Petri Nets (for Planners) - GitHub Pages · 2020. 9. 1. · A Petri net has places, transitions, and directed arcs Arcs connect places to transitions or vice versa Places contain

Introduction

Basics

Complexity &Expressivity

AnalysisTechniques

SpecialClasses ofNets

Conclusion

Petri Nets (for Planners)

B. Bonet, P. Haslum

... from various places ...

ICAPS 2011

Introduction

Basics


AnalysisTechniques


Conclusion

Introduction & Motivation

Petri Nets (PNs) is formalism for modelling discrete eventsystems

Developed by (and named after) C.A. Petri in 1960s

In general Petri nets, places are unbounded counters

advantages in expressivity and modelling convenience

questions of reachability, coverability, etc. arecomputationally harder to answer, but still decidable

Introduction

Basics


AnalysisTechniques


Conclusion

Exchange of ideas between Petri nets and planning holdspotential to benefit both areas:

Analysis methods for Petri nets are often based on ideas &techniques not common in planning:

– algebraic methods based on the state equation

– rich literature on the study of classes of nets with specialstructure

Yet, some standard planning techniques (e.g., searchheuristics) are unknown in the PN community

Introduction

Basics


AnalysisTechniques


Conclusion

Outline of the Tutorial

1 Definitions, notation and modelling

2 Decision problems, complexity and expressivity

3 Analysis techniques for general Petri nets

– Coverability

– The state equation

– Reachability

4 Petri nets with special structure

5 Conclusions

Introduction

Basics

Definitions

Ordinary Nets

Types of Nets

Vector Notation


AnalysisTechniques


Conclusion

Definitions, Notation and

Modelling

Introduction

Basics

Definitions

Ordinary Nets

Types of Nets

Vector Notation


AnalysisTechniques


Conclusion

Terminology and Intuition

A Petri net has places, transitions, and directed arcs

Arcs connect places to transitions or vice versa

Places contain zero or finite number of tokens

A marking is disposition of tokens in places

A transition is fireable if there is token at the start place ofeach input arc

When transition fires:

– it consumes token from start place of each input arc– it puts token at end place of each output arc

Execution is non-deterministic

Introduction

Basics

Definitions

Ordinary Nets

Types of Nets

Vector Notation


AnalysisTechniques


Conclusion

Example

p1 p2

t1 t2 t3

t4

p3 p4 p5

t5

t6 t7

p6 p7

2

Introduction

Basics

Definitions

Ordinary Nets

Types of Nets

Vector Notation


AnalysisTechniques


Conclusion

Formal Definition

Place/Transition (P/T) net is tuple N = (P, T,W ) where:

– P is set of places

– T is set of transitions (and P ∩ T = ∅)– W ⊆ (P × T ) ∪ (T × P )→ N

(multiset of arcs: each (x, y) has multiplicity W (x, y))

For transition t:

– preset is •t = {s : W (s, t) > 0} (input places)

– postset is t• = {s : W (t, s) > 0} (output places)

Marking is m : P → N (zero or more tokens at each place)

Introduction

Basics

Definitions

Ordinary Nets

Types of Nets

Vector Notation


AnalysisTechniques


Conclusion

Example

p1 p2

t1 t2 t3

t4

p3 p4 p5

t5

t6 t7

p6 p7

2

– marking m = 〈1, 1, 0, 0, 0, 0, 0〉– transition t6: •t6 = {p3, p5}, t6• = {p6, p7}

Introduction

Basics

Definitions

Ordinary Nets

Types of Nets

Vector Notation


AnalysisTechniques


Conclusion

Execution Semantics

A transition t is enabled or firable at marking m if

m(p) ≥W (p, t) for each p ∈ •t

Upon firing, t produces new marking m′ such that

m′(p) =

{m(p)−

consumed︷︸︸︷W (p, t) +

added︷︸︸︷W (t, p) if p ∈ •t ∪ t•

m(p) if p /∈ •t ∪ t•

Introduction

Basics

Definitions

Ordinary Nets

Types of Nets

Vector Notation


AnalysisTechniques


Conclusion

Execution Semantics

A transition t is enabled or firable at marking m if

m(p) ≥W (p, t) for each p ∈ •t

Upon firing, t produces new marking m′ such that

m′(p) =

{m(p)−

consumed︷︸︸︷W (p, t) +

added︷︸︸︷W (t, p) if p ∈ •t ∪ t•

m(p) if p /∈ •t ∪ t•

Introduction

Basics

Definitions

Ordinary Nets

Types of Nets

Vector Notation


AnalysisTechniques


Conclusion

Execution Semantics

Transition relations:

