±�}lsfh0¯amn~cVE~

fmyb+��VEo�bfm�zT�nV��@mC� V��)myb|kThpdxo���sHVLo8~c`«}lsfVEgVL~cVE~�za`�o«gjcdf´Bh��t³��)¡ kOo�kq`�g�o8beV8�cdHRTmyb¨g�h�k8dHsHo8~cmyg�dHb)dHRlo0demyb

o���VEo�bfm�zl��VxVL¯�V�kqdL��¡ k�bejlUiU�o�sf`8�8dfRTmyb'}TsHh�¯�VEb�dfRlopd+gh8kc�l��jTs�o0dHm�h8kΠA ([e])

��h�s�dHRTVx��VLo�bfmnzT��V[V�¯�V�kqdeh��

UnfNmyb+}TsfVEbeVLk8d+mnk

ΠA

(

Pref Θ∗

A

) �T¤+RTV�klg�V[dHRTV�sHVLbfjT�­dE�¤

+|h�dfmygVtdHRlo0dJdfRlV�mnklg�njlbfm�h8k�bvd�o0dfVE~�m�k�r�sfh8}=h8bfm­dHmnh�k "|U�o�`�z=V'bedfsHmyg�dL�JQ)RTmyb�R@op}T}=V�k@b���h�sJV¬To�Ui}l��V¤+RlV�kµo�k�VL¯�V�k8d

ehp�

UnfNmnb+}TsHVLg�VL~cVE~�za`�o�gjTde´£h �¶VL¯�V�kqd

fdHRlo0d¨mnb|}Tsfmn¯0opdfV[dHh

B��Q)RTmyb|g�jcde´£h �

U�o�¥�VLbekThpd[��VLo8bemnzT�nV���zTjcd�dfRTmyb[}TRTVLkTh�UWV�klh�k§gLopk¶kTh�d[z=VWbfV�V�k¶mnk

UnfA��¤+RTV�sHV�V�¯�VLk8dHb[}TsHm�¯po0dHV

dHhB~chWkTh�d|o�}T}=VLopsE�

� à � à � � � � à ��� PrefΘ∗

A

� �,�P�� � �

Q)RlV�}TsHhah���mnb|bemnUWm��yops)dfhWdfRTV�h8kTV�m�k&r�sHh�}=hqbem�dfmnh�k§���

� à � à � � � � à ��� Pref Θ∗

N , Pref Θ∗

A×CNBv Pref

Θ∗

A ×CO Pref

Θ∗

B

� ÃFà � 9 ¡ UWUWVL~TmnopdfV���sfh8U�±C�8³)opkl~�r�sHh�}=hqbem�dHm�h8k "T�¤

Q)RTmyb�sHVLbfjT��d�~chaVLb�kTh�d«V�kqdHo�m��|dfRlopd�dHRTVµ� o�g�dHh�s�bPref

Θ∗

A

opkl~Pref

Θ∗

B

}TsHh0¯qmy~cV�dHRTVOUWm�kTmnU�op�}ljT���nzlo8gH¥¢g�h�¯�VLsfmnkT�&hp�

Pref Θ∗

N

�&²Jm�s�bvd#hp�|op�n��z=VEg�opj@beV�h8kTV�U�o�`¶Rlo�¯�V«bedfsHmngd�}lsfV��T¬¢m�klg���jlbfmnh�kF��o�bm�d�¤'o�b�kThpdHmng�VL~ opz=h0¯�V�� -|kl~ bfVLgh8kl~c�n` z=VEg�o�jlbeV&dfRTVµ� o�g�dHh�s�b��)gh�UW}TjTdfVL~ beVL}lops�o0dHV��n`���UWo�`©kTh�ddHRTV�U�bfV��n¯�VLb|bfopdfmybv��`

PrefΘ∗

A = ΠA

(

PrefΘ∗

A ×CO Pref

Θ∗

B

)

o�kl~�bf`aUiUWV�dfsHmngLop�n��`8�co8b+beRTh0¤+k�zq`«dfRlV�gh�jTkqdfVLse´£V¬TopUW}T�nV[m�kµ²Jm��@� ?a�

��' � �407,@, 6 ����� ,1� 6���6�. ,1 6Q)RlV�o�z=h�¯�V�o�}T}TsHh8o8gHR�g�opk�z=V���V�klV�s�op�nmnbfVL~«dfhWUWh�sHV�gh�UW}T�nV¬�dfsHV�V´ bfRlop}=VL~�bf`cbvdHV�U�b)Rlo�¯am�kT�WU�opka`g�h�UW}=h�klV�k8d�b��WQ)RTmyb�¤+m��n�>z=VWmn�n��jlbedfs�o0dHVL~§h�k¶dHRTV�~cmybedfsHm�zljcdfVL~ªbe`cbedfV�U

