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Elementary Hornets: Nets within Nets
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Elementary Object Nets (Valk, 1998, 2003)
Nested nets (Lomazova, 2000)
Object Nets (K¨ohler und R¨olke, 2004; K¨ohler, 2007)
Hypernets (Bednarczyk u. a., 2004)
Extended Elementary Object System (Ma u. a., 2004)
Mobile systems (Lakos, 2005)
AHO systems (Hoﬀmann u. a., 2005)
Adaptive workﬂow nets (Lomazova u. a., 2006)
RN systems (Velardo und de FrutosEscrig, 2008)
Hornets (K¨ohlerBußmeier, 2009, 2014)
Michael K¨ohlerBußmeier (HAW) Elementary Hornets 3 / 23
Hornets (K¨ohlerBußmeier, 2009)
Algebraic operations on the nettokens, e.g. parallel composition: (N1kN2)
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Michael K¨ohlerBußmeier (HAW) Elementary Hornets 6 / 23
Hornets (K¨ohlerBußmeier, 2009)
Algebraic operations on the nettokens, e.g. parallel composition: (N1kN2)
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How to embed tokens from N1and N2into N3= (N1kN2)?
→All nets of kind kare deﬁned over a common superset Pk.
Michael K¨ohlerBußmeier (HAW) Elementary Hornets 6 / 23
Example: ManySorted Predicate Logic
Manysorted predicate logic Γ = (Σ,E,Ψ).
Type WorkflowNet
Operators in Σ: ·, +, k, etc.
Axioms in E:N1+N2=N2+N1etc.
Predicates in Ψ: isWellFormed(N) etc.
Type PTNet
Operators: p1, . . .,t1, . . .,∪,∩, etc.
Axioms: N1∪N2=N2∪N1etc.
Predicates: isBounded(N),isLive(N), etc.
Type ProcessesOfSafeNets
Operators: ; and ⊕
Axioms: (A;C)⊕(B;D)=(A⊕B); (C⊕D), etc.
Predicates: isClosedPrefix(N) etc.
Type BoxCalculusNets
.......
Michael K¨ohlerBußmeier (HAW) Elementary Hornets 7 / 23
Elementary Hornets (eHornets)
Now: Elementary Hornets.
eHornets are deﬁned analogously to eos.
Deﬁnition
An elementary Hornet is a tuple EH = ( b
N,U,I,d,l, µ0) such that:
1b
Nis an algebraic net, called the systemnet.
2(U,I) is a nettheory for the logic Γ.
3d:b
P→Kis the typing of the systemnet places.
4l= (b
l,lN)N∈U is the synchronisation labelling.
5µ0∈ M is the initial marking.
Remember: All nets in Ukuse places from a common superset Pk.
For elementary Hornets we consider ﬁnite place sets Pkonly.
Michael K¨ohlerBußmeier (HAW) Elementary Hornets 8 / 23
Elementary Hornets (eHornets)
Now: Elementary Hornets.
eHornets are deﬁned analogously to eos.
Deﬁnition
An elementary Hornet is a tuple EH = ( b
N,U,I,d,l, µ0) such that:
1b
Nis an algebraic net, called the systemnet.
2(U,I) is a nettheory for the logic Γ.
3d:b
P→Kis the typing of the systemnet places.
4l= (b
l,lN)N∈U is the synchronisation labelling.
5µ0∈ M is the initial marking.
Remember: All nets in Ukuse places from a common superset Pk.
For elementary Hornets we consider ﬁnite place sets Pkonly.
Michael K¨ohlerBußmeier (HAW) Elementary Hornets 8 / 23
Elementary Hornets (eHornets)
Now: Elementary Hornets.
eHornets are deﬁned analogously to eos.
Deﬁnition
An elementary Hornet is a tuple EH = ( b
N,U,I,d,l, µ0) such that:
1b
Nis an algebraic net, called the systemnet.
2(U,I) is a nettheory for the logic Γ.
3d:b
P→Kis the typing of the systemnet places.
4l= (b
l,lN)N∈U is the synchronisation labelling.
5µ0∈ M is the initial marking.
Remember: All nets in Ukuse places from a common superset Pk.
For elementary Hornets we consider ﬁnite place sets Pkonly.
Michael K¨ohlerBußmeier (HAW) Elementary Hornets 8 / 23
Elementary Hornets (eHornets)
Now: Elementary Hornets.
eHornets are deﬁned analogously to eos.