m [t〉m′ if t is enabled at m and produces m′

m [σ〉m′, for sequence σ = t1t2 · · · tn, if exists m′′ with

– m [t1〉m′′– m′′ [σ′〉m′ for σ′ = t2 · · · tn

m [〉m′ if m [t〉m′ for some t

[∗〉 is transitive closure of [〉

Marked net is (N = (P, T,W ),m0) where m0 is initialmarking

Reachable markings R(N,m0) = {m ∈ NP : m0 [∗〉m}

Firing sequences L(N,m0) = {σ ∈ T<∞ : ∃m.m0 [σ〉m}

Introduction

Basics

Definitions

Ordinary Nets

Types of Nets

Vector Notation


AnalysisTechniques


Conclusion

Execution Semantics




– m [t1〉m′′– m′′ [σ′〉m′ for σ′ = t2 · · · tn






Introduction

Basics

Definitions

Ordinary Nets

Types of Nets

Vector Notation


AnalysisTechniques


Conclusion

Execution Semantics




– m [t1〉m′′– m′′ [σ′〉m′ for σ′ = t2 · · · tn






Introduction

Basics

Definitions

Ordinary Nets

Types of Nets

Vector Notation


AnalysisTechniques


Conclusion

Execution Semantics




– m [t1〉m′′– m′′ [σ′〉m′ for σ′ = t2 · · · tn






Introduction

Basics

Definitions

Ordinary Nets

Types of Nets

Vector Notation


AnalysisTechniques


Conclusion

Example

p1 p2

p3 p4 p5

p6 p7

t1 t2 t3

t4 t5

t6 t7

2

– marking m = 〈1, 1, 0, 0, 0, 0, 0〉– enabled: t1, t2, t3

Introduction

Basics

Definitions

Ordinary Nets

Types of Nets

Vector Notation


AnalysisTechniques


Conclusion

Example

p1 p2

p3 p4 p5

p6 p7

t1 t2 t3

t4 t5

t6 t7

2

– marking m = 〈0, 1, 1, 0, 0, 0, 0〉– enabled: t3

Introduction

Basics

Definitions

Ordinary Nets

Types of Nets

Vector Notation


AnalysisTechniques


Conclusion

Example

p1 p2

p3 p4 p5

p6 p7

t1 t2 t3

t4 t5

t6 t7

2

– marking m = 〈0, 0, 1, 0, 2, 0, 0〉– enabled: t6

Introduction

Basics

Definitions

Ordinary Nets

Types of Nets

Vector Notation


AnalysisTechniques


Conclusion

Example

p1 p2

p3 p4 p5

p6 p7

t1 t2 t3

t4 t5

t6 t7

2

– marking m = 〈0, 0, 0, 0, 1, 1, 1〉– enabled: t4, t5

Introduction

Basics

Definitions

Ordinary Nets

Types of Nets

Vector Notation


AnalysisTechniques


Conclusion

Arithmetic of Functions

For two functions f, g ∈ NX :

f ≥ g if f(x) ≥ g(x) for each place x

f > g if f ≥ g and there is x such that f(x) > g(x)

f + g defined pointwise as (f + g)(x) = f(x) + g(x)

Hence, m [t〉m′ iff

m ≥W (·, t) (enable condition)

m′ = m−W (·, t) +W (t, ·)

Introduction

Basics

Definitions

Ordinary Nets

Types of Nets

Vector Notation


AnalysisTechniques


Conclusion

Ordinary Nets

A P/T net N = (P, T,W ) is ordinary iff W (p, t) ≤ 1 for all p, t

Thm: any net can be transformed into equivalent ordinary net

Introduction

Basics

Definitions

Ordinary Nets

Types of Nets

Vector Notation


AnalysisTechniques


Conclusion

Transformation Rules

1) p t2k =⇒ p

kt

k

2) p t2k + 1 =⇒ p

kt

k + 1

3) t p2k =⇒ t p

k

k

4) t p2k + 1 =⇒ t p

k + 1

k

Each rule decrease multiplicity by half and add 2 nodes

Resulting size is O(∑

x,yW (x, y)) (exponential)

Introduction

Basics

Definitions

Ordinary Nets

Types of Nets

Vector Notation


AnalysisTechniques


Conclusion

Example

p t q4

2

Introduction

Basics

Definitions

Ordinary Nets

Types of Nets

Vector Notation


AnalysisTechniques


Conclusion

Example

p t

2

q2

2

Introduction

Basics

Definitions

Ordinary Nets

Types of Nets

Vector Notation


AnalysisTechniques


Conclusion

Example

p t

2

q

2

Introduction

Basics

Definitions

Ordinary Nets

Types of Nets

Vector Notation


AnalysisTechniques


Conclusion

Example

p t

2

q

Introduction

Basics

Definitions

Ordinary Nets

Types of Nets

Vector Notation


AnalysisTechniques


Conclusion

Types of Nets

Marking m is k-bounded if m(p) ≤ k for all p ∈ P

Marked net (N,m0) is k-bounded if every reachablemarking is k-bounded

It is bounded if it is k-bounded for some k

It is safe if it is 1-bounded

Introduction

Basics

Definitions

Ordinary Nets

Types of Nets

Vector Notation


AnalysisTechniques


Conclusion

Safe Networks

Every reachable marking is 1-bounded

Marking m can be thought as state where places representsfluents:

– if m(p) = 1 then fluent p is true at m– if m(p) = 0 then fluent p is false at m

Safe networks can be used for STRIPS planning

Introduction

Basics

Definitions

Ordinary Nets

Types of Nets

Vector Notation


AnalysisTechniques


Conclusion

Direct STRIPS to PN Translations

Each atom is a place

Each (grounded) action is a transition t:

– input arcs p→ t for each precondition p

– output arcs t→ p for each positive effect p

– output arcs t→ p for each precondition p that is notdeleted nor added

Initial state gives initial marking

Goal state gives partial desired marking

Plan existence becomes ”Coverability” problem

Introduction

Basics

Definitions

Ordinary Nets

Types of Nets

Vector Notation


AnalysisTechniques


Conclusion

Example: Gripper w/ 1 Ball and 2 Arms

Ball@A Ball@B

Robby@A Robby@B

Ball@1

Ball@2

move2B

move2A

pick1@A

pick2@A

drop1@A

drop2@A

pick1@Bdrop1@B pick1@B

drop2@B pick2@B

move2B, pick1@B, move2A, drop1@A

Introduction

Basics

Definitions

Ordinary Nets

Types of Nets

Vector Notation


AnalysisTechniques


Conclusion


Ball@A Ball@B

Robby@A Robby@B

Ball@1

Ball@2

move2B

move2A

pick1@A

pick2@A

drop1@A

drop2@A


drop2@B pick2@B


Introduction

Basics

Definitions

Ordinary Nets

Types of Nets

Vector Notation


AnalysisTechniques


Conclusion


Ball@A Ball@B

Robby@A Robby@B

Ball@1

Ball@2

move2B

move2A

pick1@A

pick2@A

drop1@A

drop2@A


drop2@B pick2@B


Introduction

Basics

Definitions

Ordinary Nets

Types of Nets

Vector Notation


AnalysisTechniques


Conclusion


Ball@A Ball@B

Robby@A Robby@B

Ball@1

Ball@2

move2B

move2A

pick1@A

pick2@A

drop1@A

drop2@A


drop2@B pick2@B


Introduction

Basics

Definitions

Ordinary Nets

Types of Nets

Vector Notation


AnalysisTechniques


Conclusion


Ball@A Ball@B

Robby@A Robby@B

Ball@1

Ball@2

move2B

move2A

pick1@A

pick2@A

drop1@A

drop2@A


drop2@B pick2@B


Introduction

Basics

Definitions

Ordinary Nets

Types of Nets

Vector Notation


AnalysisTechniques


Conclusion

Safe STRIPS Problems

STRIPS problem is safe if its direct translation (N,m0) is safe

Sufficient Condition:

