Branching Immediate Observation Petri Nets A strong class with - - PowerPoint PPT Presentation
INFINITY Workshop July 7th, 2020 Branching Immediate Observation Petri Nets A strong class with simple reachability Chana Weil-Kennedy joint work with Javier Esparza and Mikhail Raskin The project has received funding from the European Research
joint work with Javier Esparza and Mikhail Raskin
The project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme under grant agreement No 787367
joint work with Javier Esparza and Mikhail Raskin
The project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme under grant agreement No 787367
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S C W R S C W R * Reachability problem: Given a Petri net , and markings and can marking reach marking in ?
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Reachability problem: Given a Petri net , and markings and can marking reach marking in ?
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Reachability problem: Given a Petri net , and markings and can marking reach marking in ?
non-elementary complexity
[Czerwinzki, Lasota, Lazic, Leroux, Mazowiecki, ’19]
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Reachability problem: Given a Petri net , and markings and can marking reach marking in ?
non-elementary complexity
[Czerwinzki, Lasota, Lazic, Leroux, Mazowiecki, ’19]
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Branching Parallel Processes (BPP)
t
[Christensen et al., ’93] [Yen, ’97] [Mayr, Weihmann, ’15] [Lasota, ’09]
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Branching Parallel Processes (BPP)
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[Christensen et al., ’93] [Yen, ’97] [Mayr, Weihmann, ’15] [Lasota, ’09]
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Branching Parallel Processes (BPP)
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2 [Christensen et al., ’93] [Yen, ’97] [Mayr, Weihmann, ’15] [Lasota, ’09]
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Branching Parallel Processes (BPP)
t
2 [Christensen et al., ’93] [Yen, ’97] [Mayr, Weihmann, ’15] [Lasota, ’09]
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Branching Parallel Processes (BPP)
t
2 [Christensen et al., ’93] [Yen, ’97] [Mayr, Weihmann, ’15] [Lasota, ’09]
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Branching Parallel Processes (BPP)
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t
[Christensen et al., ’93] [Yen, ’97] [Mayr, Weihmann, ’15] [Lasota, ’09]
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Branching Parallel Processes (BPP)
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2
t
Immediate Observation nets (IO)
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[Esparza, Raskin, W.-K., ’19] [Christensen et al., ’93] [Yen, ’97] [Mayr, Weihmann, ’15] [Lasota, ’09]
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Branching Parallel Processes (BPP)
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2
t
Immediate Observation nets (IO)
t
[Esparza, Raskin, W.-K., ’19] [Christensen et al., ’93] [Yen, ’97] [Mayr, Weihmann, ’15] [Lasota, ’09]
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Immediate Observation Conservative Branching Immediate Observation General Petri nets (BIO) (IO) (BPP) Branching Parallel Processes
non-elementary PSPACE-complete NP-complete
[Czerwinzki, Lasota, Lazic, Leroux, Mazowiecki, ’19] [Esparza, Raskin, W.-K., ’19] [Esparza, ’97]
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PSPACE-complete
Immediate Observation Conservative Branching Immediate Observation General Petri nets (BIO) (IO) (BPP) Branching Parallel Processes
non-elementary PSPACE-complete NP-complete
[Czerwinzki, Lasota, Lazic, Leroux, Mazowiecki, ’19] [Esparza, Raskin, W.-K., ’19] [Esparza, ’97]
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Unbounded Petri net classes with provably simpler reachability then the general case have semilinear reachability sets (e.g. BPP nets, reversible Petri nets…)
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BIO nets may have non-semilinear reachability set