Deﬁnition
An elementary Hornet is a tuple EH = ( b
N,U,I,d,l, µ0) such that:
1b
Nis an algebraic net, called the systemnet.
2(U,I) is a nettheory for the logic Γ.
3d:b
P→Kis the typing of the systemnet places.
4l= (b
l,lN)N∈U is the synchronisation labelling.
5µ0∈ M is the initial marking.
Remember: All nets in Ukuse places from a common superset Pk.
For elementary Hornets we consider ﬁnite place sets Pkonly.
Michael K¨ohlerBußmeier (HAW) Elementary Hornets 8 / 23
Finite superset Pk=⇒ﬁnite net universe Uk
Lemma
The number of objectnets is bounded: Uk ≤ 2(24Pk)for each type k∈K.
Sketch.
Remember:
1Pkis ﬁxed, ﬁnite set (for each k∈K).
2Each possible objectnet transition tis deﬁned by some subset of Pkfor the
preset •tand another subset for the postset t•.
3The number of synchronisation labels is ﬁnite.
4Each objectnet Nis characterised by a subset of labelled transitions.
=⇒Finite number of objectnets.
Michael K¨ohlerBußmeier (HAW) Elementary Hornets 9 / 23
Finite superset Pk=⇒ﬁnite net universe Uk
Lemma
The number of objectnets is bounded: Uk ≤ 2(24Pk)for each type k∈K.
Sketch.
Remember:
1Pkis ﬁxed, ﬁnite set (for each k∈K).
2Each possible objectnet transition tis deﬁned by some subset of Pkfor the
preset •tand another subset for the postset t•.
3The number of synchronisation labels is ﬁnite.
4Each objectnet Nis characterised by a subset of labelled transitions.
=⇒Finite number of objectnets.
Michael K¨ohlerBußmeier (HAW) Elementary Hornets 9 / 23
Finite superset Pk=⇒ﬁnite net universe Uk
Lemma
The number of objectnets is bounded: Uk ≤ 2(24Pk)for each type k∈K.
Sketch.
Remember:
1Pkis ﬁxed, ﬁnite set (for each k∈K).
2Each possible objectnet transition tis deﬁned by some subset of Pkfor the
preset •tand another subset for the postset t•.
3The number of synchronisation labels is ﬁnite.
4Each objectnet Nis characterised by a subset of labelled transitions.
=⇒Finite number of objectnets.
Michael K¨ohlerBußmeier (HAW) Elementary Hornets 9 / 23
Safeness: P/T Nets, eos,eHornets
Deﬁnition (Safe P/T Net)
A p/t net is called safe if m(p)≤1 for all places pand all reachable
markings m.
A safe p/t net has at most 2nreachable markings, where n:= P.
Deﬁnition (Safe eos, safe eHornet)
An eos (an eHornet) is safe iﬀ for all reachable markings µwe have:
(i) There is at most one nettoken on each systemnet place b
pand
(ii) each nettoken is safe.
Theorem (K¨ohlerBußmeier und Heitmann, 2010)
The reachability set of a safe eos is ﬁnite: There are at most 2O(n2)
diﬀerent markings, where n:= max{ b
P,PN1,...,PNk}.
Michael K¨ohlerBußmeier (HAW) Elementary Hornets 11 / 23
Safeness: P/T Nets, eos,eHornets
Deﬁnition (Safe P/T Net)
A p/t net is called safe if m(p)≤1 for all places pand all reachable
markings m.
A safe p/t net has at most 2nreachable markings, where n:= P.
Deﬁnition (Safe eos, safe eHornet)
An eos (an eHornet) is safe iﬀ for all reachable markings µwe have:
(i) There is at most one nettoken on each systemnet place b
pand
(ii) each nettoken is safe.
Theorem (K¨ohlerBußmeier und Heitmann, 2010)
The reachability set of a safe eos is ﬁnite: There are at most 2O(n2)
diﬀerent markings, where n:= max{ b
P,PN1,...,PNk}.
Michael K¨ohlerBußmeier (HAW) Elementary Hornets 11 / 23
Safeness: P/T Nets, eos,eHornets
Deﬁnition (Safe P/T Net)
A p/t net is called safe if m(p)≤1 for all places pand all reachable
markings m.
A safe p/t net has at most 2nreachable markings, where n:= P.