For each added atom p, there is precondition q that isdeleted such that {p, q} is mutex

Enforcing the condition:

Add ‘not-p’ atoms for each atom p

For each action that contains a deleted atom p that is notprecondition, generate two similar actions with p and not-pin precondition (respectively)

Worst-case size of pre-processing is exponential in numberof atoms that are deleted and don’t appear as preconditions

Introduction

Basics

Definitions

Ordinary Nets

Types of Nets

Vector Notation


AnalysisTechniques


Conclusion

Modelling Planning Problems

General nets can “store” multiple tokens at single place

Places can be used to represent:

– number of identical objects at location

– resource quantity

Introduction

Basics

Definitions

Ordinary Nets

Types of Nets

Vector Notation


AnalysisTechniques


Conclusion

Gripper with 3 identical balls and 2 identical arms

# in Room A # in Room B

Robby@A Robby@B

# held

# free

move2B

move2A

pick@A

drop@A

pick@B

drop@B

pick@A, pick@A, move2B, drop@B, drop@B, move2A, pick@A, move2B, drop@B

Introduction

Basics

Definitions

Ordinary Nets

Types of Nets

Vector Notation


AnalysisTechniques


Conclusion



Robby@A Robby@B

# held

# free

move2B

move2A

pick@A

drop@A

pick@B

drop@B


Introduction

Basics

Definitions

Ordinary Nets

Types of Nets

Vector Notation


AnalysisTechniques


Conclusion



Robby@A Robby@B

# held

# free

move2B

move2A

pick@A

drop@A

pick@B

drop@B


Introduction

Basics

Definitions

Ordinary Nets

Types of Nets

Vector Notation


AnalysisTechniques


Conclusion



Robby@A Robby@B

# held

# free

move2B

move2A

pick@A

drop@A

pick@B

drop@B


Introduction

Basics

Definitions

Ordinary Nets

Types of Nets

Vector Notation


AnalysisTechniques


Conclusion



Robby@A Robby@B

# held

# free

move2B

move2A

pick@A

drop@A

pick@B

drop@B


Introduction

Basics

Definitions

Ordinary Nets

Types of Nets

Vector Notation


AnalysisTechniques


Conclusion



Robby@A Robby@B

# held

# free

move2B

move2A

pick@A

drop@A

pick@B

drop@B


Introduction

Basics

Definitions

Ordinary Nets

Types of Nets

Vector Notation


AnalysisTechniques


Conclusion



Robby@A Robby@B

# held

# free

move2B

move2A

pick@A

drop@A

pick@B

drop@B


Introduction

Basics

Definitions

Ordinary Nets

Types of Nets

Vector Notation


AnalysisTechniques


Conclusion



Robby@A Robby@B

# held

# free

move2B

move2A

pick@A

drop@A

pick@B

drop@B


Introduction

Basics

Definitions

Ordinary Nets

Types of Nets

Vector Notation


AnalysisTechniques


Conclusion



Robby@A Robby@B

# held

# free

move2B

move2A

pick@A

drop@A

pick@B

drop@B


Introduction

Basics

Definitions

Ordinary Nets

Types of Nets

Vector Notation


AnalysisTechniques


Conclusion



Robby@A Robby@B

# held

# free

move2B

move2A

pick@A

drop@A

pick@B

drop@B


Introduction

Basics

Definitions

Ordinary Nets

Types of Nets

Vector Notation


AnalysisTechniques


Conclusion

Wargus Domain (Chan et al. 2007)

Gold Wood Peasant Supply

100

100

400

500250

4

Introduction

Basics

Definitions

Ordinary Nets

Types of Nets

Vector Notation


AnalysisTechniques


Conclusion

Other Types of Nets

State Machines:

– every transition has one incoming and one outgoing arci.e. |•t| = |t•| = 1 for each t ∈ T

Marked Graphs:

– every place has one incoming arc, and one outgoing arci.e. |•p| = |p•| = 1 for each p ∈ P

Free-choice Nets:

– every arc is either the only arc going from the place, oronly arc going to the transitioni.e. |p•| ≤ 1 or •(p•) = {p} for each p ∈ P

Introduction

Basics

Definitions

Ordinary Nets

Types of Nets

Vector Notation


AnalysisTechniques


Conclusion

Extensions

Inhibitor arcs (enablers):

– transition enabled when there is no token at place

Read arcs (enablers):

– do not consume tokens

Reset arcs: erase all tokens at place

Others:

– colored, hierarchical, prioritization, . . .

Introduction

Basics

Definitions

Ordinary Nets

Types of Nets

Vector Notation


AnalysisTechniques


Conclusion

Vector Notation

Two vectors associated with transition t:

W−t =

W (p1, t)...

W (p|P |, t)

W+t =

W (t, p1)...

