On the construction of pullbacks for safe Petri nets

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Submitted on 19 May 2006

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On the construction of pullbacks for safe Petri nets

Eric Fabre

To cite this version:

Eric Fabre. On the construction of pullbacks for safe Petri nets. [Research Report] RR-5722, INRIA.

2005, pp.13. �inria-00070296�

ISSN 0249-6399 ISRN INRIA/RR--5722--FR+ENG

a p p o r t

d e r e c h e r c h e

INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE

On the construction of pullbacks for safe Petri nets

Eric Fabre

N˚5722

October 1st, 2005

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8‚OP„…B R B< S LMJKB<m;†ŠL˜<ma<?aKB(O\Z[F B(_MB•tJKL˜<mLYFGJIF\„6@CF S ^MaKLN;=@;_KB•KJKB‡<maMB(D/OP<?BAuGF S 9

Nets ^{" tF S ˆOPZ BˆNBA_}

JKB<?;Apt†B JKOP<mH S O\ˆNˆ˜9"DFGJK;=LN_KB S ˆNO\Z[BAˆ˜™ ^ S B;?B S LNJKu @CF S ^KaKLN;=@;p‚†ŠaKLYDba9:LYBAˆN_M;(<maKB$D/O#<mBAu\F S 9

λNets ^

›

~ Ÿ - Ÿ y( z)(

Ÿ & Ÿ z)=

›

- 1w|t~") B<

N₀,N₁,N₂ ^Z BWJKB<?;ApKOPJK_

f_i :Ni → N₀^p i= 1,2 Z[BXJKB<

@F S ^KaKLY;?@C;Ap#;?F

N0

„…F S @C;¡OE‰QLNJK_IFP„tLNJf<mB S „1O\DBZ[B<’†BABJ

N1

O\JK_

N2

pPF S Z[B<=<mB S OWDFG@C@FGJ DAF\JK;=< S O\LYJQ< 

¢B ˆNF:FG‰C„…F S O“<?B S @CLNJKO\ˆJKB<

N = (P, T,→, P⁰)p6O\;?;=F‚DLOP<?BA_ƒ<?F@CF S ^KaKLN;=@;

g_i :N → Ni

p

i= 1,2^p

;?HKDba<matO#< •tuK

n Œ

f₁ ◦g₁ = f₂◦g₂ ⁿ

9 <mB S @LYJtO\ˆ»pŠ†B$@BAO\J<maKB-HKJML B S ;?O\ˆ^ S F\^ B S <’9F\„<maMB ^MHKˆNˆYZtO\Db‰EŒ†ŠaKBAJKB B S <?aKB S B B ‚LN;=<?;EO\JKF\<?aKB S

< S LN^KˆYB

(N3, h₁, h₂) ;mOP<?LN;=„ 9:LYJKu<?aKB";?O\@CBCDFG@C@IHM<bO#<mL B“_KLO\u S O\@p<maKB S BCB‚LN;o<m;IOHKJKLY£:HMB@CF S ^MaKLN;=@

ψ : N₃ → N ^{^ S BA;=B S} LYJKuƒ<?aKBCDAFG@C@IHM<mOP<mL

L˜<’9GpJtOP@BˆY9

h_i = g_i◦ψ^¢ BC_KBAJKFP<mB<?aKBC^KHKˆNˆYZtO\Db‰ Zf9

N₁∧^N0 N₂^{pKF S Zf9} N₁∧ N₂ ^{„…F S ;?aMF S <A}

N₀ N₁ N₂

f₁ f₂

π1 π2

g₂ g₁

N₃

N

ψ

)LYuGH

S B n

Œ

N =N₁∧ N₂

‘ <LN;”†BAˆYˆ[‰QJKFs†ŠJ"<?atOP<Ž<maKBE^KHKˆYˆNZtOPDb‰ FG^ B S OP<?LNFGJ“DAO\J"Z[BW_MBADAF\@^[FG;?B_xLNJf<?FO$^ S F:_KHKD</pQ„…FGˆNˆYFs†‡B_Zf9