Deﬁnition (Safe eos, safe eHornet)
An eos (an eHornet) is safe iﬀ for all reachable markings µwe have:
(i) There is at most one nettoken on each systemnet place b
pand
(ii) each nettoken is safe.
Theorem (K¨ohlerBußmeier und Heitmann, 2010)
The reachability set of a safe eos is ﬁnite: There are at most 2O(n2)
diﬀerent markings, where n:= max{ b
P,PN1,...,PNk}.
Michael K¨ohlerBußmeier (HAW) Elementary Hornets 11 / 23
eHornets: Number of Reachable Markings
Lemma
A safe eHornet has a ﬁnite reachability set.
Proof.
Let mbe the maximal cardinality of all place supersets: m:= max{Pk:k∈K}.
Each reachable marking is of the form µ=Pn
i=1 b
pi[Ni,Mi].
Since each nettoken [Ni,Mi] is safe, we have at most 2Pk≤2mdiﬀerent
markings Mi.
We have at most Uk ≤ 2(24Pk)≤2(24m)diﬀerent objectnets Ni.
Each systemnet place b
pis either unmarked or
marked with one of these 2(24m)·2mnettokens.
Therefore, we have at most 1+2(24m)·2m
b
Pdiﬀerent markings.
Michael K¨ohlerBußmeier (HAW) Elementary Hornets 12 / 23
eHornets: Number of Reachable Markings
Lemma
A safe eHornet has a ﬁnite reachability set.
Proof.
Let mbe the maximal cardinality of all place supersets: m:= max{Pk:k∈K}.
Each reachable marking is of the form µ=Pn
i=1 b
pi[Ni,Mi].
Since each nettoken [Ni,Mi] is safe, we have at most 2Pk≤2mdiﬀerent
markings Mi.
We have at most Uk ≤ 2(24Pk)≤2(24m)diﬀerent objectnets Ni.
Each systemnet place b
pis either unmarked or
marked with one of these 2(24m)·2mnettokens.
Therefore, we have at most 1+2(24m)·2m
b
Pdiﬀerent markings.
Michael K¨ohlerBußmeier (HAW) Elementary Hornets 12 / 23
eHornets: Number of Reachable Markings
Lemma
A safe eHornet has a ﬁnite reachability set.
Proof.
Let mbe the maximal cardinality of all place supersets: m:= max{Pk:k∈K}.
Each reachable marking is of the form µ=Pn
i=1 b
pi[Ni,Mi].
Since each nettoken [Ni,Mi] is safe, we have at most 2Pk≤2mdiﬀerent
markings Mi.
We have at most Uk ≤ 2(24Pk)≤2(24m)diﬀerent objectnets Ni.
Each systemnet place b
pis either unmarked or
marked with one of these 2(24m)·2mnettokens.
Therefore, we have at most 1+2(24m)·2m
b
Pdiﬀerent markings.
Michael K¨ohlerBußmeier (HAW) Elementary Hornets 12 / 23
eHornets: Number of Reachable Markings
Lemma
A safe eHornet has a ﬁnite reachability set.
Proof.
Let mbe the maximal cardinality of all place supersets: m:= max{Pk:k∈K}.
Each reachable marking is of the form µ=Pn
i=1 b
pi[Ni,Mi].
Since each nettoken [Ni,Mi] is safe, we have at most 2Pk≤2mdiﬀerent
markings Mi.
We have at most Uk ≤ 2(24Pk)≤2(24m)diﬀerent objectnets Ni.
Each systemnet place b
pis either unmarked or
marked with one of these 2(24m)·2mnettokens.
Therefore, we have at most 1+2(24m)·2m
b
Pdiﬀerent markings.
Michael K¨ohlerBußmeier (HAW) Elementary Hornets 12 / 23
Reachability for safe eHornets
Theorem (A lower bound known from p/t nets)
The reachability problem for safe p/t nets is PSpacecomplete.
The number of reachable markings for a safe eos is in 2O(n2).
Theorem (K¨ohlerBußmeier und Heitmann, 2010)
For safe eos the reachability problem is PSpacecomplete.
The number of reachable markings for a safe eHornets is in 2(2O(m)).
In analogy to Lipton’s famous result we obtain a lower bound:
Theorem (K¨ohlerBußmeier, 2014)
The reachability problem for safe eHornets requires exponential space
(Exponential space is suﬃcient, K¨ohlerBußmeier und Heitmann, 2015).