W (t, p|P |)

t enabled at m iff m ≥W−t

Wt = W+t −W−

t is effect of t

firing t leads to m′ = m + Wt

W =(Wt1 , Wt2 , . . . , Wt|T |

)is incidence matrix

rp: row of W corresponding to place p

Introduction

Basics

Definitions

Ordinary Nets

Types of Nets

Vector Notation


AnalysisTechniques


Conclusion

Examples

p1 t1

p2

p3

t2

t3

W =

−1 0 11 −1 01 1 −1

p1

p2

p3

p4

p5

p6

p7

t1

t2

t3

t4

t5

t6

W =

1 −1 0 0 0 0−1 1 1 −1 0 0

1 0 0 0 −1 00 1 0 0 0 −10 0 −1 1 −1 10 0 0 0 1 −1−1 0 0 0 0 1

Introduction

Basics

Definitions

Ordinary Nets

Types of Nets

Vector Notation


AnalysisTechniques


Conclusion

Representation Ambiguity and Pure Nets

Wt[i] = 0:

t

pi

or

t

pi

Pure nets have no “self loops”:•t ∩ t• = ∅ for every transition t

For pure nets, incidence matrix Wunambiguously defines the net

Any net can be transformed into apure net by splitting loops:

ta p′ tb

pi

Transformation is linear space

Introduction

Basics


Properties

Equivalence

StructuralProperties

Expressivity

Invariants

AnalysisTechniques


Conclusion

Complexity & Expressivity

Introduction

Basics


Properties

Equivalence


Expressivity

Invariants

AnalysisTechniques


Conclusion

Decision Problems for Marked Nets

Given a marked net (N,m0):

Reachability: Is there a firing sequence that ends withgiven marking m?

Coverability: Is there a firing sequence that ends withmarking m′ such that m′ ≥m for given m?

Boundedness: Does there exist a integer k such that everyreachable marking is k-bounded? m ≤ K?

Coverability and boundedness are EXPSPACE-complete

Reachability is EXPSPACE-hard, but existing algorithms arenon-primitive recursive (i.e., have unbounded complexity)

Introduction

Basics


Properties

Equivalence


Expressivity

Invariants

AnalysisTechniques


Conclusion

More Properties

Executability: Is there a firing sequence valid at m0 thatincludes transition t?

– Reduces to coverability: t is executable iff W−t is

coverable– and vice versa: reduction using a “goal transition”

Repeated Executability: Is there a firing sequence inwhich a given transition (or set of transitions) occurs aninfinite number of times?

Reachable Deadlock: Is there a reachable marking m atwhich no transition is enabled?

Liveness: Executability of every transition at everyreachable marking, i.e.,∀M : M0 [∗〉M → ∀t∃M ′,M ′′ : M [∗〉M ′ [t〉M ′′.

. . . and many more . . .

Introduction

Basics


Properties

Equivalence


Expressivity

Invariants

AnalysisTechniques


Conclusion

Equivalence Problems

Equivalence: Given two marked nets, (N1,m1) and(N2,m2), with equal (or isomorphic) sets of places, do theyhave the equal sets of reachable markings?

Trace Equivalence: Given two marked nets, (N1,m1) and(N2,m2), with equal (or isomorphic) sets of transitions, dothey have equal sets of valid firing sequences?

Language Equivalence: Trace equivalence under mappingof transitions to a common alphabet

Bisimulation: Equivalence under a bijection betweenmarkings

In general, equivalence problems are undecidable

Introduction

Basics


Properties

Equivalence


Expressivity

Invariants

AnalysisTechniques


Conclusion

Structural Properties

A structural property is independent of initial marking m0

Structural Liveness: It there a marking m such that(N,m) is live?

Structural Boundedness: Is (N,m) bounded for everyfinite initial marking m?

Repetitiveness: Is there a marking m and a firing sequenceσ valid at m such that a given transition (set of transitions)appears infinitely often in σ?

Deciding structural properties can be easier than correspondingproblem for marked net

Structural boundedness and repetitiveness are in NP

Introduction

Basics


Properties

Equivalence


Expressivity

Invariants

AnalysisTechniques


Conclusion

Complexity: Implications Of and For Expressivity

Bounded Petri nets are expressively equivalent topropositional STRIPS/PDDL

– Reachability is PSPACE-complete for both

– Recall: direct STRIPS to PN translation may blow upexponentially

General Petri nets are stictly more expressive thanpropositional STRIPS/PDDL

General Petri nets are at least as expressive as “lifted”(finite 1st order) STRIPS/PDDL

– probably also strictly more expressive (but no proof yet)

Introduction

Basics


Properties

Equivalence


Expressivity

Invariants

AnalysisTechniques


Conclusion

Counter TMs

A k-counter machine (kCM) is a deterministic finiteautomaton with k (positive) integer counters

– can increment/decrement (by 1), or reset, counters

– conditional jumps on ci > 0 or ci = 0

Note the differences:

– kCMs are deterministic: starting configurationdetermines unique execution; Petri nets have choice

– kCMs can branch on ci > 0/ci = 0; Petri nets can onlyprecondition transitions on m(pi) > 0

A kCM is k-bounded iff no counter ever exceeds k

Introduction

Basics


Properties

Equivalence


Expressivity

Invariants

AnalysisTechniques


Conclusion

Counter TMs: Results

An n-size TM can be simulated by an O(n)-size 2CM (ifproperly initialised)

– Halting (i.e., reachability) for unbounded 2CMs isundecidable

– PNs are strictly less expressive than unbounded 2CMs

An n-size and 2n space bounded TM can be simulated byO(n)-size 22

n-bounded 2CM

A 22n

-bounded n-size 2CM can be (non-deterministically!)simulated by O(n2)-size Petri net

– Reachability for Petri nets is DSPACE(2√n)-hard

Introduction

Basics


Properties

Equivalence


Expressivity

Invariants

AnalysisTechniques


Conclusion

Invariants

A vector y ∈ N|P | is P-invariant for N iff for any markingsm [∗〉m′, yTm = yTm′

P-invariant = linear combination of place markings thatis invariant under any transition firing

A vector x ∈ N|T | is a T-invariant for N iff for any firingsequence σ such that n(σ) = x and any marking m whereσ is enabled, m [σ〉m

T-invariant = multiset of transitions whose combinedeffect is zero

Introduction

Basics


AnalysisTechniques


Conclusion

Analysis Techniques

Introduction

Basics


AnalysisTechniques


Conclusion

The Coverability Tree Construction

The coverability tree of a marked net (N,m0) is an explicitrepresentation of reachable markings – but not exactly theset of reachable markings.