[Hopcroft, Pansiot, ’79] example
BIO net
VASS to Petri net classic translation 2 c1 p c2 q c3 t4 t2 t1 t3
Unbounded Petri net classes with provably simpler reachability then the general case have semilinear reachability sets (e.g. BPP nets, reversible Petri nets…)
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Unbounded Petri net classes with provably simpler reachability then the general case have semilinear reachability sets (e.g. BPP nets, reversible Petri nets…) BIO nets may have non-semilinear reachability set
[Hopcroft, Pansiot, ’79] example
BIO net
VASS to Petri net classic translation 2 c1 p c2 q c3 t4 t2 t1 t3
10
Unbounded Petri net classes with provably simpler reachability then the general case have semilinear reachability sets (e.g. BPP nets, reversible Petri nets…) BIO nets may have non-semilinear reachability set
[Hopcroft, Pansiot, ’79] example
BIO net
VASS to Petri net classic translation 2 c1 p c2 q c3 t4 t2 t1 t3
10
Unbounded Petri net classes with provably simpler reachability then the general case have semilinear reachability sets (e.g. BPP nets, reversible Petri nets…) BIO nets may have non-semilinear reachability set
[Hopcroft, Pansiot, ’79] example
BIO net
VASS to Petri net classic translation 2 c1 p c2 q c3 t4 t2 t1 t3
10
Unbounded Petri net classes with provably simpler reachability then the general case have semilinear reachability sets (e.g. BPP nets, reversible Petri nets…) BIO nets may have non-semilinear reachability set
[Hopcroft, Pansiot, ’79] example
BIO net
VASS to Petri net classic translation 2 c1 p c2 q c3 t4 t2 t1 t3
10
Unbounded Petri net classes with provably simpler reachability then the general case have semilinear reachability sets (e.g. BPP nets, reversible Petri nets…) BIO nets may have non-semilinear reachability set
[Hopcroft, Pansiot, ’79] example
BIO net
VASS to Petri net classic translation 2 c1 p c2 q c3 t4 t2 t1 t3
10
Unbounded Petri net classes with provably simpler reachability then the general case have semilinear reachability sets (e.g. BPP nets, reversible Petri nets…) BIO nets may have non-semilinear reachability set
[Hopcroft, Pansiot, ’79] example
BIO net
VASS to Petri net classic translation 2 c1 p c2 q c3 t4 t2 t1 t3
10
Unbounded Petri net classes with provably simpler reachability then the general case have semilinear reachability sets (e.g. BPP nets, reversible Petri nets…) BIO nets may have non-semilinear reachability set
[Hopcroft, Pansiot, ’79] example
BIO net
VASS to Petri net classic translation 2 c1 p c2 q c3 t4 t2 t1 t3
10
Unbounded Petri net classes with provably simpler reachability then the general case have semilinear reachability sets (e.g. BPP nets, reversible Petri nets…) BIO nets may have non-semilinear reachability set
[Hopcroft, Pansiot, ’79] example
BIO net
VASS to Petri net classic translation 2 c1 p c2 q c3 t4 t2 t1 t3
10
Unbounded Petri net classes with provably simpler reachability then the general case have semilinear reachability sets (e.g. BPP nets, reversible Petri nets…) BIO nets may have non-semilinear reachability set
[Hopcroft, Pansiot, ’79] example
BIO net
VASS to Petri net classic translation 2 c1 p c2 q c3 t4 t2 t1 t3
10
Unbounded Petri net classes with provably simpler reachability then the general case have semilinear reachability sets (e.g. BPP nets, reversible Petri nets…) BIO nets may have non-semilinear reachability set
[Hopcroft, Pansiot, ’79] example
BIO net
VASS to Petri net classic translation 2 c1 p c2 q c3 t4 t2 t1 t3
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Unbounded Petri net classes with provably simpler reachability then the general case have semilinear reachability sets (e.g. BPP nets, reversible Petri nets…) BIO nets may have non-semilinear reachability set
[Hopcroft, Pansiot, ’79] example
BIO net
VASS to Petri net classic translation 2 c1 p c2 q c3 t4 t2 t1 t3
c2 + c3 ≤ 2c1
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BIO nets reachability is a PSPACE-complete problem
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BIO nets reachability is a PSPACE-complete problem
sequences of bounded length and bounded token count.
