Transformation of Petri Nets into Context-Dependent Fusion Grammars - - PowerPoint PPT Presentation

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Transformation of Petri Nets into Context-Dependent Fusion Grammars

Hans-J¨

• rg Kreowski 1, Sabine Kuske 1 and Aaron Lye 2

University of Bremen

1 Department of Computer Science, 2 Department of Mathematics

P.O.Box 33 04 40, 28334 Bremen, Germany {kreo,kuske,lye}@informatik.uni-bremen.de

27.03.2019 13th International Conference on Language and Automata Theory and Applications (LATA)

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Motivation

We introduced fusion grammars generating hypergraphs at ICGT17. Formal framework for fusion processes in: ◮ DNA computing ◮ chemistry ◮ tiling ◮ fractal geometry ◮ visual modeling ◮ etc.

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Hypergraph

We consider hypergraphs over Σ with hyperedges like vk1 . . . v1 A wk2 . . . w1 k1 1 k2 1 where v1 · · · vk1 is a sequence of source nodes w1 · · · wk2 is a sequence of target nodes A ∈ Σ is a label. The class of all hypergraphs over Σ is denoted by HΣ.

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Fusion rule

Let F ⊆ Σ be a fusion alphabet. Let type : F → N × N. Each A ∈ F has a complement A ∈ F where type(A) = type(A). fr(A) = vk1 . . . v1 v′

1 . . .

v′

k1

A A wk2 . . . w1 w′

1

. . . w′

k2

k1 1 k2 1 k1 1 k2 1 type(A) = (k1, k2) Examples: fr(t4) = t4 t4 type(t4) = (2, 1) fr(◦) = ◦

• type(◦) = (0, 1)

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Rule application

• 1. find a matching morphism g of fr(A) in the hypergraph H

H vk1 . . . v1 v′

1 . . .

v′

k1

A A wk2 . . . w1 w′

1

. . . w′

k2

k1 1 k2 1 k1 1 k2 1

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Rule application

• 1. find a matching morphism g of fr(A) in the hypergraph H
• 2. remove the images of the two hyperedges of fr(A)

I vk1 . . . v1 v′

1 . . .

v′

k1

wk2 . . . w1 w′

1

. . . w′

k2

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Rule application

• 1. find a matching morphism g of fr(A) in the hypergraph H
• 2. remove the images of the two hyperedges of fr(A)
• 3. identify corresponding source and target vertices of the

removed edges H′ vk1 = v′

k1

. . . v1 = v′

1

wk2 = w′

k2

. . . w1 = w′

1

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Rule application

• 1. find a matching morphism g of fr(A) in the hypergraph H
• 2. remove the images of the two hyperedges of fr(A)
• 3. identify corresponding source and target vertices of the

removed edges H′ vk1 = v′

k1

. . . v1 = v′

1

wk2 = w′

k2

. . . w1 = w′

1

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Rule application

• 1. find a matching morphism g of fr(A) in the hypergraph H
• 2. remove the images of the two hyperedges of fr(A)
• 3. identify corresponding source and target vertices of the

removed edges H′ vk1 = v′

k1

. . . v1 = v′

1

wk2 = w′

k2

. . . w1 = w′

1

Rule application is denoted by H = ⇒

fr(A) H′.

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Example

fr(t4) = t4 t4 H = t4

• t4

t4

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Example

fr(t4) = t4 t4 H = t4

• t4

t4

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Example

fr(t4) = t4 t4 H = t4

• t4

t4

• I =

v2 v1 w1

• v′

2

v′

1

w′

1

t4

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Example

fr(t4) = t4 t4 H = t4

• t4

t4

• I =

v2 v1 w1

• v′

2

v′

1

w′

1

t4

• H′ =

v2 = v′

2

v1 = v′

1

w1 = w′

1

t4

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Context-dependent fusion rule and its application

PC, NC: sets of morphisms with domain fr(A) (fr(A), PC, NC) applicable to hypergraph H via a matching morphism g : fr(A) → H if