Constructed by forwards exploration:

Each enabled transition generates a successor marking.If reach m such that m >m′ for some ancestor m′ of m,replace m[i] by ω for all i s.t. m[i] >m′[i].

m′ [s = t1, . . . , tl〉m, and since m ≥m′, m [s〉m′′such that m′′ ≥m; sequence s can be repeated anynumber of times.ω means “arbitraribly large”.

Also check for regular loops (m = m′ for some ancestorm′ of m).

Every branch has finite depth.

Introduction

Basics


AnalysisTechniques


Conclusion

Example

p1

t1

p2

p3

t2

t3

(1 0 0)

(0 1 1)

t1

(0 0 2)

t2

(1 0ω)

t3

(0 1ω)

t1

(0 0ω)

t2

(1 0ω)

t3

(1ω ω)t3

. . .

t1

. . .t2

(ω ω ω)

t3

. . .t3

(1ω 0)t3

. . .

t1

. . .

t2

Introduction

Basics


AnalysisTechniques


Conclusion

Uses For The Coverability Tree

Decides coverability:

m is coverable iff m ≤m′ for some m′ in the tree (wheren < ω for any n ∈ N).If m is coverable, there exists a covering sequence oflength at most O(2n).

Decides boundedness:

(N,m0) is unbounded iff there exists a self-coveringsequence: m0 [σ〉m [σ′〉m′ such that m′ >m.I.e., (N,m0) is unbounded iff ω appears in some markingin the coverability tree.If (N,m0) is unbounded, there exists a self-coveringsequence of length at most O(2n).

In general, does not decide reachability.

Except if (N,m0) is bounded.

Introduction

Basics


AnalysisTechniques


Conclusion

The State Equation

The firing count vector (a.k.a. Parikh vector) of a firingsequence σ = ti1 , . . . , til is a |T |-dimensional vectorn(σ) = (n1, . . . , n|T |) where ni ∈ N is the number ofoccurrences of transition ti in σ.

If m0 [σ〉m′, then

m′ = m0+w(ti1)+. . .+w(til) = m0+∑

j=1...|T |

w(tj)n(σ)[j],

i.e., m′ = m0 +Wn(σ).

m is reachable from m0 only if Wn = (m−m0) has asolution n ∈ N|T |.This is a necessary condition but not sufficient.

A solution n is realisable iff, in addition, n = n(σ) forsome valid firing sequence σ.

Introduction

Basics


AnalysisTechniques


Conclusion

The State Equation & Invariance

y ∈ N|P | is a P-invariant iff it is a solution to yTW = 0.

yTm = yTm0 for any m reachable from m0.

x ∈ N|T | is a T-invariant iff it is a solution to Wx = 0.

m [σ〉m whenever n(σ) = x and σ enabled at m.

Any (positive) linear combination of P-/T-invariants is aP-/T-invariant.

The reverse dual of a net N is obtained by swapping placesfor transitions and vice versa, and reversing all arcs.

The incidence matrix of the reverse dual is the transposeof the incidence matrix of N .A P-(T-)invariant of N is a T-(P-)invariant of the reversedual.

Introduction

Basics


AnalysisTechniques


Conclusion

Example: P-Invariants

1100110

T

1 −1 0 0 0 0−1 1 1 −1 0 0

1 0 0 0 −1 00 1 0 0 0 −10 0 −1 1 −1 10 0 0 0 1 −1−1 0 0 0 0 1

=

0000000

T

1011012

T

1 −1 0 0 0 0−1 1 1 −1 0 0

1 0 0 0 −1 00 1 0 0 0 −10 0 −1 1 −1 10 0 0 0 1 −1−1 0 0 0 0 1

=

0000000

T

p1

p2

p3

p4

p5

p6

p7

t1

t2

t3

t4

t5

t6

Introduction

Basics


AnalysisTechniques


Conclusion

Example: T-Invariants

1 −1 0 0 0 0−1 1 1 −1 0 0

1 0 0 0 −1 00 1 0 0 0 −10 0 −1 1 −1 10 0 0 0 1 −1−1 0 0 0 0 1

001100

=

000000

1 −1 0 0 0 0−1 1 1 −1 0 0

1 0 0 0 −1 00 1 0 0 0 −10 0 −1 1 −1 10 0 0 0 1 −1−1 0 0 0 0 1

110011

=

000000

p1

p2

p3

p4

p5

p6

p7

t1

t2

t3

t4

t5

t6

Introduction

Basics


AnalysisTechniques


Conclusion

Minimal Invariants

The support of a P-/T-invariant y is the set {i |y[i] > 0}.An invariant has minimal support iff no invariants support isa strict subset.

The number of minimal support P-/T-invariants of a netis finite, but may be exponential.All P-/T-invariants are (positive) linear combinations ofminimal support P-/T-invariants.

A P-/T-invariant y is minimal iff no y′ < y is invariant.

A minimal invariant need not have minimal support.For each minimal support, there is a unique minimalinvariant.

Algorithms exist to generate all minimal supportP-/T-invariants of a net.

Introduction

Basics


AnalysisTechniques


Conclusion

The Fourier-Motzkin Algorithm for P-Invariants

1 Initialise B = [W : In] (n = |P |).

2 For j = 1, . . . , |T |1 Append to B all rows resulting from positive linear

combinations of pairs of rows in B that eliminate columnj.