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If M0
* M
M0
tk1
1 M1
tk2
2 M2 → …
tkl
l Ml = M
then ∃ markings M1, M2, …, Ml ∃ transitions t1, t2, …, tl ∃ constants k1, k2, …kl ≥ 0
Main Theorem
In a BIO net with n places, and transitions producing ≤ γ tokens
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If M0
* M
M0
tk1
1 M1
tk2
2 M2 → …
tkl
l Ml = M
then ∃ markings M1, M2, …, Ml ∃ transitions t1, t2, …, tl ∃ constants k1, k2, …kl ≥ 0
Main Theorem
In a BIO net with n places, and transitions producing ≤ γ tokens such that l ∈ O(|M|n)n l
bound on (accelerated) length
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If M0
* M
M0
tk1
1 M1
tk2
2 M2 → …
tkl
l Ml = M
then ∃ markings M1, M2, …, Ml ∃ transitions t1, t2, …, tl ∃ constants k1, k2, …kl ≥ 0
Main Theorem
In a BIO net with n places, and transitions producing ≤ γ tokens such that l ∈ O(|M|n)n l
bound on (accelerated) length
and ∀i, Mi ∈ O(|Mo||M|nγ)n
bound on token count
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If M0
* M
M0
tk1
1 M1
tk2
2 M2 → …
tkl
l Ml = M
then ∃ markings M1, M2, …, Ml ∃ transitions t1, t2, …, tl ∃ constants k1, k2, …kl ≥ 0
Main Theorem
In a BIO net with n places, and transitions producing ≤ γ tokens such that l ∈ O(|M|n)n l
bound on (accelerated) length
and ∀i, Mi ∈ O(|Mo||M|nγ)n
bound on token count
token count accelerated length
|M| |M0|
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If M0
* M
M0
tk1
1 M1
tk2
2 M2 → …
tkl
l Ml = M
then ∃ markings M1, M2, …, Ml ∃ transitions t1, t2, …, tl ∃ constants k1, k2, …kl ≥ 0
Main Theorem
In a BIO net with n places, and transitions producing ≤ γ tokens such that l ∈ O(|M|n)n l
bound on (accelerated) length
and ∀i, Mi ∈ O(|Mo||M|nγ)n
bound on token count
token count accelerated length
|M| |M0|
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NPSPACE algorithm for reachability
* M ?
If M0
* M
M0
tk1
1 M1
tk2
2 M2 → …
tkl
l Ml = M
then ∃ markings M1, M2, …, Ml ∃ transitions t1, t2, …, tl ∃ constants k1, k2, …kl ≥ 0 In a BIO net with n places, and transitions producing ≤ γ tokens such that l ∈ O(|M|n)n and ∀i, Mi ∈ O(|Mo||M|nγ)n
Main Theorem
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NPSPACE algorithm for reachability
* M ?
Guess the first marking M1 Check that there such that M0
tk
Guess the next marking M2 …
If M0
* M
M0
tk1
1 M1
tk2
2 M2 → …
tkl
l Ml = M
then ∃ markings M1, M2, …, Ml ∃ transitions t1, t2, …, tl ∃ constants k1, k2, …kl ≥ 0 In a BIO net with n places, and transitions producing ≤ γ tokens such that l ∈ O(|M|n)n and ∀i, Mi ∈ O(|Mo||M|nγ)n
Main Theorem
M0
q p q
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M0
q p q
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r
M1
q q r r
M0
q p q
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M0
q r p
M1
q q q r r
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M2
q q q r r r
M0
q r p
M1
q q q r r
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M0
q r p
M1 M2
q q q q q q r r r r r
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M3
q q q r r
M0
q r p
M1 M2
q q q q q q r r r r r
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M0
q r p
M1 M2
q q q q q q r r r r r
M3
q q q r r
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M
r r
M0
q r p
M1 M2
q q q q q q r r r r r
M3
q q q r r
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M
r r
M0
q r p
M1 M2
q q q q q q r r r r r
M3
q q q r r
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M
r r
M0
q r p
M1 M2
q q q q q q r r r r r
M3
q q q r r
final tokens
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M
r r
M0
q r p
M1 M2
q q q q q q r r r r r
M3
q q q r r
final tokens
helper tokens
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M
r r
M0
q r p
M1 M2
q q q q q q r r r r r
M3
q q q r r
final tokens helper tokens
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M
r r
M0
q r p
M1 M2
q q q q q q r r r r r
M3
q q q r r
final tokens helper tokens
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M
r r
M0
q r p
M1 M2
q q q q q q r r r r r
M3
q q q r r
final tokens helper tokens
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M0 M
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M0 M