N = A ×XN B ×

YN CbfRTh�¤+k

mnk§²Jm��8jTsfV��T�x¡£d[Rlo8b|dfRTsHV�Vigh�UW}=h8kTV�kqdHb¨¤+RTmygHR¶h�kl��`&m�kqdfVLsHo8g�d¨dHRTsfh8jT��RµdfRTVLm�s[mnk8dfVLse� o8gVEb¨z=Vdv¤�V�VLkVEo�g�R&h�dfRTV�sE�)Q)Rqj@b��ldfRlV�sHV�myb|kTh«hpdHRTV�s¨mnk8dfVLsHo8g�dHm�h8k&z=Vdv¤�V�V�kµdfRTV�gh8Ui}=h8kTV�kqdHb

Aopk@~BdHRlopk&dHRTV

mnkqdfV�sf� o�g�VX��o�kl~§bfmnUimn�yopsH��`8�5dHRTV�sHVWmnb�kTh&hpdHRTV�s�m�kqdfVLsHo8g�dfmnh�k§z=V�d ¤'V�V�k§dHRTV�g�h�UW}=h�klV�k8d�b

Bo�kl~C

K�KML�NPORQ@SUT

Page 21: Modular construction of finite and complete prefixes of Petri net … · 2016. 12. 27. · Modular construction of nite and complete pre xes of Petri net unfoldings Agnes Madalinski,

�L� �� �C�� � � � ����� �� ���

dfR@opkWdHRTV¨m�kqdfVLse� o8gVY�J¡ k�dfRTV¨}TsHh�}=hqbem�dfmnh�klb>z=V��nh0¤��8m­d�mybtbfRTh0¤+kidHRlo0dtdfRlV¨gLopkTh8kTmyg�op�T}lsfV��T¬�� o�gdfh8sHb

hp�5dHRTmnb�dHRTsfVLV¨gh8Ui}=h8kTV�kqd�be`cbedfV�U¦g�opk�z=V¨g�h�UW}TjcdHVL~�sfVLg�jTs�bemn¯�V��n`���¤+m­dHRWdHRTVxbeh�´£gLop�n��VE~�}TsHh�}lo��8o0dHm�h�ksHjT��Vxopk@~idHRTV¨UWV�sH��V|sHjT�nV���Q)RlVLbfV|sHjT�nVLb�g�opk«z=Vxgh8U#zTm�klVL~Wdfh#V�¬qdHV�kl~WdHRTV[gh�UW}Tjcd�o0dHm�h8k�}TsHhcgVE~cjTsHVdfh«opkq`�be`cbedfVLU/¤+m�dfRµo�dfsHV�V�beRlo�}=V��

A CB

B ′

A CYX B

²Jmn��jTsHV����>wxmybvdHsfmnzTjcdHVL~&bf`abedfVLUN = A×X

N B ×YN C�

� à � à � � � � à ����� � � à � � # �c� � à � � � . ¿�� � �� N = A×XN B×

YN C = A×X

N B′ � �P� �U� B′ = B×Y

N C�

� � 3�� � ��� � � ��P� � ���

A = A×XN ΠX

([

B ×YN ΠY (C)

])

A′ = A×XN ΠX (B′)

�� ��?� �B@ R � � �=� � �-� �� ��� � � �� � ���=� �6���� ���R

PrefΘ∗

A = ΠA

(

Pref Θ∗

A

)

PrefΘ∗

A′ = ΠA

(

Pref Θ∗

A′

)

* � � �Pref

Θ∗

A = PrefΘ∗

A′

¡ k.hpdHRTV�s#¤�h�s�~TbL�FdHRTV�bfjTUWU�opsH`¶kTVd�b#gLopkªz=V�gh8UW}TjcdfVE~ªsfVEgjTs�bemn¯�V��n`8��¤+RTmng�R.}TsHh�¯amy~cVLb�oµ¤'oE`¶dfhh�zTdHopmnk&dfRlV�g�o�kTh�kTmyg�o��F}TsHV�T¬�� o�gdfh8s|mnk

A¤+m­dHR¶beU�o����JbfgLop�nV�g�h�UW}Tjcd�o0dHm�h8klb#±�m�dxmyb¨jTkTkTVEgVLbHbHopsH`«dfh