O\JBA£QHtOPˆNLNAOP<mLYFGJ 6‡FGJK;?LY_KB S <?aKBl^ S F‚_MHKD<`JMB<

N1×N N2

ptO\JM_x<maKB-OP;?;?F:DALNOP<mB_D/O\JMFGJKLNDAO\ˆ—^ S F#U=BD<mLYFGJK;

π_i : N₁×N N₂ → Ni

p

i = 1,2^{‘’J uGBJKB S O\ˆ´p} N₁×N N₂ O\JK_3<maMB

π_i _KFƒJKF\< ;mOP<?LN;o„9 <maKB“^KHMˆNˆNZKO\Db‰

DAFGJM_KLY<?LNFGJp

f₁ ◦π₁ 6= f₂ ◦π₂^{WFs†‡B B S} p‡Zf9BA£QHtO\ˆYLNALYJKuj<?aKBA@pTFGJKBxuGB<m;I<?aKB_KBA;=LS BA_ S B;?HKˆ˜</

(N, e) BA£QHtOPˆNLNBA;

f₁◦π₁ ^OPJK_ f₂ ◦π₂ ^L (f1◦π₁)◦e= (f2◦π₂)◦ep[O\JK_ƒ„…F S OPJQ9F\<?aKB S DAO\JK_KLY_tOP<mB

(N₃, h) ^{<maKB S} B B‚LN;o<m;WOCHKJMLN£QHKB

ψ :N₃ → N ;=HKDbaj<?atOP<

h =e◦ψ ^{•KuK[h ‘} <XLY;Š;=<

S

O\LNu\aQ<=„…F

S

†‡O

S

_<?F

DbaKBADb‰x<?atOP<

(N, π₁◦e, π₂◦e)9‚LYBAˆY_K;(<maKB-_MBA;?LS B_^MHKˆNˆYZtO\Db‰6 tF S _KB<bO\LYˆN;Ap:†‡B S B„…B S <?aKB S BAO\_KB S <?F n ´p DbatO\^ %E™oh‚pM<?aK@ n pKF S <?F#95 »pt;?BD\ e†ŠaKB S B <?aKLN;`DFGJK;o< S HMD<mLYFGJLN;ŠO\ˆY;?F HK;=BA_

"!

±$#»±=²&%('1³

B

#´±o²»¸˜¹µo± )Ž¾+*,,±?¾# ¾-#

!

¹.#”¹0/1/K³

B

#´±o² ¹µ #´³32

B

¾)¹ ²´±µo¹4#&*#²»±65$º879#

!

±(³

B

#´±o²:%(±;5s³¿¹ ²:7

B

±<#

N0=?>

!

³¿µ

!

³¿¾

B

20#

# !

±Žµo¹¾1±·

"!

±@#

> 2 B

±A#´¾

N1,N2

%(¹B7W¾#´³3/1/

!

¹BC±³

B

#»±=²´¹µ #´³32

B

¾D2*#´¾»³15s±

N0

º87E%(±?¹

B

¾D2¸K¾&7 B µ !

²:2

B

³1Fo±65G#´²´¹

B

¾»³H#´³32 B ¾ =

¹¾

³3#

>

³3/1/Qº\±I2º8C/³32J*#¾—³ B # !

±I4*/1/½ºP¹µLKE5s±AM

B

³3#»³12

B

,³1C±

B

¹0#"#

!

±”±

B

592 ¸N#

!