Michael K¨ohlerBußmeier (HAW) Elementary Hornets 14 / 23
Reachability for safe eHornets
Theorem (A lower bound known from p/t nets)
The reachability problem for safe p/t nets is PSpacecomplete.
The number of reachable markings for a safe eos is in 2O(n2).
Theorem (K¨ohlerBußmeier und Heitmann, 2010)
For safe eos the reachability problem is PSpacecomplete.
The number of reachable markings for a safe eHornets is in 2(2O(m)).
In analogy to Lipton’s famous result we obtain a lower bound:
Theorem (K¨ohlerBußmeier, 2014)
The reachability problem for safe eHornets requires exponential space
(Exponential space is suﬃcient, K¨ohlerBußmeier und Heitmann, 2015).
Michael K¨ohlerBußmeier (HAW) Elementary Hornets 14 / 23
Reachability for safe eHornets
Theorem (A lower bound known from p/t nets)
The reachability problem for safe p/t nets is PSpacecomplete.
The number of reachable markings for a safe eos is in 2O(n2).
Theorem (K¨ohlerBußmeier und Heitmann, 2010)
For safe eos the reachability problem is PSpacecomplete.
The number of reachable markings for a safe eHornets is in 2(2O(m)).
In analogy to Lipton’s famous result we obtain a lower bound:
Theorem (K¨ohlerBußmeier, 2014)
The reachability problem for safe eHornets requires exponential space
(Exponential space is suﬃcient, K¨ohlerBußmeier und Heitmann, 2015).
Michael K¨ohlerBußmeier (HAW) Elementary Hornets 14 / 23
Reachability for safe eHornets
Theorem (A lower bound known from p/t nets)
The reachability problem for safe p/t nets is PSpacecomplete.
The number of reachable markings for a safe eos is in 2O(n2).
Theorem (K¨ohlerBußmeier und Heitmann, 2010)
For safe eos the reachability problem is PSpacecomplete.
The number of reachable markings for a safe eHornets is in 2(2O(m)).
In analogy to Lipton’s famous result we obtain a lower bound:
Theorem (K¨ohlerBußmeier, 2014)
The reachability problem for safe eHornets requires exponential space
(Exponential space is suﬃcient, K¨ohlerBußmeier und Heitmann, 2015).
Michael K¨ohlerBußmeier (HAW) Elementary Hornets 14 / 23
Source of Complexity: Universe of Object Nets
Remember: A safe eHornet has at most 1+2(24m)·2m
b
Preachable
markings.
Main Observation
The number of possible object nets dominates: U(m)=2(24m).
Approach of the Paper
Restrict the structure of object nets to improve complexity.
With 2Poly(m)object nets =⇒Reachability PSpacecomplete.
Candidates
StateMachines, FreeChoice, acyclic Nets, etc.
Michael K¨ohlerBußmeier (HAW) Elementary Hornets 16 / 23
Source of Complexity: Universe of Object Nets
Remember: A safe eHornet has at most 1+2(24m)·2m
b
Preachable
markings.
Main Observation
The number of possible object nets dominates: U(m)=2(24m).
Approach of the Paper
Restrict the structure of object nets to improve complexity.
With 2Poly(m)object nets =⇒Reachability PSpacecomplete.
Candidates
StateMachines, FreeChoice, acyclic Nets, etc.
Michael K¨ohlerBußmeier (HAW) Elementary Hornets 16 / 23
Source of Complexity: Universe of Object Nets
Remember: A safe eHornet has at most 1+2(24m)·2m
b
Preachable
markings.
Main Observation
The number of possible object nets dominates: U(m)=2(24m).
Approach of the Paper
Restrict the structure of object nets to improve complexity.
With 2Poly(m)object nets =⇒Reachability PSpacecomplete.
Candidates
StateMachines, FreeChoice, acyclic Nets, etc.
Michael K¨ohlerBußmeier (HAW) Elementary Hornets 16 / 23
Source of Complexity: Universe of Object Nets
Remember: A safe eHornet has at most 1+2(24m)·2m
b
Preachable
markings.
Main Observation
The number of possible object nets dominates: U(m)=2(24m).
Approach of the Paper
Restrict the structure of object nets to improve complexity.
With 2Poly(m)object nets =⇒Reachability PSpacecomplete.
Candidates
StateMachines, FreeChoice, acyclic Nets, etc.
Michael K¨ohlerBußmeier (HAW) Elementary Hornets 16 / 23
Source of Complexity: Universe of Object Nets
Remember: A safe eHornet has at most 1+2(24m)·2m
b
Preachable
markings.