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M0 M
p q q q q q q r r r r r r r r r r q q p p p p
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M0 M
p q q q q q q r r r r r r r r r r q q p p p p
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M0 M
p q q q q q q r r r r r r r r r r q q p p p p
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M0 M
p q q q q q q r r r q q q q p p q q
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M0 M M4 M7
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M0 M M4 M7
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M0 M
M4 M8
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M0 M
M4 M8
per intermediate marking
≤ |M|
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M0 M
M4 M8
per intermediate marking
≤ |M|
per intermediate marking
≤ n2
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M0 M
M4 M8
per intermediate marking
≤ |M|
accelerated length O(# such configurations)
per intermediate marking
≤ n2
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flat
[Leroux, Sutre, ’05]
sequence such that iff
∃ t*
1 t* 2 …t* l
M0
* M
M0
tk1
1 tk2 2 …tkl l M
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flat BPP , IO nets
[Leroux, Sutre, ’05]
sequence such that iff
∃ t*
1 t* 2 …t* l
M0
* M
M0
tk1
1 tk2 2 …tkl l M
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flat BPP , IO nets
[Leroux, Sutre, ’05]
sequence such that iff
∃ t*
1 t* 2 …t* l
M0
* M
M0
tk1
1 tk2 2 …tkl l M
pre*-flat
sequence such that iff
∀M, ∃ t*
1 t* 2 …t* l
M0
* M
M0
tk1
1 tk2 2 …tkl l M
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flat BPP , IO nets
If M0
* M
M0
tk1
1 M1
tk2
2 M2 → …
tkl
l Ml = M
then ∃ markings M1, M2, …, Ml ∃ transitions t1, t2, …, tl ∃ constants k1, k2, …kl ≥ 0 In a BIO net with n places, and transitions producing ≤ γ tokens such that l ∈ O(|M|n)n and ∀i, Mi ∈ O(|Mo||M|nγ)n
Main Theorem [Leroux, Sutre, ’05]
sequence such that iff
∃ t*
1 t* 2 …t* l
M0
* M
M0
tk1
1 tk2 2 …tkl l M
pre*-flat
sequence such that iff
∀M, ∃ t*
1 t* 2 …t* l
M0
* M
M0
tk1
1 tk2 2 …tkl l M
30
flat BPP , IO nets
If M0
* M
M0
tk1
1 M1
tk2
2 M2 → …
tkl
l Ml = M
then ∃ markings M1, M2, …, Ml ∃ transitions t1, t2, …, tl ∃ constants k1, k2, …kl ≥ 0 In a BIO net with n places, and transitions producing ≤ γ tokens such that l ∈ O(|M|n)n and ∀i, Mi ∈ O(|Mo||M|nγ)n
Main Theorem [Leroux, Sutre, ’05]
sequence such that iff
∃ t*
1 t* 2 …t* l
M0
* M
M0
tk1
1 tk2 2 …tkl l M
pre*-flat
sequence such that iff
∀M, ∃ t*
1 t* 2 …t* l
M0
* M
M0
tk1
1 tk2 2 …tkl l M
BIO nets
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flat BPP , IO nets model checking tools with acceleration techniques e.g. FAST [Bardin, Finkel, Leroux, Petrucci, ’03]
If M0
* M
M0
tk1
1 M1
tk2
2 M2 → …
tkl
l Ml = M
then ∃ markings M1, M2, …, Ml ∃ transitions t1, t2, …, tl ∃ constants k1, k2, …kl ≥ 0 In a BIO net with n places, and transitions producing ≤ γ tokens such that l ∈ O(|M|n)n and ∀i, Mi ∈ O(|Mo||M|nγ)n
Main Theorem [Leroux, Sutre, ’05]
sequence such that iff
∃ t*
1 t* 2 …t* l
M0
* M
M0
tk1
1 tk2 2 …tkl l M
pre*-flat
sequence such that iff
∀M, ∃ t*
1 t* 2 …t* l
M0
* M
M0
tk1
1 tk2 2 …tkl l M
BIO nets
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IO Conservative General Petri nets BIO BPP non-elementary
[Czerwinzki, Lasota, Lazic, Leroux, Mazowiecki, ’19]
PSPACE-complete
[Esparza, Raskin, W.-K., ’19]
NP-complete
[Esparza, ’97]
PSPACE-complete
[Esparza, Raskin, W.-K., ’20]
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Other results:
IO Conservative General Petri nets BIO BPP non-elementary
[Czerwinzki, Lasota, Lazic, Leroux, Mazowiecki, ’19]
PSPACE-complete
[Esparza, Raskin, W.-K., ’19]
NP-complete
[Esparza, ’97]
PSPACE-complete
[Esparza, Raskin, W.-K., ’20]
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Other results:
IO Conservative General Petri nets BIO BPP non-elementary
[Czerwinzki, Lasota, Lazic, Leroux, Mazowiecki, ’19]
PSPACE-complete
[Esparza, Raskin, W.-K., ’19]
NP-complete
[Esparza, ’97]
PSPACE-complete
[Esparza, Raskin, W.-K., ’20]
Future: Investigate consequences in chemical reaction networks, formal languages, etc.