±=¾»±

B

2#»±=¾=·

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N₁^xN₂

f₂ ^oπ₂ f₁ ^oπ₁ N₃

N₀ N ^e

ψ h

)LNuGH S B$h‚Œ N

f₁◦π₁ f₂◦π₂

$ ((

Nets

6‡FGJK;=LN_KB S <’†‡F3JKB<m;

Ni = (Pi, T_i,→i, P_i⁰), i = 1,2 ^{S BˆOP<?BA_ Zf9} <’†‡F3@CF S ^MaKLN;=@;

f, g : N1 → N2



¢B†(O\Jf<-<?FxZKHKLYˆN_j<?aKBIBA£QHtOPˆNLNB

S (N, e) ^FP„ f

O\JK_

g

p

O"JKB<

N O\JK_3Ox@CF

S

^KaMLN;?@

e : N → N₁

;mOP<?LN;o„9‚LYJKu

f◦e =g◦ep‚O\JK_x;=HKDbax<matO#<„…F S O\Jf9F\<maKB S DAO\JK_KLY_tOP<?BX^tO\LS

(N₃, h)^{<maKB S BXB} ‚LN;=<?;(OHMJKLN£QHKB

@F S ^KaKLY;?@

ψ :N3 → N ;mOP<?LN;o„9‚LYJKu

h =e◦ψ ^{•KuK r }

N₂ N₁

N₃

N ψ h

e

g f

)LYuGH

S B r

Œ

(N, e) ^N f g

!"$#&%'( !

Sets

¢B S BAD/OPˆNˆaKB S B$<’†‡FCDAˆNO\;?;=LNDAO\ˆ S B;?HKˆ˜<m;`<maKOP<`†ŠLNˆYˆ—Z[B LYJK;=< S HK@CBAJf<mO\ˆLNJ<?aKB$;?BA£QHKBˆ´

- ) $ )$ . /

| (#1 › {

¢B DAF\JK;?LY_KB

S

<maMBDAOP<mBuGF

S

9xF\„;=B<m;E†ŠLY<?a

„…HKJKD<mLNF\JK;EO\;Š@F

S

^KaKLY;?@C; 1F

S

BA£QHKL

O\ˆNBJQ<?ˆY9"^ FGLYJf<mBA_

;?B<‡†ŠLY<?aC<?F\<bOPˆK„…HKJKD<?LNFGJM; B<

T₁, T₂ Z[B`<’†‡F;?B<m; S BAˆNOP<mB_Zf9C^tO S <mLOPˆM„…HKJKD<mLNF\JK;

f, g:T₁ →T₂^{ € aKB}

BA£QHtO\ˆYLNB S F\„

f ^OPJK_ g LY;(<maKB$^tO\LS

(T, e) ^{†ŠaKB S B}

T = {t₁ ∈T₁ :f(t₁) =g(t₁) ^{F S Z FP<ma} f ^O\JK_ g ^{O S} B-HKJK_KB•tJKB_ OP<

t₁} ^´h

O\JK_

e LY;<?aKB"DAO\JKFGJKLYD/O\ˆTLYJ#U=BAD<mLNF\JF\„

T ^LYJQ<?F T₁ ^†B ˆNˆŽHK;?B"<?aKB";=aKF S <matO\JM_

t₁ ∈ T LYJK;=<?B/O\_F\„

t ∈ T, t₁ = e(t)I‘’J3<maMBC;=B<=<mLNJMuF\„‡^[FGLYJQ<?BA_3;?B<m;p†ŠaMB S B“„…HKJKD<mLYFGJK;-^[FGLNJf<l<?F<?aKB ;?^[BADLO\ˆ O\ˆYHKB