• 1. ∀c ∈ PC

C fr(A) H = c g ∃h and h is injective on the set

• f hyperedges.
• 2. ∀c ∈ NC

C fr(A) H = c g ∃h

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Multiplication

Generalization of dublication used in DNA computing. Let C(H) be the set of all connected components of H. Define multiplicity m: C(H) → N. H = ⇒

m m · H =

• C∈C(H)

m(C) · C

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Multiplication

Generalization of dublication used in DNA computing. Let C(H) be the set of all connected components of H. Define multiplicity m: C(H) → N. H = ⇒

m m · H =

• C∈C(H)

m(C) · C Example: H =

t4

• t4

t4

• =

C1 + C2 = ⇒

m

2 · C1 + 3 · C2 for m(C1) = 2, m(C2) = 3 =

t4

• t4
• t4

t4

• t4

t4

• t4

t4

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Context-dependent fusion grammar CDFG = (Z, F, M, T, P)

◮ Z ∈ HF∪F∪M∪T finite start hypergraph F, M, T ⊆ Σ, fusion, marker, terminal alphabet (all finite) M ∩ (F ∪ F) = ∅, T ∩ (F ∪ F) = ∅ = T ∩ M P finite set of context-dependent fusion rules

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Context-dependent fusion grammar CDFG = (Z, F, M, T, P)

◮ Z ∈ HF∪F∪M∪T finite start hypergraph F, M, T ⊆ Σ, fusion, marker, terminal alphabet (all finite) M ∩ (F ∪ F) = ∅, T ∩ (F ∪ F) = ∅ = T ∩ M P finite set of context-dependent fusion rules ◮ A direct derivation is either H = ⇒

cdfr H′

for some cdfr ∈ P or H = ⇒

m m · H = C∈C(H)

m(C) · C for some multiplicity m: C(H) → N. ◮ Derivations are defined by the reflexive and transitive closure.

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Context-dependent fusion grammar CDFG = (Z, F, M, T, P)

◮ Z ∈ HF∪F∪M∪T finite start hypergraph F, M, T ⊆ Σ, fusion, marker, terminal alphabet (all finite) M ∩ (F ∪ F) = ∅, T ∩ (F ∪ F) = ∅ = T ∩ M P finite set of context-dependent fusion rules ◮ A direct derivation is either H = ⇒

cdfr H′

for some cdfr ∈ P or H = ⇒

m m · H = C∈C(H)

m(C) · C for some multiplicity m: C(H) → N. ◮ Derivations are defined by the reflexive and transitive closure. ◮ The generated language L(CDFG) = {remM(Y ) | Z

∗

= ⇒ H, Y ∈ C(H)∩(HT∪M−HT)}, where remM(Y ) removes all marker hyperedges from Y .

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Petri Net PN = (P, T, F, W, M0)

◮ disjoint finite sets P and T ◮ flow relation F ⊆ (P × T) ∪ (T × P) ◮ weight function W: F → N>0 ◮ marking M: P → N

p1 p2 p3 p4 p5 t1 t3 t2 t4 2

◮ •t and t• denote the set of pre- and post-places of t ∈ T.

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Petri Net PN = (P, T, F, W, M0)

◮ disjoint finite sets P and T ◮ flow relation F ⊆ (P × T) ∪ (T × P) ◮ weight function W: F → N>0 ◮ marking M: P → N

p1 p2 p3 p4 p5 t1 t3 t2 t4 2

◮ •t and t• denote the set of pre- and post-places of t ∈ T. ◮ t is enabled in M if M(p) ≥ W(p, t) for each p ∈ •t. ◮ M[tM′: in each p ∈ •t W(p, t) tokens are consumed M[tM′: and to each p ∈ t• W(t, p) tokens are added.

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Petri Net PN = (P, T, F, W, M0)

◮ disjoint finite sets P and T ◮ flow relation F ⊆ (P × T) ∪ (T × P) ◮ weight function W: F → N>0 ◮ marking M: P → N

p1 p2 p3 p4 p5 t1 t3 t2 t4 2

◮ •t and t• denote the set of pre- and post-places of t ∈ T. ◮ t is enabled in M if M(p) ≥ W(p, t) for each p ∈ •t. ◮ M[tM′: in each p ∈ •t W(p, t) tokens are consumed M[tM′: and to each p ∈ t• W(t, p) tokens are added. ◮ Reach(PN) = {M′′ | M0[∗M′′} is the set of markings reachable from M0.