Rlo�kl~c�nVB′³�

� ÃFÃ�� 9 YxklV#Rlo�bUnfB′ = UnfB ×

YO UnfC

zq`.±="q³��5¤+RTVLklgVΠB (UnfB′) = UnfB ×

YO ΠY (UnfC)jlbfm�kl�µ}lsfh8}=V�sfdfmnVLb�hp�+}lsfhpuvVLgdfmnh�klb�h�k.jTkT�¼h8�y~cm�kT�qb����µ²lsHh�U r�sHh�}=hqbem�dfmnh�k �&h�klV«Rlo�bΠY (UnfC) =

ΠY

(

UnfΠY (C)

) �'behΠB (UnfB′) = UnfB ×

YO ΠY

(

UnfΠY (C)

)

= ΠB

(

UnfB ×YO UnfΠY (C)

)

=

ΠB

(

UnfB×YNΠY (C)

) �¨Q)RTmyb¨sfVEbejT��dHb¨mnkΠX (UnfB′) = ΠX

(

UnfB×YNΠY (C)

) �|¡ kOhpdHRTV�sx¤�h�s�~TbL�TdHRTVsHjTklb|h��

B′ = B ×YN Copkl~µhp�

B ×YN ΠY (C)

o�sfV�mn~TV�k8dHmngLop����sHh�UºdfRTV#}=VLsHbf}=VLgdfmn¯�V�h��>mnk8dfVLse� o8gVX�

zTjcd�hp��g�h�jTs�bfV'dfRTV¨bfVLg�h�kl~ikTV�dtmyb>�8V�kTVLsHo����n`#bfUWo����nV�s>dHRlopkWdfRTV+�@sHbed�h8kTV|bfm�klg�V)dHRTV|}TsHm�¯po0dHV)}T�yo�g�VLb>h��C¯0o�kTmnbfR&mnk&dfRTV�gh8klbvdHsHjlg�dfmnh8k�h��JdHRTV#bfjTUWU�opsH`�kTV�dL� -¨bxo�g�h�klbfV �qjTVLklgV��lh�klV�Rlo8b

ΠA (UnfA) =

ΠA (UnfA′)�T¤+RTmygHR&V�kqdHo�m��yb

Pref Θ∗

A= Pref Θ∗

A′

�¤

� à � à � � � � à ����� ¿ � #=¿ � �. ¿�� � � N = A ×X

N B ×YN C

�� �B = ΠX (A) ×X

N B ×YN ΠY (C)

�* � � �

ΠB

(

Pref Θ∗

N

)

v PrefΘ∗

B = ΠB

(

Pref Θ∗

B

) �

^ah�dfRTV[d ¤�hi�yo0dHV�s�op�5bejlUiU�o�sf`WkTV�dHb+opsHVxbfj�2«gmnV�kqd'dfhWgh8UW}TjcdfVxdHRTV�gLopkTh8kTmngLop�=}TsHV�T¬�� o�gdfh�s)m�k�dHRTVgVLk8dHsHo��Fgh�UW}=h�klV�k8dE� � ÃFÃ�� 9 YxkTV�g�o�k bfRTh�¤ dHRlo0d

ΠB (UnfN ) = ΠB (UnfB)��sfh8U(r�sHh�}=h8bfm�dfmnh�k �Oo�b���h��n��h0¤|bL�OQ)RTV

bfjTUWUWo�sf`�kTVd�b�sHVLg�V�mn¯�VL~imnkBh�k�m�dHbtm�kqdfV�sf� o�g�VLb

Xopkl~

Y��mn¯�V+V�¬co8g�dH��`#dfRTV¨mnkc��h�sHU�o0dfmnh�kWh��

Aopkl~

C��sHVLbf}=VLg�dHm�¯�VL��`8�=dfRlopd[myb[kTVLgVEbfbHopsH`�dfh

BdHh�sfVEgh0¯�V�sxm­d�b[z=V�Rlo�¯am�h8jTsHb¨mnk

N��Q)RTmybxUWVLo�klb¨dfR@o0d

Nopk@~BRlo�¯�V�my~cV�k8dHmyg�op�FsHjTklb'��sHh�U/dfRTV�}=h8m�kqd+h��J¯am�VL¤-h��

B�

� A ( 5F $ 313 6 84504.1,-.1,:504< 24+;)�2 $ N $ 5 .1) L +F 31< ) L F L ? )�3 = . L N�, @

=?> K@=�A

Page 22: Modular construction of finite and complete prefixes of Petri net … · 2016. 12. 27. · Modular construction of nite and complete pre xes of Petri net unfoldings Agnes Madalinski,