Main Observation
The number of possible object nets dominates: U(m)=2(24m).
Approach of the Paper
Restrict the structure of object nets to improve complexity.
With 2Poly(m)object nets =⇒Reachability PSpacecomplete.
Candidates
StateMachines, FreeChoice, acyclic Nets, etc.
Michael K¨ohlerBußmeier (HAW) Elementary Hornets 16 / 23
StateMachines as ObjectNets
Deﬁnition
eSmHornet := a Hornet with statemachines as objectnets:
∀N∈ Uk:∀t∈TN:•t ≤ 1∧ t• ≤ 1
Proposition
For a safe eSmHornet the number of object nets is bounded:
Uk ≤ 2O(Pk4)
In the general case of safe Hornets the number was doubleexponential:
Uk ≤ 2(24Pk)
Michael K¨ohlerBußmeier (HAW) Elementary Hornets 17 / 23
StateMachines as ObjectNets
Deﬁnition
eSmHornet := a Hornet with statemachines as objectnets:
∀N∈ Uk:∀t∈TN:•t ≤ 1∧ t• ≤ 1
Proposition
For a safe eSmHornet the number of object nets is bounded:
Uk ≤ 2O(Pk4)
In the general case of safe Hornets the number was doubleexponential:
Uk ≤ 2(24Pk)
Michael K¨ohlerBußmeier (HAW) Elementary Hornets 17 / 23
Proof.
Each object net Nin Ukis characterised by its set of labelled transitions.
To each transition tassign a channel c∈Ckor ⊥k:
LT k=Tk×(Ck∪ {⊥k})
For StateMachines the number of possible transitions is:
Tk=1 + Pk
1
 {z }
preset
·1 + Pk
1
 {z }
postset
≤(1 + Pk)2≤const · Pk2
We cannot use more channels than transitions, i.e. Ck∪ {⊥k} ≤ Tk.
LT k=Tk · (Ck+ 1) ≤ Tk·Tk=Tk2
. . . ≤const · Pk22=const · Pk4
Therefore, we have at most 2O(Pk4)diﬀerent object nets. qed.
Michael K¨ohlerBußmeier (HAW) Elementary Hornets 18 / 23
Proof.
Each object net Nin Ukis characterised by its set of labelled transitions.
To each transition tassign a channel c∈Ckor ⊥k:
LT k=Tk×(Ck∪ {⊥k})
For StateMachines the number of possible transitions is:
Tk=1 + Pk
1
 {z }
preset
·1 + Pk
1
 {z }
postset
≤(1 + Pk)2≤const · Pk2
We cannot use more channels than transitions, i.e. Ck∪ {⊥k} ≤ Tk.
LT k=Tk · (Ck+ 1) ≤ Tk·Tk=Tk2
. . . ≤const · Pk22=const · Pk4
Therefore, we have at most 2O(Pk4)diﬀerent object nets. qed.
Michael K¨ohlerBußmeier (HAW) Elementary Hornets 18 / 23
Proof.
Each object net Nin Ukis characterised by its set of labelled transitions.
To each transition tassign a channel c∈Ckor ⊥k:
LT k=Tk×(Ck∪ {⊥k})
For StateMachines the number of possible transitions is:
Tk=1 + Pk
1
 {z }
preset
·1 + Pk
1
 {z }
postset
≤(1 + Pk)2≤const · Pk2
We cannot use more channels than transitions, i.e. Ck∪ {⊥k} ≤ Tk.
LT k=Tk · (Ck+ 1) ≤ Tk·Tk=Tk2
. . . ≤const · Pk22=const · Pk4
Therefore, we have at most 2O(Pk4)diﬀerent object nets. qed.
Michael K¨ohlerBußmeier (HAW) Elementary Hornets 18 / 23
Proof.
Each object net Nin Ukis characterised by its set of labelled transitions.
To each transition tassign a channel c∈Ckor ⊥k:
LT k=Tk×(Ck∪ {⊥k})
For StateMachines the number of possible transitions is:
Tk=1 + Pk
1
 {z }
preset
·1 + Pk
1
 {z }
postset
≤(1 + Pk)2≤const · Pk2
We cannot use more channels than transitions, i.e. Ck∪ {⊥k} ≤ Tk.