2 Remove from B all rows with non-zero jth element.

3 B = [0 : D], where the rows of D are P-invariants.

Introduction

Basics


AnalysisTechniques


Conclusion

Example

B =

1 −1 0 0 0 0 | 1 0 0 0 0 0 0−1 1 1 −1 0 0 | 0 1 0 0 0 0 0

1 0 0 0 −1 0 | 0 0 1 0 0 0 00 1 0 0 0 −1 | 0 0 0 1 0 0 00 0 −1 1 −1 1 | 0 0 0 0 1 0 00 0 0 0 1 −1 | 0 0 0 0 0 1 0−1 0 0 0 0 1 | 0 0 0 0 0 0 1

Introduction

Basics


AnalysisTechniques


Conclusion

Example

B =

1 −1 0 0 0 0 | 1 0 0 0 0 0 0−1 1 1 −1 0 0 | 0 1 0 0 0 0 0

1 0 0 0 −1 0 | 0 0 1 0 0 0 00 1 0 0 0 −1 | 0 0 0 1 0 0 00 0 −1 1 −1 1 | 0 0 0 0 1 0 00 0 0 0 1 −1 | 0 0 0 0 0 1 0−1 0 0 0 0 1 | 0 0 0 0 0 0 1

0 0 1 −1 0 0 | 1 1 0 0 0 0 00 −1 0 0 0 1 | 1 0 0 0 0 0 10 1 1 −1 −1 0 | 0 1 1 0 0 0 00 0 0 0 −1 1 | 0 0 1 0 0 0 1

Introduction

Basics


AnalysisTechniques


Conclusion

Example

B =

0 1 0 0 0 −1 | 0 0 0 1 0 0 00 0 −1 1 −1 1 | 0 0 0 0 1 0 00 0 0 0 1 −1 | 0 0 0 0 0 1 00 0 1 −1 0 0 | 1 1 0 0 0 0 00 −1 0 0 0 1 | 1 0 0 0 0 0 10 1 1 −1 −1 0 | 0 1 1 0 0 0 00 0 0 0 −1 1 | 0 0 1 0 0 0 1

0 0 0 0 0 0 | 1 0 0 1 0 0 10 0 1 −1 −1 1 | 1 1 1 0 0 0 1

Introduction

Basics


AnalysisTechniques


Conclusion

Example

B =

0 0 −1 1 −1 1 | 0 0 0 0 1 0 00 0 0 0 1 −1 | 0 0 0 0 0 1 00 0 1 −1 0 0 | 1 1 0 0 0 0 00 0 0 0 −1 1 | 0 0 1 0 0 0 10 0 0 0 0 0 | 1 0 0 1 0 0 10 0 1 −1 −1 1 | 1 1 1 0 0 0 1

0 0 0 0 −1 1 | 1 1 0 0 1 0 00 0 0 0 −2 2 | 1 1 1 0 1 0 1

Introduction

Basics


AnalysisTechniques


Conclusion

Example

B =

0 0 0 0 1 −1 | 0 0 0 0 0 1 00 0 0 0 −1 1 | 0 0 1 0 0 0 10 0 0 0 0 0 | 1 0 0 1 0 0 10 0 0 0 −1 1 | 1 1 0 0 1 0 00 0 0 0 −2 2 | 1 1 1 0 1 0 1

0 0 0 0 0 0 | 0 0 1 0 0 1 10 0 0 0 0 0 | 1 1 0 0 1 1 00 0 0 0 0 0 | 1 1 1 0 1 2 1

Introduction

Basics


AnalysisTechniques


Conclusion

Example

B =

0 0 0 0 0 0 | 1 0 0 1 0 0 10 0 0 0 0 0 | 0 0 1 0 0 1 10 0 0 0 0 0 | 1 1 0 0 1 1 00 0 0 0 0 0 | 1 1 1 0 1 2 1

P-invariants:

z1 = (1 0 0 1 0 0 1)

z2 = (0 0 1 0 0 1 1)

z3 = (1 1 0 0 1 1 0)

z4 = (1 1 1 0 1 2 1)

p1

p2

p3

p4

p5

p6

p7

t1

t2

t3

t4

t5

t6

Introduction

Basics


AnalysisTechniques


Conclusion

The State Equation & Structural Properties

N is structurally bounded iff yTW ≤ 0 has a solutiony ∈ N|P | such that y[i] ≥ 1 for i = 1, . . . , |P |.y is a linear combination of all place markings that isinvariant or decreasing under any transition firing.

N is repetitive w.r.t. transition t iff Wx ≥ 0 has a solutionx ∈ N|T | such that x[t] > 0.

x is a multiset of transitions, including t at least once,whose combined effect is zero or increasing.Can always find some initial marking m0 from which x isrealisable.

Introduction

Basics


AnalysisTechniques


Conclusion

Example

p1

p2

p3

p4

p5

p6

p7

t1

t2

t3

t4

t5

t6

z1 =(

1 0 0 1 0 0 1)

andz4 =

(1 1 1 0 1 2 1

)are P-invariants of the net.

y = z1 + z4 =(

2 1 1 1 1 2 2)

is also aP-invariant.

yTW = 0 and y ≥ 1: The net is structurally bounded.

Introduction

Basics


AnalysisTechniques


Conclusion

Reachability

Decidability of the (exact) reachability problem for generalPetri nets was open for some time.

Algorithm proposed by Sacerdote & Tenney in 1977incorrect (or gaps in correctness proof).Correct algorithm by Mayr in 1981.Simpler correctness proof (for essentially the samealgorithm) by Kosaraju in 1982.

Other algorithms have been presented since.

All existing algorithms have unbounded complexity.

Fun fact: A 2-EXP algorithm was proposed in 1998, butlater shown to be incorrect.

Introduction

Basics


AnalysisTechniques


Conclusion

Reachability: Preliminaries

m is semi-reachable from m0 iff there is a transitionsequence s = ti1 , . . . , tin such thatm = m0 + w(ti1) + . . .+ w(tin).

s is does not have to be valid (firable) at m0.m is semi-reachable from m0 iff Wn = (m−m0) has asolution n ∈ N|T |.

