Hornets: Nets within Nets combined with Net Algebra

Hornets: Nets within Nets combined with Net Algebra Michael Köhler-Bußmeier Department for Informatics University of Hamburg

Application and Theory of Petri Nets

Michael K¨ ohler-Bußmeier (UHH)

Hornets: Nets within Nets combined with Net Algebra

ICATPN’09

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Outline 1

Nets within Nets

2

Elementary Object Systems (EOS)

3

Object Nets

4

Higher-Order Object Nets

5

A Workflow Management Network

6

Decidability Results



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Outline 1

Nets within Nets

2


3

Object Nets

4


5


6




ICATPN’09

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Nets within Nets s1 t1

s1’

t

s

s’

Elementary Object Nets [Val98, Val03] Object Nets [Köh07] Recursive nets [HP99] Nested nets [Lom00] Hypernets [BBPP04] Mobile systems [Lak05] AHO systems [HEM05] Adaptive workflow nets [LvHO+ 06] RN systems [VdFE08]



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Outline 1

Nets within Nets

2


3

Object Nets

4


5


6




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Definition An elementary object system has two nesting levels.

Definition b N , d, Θ, µ0 ) An elementary object system is a tuple OS = ( N, 1 2 3

b is a P/T-net, called the system-net. N N = {N1 , . . . , Nn } is a set of P/T-nets, called object-nets. b → {•} ∪ N is the typing of the system-net places. d :P

4

Θ ⊆ T defines synchronised transitions (system/object).

5

µ0 ∈ M is the initial marking.



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Example p1

p4

p2

p5

t

p6

p3

Places p with different types: d(p1 ) = d(p1 ) = d(p4 ) = N1 d(p3 ) = d(p5 ) = d(p6 ) = N2



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Example p1

a1

p2 p3

a1

a2 b2

p4

b1

t1 a1

t1

t1

t2

b1

p5

t

b1

p6

c2

Marking of the system net: µ = + + + Michael K¨ ohler-Bußmeier (UHH)

p1 p1 p2 p3


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Example p1

a1

p2 p3

a1

a2 b2

p4

b1

t1 a1

t1

t1

t2

b1

p5

t

b1

p6

c2

Tokens are net-tokens, which have independent (nested) markings: µ = + + + Michael K¨ ohler-Bußmeier (UHH)

p1 [0] p1 [a1 + b1 ] p2 [a1 ] p3 [a2 + b2 ]


ICATPN’09

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Example p1

a1

p2 p3

a1

a2 b2

p4

b1

t1 a1

t1

t1

t2

b1

p5

t

b1

p6

c2

Firing of t selects a submarking λ of µ: λ = + + + Michael K¨ ohler-Bußmeier (UHH)

··· p1 [a1 + b1 ] p2 [a1 ] p3 [a2 + b2 ]


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Example p1

a1

p2 p3

p4

b1

t1 a1

a1

t1

b1

t1

b1

p5

< t1, t2 >

a1

a2 b2

t1

t

b1 a2

t2

c2

b2

t2

p6

c2

The markings of the net-tokens are added.



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Example p1

a1

p2 p3

a1

a2 b2

p4

b1

t1 a1

a1

t1

t1

b1

b1

t

b1 a2

t2

t1

a1

c2

b2

t2

b1

p5

< t1, t2 >

a2 c2

t1

b2

t2

p6

c2

Synchronisation θ = t[t1 , t2 ] of t with t1 and t2 .



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Example p1

a1

b1

t1

a1

a1

p2 p3

a1

t1

b1

t1

a1 b1

t1

a2 b2

t1

t

b1 a2

t2

c2

b2

t2

a2

t2

b2 a2

c2

b2

t2

b1

p4

b1

< t1, t2 >

a1

t1

c2

a2 b2

t2

p5 c2

p6 c2

The resulting markings are distributed, resulting in ρ.