LT k=Tk · (Ck+ 1) ≤ Tk·Tk=Tk2
. . . ≤const · Pk22=const · Pk4
Therefore, we have at most 2O(Pk4)diﬀerent object nets. qed.
Michael K¨ohlerBußmeier (HAW) Elementary Hornets 18 / 23
Proof.
Each object net Nin Ukis characterised by its set of labelled transitions.
To each transition tassign a channel c∈Ckor ⊥k:
LT k=Tk×(Ck∪ {⊥k})
For StateMachines the number of possible transitions is:
Tk=1 + Pk
1
 {z }
preset
·1 + Pk
1
 {z }
postset
≤(1 + Pk)2≤const · Pk2
We cannot use more channels than transitions, i.e. Ck∪ {⊥k} ≤ Tk.
LT k=Tk · (Ck+ 1) ≤ Tk·Tk=Tk2
. . . ≤const · Pk22=const · Pk4
Therefore, we have at most 2O(Pk4)diﬀerent object nets. qed.
Michael K¨ohlerBußmeier (HAW) Elementary Hornets 18 / 23
Proof.
Each object net Nin Ukis characterised by its set of labelled transitions.
To each transition tassign a channel c∈Ckor ⊥k:
LT k=Tk×(Ck∪ {⊥k})
For StateMachines the number of possible transitions is:
Tk=1 + Pk
1
 {z }
preset
·1 + Pk
1
 {z }
postset
≤(1 + Pk)2≤const · Pk2
We cannot use more channels than transitions, i.e. Ck∪ {⊥k} ≤ Tk.
LT k=Tk · (Ck+ 1) ≤ Tk·Tk=Tk2
. . . ≤const · Pk22=const · Pk4
Therefore, we have at most 2O(Pk4)diﬀerent object nets. qed.
Michael K¨ohlerBußmeier (HAW) Elementary Hornets 18 / 23
The Reachability Set of a safe eSmHornet (1)
Proposition
A safe eSmHornet has at most 2O(n5)reachable markings, where nis
the maximum of all Pkand b
P.
Let mdenote the maximum of all Pk.
Proof: Each reachable marking is of the form µ=Pl
i=1 b
pi[Ni,Mi].
Nettokens are safe: At most 2Pk≤2mdiﬀerent markings Mi.
Let U(m)be the number of object nets N.
System net is safe: Each system net place b
pis either unmarked or
marked with one of the U(m)·2mnettokens:
Number of reachable markings:
(1 + U(m)·2m)
b
P
Michael K¨ohlerBußmeier (HAW) Elementary Hornets 19 / 23
The Reachability Set of a safe eSmHornet (1)
Proposition
A safe eSmHornet has at most 2O(n5)reachable markings, where nis
the maximum of all Pkand b
P.
Let mdenote the maximum of all Pk.
Proof: Each reachable marking is of the form µ=Pl
i=1 b
pi[Ni,Mi].
Nettokens are safe: At most 2Pk≤2mdiﬀerent markings Mi.
Let U(m)be the number of object nets N.
System net is safe: Each system net place b
pis either unmarked or
marked with one of the U(m)·2mnettokens:
Number of reachable markings:
(1 + U(m)·2m)
b
P
Michael K¨ohlerBußmeier (HAW) Elementary Hornets 19 / 23
The Reachability Set of a safe eSmHornet (1)
Proposition
A safe eSmHornet has at most 2O(n5)reachable markings, where nis
the maximum of all Pkand b
P.
Let mdenote the maximum of all Pk.
Proof: Each reachable marking is of the form µ=Pl
i=1 b
pi[Ni,Mi].
Nettokens are safe: At most 2Pk≤2mdiﬀerent markings Mi.
Let U(m)be the number of object nets N.
System net is safe: Each system net place b
pis either unmarked or
marked with one of the U(m)·2mnettokens:
Number of reachable markings:
(1 + U(m)·2m)
b
P
Michael K¨ohlerBußmeier (HAW) Elementary Hornets 19 / 23
The Reachability Set of a safe eSmHornet (1)
Proposition
A safe eSmHornet has at most 2O(n5)reachable markings, where nis
the maximum of all Pkand b
P.
Let mdenote the maximum of all Pk.
Proof: Each reachable marking is of the form µ=Pl
i=1 b
pi[Ni,Mi].
Nettokens are safe: At most 2Pk≤2mdiﬀerent markings Mi.
Let U(m)be the number of object nets N.