If m is semi-reachable from m0, then m + a is reachablefrom m0 + a for some sufficiently large a ≥ 0.

Introduction

Basics


AnalysisTechniques


Conclusion

A controlled net is a pair of a marked net(N = 〈P, T, F 〉,m0) and an NFA (A, q0) over alphabet T .

A defines a (regular) subset of (not necessarily firable)transition sequences.Define reachability/coverability/boundedness for (N,m0)w.r.t. A in the obvious way.The coverability tree construction is easily modified toconsider only sequences accepted by A.

The reverse of N , NRev (w.r.t. A) is obtained by reversingthe flow relation (and arcs in A).

W (NRev) = −W (N).

Introduction

Basics


AnalysisTechniques


Conclusion

Reachability: A Sufficient Condition

In (N,m0) w.r.t. (A, q0), if

(a) (m∗, q∗) is semi-reachable from (m0, q0),(b) (m0 + a, q0) is reachable from (m0, q0), for a ≥ 1,(c) (m∗ + b, q∗) is reachable from (m∗, q∗) in NRev w.r.t. A,

for b ≥ 1,(d) (b− a, q∗) is semi-reachable from (0, q∗),

then (m∗, q∗) is reachable from (m0, q0).

The conditions above are effectively checkable:

(b) & (c) by coverability tree construction,(a) & (d) through the state equation.

Introduction

Basics


AnalysisTechniques


Conclusion

(m0, q0)

+a

(m0 + ka, q0)

+a(m∗ + ka, q∗)

b− a(m∗ + kb, q∗)

b− a−b

(m∗, q∗)

−b

Introduction

Basics


AnalysisTechniques


Conclusion

Reachability: The Mayr/Kosaraju Algorithm

Consider a controlled net (N,A) of the form,

A1 A2 Ak

q0

m0 mout1 min

2 mout2 min

k

q∗

m∗

ti1

with constraints min/outi [j] = x

i/oi,j or m

in/outi [j] ≥ yi/o

i,j ≥ 0.

If the sufficient reachability condition holds for each(min

i , qini ) and (mout

i , qouti ) w.r.t Ai, then (m∗, q∗) is

reachable from (m0, q0).

Let ∆(Ai) = {m |m = Wn(s), s ∈ L(Ai)}.Let Γ = {min

i ,mouti ,ni |mi+1

in −miout =

w(tii),miout −mi

in ∈ ∆(Ai), and constraints hold}.If (m0, q0) [s〉 (m∗, q∗), s defines an element in Γ.

Introduction

Basics


AnalysisTechniques


Conclusion

Γ is a semi-linear set: consistency (non-emptiness) isdecidable via Pressburger arithmetic.

If Γ is consistent, but the sufficient condition does not holdin some Ai, then Ai can be replaced by a new “chain” ofcontrollers, A1

i , . . . , Alii , each of which is “simpler”:

more equality constraints (min/out

il= x

i/o

il,j), or

same equality constraints and smaller automaton.

There can be several possible replacements(non-deterministic choice).

If (m∗, q∗) is not reachable from (m0, q0), every choice(branch) eventually leads to an inconsistent system.

Introduction

Basics


AnalysisTechniques


Conclusion

Special Classes of Nets

Introduction

Basics


AnalysisTechniques


Conclusion

Special Classes of Nets

State Machines:

every transition has one incoming and one outgoing arci.e. |•t| = |t•| = 1 for each t ∈ T .

Marked Graphs:

every place has one incoming arc, and one outgoing arci.e. |•p| = |p•| = 1 for each p ∈ P .

Free-choice Nets:

every arc is either the only arc going from the place, oronly arc going to the transitioni.e. |p•| ≤ 1 or •(p•) = {p} for each p ∈ P .

Introduction

Basics


AnalysisTechniques


Conclusion

Marked Graphs

An ordinary Petri net with |•p| = |p•| = 1 for each place p isa T-graph, or marked graph.

Abstracting away places leaves a directed graph:

Called the underlying graph (usually denoted G).A marking of the net is a marking of the edges of G.

Marked graphs model “decision-free” concurrent systems.

Several properties of marked graphs are decidable inpolynomial time:

Structural liveness and boundedness.Liveness and boundedness for a given initial marking.

Simple condition for realisability (and thus reachability).

Introduction

Basics


AnalysisTechniques


Conclusion

Example: Marked and Underlying Graphs

SD

RA

RD

SA

SD

RA

RD

SA

••

U

S

P

C

US

P

C••

•

Introduction

Basics


AnalysisTechniques


Conclusion

Some Properties of Marked Graphs

Theorem: The total number of tokens on every directedcircuit in the underlying graph is invariant.

Theorem: The maximum number of tokens an edge a→ bin (G,m0) can ever have is equal to the minimum numberof tokens m0 places on any directed circuit that containsthis edge.

Theorem: A marked graph (G,m0) is live iff m0 places atleast one token on every directed circuit of G.

Theorem: A live marked graph (G,m0) is k-bounded iffevery place (edge in G) belongs to a directed circuit and m0

places at most k tokens on every directed circuit of G.

Theorem: A marked graph net has a live and boundedmarking iff G is strongly connected.

Introduction

Basics


AnalysisTechniques


Conclusion

Free Choice Nets

An ordinary Petri net such that |p•| ≤ 1 or •(p•) = {p} foreach place p, is a free choice net.

Equivalently: If p• ∩ p′• 6= ∅ then |p•| = |p′•| = 1, for allp, p′ ∈ P .

Extended free choice net: If p• ∩ p′• 6= ∅ then p• = p′•, forall p, p′ ∈ P .

An extended free choice net can be transformed to a basicfree choice net, adding at most a linear number of placesand transitions.

Note: Marked graphs and state machines are also freechoice nets.