������������ ���������������������������! ��D

H'`&bemnUWm��yo�sxopsH��jTUWVLk8dHb¨o8b|mnkOr�sHh�}=hqbem�dfmnh�k "T�=V�� �l�xo«g�jcde´£h �ªV�¯�V�kqdxmnkΠB (UnfN )

U�oE`�hTg�gjlsz=V���h�sHV>h�kTV>mnk

ΠB (UnfB)��m­d�mybFV�¯amn~TV�k8dFdHRlo0d

ΠB

(

Pref Θ∗

N

) myb��Emnk���V�kTVLsHo��C�EbeU�o����nV�s�dfRlo�kPref Θ∗

B

�¤

Q)RTV�UWhc~cjl�no�s¨gh�k@bvdHsfjlgdfmnh�k&hp��g�o�kTh�kTmyg�o��F}TsfV��T¬�� o�gdfh8sHb|~cVEbfg�sfmnz=VL~&o�z=h�¯�V�h�kOopkµV��nV�UWV�kqdHo�sf`bedHsfmnkT�.h���gh8UW}@h8kTV�kqdHb«V�¬qdHV�kl~lb«klopdfjTs�op�n�n` dHh kTV�dHb�Rlo�¯qmnkT�©oªdfsHV�V¶bfRlop}=V8��X�h�UW}Tjcd�o0dHm�h8klb�opsHVh8sf�qopkTmybfVL~&o�b¨��h��n��h0¤|b��[Q)RTV#�lsHbed¨dHRTm�kl��mnb¨dfh�h�zcd�opmnk¶bejTUWU�opsH`�kTV�dHb¨��sfh8U op�n��kTVLm��8Rqz=h�s�b¨opdxVLo8gHRg�h�UW}=h�klV�k8d�b��x^ajTUWU�opsH`�kTVd�bxopsHV��lsHbed[gh�UW}TjTdfVL~ObedHo�sedHmnkT��o0d¨dfRTV���VEoE¯�VEb|hp�>dfRTVibvdHsfjlgdfjTsHV��=o�kl~}lsfh8��sHVLbfbfm�kT�ªmnkq¤)ops�~Tb��+jlbfm�kT� dfRTV¶}TsHh�}@op�8o0dHm�h8k�sHjT��V8� ^c}=VLgm��@g�o��n��`8�|o0d�VEo�g�R»gh�UW}=h8kTV�kqdL�+¤+RTVLkbfjTUWU�opsH`&kTVd�b�opsHV�o�¯0o�m��yopzT�nV#h�k

n − 1mnk8dHV�sf� o�g�VLbxh8jcd[h���dfRlV

nhp��dHRTmnb�gh�UW}=h8kTV�kqdL�5dfRTViUWV�sH��V

sHjT�nVµmnb�op}T}T�nmnVL~ ± mC� V��Bmyb�sHV�}l�no8gVL~ za`

Bo�b«o�z=h�¯�V�³���dfRTVLk dfRTVO}TsHh�}lo��8o0dHm�h�k sHjT�nVµ`qmnV��y~TbWdHRTV

bfjTUWU�opsH`«kTV�d+dHh«beVLkl~�h�k�dfRTV��yo�bed+zlsHo�klgHRF��Q)RTmnb|}TRlo�bfV[dHV�sHUWm�klopdfVEb)¤+RlV�k&VLo8gHRµgh8Ui}=h8kTV�kqd|Rlo�bsHVLg�V�mn¯�VL~#oxbfjTUWU�opsH`�kTV�dJ��sHh�U·VLo8gHR�hp�@m­d�bJkTV�mn��Raz=h�s�b���Q)RTVLk#dfRTV)UWV�sf�8V�hp�=op�n�qmnklg�h�UWm�kl�[bfjTUiU�o�sf`klVdHb�o0d#VLo�g�R.g�h�UW}=h�kTVLk8d�op�n��h0¤|b�dfh¶gh�UW}TjTdfV�dfRlV�g�o�kTh�kTmyg�o��>}lsfV�l¬¶� o8g�dHh�s�hp�)dfRTmyb#g�h�UW}=h�klV�k8dE�+¨hpdHV�dHRlo0dxo0dxop�n��bedfVL}lb+mnk&dfRTmyb¨gh8UW}TjcdHopdfmnh�kµopsHV��nhcgLop��opkl~&~ch«kThpd¨U�op¥�V�jlbfV�hp��opkq`�¥akTh�¤+�nVL~T��Vo�z=h�jcd)dHRTV����nh�z@op��bf`cbvdHV�U&�

� � ���� �������������

Q)Rlmnb�}lo�}=V�s�Rlo8b�}TsfVEbeVLk8dHVL~ªo&kTh0¯�VL�top}T}lsfhqo�g�R¶dfh¶o�~l~csfVEbfb�dfRTV�g�h�klbedfsHjlg�dHmnh�k¢h��'�lkTm�dfV�gh�UW}T�nVdHV}lsfV��T¬cVLb[h��'r,+¦jTkc��h8�n~Tm�kT�qb��W¡£d#be}=VEgmyop�nmnbfVLbxdHhµ~cmnbedHsfmnzTjcdHVL~¢bf`cbvdHV�U�bL�FmC� V��Wr3+¦V¬c}TsHVLbHbeVE~¶o8b[klVde´¤'h�sH¥cb+hp��gh8UW}=h�kTV�kqdHbL�=opkl~&}TsHh0¯qmy~cVEb)dfRlV�²JX�r»m�kµ� o�gdfh8sfmybeVE~���h�sHU&�ldfRlopd¨myb¨m­d[~cV�dfVLsfUWmnkTVLb|�lkTm�dfVz=VLRlo�¯qmnh�jlsHb�hp�FVLo�g�R«g�h�UW}=h�kTVLk8dtdfRlopdL�q}ljcd�dfh��8VdHRTV�sE�qopsHV|bfj�2«gmnV�kqdtdHh#zTjTmn�n~�o�²JX�r©hp��dfRTVx���nh�zlo��bf`cbedfV�U&�¨Zµh8sfVLh�¯�VLsL�cdHRTV�g�h�klbedfsHjlgdfmnh�kµhp�>dfRTVEbeV�� o8g�dfh8sHb|h���dHRTV�²JX�r»dHo�¥�VLb+dHRTV���h8sfU�h��>UWhc~cjl�no�sg�h�UW}Tjcd�o0dHm�h8klb��c}=V�sf��h�sHUWVL~�o0d)dfRTV�bHg�o���V[hp��oibfm�kl���nV�gh�UW}=h8kTV�k8dE��^a}=VEgm��@g�o����n`��Tgh8UW}=h�kTVLk8dHb+gh�Ui´U�jTkTmngLo0dHV#bejlUiU�o�sf`�kTVd�b+dfh�dHRTV�mns¨kTV�mn��Raz=h�jTs�bL�cdfRlopd¨sHV�}TsHVLbfV�kqd+dfRTV#z=V�Rlo�¯am�h8jTs�b'dfRlopd|dHRTV�`&op�n�nh�¤h8k�dfRTVLm�s|mnk8dfVLse� o8gV�¤+m�dfR�dHRTmnb+klV�mn��Rqz=h8jTsL�

-|}T}@opsHV�k8dH��`8�pdHRTV|g�h�UW}Tjcd�o0dfmnh8kWbHgHRTVLUiV+sHV �qjTmnsHVLb>beVL¯�V�s�op�csfh8jTkl~Tb>hp�5jTkc��h��y~cmnkT��mnkWVLo8gHR�gh8Ui}=h�´klV�k8dE�0h�klV�¤+m�dHR�dHRTV'gh�UW}=h8kTV�k8d�op�nh�kTV8��opklhpdfRlV�s�h8kTVt¤+m­dHR�dfRTV'gh8UW}=h�kTV�kqdJgh8jT}T�nVL~�dHh|dfRTV�sHVLgVLm�¯�VE~bfjTUWU�opsH`¶kTVdE��Q)RTVLbfV«beVEgh8kl~§h8}=V�s�o0dHm�h8k¢g�o�kªhp�+gh8jTs�beVWz=V�klV�Td���sHh�U�dfRTV��ls�bvd#h�kTV8��¤+RTmng�RªsHV´~TjlgVEb|dfRTV�h�¯�VLsfRTVEo�~��+¡ k8dfVLsfVEbvdHm�kl���n`��@h8klgV#bfjTUWU�opsH`�kTV�dHbxo�sfV#gh8UW}TjcdfVE~��5bfh�UWV�UWhc~cV���´ gHRTVEgH¥amnkT�d�o�bf¥cb¨U�o�`&z=VLg�h�UWVibfm�UW}T�nV�sE��²lh�s[V¬To�Ui}l��V�¤+RTVLk¶h8kTVigh�UW}=h8kTV�kqd[mnb[sHV�}T�yo�gVE~µzq`Oo�kThpdHRTV�s+¸vmnUi´}l��VLUiVLk8d�o0dfmnh8kF� ¹�dfRTV�z=VLRloE¯amnh�jTs�b[hp�'dfRTV�kTVL¤ ¯�V�s�bfm�h8k§m�k¢dfRTV«���nh�zlo��>be`cbedfVLU�gLopk¢z=V«~cVdHV�sHUWm�kTVE~mnUWUWVL~cmyo0dHV��n`��'¤+m�dfRTh8jcd«sfV�´BjTkc��h8�n~Tm�kT�¢dfRTVµ�8��h8zlop�+bf`cbvdHV�U&� Q)RTmybW}=hqbfbfmnzTm��nm�d ` dfhªsHV´£jlbfV�}TsfVL¯am�h8jlbg�h�UW}Tjcd�o0dHm�h8klb�op�¼dfVLs�o��nhcg�o��>be`cbedfVLU]jT}5~To0dHVWmyb�hp�)gh8jTsHbfViV¬adfsHV�UWV��n`¶~cVLbfm�s�opzl��V8�Fo�kl~¶h8kTVWhp��dfRTVUWh�dfmn¯0o0dHmnh�klb���h8s)dfRlmnb|o�}T}TsHh8o�g�RF�

-|��dfRlh�jT�8R&dfRTV�dHVLg�RTkTm(�qjTV#¤'o�b|VLbHbfV�k8dHmno����n`�~cVEbfg�sfmnz=VL~���h�sxo�}@opmns|h���gh8Ui}=h8kTV�kqdHbL�lm�dxklopdfjTs�op�n��`V�¬adfV�k@~Tb�dHh�dHsfVLV´ beRlo�}=VL~�bf`abedHV�U�b��>Q)RTV�kTV¬ad+V �5h�sfdHb'h8k�dfRTmyb'dfRlV�UWV�¤+m��n�5��hcgj@b'h8k«dHRTV�gh8klbfmn~cVLsHop´dHmnh�k�hp�lh�dfRTVLs�o8~cV �qjlopdfV�h�s�~cV�s�bL�0opkl~�dfhxdfRTV+~cV�sHmn¯0o0dHm�h8k�hp�=o¨UWh�sHV'op}l}Tsfh8}TsHmno0dHVtkThpdHm�h8k#hp�=o�~cV��qjlo0dHVh8sH~TV�sE��Q)RTV���V�klV�sHo��nm#!Eo0dfmnh�k«dHhW��VLkTV�s�op�5kTV�dHb|o�b)mnk8dHV�sf� o�g�VLb)¤+mn����op�ybfhiz=V�V�¬TopUWm�klVL~��

� ��������/��+��1��|" �+���J�

Z¶opkq`«dfRlo�kT¥cb'dHh��¨myg�dHh�s��RTh8UWV�kT¥�hWo�kl~ Aªop��dfVLs�>h��8��VLs'��h�s+mnk8dHV�sHVLbedfmnkT��~cmybfg�jlbfbfmnh�klbL�

���+�0�)���+�� '�+�

/n��1�r>oph��nh H)op�y~To�kF��^qdHV� o�k �¨o8opsE�5opkl~�H)opsHzlo�sHo��� 8kTmn�l�[wxmybvdHsfmnzTjcdHVL~��|kc��h8�n~cmnkT��hp�trJVdHsHmE+¨Vd�b��¡ k � ����� � �=�T}lo���VLb��E�p���=���l���c�������l�

/ �@1��8VLo�kc´£Zµmyg�RTV��FX�h8jT¯asfVLjTsL�c^ �Lzlo�bedfmnV�k $ sfmn¯�VdE�copkl~�wxV�kTmyb'r�h�m�dfsHV�klo�jl~����|kc��h8�n~cmnkT�ih���}TsHhc~cjlgdHbh���be`aUWUWVdfsHmyg�op��r�VdfsHmFkTV�dHbL���� � ��������0�� �! J��� � ����� � �����/���=� � �R��=�p��?�7 �n�E�c���=���C"T�c�����T�8�

K�KML�NPORQ@SUT

Page 23: Modular construction of finite and complete prefixes of Petri net … · 2016. 12. 27. · Modular construction of nite and complete pre xes of Petri net unfoldings Agnes Madalinski,