ICATPN’09

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Example p1

a1

t1

b1 a1

p2

t

< t1, t2 >

p3

a2

t1

t2

b2

a2 b2

t2

b1

p4 p5

c2

p6 c2

Successor marking µ0 = µ − λ + ρ: t[t1 ,t2 ](λ,ρ)

µ −−−−−−−→ µ0 OS



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Outline 1

Nets within Nets

2


3

Object Nets

4


5


6




ICATPN’09

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Object Net Systems (ONS) Extending Eos to Object Net Systems: 1

Unbounded nesting of markings

2

Firing may increase the marking’s nesting depth.

3

Moving of net-tokens along channels Net N2

y

p1::N1

p2::N2 y

p1::N1

p2::N2

q2::N3

q1::N3

x

t !c:N2

x

p3::N1

x

t !c:N2

x

p3::N1 Net N1 ?c::N2 z v

r::N2

Net N1 ?c::N2 z v

Net N2 q1::N3

r::N2



q2::N3

ICATPN’09

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Object Net Systems (ONS) Extending Eos to Object Net Systems: 1

Unbounded nesting of markings

2

Firing may increase the marking’s nesting depth.

3

Moving of net-tokens along channels Net N2

y

p1::N1

p2::N2 y

p1::N1

p2::N2

q2::N3

q1::N3

x

t !c:N2

x

p3::N1

x

t !c:N2

x

p3::N1 Net N1 ?c::N2 z v

r::N2

Net N1 ?c::N2 z v

Net N2 q1::N3

r::N2



q2::N3

ICATPN’09

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Outline 1

Nets within Nets

2


3

Object Nets

4


5


6




ICATPN’09

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Hornets: Algebraic Net Operations Algebraic operations on the net-tokens, e.g. parallel composition: (N 1 kN2 )

q u Hy r H j H (xky) t p * u x



ICATPN’09

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=

q u Hy r H j H (xky) t p * u x

i1- - iu- - riv - - i N1 i f1 a c b



ICATPN’09

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i2- - ris - - i N2 i f2 e d } Z Z

=

q Z Zu Hy r H j H (xky) t p * u x

i1- - iu- - riv - - i N1 i f1 a c b



ICATPN’09

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u - rhv- i1 - h h N3 a c b q i 3 h Hy r* H j H (xky) u i2 @ R h t p * rs - e d - h x



ICATPN’09

f1 h @ R - h f f2 h

11 / 26

Net Algebra Hornets are related to Adaptive Workflows [LvHO+ 06]: Hornets adaptive workflows

= =

object nets workflows

ri-N1 - i-N2 - i

i = i1

f1 = i 2

(N1 · N2 )


f = f2

+ +

net algebra composition

f =f1

3 N1 Q =f2 s Qj rj i =iQ 1 s Q N2 3 =i2 (N1 + N2 )

+

nets-within-nets

f i1h N1 - h1 Q 3 s Q - hf rh i f23 Q s Q h2- N2 - h i


(N1 kN2 )

ICATPN’09

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Outline 1

Nets within Nets

2


3

Object Nets

4


5


6




ICATPN’09

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Example: Workflow Management Network Each workflow is executed in a network of workflow agents. Workflows are composed from sub-workflows. Each agent either (a) executes its workflow itself or (b) decomposes it and delegates the sub-workflows to other agents. all workflows that the agent can handle by delegation

root WFMS agent WFMS 1 subordinate WFMS agent

WF1, WF2, (WF1 || WF2), .... WF1, ...

workflows that the agent can handle on its own

WFMS 11

WF2, .... WF2, ...

....


....

WFMS 12

WF1, WF3, (WF1 + WF3), .... WF1, WF3, ...

....