A fundamental property of free choice nets: if •t ∩ •t′ 6= ∅then whenever t is enabled, so is t′.

Introduction

Basics


AnalysisTechniques


Conclusion

Example: Free Choice Nets

p1

p5p4

p7

p2

p3

p6

p2p1

p4p3

Introduction

Basics


AnalysisTechniques


Conclusion

Decomposition of Free Choice Nets

A subnet of N = (P, T,W ) is a net N ′ = (P ′, T ′,W ′) withP ′ ⊆ P , T ′ ⊆ T and W ′ = W|(P ′∪T ′).

The P-subnet induced by S ⊆ P is (S, •S ∪ S•,W ′)That is, the subnet consisting of S and all transitionincident on a place in S.

The T-subnet induced by U ⊆ T is (•U ∪ U•, U,W ′)That is, the subnet consisting of U and all places incidenton a transition in U .

A P-component is a strongly connected P-subnet such that|•t|, |t•| ≤ 1, for all t.

A P-component is a state machine.

A T-component is a strongly connected T-subnet such that|•p|, |p•| ≤ 1, for all p

A T-component is a marked graph.

Introduction

Basics


AnalysisTechniques


Conclusion

Decomposition of Free Choice Nets

Theorem: A live free choice net (N,m0) is 1-bounded(safe) iff it is covered by P-components, each of which has asingle token at m0.

Theorem: A live and safe free choice net (N,m0) iscovered by T-components, and for each T-component, N ′,there is a reachable marking m such that (N ′,m|N ′) is liveand safe.

Introduction

Basics


AnalysisTechniques


Conclusion

Example: Decomposition into State Machines

p1

p5p4

p7

p2

p3

p6

p1

p5p4

p7

p2

p3

p6

Introduction

Basics


AnalysisTechniques


Conclusion

Example: Decomposition into Marked Graphs

p1

p5p4

p7

p2

p3

p6

p1

p5p4

p7

p2

p3

p6

Introduction

Basics


AnalysisTechniques


Conclusion

Siphons and Traps

A siphon is a subset S of places such that •S ⊆ S•.Every transition that outputs a token to a place in S alsoconsumes a token from a place in S.If m places no token in S, no marking reachable from mdoes either.

A trap is a subset S of places such that S• ⊆ •S.

Every transition that consumes a token from a place in Salso outputs a token to a place in S.If m places at least one token in S, so does every markingreachable from m.

Theorem: A free choice net (N,m0) is live iff every siphoncontains a marked trap.

Introduction

Basics


AnalysisTechniques


Conclusion

Example: Siphons and Traps

p1

p5p4

p7

p2

p3

p6

The only siphon is P .

P is also a trap.

p2p1

p4p3

{p2} and {p3, p4} aresiphons.

{p1} and {p3, p4} are traps.

Introduction

Basics


AnalysisTechniques


Conclusion

Some Complexity Results for Free Choice Nets

Liveness for marked free choice nets is decidable inpolynomial time.

Boundedness of live free choice nets is decidable inpolynomial time.

A number of properties of live and bounded free choice netsare decidable in polynomial time, e.g.,

Transition executability and repeated executability.The “home state” property (markings that can always bere-reached).

Reachability in free choice nets is NP-hard.

Introduction

Basics


AnalysisTechniques


Conclusion

Characterisation by Derivation Rules

Initial net:

Rule #1: Add a new place p′ with r(p′) =∑

p∈P λpr(p)

and |p′•| = 1.

Rule #2: Replace place p with a connected P-graph N ′,and connect each input and output of p to at least oneplace in N ′.

Must have |•p| > 1 and |p•| > 1, except for initial net.Every place p′ ∈ N ′ must appear on a path in theresulting net that enters and leaves N ′.

Theorem: The class of nets obtained by applying the aboverules to the initial net is exactly the class of structurally liveand structurally bounded free choice nets.

Introduction

Basics


AnalysisTechniques


Conclusion

Reachability: Acyclic Nets

Recall: m0 [s〉m implies∃n ∈ N|T | : Wn = (m−m0).

A solution n is realisable iffn = n(s) for some valid firingsequence s.

Theorem: For an acyclic net,every solution to Wn = (m−m0)is realisable.

Reachability in acyclic nets isNP-hard.

tx px

p1 t1 p′1

p2 t2 p′2

......

...

pn tn p′n

Introduction

Basics


AnalysisTechniques


Conclusion

Reachability: Marked Graphs

Theorem: In a live marked graph, m is reachable from m0

iff m0 and m place the same total number of tokens onevery fundamental circuit of the underlying graph.

A fundamental circuit is obtained by adding one edge to aspanning tree.The directed fundamental circuits of a marked graph are afull set of linearly independent P-invariants.

SD

RA

RD

SA

••

SD

RA

RD

SA

••

SD

RA

RD

SA

••

Introduction

Basics


AnalysisTechniques


Conclusion

Conclusions

Introduction

Basics


AnalysisTechniques


Conclusion

Summary & Conclusions

Petri nets: Intuitive, graphical modelling formalism, closelyrelated to planning.

Petri net theory offers a different set of tools:

Algebraic methods (based on the state equation).Characterisation and study of classes of nets with specialstructure.

Planning also has tools potentially applicable to Petri nets.

Introduction

Basics


AnalysisTechniques


Conclusion

The Many Things We Haven’t Talked About

Extensions of basic Place-Transition nets:

Read arcs, reset arcs and inhibitor arcs.Colored Petri nets, timed nets, stochastic nets, etc.

Other properties of Petri nets (and related decisionproblems):

Model checking (tense logics, process calculi).Language (trace) properties.

Heaps more results concerning different Petri net subclasses.

Documents

Petri Nets (for Planners) - GitHub Pages · 2020. 9. 1. · A Petri net has places, transitions, and directed arcs Arcs connect places to transitions or vice versa Places contain