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Transformation of Petri Nets into Context-Dependent Fusion Grammars

PN = (P, T, F, W, M0) ❀ CDFG(PN) = (ZPN, FPN, MPN, TPN, PPN)

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Transformation of Petri Nets into Context-Dependent Fusion Grammars

PN = (P, T, F, W, M0) ❀ CDFG(PN) = (ZPN, FPN, MPN, TPN, PPN) ◮ FPN = {◦} ∪ T, MPN = {µ}, TPN = {•} type(◦) = (0, 1), type(t) = (|•t|, |t•|) for each t ∈ T

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Transformation of Petri Nets into Context-Dependent Fusion Grammars

PN = (P, T, F, W, M0) ❀ CDFG(PN) = (ZPN, FPN, MPN, TPN, PPN) ◮ FPN = {◦} ∪ T, MPN = {µ}, TPN = {•} type(◦) = (0, 1), type(t) = (|•t|, |t•|) for each t ∈ T Both ◦ and • represent token, because F ∩ T = ∅

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Transformation of Petri Nets into Context-Dependent Fusion Grammars

PN = (P, T, F, W, M0) ❀ CDFG(PN) = (ZPN, FPN, MPN, TPN, PPN) ◮ FPN = {◦} ∪ T, MPN = {µ}, TPN = {•} type(◦) = (0, 1), type(t) = (|•t|, |t•|) for each t ∈ T Both ◦ and • represent token, because F ∩ T = ∅ ◮ ZPN = hg(P, T, M0) +

t∈T

Ct + C• +

t∈T

Dt

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Transformation of Petri Nets into Context-Dependent Fusion Grammars

PN = (P, T, F, W, M0) ❀ CDFG(PN) = (ZPN, FPN, MPN, TPN, PPN) ◮ FPN = {◦} ∪ T, MPN = {µ}, TPN = {•} type(◦) = (0, 1), type(t) = (|•t|, |t•|) for each t ∈ T Both ◦ and • represent token, because F ∩ T = ∅ ◮ ZPN = hg(P, T, M0) +

t∈T

Ct + C• +

t∈T

Dt

◮ hg(P, T, M0) represents the initial Petri net ◮ Ct is used for firing transitions t according to W

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Transformation of Petri Nets into Context-Dependent Fusion Grammars

PN = (P, T, F, W, M0) ❀ CDFG(PN) = (ZPN, FPN, MPN, TPN, PPN) ◮ FPN = {◦} ∪ T, MPN = {µ}, TPN = {•} type(◦) = (0, 1), type(t) = (|•t|, |t•|) for each t ∈ T Both ◦ and • represent token, because F ∩ T = ∅ ◮ ZPN = hg(P, T, M0) +

t∈T

Ct + C• +

t∈T

Dt

◮ hg(P, T, M0) represents the initial Petri net ◮ Ct is used for firing transitions t according to W

◮ PPN = {consume, replace} ∪ {fire(t), delete(t) | t ∈ T}

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Transformation of Petri Nets into Context-Dependent Fusion Grammars

PN = (P, T, F, W, M0) ❀ CDFG(PN) = (ZPN, FPN, MPN, TPN, PPN) ◮ FPN = {◦} ∪ T, MPN = {µ}, TPN = {•} type(◦) = (0, 1), type(t) = (|•t|, |t•|) for each t ∈ T Both ◦ and • represent token, because F ∩ T = ∅ ◮ ZPN = hg(P, T, M0) +

t∈T

Ct + C• +

t∈T

Dt

◮ hg(P, T, M0) represents the initial Petri net ◮ Ct is used for firing transitions t according to W

◮ PPN = {consume, replace} ∪ {fire(t), delete(t) | t ∈ T}

◮ consume is used for consuming tokens ◮ fire(t) is used for simulating a firing of t