² ^F\„O

;?B<W<?FC@CB/OPJ ’HKJK_KB•tJKB_p ´h ‡<bOP‰\BA;`<?aKB$;?LN@C^KˆYBA;=<`„…F S @

f(t₁) =g(t₁)^

)-L BAJ3O\JMF\<maKB S D/O\JM_KLN_tO#<mBI^tOPLS

(T3, h)^p—<maMB HKJKLN£QHKB @CF S ^KaKLN;=@ …^tO S <mLOPˆ„…HKJMD<mLYFGJ

ψ : T₃ → T

LN;`FGZ‚<bO\LYJKBA_Zf9

ψ =e⁻¹◦h

1L˜<ŠLN;`BAO\;=9x<?FDbaMBADb‰x<maKOP<

Im(h)⊆T



- ) $ )+ * Ÿ› /

| (1

› {

¢BJKFs†*DAFGJM;?LN_MB

S

<maKBD/OP<?BAuGF

S

9$F\„t;=B<m;¡†ŠLY<ma

„…HKJKD<mLNF\JK;A

€

aKB‡DF‚B£QHtO\ˆNLYAB

S

_KLNO\u

S

O\@‹DAF

S?S

B;?^[FGJK_K;

<mF`•tuK r †ŠL˜<ma O\ˆNˆ\O S?S Fs†Š; S B B S ;=BA_ B<

S₂, S₁ ^Z B<’†FW;=B<?; S BAˆO#<mBA_$Zf9l<?F\<bO\ˆP„…HKJKD<?LNFGJM;

F, G:S₂ →S₁^p

O\JK__KBAJMF\<mB$Zf9

(S, E) <maMB DF‚B£QHtO\ˆNLYAB S F\„

F ^O\JK_ G^{ €} aMBDFGJK;=< S HKD<mLYFGJLN;ŠO“ZKLY<`@CF S B-DAFG@C^KˆNB  /10 @2/13

! "#

a=b=c _ _ _

d _ S₂

S

S₁ a b c d

E G F

)LNuGH S B 8KΠN "0

F G

VWB•tJKB$<maMB S BAˆNOP<mLYFGJ

R F\JBˆNB@BJQ<?;ŠF\„

S₁ ^Zf9

p₁R p⁰₁ ⇔ ∃p2 ∈S₂, {p1, p⁰₁}={F(p2), G(p2)} ^r

O\JK_“DAF\JK;?LY_KB

S

<?aKB`BA£QHKL

OPˆNBAJMDAB

S

BAˆO#<mLNF\J

≡^{uGBJKB S} OP<mB_“ZQ9

R^¡¢ B`_KBAJKFP<mB`Zf9

[p₁]<maMB`DAˆ OP;?;”F\„

p₁ ^{„…F S} ≡^

€ aKBAJ

S = {[p₁] :p₁ ∈S₁} ⁸

O\JK_<maKB-„…HKJMD<mLYFGJ

E :S₁ →S LY;`;?LY@^Kˆ˜9<?aKB$£QHKF\<mLYBAJf<EFG^[B S O#<mLNF\Jp

E(p1) = [p1]^

)-L BAJƒO\JKF\<?aKB S DAO\JK_KLY_tOP<mB$^tOPLS

(S3, H)p‚<maKB HKJKLY£:HMB @CF S ^KaKLN;=@ …<?F\<bOPˆ—„…HKJKD<mLNF\J

Ψ :S₃ →S ^LN;

FGZM<mO\LNJKB_ Zf9

Ψ =H ◦E^{−1p F S} LNJƒF\<maMB S †F S _K;lZf9

∀[p₁]∈ S,Ψ([p₁]) = H(p₁)X‘’JM_KBAB_pLY<WLN;EB/OP;=9

<mFCDbaKBADb‰x<?atOP<

H LN;`JKBDAB;?;mO S LNˆ˜9DAˆNO\;?;`LYJ O S LOPJQ<A

!"" ( & !

Nets

B<

(N, e) _KBJKF\<?B$<?aKB$_KBA;=LS BA_BA£QHtO\ˆYLNAB S pM†ŠL˜<ma

N = (P, T,→, P⁰) ^O\JK_ e:N → N₁^

{}|

&

y(…z(Ÿ & y ) klJ< S O\JK;=LY<?LNFGJ;?B<?;Ap

f, g :T₁ →T₂ ^{O S B^tO S <?L}O\ˆQ„…HKJKD<mLYFGJp\;=Fl†‡B(O\_KFG^M<)_KB•KJKLY<?LNFGJ´h

„…F S

T ^O\JM_ e ^FGJ T^

…|t~ › y ) k-JC^KˆNO\DAB`;=B<m;p:<?aKBŠ_KB•KJKLY<?LNFGJ“LN;TO$ZKL˜<T@F S B(DAF\@^KˆYB  € aKBŠ@CF S ^MaKLN;=@ _KB•tJKLY<?LNFGJ“LYJ