System net is safe: Each system net place b
pis either unmarked or
marked with one of the U(m)·2mnettokens:
Number of reachable markings:
(1 + U(m)·2m)
b
P
Michael K¨ohlerBußmeier (HAW) Elementary Hornets 19 / 23
The Reachability Set of a safe eSmHornet (1)
Proposition
A safe eSmHornet has at most 2O(n5)reachable markings, where nis
the maximum of all Pkand b
P.
Let mdenote the maximum of all Pk.
Proof: Each reachable marking is of the form µ=Pl
i=1 b
pi[Ni,Mi].
Nettokens are safe: At most 2Pk≤2mdiﬀerent markings Mi.
Let U(m)be the number of object nets N.
System net is safe: Each system net place b
pis either unmarked or
marked with one of the U(m)·2mnettokens:
Number of reachable markings:
(1 + U(m)·2m)
b
P
Michael K¨ohlerBußmeier (HAW) Elementary Hornets 19 / 23
The Reachability Set of a safe eSmHornet (1)
Proposition
A safe eSmHornet has at most 2O(n5)reachable markings, where nis
the maximum of all Pkand b
P.
Let mdenote the maximum of all Pk.
Proof: Each reachable marking is of the form µ=Pl
i=1 b
pi[Ni,Mi].
Nettokens are safe: At most 2Pk≤2mdiﬀerent markings Mi.
Let U(m)be the number of object nets N.
System net is safe: Each system net place b
pis either unmarked or
marked with one of the U(m)·2mnettokens:
Number of reachable markings:
(1 + U(m)·2m)
b
P
Michael K¨ohlerBußmeier (HAW) Elementary Hornets 19 / 23
The Reachability Set of a safe eSmHornet (2)
The number of diﬀerent markings in the safe eSmHornet is:
(1 + U(m)·2m)
b
P≤1+2(const·m4)·2m
b
P
=1+2(const·m4+m)
b
P
≤2(const·m4)
b
P
= 2(const·m4·
b
P)
Since ndenotes the maximum of mand b
Pwe obtain:
2(const·m4·
b
P)≤2(const·n4·n)= 2(const ·n5)
The number of reachable markings is in 2O(n5), i.e. in 2O(Poly(n)) qed.
Michael K¨ohlerBußmeier (HAW) Elementary Hornets 20 / 23
The Reachability Set of a safe eSmHornet (2)
The number of diﬀerent markings in the safe eSmHornet is:
(1 + U(m)·2m)
b
P≤1+2(const·m4)·2m
b
P
=1+2(const·m4+m)
b
P
≤2(const·m4)
b
P
= 2(const·m4·
b
P)
Since ndenotes the maximum of mand b
Pwe obtain:
2(const·m4·
b
P)≤2(const·n4·n)= 2(const ·n5)
The number of reachable markings is in 2O(n5), i.e. in 2O(Poly(n)) qed.
Michael K¨ohlerBußmeier (HAW) Elementary Hornets 20 / 23
The Reachability Set of a safe eSmHornet (2)
The number of diﬀerent markings in the safe eSmHornet is:
(1 + U(m)·2m)
b
P≤1+2(const·m4)·2m
b
P
=1+2(const·m4+m)
b
P
≤2(const·m4)
b
P
= 2(const·m4·
b
P)
Since ndenotes the maximum of mand b
Pwe obtain:
2(const·m4·
b
P)≤2(const·n4·n)= 2(const ·n5)
The number of reachable markings is in 2O(n5), i.e. in 2O(Poly(n)) qed.
Michael K¨ohlerBußmeier (HAW) Elementary Hornets 20 / 23
eSmHornets: Complexity of the Reachability Problem
Lemma
For safe eSmHornets there exists a nondeterministic algorithm that
decides the reachability problem within polynomial space:
Reach ∈NSpace On5
where nis the maximum of all Pkand b
P.
Proof.
(Sketch) We know that we have at most 2O(n5)diﬀerent markings.
It is suﬃcient to guess a ﬁring sequence of this length, which needs at
most On5bits.
Michael K¨ohlerBußmeier (HAW) Elementary Hornets 21 / 23
eSmHornets: Complexity of the Reachability Problem
Lemma
For safe eSmHornets there exists a nondeterministic algorithm that
decides the reachability problem within polynomial space:
Reach ∈NSpace On5
where nis the maximum of all Pkand b
P.
Proof.
(Sketch) We know that we have at most 2O(n5)diﬀerent markings.