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Example hg(P, T, M0)

p1 p2 p3 p4 p5 t1 t3 t2 t4 2

❀

µ 1 2 3 4 5

• ◦
• t1

t3 t2 t4

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Example hg(P, T, M0)

p1 p2 p3 p4 p5 t1 t3 t2 t4 2

❀

• µ
• 1

2 3 4 5 t1 t3 t2 t4

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Example ZPN

hg(P, T, M0) =

• µ
• 1

2 3 4 5 t1 t3 t2 t4

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Example ZPN

hg(P, T, M0) =

• µ
• 1

2 3 4 5 t1 t3 t2 t4

Ct1 = Ct2 = Ct3 = Ct4 =

• t1

t1

• t2

t2

• t3

t3

• t4

t4

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Example fire(t4) = (fr(t4), PC, NC)

where fr(t4) = t4 t4 Positive context according to W: PC = { fr(t4) → t4

• t4

} Negative context: NC = { fr(t4) → t4

• t4

, fr(t4) → t4

• t4

}

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consume = (fr(◦), PC, NC)

fr(◦) =

• PC =

{ fr(◦) → ◦

• }

NC = ∅

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Example derivation (fire and consume)

• µ
• 1

2 3 4 5 t1 t3 t2 t4

• t4

t4

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Example derivation (fire and consume)

• µ
• 1

2 3 4 5 t1 t3 t2 t4

• t4

t4

= ⇒

fire(t4)

• µ
• t1

t3 t2 t4

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Example derivation (fire and consume)

• µ
• 1

2 3 4 5 t1 t3 t2 t4

• t4

t4

= ⇒

fire(t4)

• µ
• t1

t3 t2 t4

2

= ⇒

consume

• µ

t1 t3 t2 t4

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Derivation schema

Z = ⇒

m hg(P, T, M0) + C• +

• t∈T

m(Ct) · Ct +

• t∈T

Dt = ⇒

fire(t1) X1 b1

= ⇒

consume hg(P, T, M1) + C• +

• t∈T

mt2...tk(Ct) · Ct +

• t∈T

Dt = ⇒

fire(t2) X2 b2

= ⇒

consume . . .

= ⇒

fire(tn) Xn bn

= ⇒

consume hg(P, T, Mn) + C• +

• t∈T

Ct +

• t∈T

Dt

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Transformation of Petri Nets into Context-Dependent Fusion Grammars

PN = (P, T, F, W, M0) ❀ CDFG(PN) = (ZPN, FPN, MPN, TPN, PPN) ◮ FPN = {◦} ∪ T, MPN = {µ}, TPN = {•} type(◦) = (0, 1), type(t) = (|•t|, |t•|) for each t ∈ T Both ◦ and • represent token, because F ∩ T = ∅ ◮ ZPN = hg(P, T, M0) +

t∈T

Ct + C• +

t∈T

Dt

◮ hg(P, T, M0) represents the initial Petri net ◮ Ct is used for firing transitions t according to W

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Transformation of Petri Nets into Context-Dependent Fusion Grammars

PN = (P, T, F, W, M0) ❀ CDFG(PN) = (ZPN, FPN, MPN, TPN, PPN) ◮ FPN = {◦} ∪ T, MPN = {µ}, TPN = {•} type(◦) = (0, 1), type(t) = (|•t|, |t•|) for each t ∈ T Both ◦ and • represent token, because F ∩ T = ∅ ◮ ZPN = hg(P, T, M0) +

t∈T

Ct + C• +

t∈T

Dt

◮ hg(P, T, M0) represents the initial Petri net ◮ Ct is used for firing transitions t according to W ◮ C• =

• is used for terminating tokens

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Transformation of Petri Nets into Context-Dependent Fusion Grammars

PN = (P, T, F, W, M0) ❀ CDFG(PN) = (ZPN, FPN, MPN, TPN, PPN) ◮ FPN = {◦} ∪ T, MPN = {µ}, TPN = {•} type(◦) = (0, 1), type(t) = (|•t|, |t•|) for each t ∈ T Both ◦ and • represent token, because F ∩ T = ∅ ◮ ZPN = hg(P, T, M0) +

t∈T

Ct + C• +

t∈T

Dt

◮ hg(P, T, M0) represents the initial Petri net ◮ Ct is used for firing transitions t according to W ◮ C• =

