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s = r◦l−1 ,9¨­MV�-MV:»OQMVU 1 : y�M'�a3T5R7a5.36�43E7�1�:l5CS^3EOHMVUE~q8�U^365.M'N.NRM

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¤�1�`qNROQMV¶q:H5RMVUT3',�Ϊ5RMª\XYLMVU]STM·3T«'�H:H�¦OHMVU£�M'��M'N.:¢NC��S6S�3�S^5C_]`<y�M'OQ1Q_]`¢:H5C_]`a38��Q�POQ5.MªWXYHNRMV5.3T�H:q��MV:¢ÍqYlM'U^36U68���MV:a�9«V¤Ø84U��/j�:H:HM':»~HUT1-YHNRMV¡»N.1aSE84NRN.MªÎÀ� x b WXYqN.M'5¥36�H:H��M':5R:qS %$� x b /21QOQM'N.N 8�YH��M'YH5RNROQMV3�¤jM'U6OQM':PÜH84YLMVU�:q5R_]`a3��H¡»��M'��MV`qU^3v,

��� ���*� � �� ���05� � 05� � ��� ��05��� �����Î�8�S�MVU]S^3TM OHMVU<`H5RMVU<7�1-UT�-M'S^3TMVNRN.3TMV:¿µ�U]84~q`-36U68�:qS^��1�U6¡»843T5R1�:qS6ST}$S^3TM'¡»M;Yq8�ST5.M'U^3<84�Q��OQMV¡%$5R:H�-N.M b ���lS^`H1-�Q3 b �EMVUT� 84`qUTM':P�¢OQ5.M�8�:qOQMVU6MV: YlM'5ROHMV:�7�MVUT��1�NR��M': y�MV¤�MV5RNCS MV5R: M'5.�-MV:HMvS�)U68�:qS^��1�U6¡»843T5R1�:qST¡»1QOQMVNRNÇ,$�SX¤�5.U]O-y�M'¤jM'5.NCS�M'5.:g�$�HU6«VMVUX\ÀYlM'UTYqN.5C_]� ÍHYLMVUÀOH5.M;¤�M'STMV:a3TNR5 b_]`HM':��jMvS�3]84:qOQ3TMV5RNRM»OHMVU§«'�H��U6�H:qOQM'N.5RMV�-MV:qOHMV:­µ�U]84~q`HMV:�ST19¤�5.M¢OQM'U�y�MV¤�MV5RN.5R��M':Z�)U]84:qS b��1-UT¡¢89365.1-:qST¡AMv_]`q84:q5RST¡»MV:Á�-MV��M'YlM':PÜs¤j1-YlM'5EOH8�S;WÀ�H��M':H¡»MVU6�Á84�Q��µ�MV¡»MV5R:qS684¡»��M'5¥36MV:¡»5.3X�H:qOg¼X:a3TM'U6S6_]`H5RM'OQM':�«'��¬® ¯�°H±�²H³�NR5.M'�43v,

Ã × k9mQhrn"#a�¨���`HUTM':qO�¬® ¯�°q±�²H³QS `H5.M'U68�U6_]`q5RS6_]`HM�µ�U]84~H`qMV: 8��Q�ªOQM'¡b|X}$~lM'UT�-U68�~H`H�-1�:H«'MV~Q3¦Yl8�ST5.M bU6MV:PÜ�NR5.M'�43�OHMV:�µ�U]84~H`qMV:�OQMvS�WÀN.�-MVYHU]845CST_]`qMV:�µ�U]84~H`H�-U68�¡»¡»843T5R� b %$}QS^3TM'¡»S¢OHMVUg��¼

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��M'UTNR5.:Hi;��Ü �$�HU6« "���� Ü OH8-S;��M'NR���Q¶q�-MVU6M»/Á1$OHMVNRNjOQMvS§¡¢84U6�$5.M'U^36MV:Z��M'UT5C_]`a3TM·36MV:­YH5R:q�4U6MV:µ�U]84~q`HMV:Z«V�zµ�U6�H:qOQM-,P��5R:zS^1-NR_]`qMVU§µ�U68�~H`Z5RS^3§MV5R:�� �q~lM'N