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The Workflow Agent delegation = (N1 || N2) info management compose3 NOR N = (N1; N2)

compose

update

N = (N1 || N2) OR N = (N1; N2) OR N = (N1 + N2)

N1; N2

N

N

in & propagate downwards

receive

N

N

proposed nets

check

subordinate WFMS agents

A

N

atomic well formed workflows

N

insert

N

propagate basic delegation info upwards

A1:enactWF(N1); A2:enactWF(N2)

wf instance N

:upd(N)

enact :enactWF(N)

N [N1,true]

[N,true]

start execution

N = (N1 || N2) OR N = (N1; N2) OR N = (N1 + N2)

"in doubt - deligate" (decompose)

running workflows N1

N

log


delegate and distribute N = (N1 || N2) OR N = (N1; N2) OR N = (N1 + N2)

[N,false]

local workflow repository

:upd(N)

[N1,A1]; [N2,A2] A1; A2

[N,true]

[N,b]

[N, true]

workflow repository management

[N,b]

[N1,b1]; [N2,b2]

b= is_local(N,this)

propagate delegation info upwards

[N,A]

A

N = (N1 || N2) OR N = (N1; N2) compose2 OR N = (N1 + N2) b= (b1 AND b2)

N

analysis

delegation info

[N,A]

N

E:secure_in(N)

[N,A]

A:upd(N)

well formed workflows

:secure_in(N)

:inWF(N)

OR N = (N1 + N2)

[N1,A]; [N2,A]

update delegation info

N

N

N:start()

stop execution

workflow execution

N:stop()


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Refinement: Workflow Analysis Refinement of the transition check in the module analysis: proposed nets

workflow nets

N N

N

& isFinalPlace(f,N)

N

& isConnected(i,f,N)

isLive(N)

N

N

N

bounded

N

isStructBounded(N)

N

well formed workflows

N

N

workflows

N

N

check

N

isInitialPlace(i,N)

C = controler(N)

N

N || C

N N

N

N N

not isStrucBounded(N)


N

isWFBounded(N)

not isLive(N)

N

not isWFBounded(N)

rejected nets


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Outline 1

Nets within Nets

2


3

Object Nets

4


5


6




ICATPN’09

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Decidability Results: Net Classes

' &


P/T-Nets Reachability

$ %


ICATPN’09

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Decidability Results: Net Classes

'

' &

&


Transfer Nets P/T-Nets Reachability Boundedness

$

$ %


%

ICATPN’09

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Decidability Results: Net Classes '

'

' &

&

&


$

Reset Nets Transfer Nets P/T-Nets Reachability Boundedness

$

$ %

Coverability


%

%

ICATPN’09

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Decidability Results: Net Classes '

'

'

' &

&

&

&


$

Selbstmodifying Nets

$

Reset Nets Transfer Nets P/T-Nets Reachability Boundedness

$

$ %

Coverability


%

%

ICATPN’09

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Hornets Theorem Each counter machine can be simulated by a Hornet.

Proof. (Obvious.) Define the net-type Nat with the net-constant zero and the unary operator suc.



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Object Nets Theorem (Köhler and Rölke 2004, ICATPN’04) Each counter machine can be simulated by an Object Net System.

Proof. Let NZERO and NSUC be object nets. We use the marking’s nesting depth to encode a counter: cod(0) = NZERO [] cod(n + 1) = NSUC [pos + cod(n)] The test for zero, incrementation and decrementation can be simulated.



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EOS: Weak Simulation of Counter Machines Theorem (Köhler 2007) Let A be a counter automaton. The Eos OS(A) has the following weak simulation property: ∗

(q, n1 , . . . , nk ) − → (q 0 , n10 , . . . , nk0 ) A

iff ∗

µ(q, n1 , . . . , nk ) −−−−→ µ(q 0 , n10 , . . . , nk0 ) OS(A)

is-pos

inc1

dec1

counter

inc1

reduce

2

control

=0

a

recreate

r

is-pos n

counter


dec1

q

>0


s ICATPN’09

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EOS Boundedness / Coverability Theorem Boundedness and Coverability are decidable for Eos. Boundedness is decidable iff < is a decidable, monotonous well-quasiorder. Coverability is decidable iff ≤ is [Finkel and Schnoebelen, 2001]. m  1 < m2  y∗ m10