• is used for terminating tokens

◮ Dt is used for deleting transition t

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Example ZPN

hg(P, T, M0) =

• µ
• 1

2 3 4 5 t1 t3 t2 t4

C• =

• Ct1 =

Ct2 = Ct3 = Ct4 =

• t1

t1

• t2

t2

• t3

t3

• t4

t4

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Example ZPN

hg(P, T, M0) =

• µ
• 1

2 3 4 5 t1 t3 t2 t4

C• =

• Ct1 =

Ct2 = Ct3 = Ct4 =

• t1

t1

• t2

t2

• t3

t3

• t4

t4

Dt1 = Dt2 = Dt3 = Dt4 =

t1 t2 t3 t4

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Transformation of Petri Nets into Context-Dependent Fusion Grammars

PN = (P, T, F, W, M0) ❀ CDFG(PN) = (ZPN, FPN, MPN, TPN, PPN) ◮ FPN = {◦} ∪ T, MPN = {µ}, TPN = {•} type(◦) = (0, 1), type(t) = (|•t|, |t•|) for each t ∈ T Both ◦ and • represent token, because F ∩ T = ∅ ◮ ZPN = hg(P, T, M0) +

t∈T

Ct + C• +

t∈T

Dt

◮ hg(P, T, M0) represents the initial Petri net ◮ Ct is used for firing transitions t according to W ◮ C• =

• is used for terminating tokens

◮ Dt is used for deleting transition t

◮ PPN = {consume, replace} ∪ {fire(t), delete(t) | t ∈ T}

◮ consume is used for consuming tokens ◮ fire(t) is used for simulating a firing of t

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Transformation of Petri Nets into Context-Dependent Fusion Grammars

PN = (P, T, F, W, M0) ❀ CDFG(PN) = (ZPN, FPN, MPN, TPN, PPN) ◮ FPN = {◦} ∪ T, MPN = {µ}, TPN = {•} type(◦) = (0, 1), type(t) = (|•t|, |t•|) for each t ∈ T Both ◦ and • represent token, because F ∩ T = ∅ ◮ ZPN = hg(P, T, M0) +

t∈T

Ct + C• +

t∈T

Dt

◮ hg(P, T, M0) represents the initial Petri net ◮ Ct is used for firing transitions t according to W ◮ C• =

• is used for terminating tokens

◮ Dt is used for deleting transition t

◮ PPN = {consume, replace} ∪ {fire(t), delete(t) | t ∈ T}

◮ consume is used for consuming tokens ◮ fire(t) is used for simulating a firing of t ◮ replace is used for termination

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Transformation of Petri Nets into Context-Dependent Fusion Grammars

PN = (P, T, F, W, M0) ❀ CDFG(PN) = (ZPN, FPN, MPN, TPN, PPN) ◮ FPN = {◦} ∪ T, MPN = {µ}, TPN = {•} type(◦) = (0, 1), type(t) = (|•t|, |t•|) for each t ∈ T Both ◦ and • represent token, because F ∩ T = ∅ ◮ ZPN = hg(P, T, M0) +

t∈T

Ct + C• +

t∈T

Dt

◮ hg(P, T, M0) represents the initial Petri net ◮ Ct is used for firing transitions t according to W ◮ C• =

• is used for terminating tokens

◮ Dt is used for deleting transition t

◮ PPN = {consume, replace} ∪ {fire(t), delete(t) | t ∈ T}

◮ consume is used for consuming tokens ◮ fire(t) is used for simulating a firing of t ◮ replace is used for termination ◮ delete(t) is used for deleting transition t

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Example delete(t4) = (fr(t4), PC, NC)

fr(t4) = t4 t4 PC = ∅ NC = { fr(t4) → t4 t4 t4 } where t4 maps into the left connected component. Matching morphism g cannot map the t4-hyperedge into Ct4.

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replace = (fr(◦), PC, NC)

fr(◦) =

• PC = { fr(◦) → ◦

+ •

• }

NC = { fr(◦) → ◦

• + ◦

, fr(◦) → ◦ + ◦

• }

where ◦ maps into the left connected component

• maps into the right connected component.