G = 〈V, E, L, s, t, l〉84�qS

«'¤jM'5lOQ5CSRy��H:H�a3TM': MV:qOHN.5C_]`HMV:�/2MV:H�-MV:V

7-1�: �<:H1�3TM':¢�H:qOE

7-1�: �;84:a36MV:PÜ-«V¤�MV5Q3614368 bNRMV:©WXYHYH5RNCOQ�H:H�-MV:s, t : E → V

�s

S�36MV`a3X��ÍHU - ST1��qU6_VM>/;Yq«V¤;,�-'�ª�HMVNRNRM>/VÜt

��ÍqU -Ú368�UT�-M·3</Yq«V¤;,�-TKL5.M'N /6� Ü$M'5.:HM'UX/2MV:q��M�7�1�:�/Á84U6�$5.M'UT�q:H��M':¿� -�Ê 8�YlM'NRS</T�

L�H:qO�MV5R:HMVU�3T1�368�N.M':�WÀY bYq5.NCOQ�H:H�

l : V ∪ E → L, ��5R: /21-UT~q`H5RST¡¦�lS

m : G → H«V¤�5CST_]`HM':�«V¤�MV5ØS^1-NR_]`qMV:

µ�U]84~q`HMV:G = 〈VG, EG, LG, sG, tG, lG〉

�H:lOH = 〈VH , EH , LH , sH , tH , lH〉

5RS^3jM'5.:�E8-84U<7�1-:©~q8�U^365.M'N.NRMV:­WXYHYq5.NCOQ�H:H�-MV:

m = 〈mV : VG → VH , mE : EG → EH〉ÜP��ÍHU

Oq8�SØ��5RN.3�g+�,∀v ∈ dom(mV ) : lG(v) = lH(mV (v))>Q,∀e ∈ dom(mE) : lG(e) = lG(mE(e))=H,∀e ∈ dom(mE) :� sG(e) ∈ dom(mV ) ∧mV (sG(e)) = sH(mE(e))� tG(e) ∈ dom(mV ) ∧mV (tG(e)) = tH(mE(e))

ÃA%'%Vk9��Ær& %v� #ak9&)�"(�� KL�HUÀΪM·¶q:q5¥365.1-:¿OQMVU ��"V"T�!�R?0'G"T�R.1�!'G��� iF <¤jM'U6OQM':2OQ5.M �jM'��U65 �sM�� ��J�9��=B"V'G� �q:qO ��� �/.[?0�2=mN1'G� � � �B�H= "T'G� `HMVU]84:H�-MV«'1���M':�i;#[gQ�5.:HM %$5R��:l893T�qUSig = (S, OP )Yq5.NCOQM·3�OHMV: ST}$:a3684�a365RS6_]`HMV: £X84`q¡AM':z��ÍHU¢M'5.:qM��<NC8�S6S^M

Alg(Sig)7-1�: WÀN.�-MVYHU6MV: «V�HU

%Q5.�-:q8936�HU',#%$5.M;YLM'S^3TM'`a3ª84�qSØM'5.:qMVUÀ/2M':H��MS

7-1�: � � �!"K.1�3J �H:qO�MV5R:HMVUª/2MV:H�-MOP

7-1�:$ Z3.0�2= "T�R� �;),) � %*?2�B�Q.1� ,/f-M'OHM�WÀN.�-MVYHU]8ª8��qS

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«'¤jM'5Sig b WÀN.�-MVYHU6MV:ª5RS^3 MV5R:HM¹l84¡»5RN.5RME7�1-:�1-~lM'U6843T5R1�:qST7�M'U^36U6����NR5R_]`qMV:ÀWÀYHYH5RNROQ�q:H��M':;«V¤�5 bS6_]`HM':gOHMV:�YlM'5ROQM':2� U]�4�-MVU6¡AM':H��M':P,%QMV5RMV:Á:$�H:

AMV5R:HM WÀN.�-MVYHU]8»«V�¿MV5R:HMVU %$5R��:q843T�HU

AS = (S, OP )�H:qO

GM'5.:¿¡»8�U b�$5RMVUT3TM'U)�-MVU65R_]`a3TMV3TM'U YH5R:q�4U6MVUrµ�U68�~H`P,v��5R:A�)�H~LMVN

GAS = 〈V, E, L, s, t, l, A, at, av〉¡A5.3

M'5.:qMVU�3T1�3684NRMV:­WXYHYq5.NCOQ�H:H�at : L → S

�at

S^3TM'`-3���ÍHU)-�893^36UT5RYH�Q36M¦3�}$~LM>/»Yq«V¤;, - W�3T3TU65 bYq�Q3^3�}$~�/6���q:qO M'5.:HM'U36143]84NRMV:�WÀYHYH5RNROQ�q:H�av : V ∪E → AV