It is suﬃcient to guess a ﬁring sequence of this length, which needs at
most On5bits.
Michael K¨ohlerBußmeier (HAW) Elementary Hornets 21 / 23
eSmHornets: Complexity of the Reachability Problem
Corollary
The reachability problem for safe eSmHornets can be solved within
polynomial space.
Proof.
Use Savitch’s technique.
Corollary
The reachability problem for safe eSmHornets is PSpacecomplete.
Proof.
It is in PSpace.
It already is PSpacecomplete for safe P/T nets, which is a subclass
of safe eSmHornets.
Michael K¨ohlerBußmeier (HAW) Elementary Hornets 22 / 23
eSmHornets: Complexity of the Reachability Problem
Corollary
The reachability problem for safe eSmHornets can be solved within
polynomial space.
Proof.
Use Savitch’s technique.
Corollary
The reachability problem for safe eSmHornets is PSpacecomplete.
Proof.
It is in PSpace.
It already is PSpacecomplete for safe P/T nets, which is a subclass
of safe eSmHornets.
Michael K¨ohlerBußmeier (HAW) Elementary Hornets 22 / 23
eSmHornets: Complexity of the Reachability Problem
Corollary
The reachability problem for safe eSmHornets can be solved within
polynomial space.
Proof.
Use Savitch’s technique.
Corollary
The reachability problem for safe eSmHornets is PSpacecomplete.
Proof.
It is in PSpace.
It already is PSpacecomplete for safe P/T nets, which is a subclass
of safe eSmHornets.
Michael K¨ohlerBußmeier (HAW) Elementary Hornets 22 / 23
eSmHornets: Complexity of the Reachability Problem
Corollary
The reachability problem for safe eSmHornets can be solved within
polynomial space.
Proof.
Use Savitch’s technique.
Corollary
The reachability problem for safe eSmHornets is PSpacecomplete.
Proof.
It is in PSpace.
It already is PSpacecomplete for safe P/T nets, which is a subclass
of safe eSmHornets.
Michael K¨ohlerBußmeier (HAW) Elementary Hornets 22 / 23
Summary
Hornets: algebraic operations on nested objectnets
Safe eHornets:
Safe eHornets have ﬁnite state spaces.
Reachability requires exponential space.
Structural Restrictions of Object Nets:
StateMachines as Object Nets: eSmHornets
The reachability problem is PSpacecomplete for safe eSmHornets.
Generalisation deﬁned in the paper:
Generalisation of StateMachines: fanbounded Hornets.
The reachability problem is still PSpacecomplete for safe,
fanbounded eHornets.
Michael K¨ohlerBußmeier (HAW) Elementary Hornets 23 / 23
Summary
Hornets: algebraic operations on nested objectnets
Safe eHornets:
Safe eHornets have ﬁnite state spaces.
Reachability requires exponential space.
Structural Restrictions of Object Nets:
StateMachines as Object Nets: eSmHornets
The reachability problem is PSpacecomplete for safe eSmHornets.
Generalisation deﬁned in the paper:
Generalisation of StateMachines: fanbounded Hornets.
The reachability problem is still PSpacecomplete for safe,
fanbounded eHornets.
Michael K¨ohlerBußmeier (HAW) Elementary Hornets 23 / 23
Summary
Hornets: algebraic operations on nested objectnets
Safe eHornets:
Safe eHornets have ﬁnite state spaces.
Reachability requires exponential space.
Structural Restrictions of Object Nets:
StateMachines as Object Nets: eSmHornets
The reachability problem is PSpacecomplete for safe eSmHornets.
Generalisation deﬁned in the paper:
Generalisation of StateMachines: fanbounded Hornets.
The reachability problem is still PSpacecomplete for safe,
fanbounded eHornets.
Michael K¨ohlerBußmeier (HAW) Elementary Hornets 23 / 23
Summary
Hornets: algebraic operations on nested objectnets
Safe eHornets:
Safe eHornets have ﬁnite state spaces.
Reachability requires exponential space.
Structural Restrictions of Object Nets:
StateMachines as Object Nets: eSmHornets
The reachability problem is PSpacecomplete for safe eSmHornets.
Generalisation deﬁned in the paper:
Generalisation of StateMachines: fanbounded Hornets.
The reachability problem is still PSpacecomplete for safe,
fanbounded eHornets.
Michael K¨ohlerBußmeier (HAW) Elementary Hornets 23 / 23
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