Nets

O\D<?HtO\ˆYˆY9x;o<bOP<?BA;ŠLNJ 6 8 <matO#<

φ^op:^•t₂ →^•t₁ ^O\JK_ φ^op:t₂^• →t₁^{• O S} Bl<?F\<bO\ˆ„…HMJKD<?LNFGJK;pK„…F S

t₂ =φ(t₁)^p

†ŠaKLNDbaF S LNBAJf<?;EHM;Š<?F“DAFP™ BA£QHtO\ˆYLNAB S ;`LYJ

Sets8‚F“ˆNB<

t ^{Z[B O < S} O\JM;?LY<?LNFGJFP„

TpM†ŠL˜<ma

t₁ =e(t)∈T₁^

W;?;=HK@B$• S ;=<E<matOP<

f, g ^{O S} B _MB•tJKB_ OP<

t₁^p[O\JK_ f(t1) =g(t1) =t₂ ∈T₂^‡¢ B <mO\‰\B$„…F S

e^{op LYJ •}t₁

<maKB DF‚B£QHtO\ˆNLYAB S F\„

f^op, g^op : ^•t₂ → ^•t₁^{ 4 £[ r} `<maQHK;-_KB•tJKBA;

R^•t1p—<?aKBIBA£QHKL O\ˆNBJKDAB S BAˆNOP<mLYFGJ

≡^•t1

O\JK_^KˆOPDAB-DAˆNO\;?;=BA;

[p1]^•t1WJK_ƒ;=LN@CLNˆNO S ˆ˜9“LNJx<maMB ^[FG;o<=™’;=B<EF\„

t₁^

¢'aMBAJ

f, g

O S

B(Z[F\<ma HMJK_KB•KJKBA_OP<

t₁

p\†B(<bO\‰PB(„…F

S

e^op

LYJ •

t₁

1F

S

t₁^•

<maMB`DAF:BA£QHtO\ˆYLNAB

S

F\„6„…HKJMD<mLYFGJK;

f^op, g^{op „S FG@} <?aKB$BA@C^M<’9;?B</‡8‚F

e^op LN;Š;?LY@^Kˆ˜9<?aKB$LN_KBJf<mLY<’9\

‘’J;?HM@@O S 9\p‚<?aKB$^KˆO\DB$;?B<

P ^FP„ N ^LY;ŠO ;?HKZK;=B<WF\„

2^{P1 uGL BAJZQ9}

P = {[p₁]^•t1 :t₁ ∈T, p₁ ∈^•t₁} ∪ {[p₁]^t1• :t₁ ∈T, p₁ ∈t₁^•} ^´e

O\JK_ <maMB

S

BAˆNOP<mLYFGJ

e

FGJ“^Kˆ OPDABA;)LN;;=LN@C^KˆY9 uGL

BAJ“Zf9

p←→^e p₁

L

p₁ ∈p

”k-ZM;?B

S

B`<maKOP<TOl^KˆNO\DAB

p₁ ∈P₁

JKF\<EDAF\JKJKBAD<mB_ƒ<mFCOI<

S

O\JK;=LY<mLYFGJF\„

T atOP;EJMFDFGHKJf<mB

S

^tO

S

<ELNJ

P^

Ž› |

$

t₁, t⁰₁ ∈T^{' N} p₁, p⁰₁ ⊆t⁰₁^• ∩ ^•t₁

p₁ ≡^t01• p⁰₁ ⇐⇒ p₁ ≡^•t1 p⁰₁ ^05!

@A@CBED'FGH'H