�av

S^3TMV`a3���ÍHU - 843^3TU65RYH�Q3TM798�N.�HM0/;YH«'¤;,�- W�3T3TU65.Yq�Q3�¤jM'U^3 /T�·Ü$¤j1-YlM'5

AV =⊎

s∈S

AsOH5.M<OQ5CS$y��H:q�-36M;�EMVU6MV5R:H5R���H:H�»84NRNRMVU

�)U]�4��M'UT¡»M':H��M':�7-1�:A

YH5RNROQMV3'Üs`HM'5���3 = "T"T�!�R?0'G"T�R.0�!"K.1� w �8=2Z3# Üs¤�MV:H:2��ÍqUal : AV → S¡»5.3

∀v ∈ As : al(v) = s��1-N.�-MV:qOQMvS�ÎÀ5C84�-U68�¡»¡¸��1�¡»¡ �Q3T5RMVUT3�g

�L

� �S

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� �AV

at

l al

av

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GAGG = 〈O,⊥, P ; a, s, t : O → O, p : O\{⊥} → P, aP : P → P ∪ {⊥}〉

YLM'S^3TM'`-3E84�lS «'¤jM'5$OH5RSRy��H:H�a3TM':PÜ�MV:qOHN.5C_]`HMV: Üv:q5R_]`a3TNRMVM'UTM':¦/2M':H��M':O

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⊥∈ OÜL��M':q84:H:a3 2 �B"V"K� % ÜLST19¤�5RM;��ÍH:Q��36143]84NRMV:¿WÀYHYH5RNROQ�q:H��M':

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8�YqS�36U68-_ 365.1-:�/ � /��YlM':HMV:q8�YqS^3TU]84�a3T5R1�:�/6� ÜaS^1¢OH8-STS���1-N.�-MV:qOHM<��MvOQ5.:q���H:H�-MV:�MVUT��ÍHNRN¥3ÀST5.:lO g+-,;����5R��M':qST_]`l89� 3TM':2OQM'S���143T3T1-¡ b ��NRMV¡»MV:a3]S6�

a(⊥) = s(⊥) = t(⊥) =⊥>H,;� �<:H1�3TMV:g1QOQM'UX�;84:a3TM��∀o ∈ O : s(o) =⊥⇔ t(o) =⊥=q,;� �<:H1�3TMV:g¡»5.3À�<:H1436MV:g84NCS�WXYqS^3TU]84�a3T5R1�:L�∀o ∈ O : s(o) =⊥, a(o) 6=⊥⇒ s(a(o)) =⊥Jl,;� �;84:a36M�¡»5.3ª�<:H1�3TMV:g84NCS�WXYlS�36U68��-365.1-:l�∀o ∈ O : s(o) 6=⊥, a(o) 6=⊥, s(a(o)) =⊥⇒ a(s(o)) = a(t(o)) = a(o)[H,;� �;84:a36M�¡»5.3ª�;84:a36M�8�NRS�WÀYqS�36U68��a3T5R1�:l�∀o ∈ O : s(o) 6=⊥, s(a(o)) 6=⊥⇒ a(s(o)) = s(a(o)) ∧ a(t(o)) = t(a(o))´q,;� �;84:a36M�Ü �ª�HM'N.NRM;�H:qOÁKL5RMVNP8��Q�"�-N.M'5R_]`qMVUØ��YLMV:HM��∀o ∈ O : s(o) 6=⊥⇒ p(o) = p(s(o)) = p(t(o))�Q,;��NR5.:HMv84U6M<��YlM':HMV:lS�36UT�H�a36�HU]�∀p1, p2 ∈ P : ap(p1) = ap(p2)⇒ p1 = p2�q,;� WXYlS�36U68��-365.1-:g:a�qU�5.:gOQ5RM;:q��_]`qS^3T`3j�`HM'UTM���YLMV:HM��∀o ∈ O : a(o) 6=⊥⇒ p(a(o)) = ap(p(o))�q,;�xKL}$�$N.M':Q��U6MV5R`HMV5.3ÀOQMVUXWÀYHYH5RNROH�H:H��M':

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src�H:qO

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〈VG, EG〉Üs¤j1-YlM'5

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5CS�3M'5.:¢�E8-84U〈class, card〉

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MV5R:©�E8�8�U〈pe, HE〉

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∀eb ∈ EBm¡A5.3

eb = 〈pe, HE〉, pe = 〈pet, psrc, pdst, card〉,

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vbs = 〈psrc, hsrc〉 ∧ vbd = 〈pdst, hdst〉.

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∀eb ∈ EBm¡»5.3

eb = 〈pe, HE〉, pe = 〈pet, psrc, pdst, 〈l, h〉〉 :

l ≤ |HE| ≤ h.

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vb = 〈pv, HV 〉, pv = 〈pclass, 〈l, h〉〉 :

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