�p� �� �C�� � � � ����� �� ���

/ "�1���hch8bedB4�kT�8V����¼sHmnVdE� H'sHo�klgHRlm�kT�¢r�sfhTgVLbHbfVLb�hp�[rJVdHsfm +|V�dHbL� ��� �!�9������� ��� �R0�|�p��� 7�?�7 ��7�Dc����9DCDT���E+|VL¤|be�nVdfdfV�sH¡ kc��h �."��l�c�8�T�

/ � 1��8o�¯am�VLs 4tbf}lo�s !Eo�o�kl~«^adfV� o�k�{+h8UiVLsL� -|k«jlkc��h��y~cmnkT��op�n��h8sfm�dfRTU���h8s�bf`aklgHRlsfh8kTh�jlb�}TsfhT~cjlg�d�b�h��dfs�opklbfm�dfmnh�kµbf`abedfVLU�b��'¡ k ���2 ����� � �=��� �� � ���9��� ��� � � � ��� � ��� � ���!��� � � � * �P� �����0�5¯�h��njTUWV��E�����hp� � � � �@�T}lop�8VLb|� �c���T�5�9D DCDT�

/ 7�1��8o�¯am�VLs�4�be}lo�s !EoT�|^qdfV�� opk»{ �UWV�sE�+opk@~ Aªop��dfVLs �>h��8��VLsL� -¨k ¡ UW}TsHh0¯�V�UWV�kqd«h���ZOg�Zµmn���yo�k*� b�|kc��h8�n~Tm�kT� -|�n��h8sfm�dHRTU&��¡ k;� ����� �� � � ?� �9�� ��� ��@ � � �0 � �� !� ���@�l}lop�8VLb+���C7 ��"T���T�T�p���q�c�

/ ��1 4|��²lopzlsfV8�x²lo8g�dfh8sfm"!Lopdfmnh�kµhp� �¨kc��h��y~cmnkT�8b¨��h�s[wxmybvdHsfmnzTjcdHVL~¶Q)mn��VW^a`cbedfV�U�bL�=r>opsfd����|{+VE~cjlg�VL~¡ k8dHV�s�o�g�dHm�h8kµX'o�bfV��>Q�VLg�RTkTmyg�o��F{+V�}=h8sed��@7�� DT�q¡v{|¡e^�-�� -|}TsHm�������� "l�

/8?@1 4|��²lo�zTsfV8�i²lo�gdfh8sfm"!Lopdfmnh�k¶hp� �¨kc��h��y~cmnkT�8b[��h8s�wxmnbedfsHm�zljcdfVE~¢Q)mn��V«^a`cbedfV�U�bL�Fr�o�sed#�$� $ V�klV�s�op�X'o�bfV��>Q�VLg�RTkTmyg�op�F{|V�}=h�sfd��L�8���l�q¡v{+¡e^�-#�lZOo�`������p�@�

/ ��1 4|�=²@opzTsHV��¨Yxk&dfRlViX�h8klbedfsHjlg�dHm�h8k&hp�>r�jl���nzlo�g�¥cb+��h8s[^co0��V#rJVdHsHm +|Vd�b��+¡ k � � �:� � � �R���=���P �� �* � � �����F� �B�I�� �!� �0��� �� � ��?� � �����9�C��� :�U� � ��� � ���!��� � � � � � * � ��� � � � * ��� � � � �,���P�%�� � � ���jlkTV��p�8���l�

/ D�1 4|�=²@opzlsfV8� -�� H�VLkq¯�VLkTmnbedfV8�5^5� �¨o8opsE�@opkl~¶Xx�=�qops�~��)wxmybvdHsfmnzTjcdHVL~OZµh8kTm­dHh�sHm�kl��hp��X�h8klgjTsHsHV�k8dopkl~ -¨bf`qk@gHRTsHh�kTh8jlb�^a`cbedfVLUWbL���������!� ��5� ��� � ����� ��<�� �2�� �@ � � �0 � � ��� �=��5� R !�P���>}lo���VLb" "��a�p�l�cZOoE`��p�8�C7c�

/n�L��1 � ��RTh�UWV�kl¥�hl� ���9� � � � � ��� � �������6@ �R� ��� � � �!�� � � ��9������������;�5���=� ���� ��©r�Rlw dHRTVLbfmnbL��|kTmn¯�V�s�bfm­dv`«h��E+|V�¤)g�o�bedf�nV�jT}=h8k&Qt`akTV��@�p�8�8�T�

/n����1 �¨myg�dHh�s��RTh�UWV�kT¥�h@�TZOo�g�m�Veu �¨h�jTdfkq`8�lopkl~:A¢o��­dHV�s�>h��8��VLsL�tX'o�kTh�klmngLop��r�sHV�T¬cVLb+hp��rJVdHsfm�+|V�d�|kc��h8�n~Tm�kT�qb������ ���9������� ��� �R �� ��� � ��� ' � � � � ���R��� ! �T}lo���VLb DC7 �5���L�T�TY[gdfh8z=V�s|����� "l�

/n�E�@1 �W� ��qZOgLZµmn���yopkF� � ����� ��� � � ���9�C��� � � ��� � �����:�J�� ���� ��������- �4?� � � � � ����� �%�@ !�=����� ����� �� ���r�Rlw·dHRTVLbfmnbL�F�9D Dq�c�

/n�9"�1 �W�* t��ZOgLZµm��n�yopkF� �xbemnkT�µjTkc��h��y~cmnkT�8b�dHhµo�¯�h�my~¶dfRTV�bvd�o0dfV�V¬c}T�nhqbemnh�k§}lsfh8zT��VLU�m�k¢dHRTV�¯�VLsfm��T´g�opdfmnh�kOh���o�bf`aklgHRTsHh�klh�jlb¨g�m�s�gjTm�dHbL��¡ kF�5�U� � � ���2 ���!� � �=��� ����B��� � *�P� � ��� � ����� � ��� �6�=� �R� ���!� � � ���=���@�l}lop��VEb[�E�p���5�@?�?a�c��jT�n`¶�9D Dq�c�

/n����1�Z.� +|mnV��ybfV�kF� $ �Fr��nhpdH¥qmnkF�Jopkl~ $ � A m�k@be¥�VL�C��r�VdHsfm,+¨Vd�b���4�¯�VLk8d�^adfsHjlg�dHjTsfVEb�opk@~¢w¨h8U�opmnklb��r�o�sed+¡�� * �P� ��� � � ����� � � ��� � �=��� � ���F��"l±e�E³ � ��7 �=�����T�c�8o�kqjlopsH`µ�9D8���l�

/n��7@1 $ ��`akTk>A mnklbf¥�V��B�:- +|VL¤¦wxV�lkTm�dHm�h8k.hp�+ZOh�sH}TRTmybeU�h8kªrJV�dfsHm3+|V�dHbL��¡ k � *:� � ��� � ' � �5��� �� �R� �������� �U� ?� ��� ����� �� R��� � �U� * � �R����� �� �R��I� � ��� = J�U� � ����� � � ��� � �=� � �R��c}lop�8VLb¨�L�8���5��70�T� Fh8kl~ch�kF� ���W�F�9D����l�T^a}TsHm�kT�8V�sf´ �>VLsf�yop�l�

/n�L��1 $ ��`akTk A m�klbf¥�V��B�ªX'o0dfVL��h8sfmnVLb�hp�|UWhc~cVL�nb���h�sigh8klgjTsHsHV�klg�`��µ¡ k � � �0��� ��:��� � ��� � ���!����� ��� �� ��!� � � �=� � ��� ������ �P��<����R !�� ���l}lo���VLb|�0�8���c�p��?a� Fh8kl~ch8kF� ���W�F�9D8�C7c�T^a}TsHm�kl��V�sf´ �>V�sH�nop�l�

=?> K@=�A

Page 24: Modular construction of finite and complete prefixes of Petri net … · 2016. 12. 27. · Modular construction of nite and complete pre xes of Petri net unfoldings Agnes Madalinski,

Unité de recherche INRIA RennesIRISA, Campus universitaire de Beaulieu - 35042 Rennes Cedex (France)

Unité de recherche INRIA Futurs : Parc Club Orsay Université - ZAC des Vignes4, rue Jacques Monod - 91893 ORSAY Cedex (France)

Unité de recherche INRIA Lorraine : LORIA, Technopôle de Nancy-Brabois - Campus scientifique615, rue du Jardin Botanique - BP 101 - 54602 Villers-lès-Nancy Cedex (France)

Unité de recherche INRIA Rhône-Alpes : 655, avenue de l’Europe - 38334 Montbonnot Saint-Ismier (France)Unité de recherche INRIA Rocquencourt : Domaine de Voluceau - Rocquencourt - BP 105 - 78153 Le Chesnay Cedex (France)

Unité de recherche INRIA Sophia Antipolis : 2004, route des Lucioles - BP 93 - 06902 Sophia Antipolis Cedex (France)

ÉditeurINRIA - Domaine de Voluceau - Rocquencourt, BP 105 - 78153 Le Chesnay Cedex (France)������������� ����� ������� ��� ���

ISSN 0249-6399