Can only be applied to ◦ and C• =

• if no ◦-hyperedges are

attached to the target of the ◦-hyperedge.

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Example derivation (delete and replace)

• µ

t1 t3 t2 t4 t1 t2 t3 t4

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Example derivation (delete and replace)

• µ

t1 t3 t2 t4 t1 t2 t3 t4

• 4

= ⇒

delete(t)

• µ

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Example derivation (delete and replace)

• µ

t1 t3 t2 t4 t1 t2 t3 t4

• 4

= ⇒

delete(t)

• µ
• 4

= ⇒

replace

• µ

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Derivation schema

Z = ⇒

m hg(P, T, M0) + C• +

• t∈T

m(Ct) · Ct +

• t∈T

Dt = ⇒

fire(t1) X1 b1

= ⇒

consume hg(P, T, M1) + C• +

• t∈T

mt2...tk(Ct) · Ct +

• t∈T

Dt = ⇒

fire(t2) X2 b2

= ⇒

consume . . .

= ⇒

fire(tn) Xn bn

= ⇒

consume hg(P, T, Mn) + C• +

• t∈T

Ct +

• t∈T

Dt = ⇒

m′ hg(P, T, Mn) + k · C• +

• t∈T

Dt with k =

• p∈P

Mn(p) = ⇒

delete(t1) · · ·

= ⇒

delete(t|T|) X k

= ⇒

replace H

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Derivation schema

Z = ⇒

m hg(P, T, M0) + C• +

• t∈T

m(Ct) · Ct +

• t∈T

Dt = ⇒

fire(t1) X1 b1

= ⇒

consume hg(P, T, M1) + C• +

• t∈T

mt2...tk(Ct) · Ct +

• t∈T

Dt = ⇒

fire(t2) X2 b2

= ⇒

consume . . .

= ⇒

fire(tn) Xn bn

= ⇒

consume hg(P, T, Mn) + C• +

• t∈T

Ct +

• t∈T

Dt = ⇒

m′ hg(P, T, Mn) + k · C• +

• t∈T

Dt with k =

• p∈P

Mn(p) = ⇒

delete(t1) · · ·

= ⇒

delete(t|T|) X k

= ⇒

replace H

remM(H) is a hypergraph representation of a marking

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Main result

Theorem

Let PN = (P, T, F, W, M0) be a Petri net. Let CDFG(PN) be the corresponding context-dependent fusion grammar. Then L(CDFG(PN)) = gr(Reach(PN)).

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Main result

Theorem

Let PN = (P, T, F, W, M0) be a Petri net. Let CDFG(PN) be the corresponding context-dependent fusion grammar. Then L(CDFG(PN)) = gr(Reach(PN)).

Lemma

M1[tM2 if and only if hg(P, T, M1)+Ct = ⇒

fire(t) X b

= ⇒

consume hg(P, T, M2)

with b =

p∈•t

W(p, t).

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Conclusion

Contributions: ◮ We have introduced context-dependent fusion grammars. ◮ We have transformed Petri nets into these grammars.

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Conclusion

Contributions: ◮ We have introduced context-dependent fusion grammars. ◮ We have transformed Petri nets into these grammars. Future work: ◮ Can Petri nets be transformed into fusion grammars without context conditions? Are only positive or only negative context conditions powerful enough to cover Petri nets? ◮ At ICGT18 we introduced splicing/fusion grammars enhancing fusion grammars by the inversion of fusions. How does a natural transformation of context-dependent fusion grammars into splicing/fusion grammars look like?

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Conclusion

Contributions: ◮ We have introduced context-dependent fusion grammars. ◮ We have transformed Petri nets into these grammars. Future work: ◮ Can Petri nets be transformed into fusion grammars without context conditions? Are only positive or only negative context conditions powerful enough to cover Petri nets? ◮ At ICGT18 we introduced splicing/fusion grammars enhancing fusion grammars by the inversion of fusions. How does a natural transformation of context-dependent fusion grammars into splicing/fusion grammars look like? Thank you! Questions?