=⇒

m 1 < m 2   y∗ y∗ 0 m1 < m20

One can show: Firing is monotonous for Eos wrt. the wqo : P b a [A ] ∧ αβ ⇐⇒ α = m Pni =1 bi i β = i =1 bi [Bi ] ∧ ∀1 ≤ i ≤ m : b ai = b bi ∧ A i ≤ B i Michael K¨ ohler-Bußmeier (UHH)


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Summary: Decidability Hierarchy '

'

'

Reset-Nets Transfer-Nets, Eos

'

P/T-Nets, GSM-ONS

&

&

&

&

$

Selbstmodifying Nets, ONS, Hornets


Reachability Boundedness

$

$

$ %

Coverability


%

%

ICATPN’09

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Conclusion Nets within Nets Elementary Object Systems (EOS) Object Nets, Channels for in/out transfers Hornets: Higher-Order Object Nets Relationship to adaptive workflow nets Example: Workflow Management (Un)Decidability Results



ICATPN’09

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Marek A. Bednarczyk, Luca Bernardinello, Wieslaw Pawlowski, and Lucia Pomello. Modelling mobility with Petri hypernets. In Josè Luiz Fiadeiro, Peter D. Mosses, and Fernando Orejas, editors, Recent Trends in Algebraic Development Techniques (WADT 2004), volume 3423 of Lecture Notes in Computer Science, pages 28–44. Springer-Verlag, 2004. Kathrin Hoffmann, Hartmut Ehrig, and Till Mossakowski. High-level nets with nets and rules as tokens. In Application and Theory of Petri Nets and Other Models of Concurrency, volume 3536 of Lecture Notes in Computer Science, pages 268 – 288. Springer-Verlag, 2005. Serge Haddad and Denis Poitrenaud. Theoretical aspects of recursive Petri nets. In S. Donatelli and J. Kleijn, editors, Proceedings of the 20th International Conference on Application and Theory of Petri Nets, volume 1639 of Lecture Notes in Computer Science, pages 228–247. Springer-Verlag, 1999. Michael K¨ ohler. Reachable markings of object Petri nets. Fundamenta Informaticae, 79(3-4):401 – 413, 2007. Charles Lakos. A Petri net view of mobility. In Formal Techniques for Networked and Distributed Systems (FORTE 2005), volume 3731 of Lecture Notes in Computer Science, pages 174–188. Springer-Verlag, 2005.



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Irina A. Lomazova. Nested Petri nets – a formalism for specification of multi-agent distributed systems. Fundamenta Informaticae, 43(1-4):195–214, 2000. Irina A. Lomazova, Kees M. van Hee, Olivia Oanea, Alexander Serebrenik, Natalia Sidorova, and Marc Voorhoeve. Nested nets for adaptive systems. In Application and Theory of Petri Nets and Other Models of Concurrency, Lecture Notes in Computer Science, pages 241–260. Springer-Verlag, 2006. R¨ udiger Valk. Petri nets as token objects: An introduction to elementary object nets. In J¨ org Desel and Manuel Silva, editors, Application and Theory of Petri Nets, volume 1420 of Lecture Notes in Computer Science, pages 1–25, 1998. R¨ udiger Valk. Object Petri nets: Using the nets-within-nets paradigm. In J¨ org Desel, Wolfgang Reisig, and Grzegorz Rozenberg, editors, Advanced Course on Petri Nets 2003, volume 3098 of Lecture Notes in Computer Science, pages 819–848. Springer-Verlag, 2003. Fernando Rosa Velardo and David de Frutos-Escrig. Name creation vs. replication in petri net systems. Fundam. Inform., 88(3):329–356, 2008.



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Hornets: Nets within Nets combined with Net Algebra

Hornets: Nets within Nets combined with Net Algebra

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