On the Upward/Downward Closures of Petri Nets Mohammed Faouzi Atig 1 - - PowerPoint PPT Presentation

On the Upward/Downward Closures of Petri Nets

Mohammed Faouzi Atig1, Roland Meyer2, Sebastian Muskalla2 and Prakash Saivasan2 August 25, MFCS 2017, Aalborg

1 Uppsala University, Sweden mohamed_faouzi.atig@it.uu.se 2 TU Braunschweig, Germany {roland.meyer, s.muskalla, p.saivasan}@tu-bs.de

Goal

Goal Study the upward and downward closures of Petri net coverability languages

1

Goal

Goal Study the upward and downward closures of Petri net coverability languages Why? Petri nets are an important model for concurrent systems Upward and downward closures are useful approximations for verification purposes

1

Goal

Goal Study the upward and downward closures of Petri net coverability languages Upward/Downward closures in general Good: Always simply regular Bad: Representations might be not be effectively computable

• r very large

1

Goal

Goal Study the upward and downward closures of Petri net coverability languages Upward/Downward closures in general Good: Always simply regular Bad: Representations might be not be effectively computable

• r very large

1

Goal

Goal Study the upward and downward closures of Petri net coverability languages Upward/Downward closures in general Good: Always simply regular Bad: Representations might be not be effectively computable

• r very large

1

Goal

Goal Study the upward and downward closures of Petri net coverability languages Here: Closures effectively regular Want to construct finite state automata (FSA) as representations Time needed for the construction? Size of the minimal FSAs?

1

Goal

Goal Study the upward and downward closures of Petri net coverability languages Here: Closures effectively regular Want to construct finite state automata (FSA) as representations ↰ Time needed for the construction? ↰ Size of the minimal FSAs?

1

Petri Net Coverability Languages and their Closures

(Labeled) Petri Nets

p0 t0

ε

p1 t1

ε

p2 t2

ε

p3 tf

a

pf 2 2 2

N = ( a { p0, p1, p2, p3, pf }

• places P

, { t0, t1, t2, tf }

• transitions T

, F )

2

(Labeled) Petri Nets

p0 t0

ε

p1 t1

ε

p2 t2

ε

p3 tf

a

pf 2 2 2

N = ( a { p0, p1, p2, p3, pf }

• places P

, { t0, t1, t2, tf }

• transitions T

, F )         1         = M0

2

(Labeled) Petri Nets

p0 t0

ε

p1 t1

ε

p2 t2

ε

p3 tf

a

pf 2 2 2

N = ( a { p0, p1, p2, p3, pf }

• places P

, { t0, t1, t2, tf }

• transitions T

, F )         1         = M0 [t0⟩

2

(Labeled) Petri Nets

p0 t0

ε

p1 t1

ε

p2 t2

ε

p3 tf

a

pf 2 2 2

N = ( a { p0, p1, p2, p3, pf }

• places P

, { t0, t1, t2, tf }

• transitions T

, F )         1         = M0 [t0⟩         23         = Mf

2

(Labeled) Petri Nets

p0 t0

ε

p1 t1

ε

p2 t2

ε

p3 tf

a

pf 2 2 2

N = ( a { p0, p1, p2, p3, pf }

• places P

, { t0, t1, t2, tf }

• transitions T

, F )         1         = M0 [t0t1t1⟩         23         = Mf

2

(Labeled) Petri Nets

p0 t0

ε

p1 t1

ε

p2 t2

ε

p3 tf

a

pf 2 2 2

N = ( a { p0, p1, p2, p3, pf }

• places P

, { t0, t1, t2, tf }

• transitions T

, F )         1         = M0 [t0t1t1t2t2t2t2⟩         23         = Mf

2

(Labeled) Petri Nets

p0 t0

ε

p1 t1

ε

p2 t2

ε

p3 tf

a

pf 2 2 2

N = ( a { p0, p1, p2, p3, pf }

• places P

, { t0, t1, t2, tf }

• transitions T

, F )         1         = M0 [t0t1t1t2t2t2t2tftftftftftftftf⟩         23         = Mf

2

(Labeled) Petri Nets

p0 t0

ε

p1 t1

ε

p2 t2

ε

p3 tf

a

pf 2 2 2

N = ( a { p0, p1, p2, p3, pf }

• places P

, { t0, t1, t2, tf }

• transitions T

, F )         1         = M0 [t0t1t1t2t2t2t2tftftftftftftftf⟩         23         = Mf         8         ⩾

2

(Labeled) Petri Nets

p0 t0

ε

p1 t1

ε

p2 t2

ε

p3 tf

a

pf 2 2 2

N = ( {a}, { p0, p1, p2, p3, pf }

• places P

, { t0, t1, t2, tf }

• transitions T

, F, λ )         1         = M0 [t0t1t1t2t2t2t2tftftftftftftftf⟩         23         = Mf         8         ⩾

2

(Labeled) Petri Nets

p0 t0

ε

p1 t1

ε

p2 t2

ε

p3 tf

a

pf 2 2 2

N = ( {a}, { p0, p1, p2, p3, pf }

• places P

, { t0, t1, t2, tf }

• transitions T

, F, λ )         1         = M0 [t0t1t1t2t2t2t2tftftftftftftftf⟩         23         = Mf         8         ⩾

→

λ εεεεεεεaaaaaaaa = a8

2

Coverability Language

(N, M0, Mf) Petri net with initial and final marking Coverability language L ( N, M0, Mf ) = { λ(σ)

• M0[σ⟩M, M ⩾ Mf

}

3

Coverability Language

(N, M0, Mf) Petri net with initial and final marking Coverability language L ( N, M0, Mf ) = { λ(σ)

• M0[σ⟩M, M ⩾ Mf

} In the example: L ( N, M0, Mf ) = { a8}

3

Upward and Downward Closure

Subword relation v⪯ w iff v obtained from w by deleting letters iff w obtained from v by inserting letters

4

Upward and Downward Closure

Subword relation v⪯ w iff v obtained from w by deleting letters iff w obtained from v by inserting letters Upward closure L↑ = {w | ∃v ∈ L: v⪯ w}

4

Upward and Downward Closure

Subword relation v⪯ w iff v obtained from w by deleting letters iff w obtained from v by inserting letters Upward closure L↑ = {w | ∃v ∈ L: v⪯ w} Downward closure L↓ = {w | ∃v ∈ L: w⪯ v}

4

Upward and Downward Closure

Subword relation v⪯ w iff v obtained from w by deleting letters iff w obtained from v by inserting letters Upward closure L↑ = {w | ∃v ∈ L: v⪯ w} Downward closure L↓ = {w | ∃v ∈ L: w⪯ v} In the example: L ( N, M0, Mf ) ↑ = { ak k ⩾ 8 } L ( N, M0, Mf ) ↓ = { ak k ⩽ 8 }

4

Computing the Upward Closure

Computing the Upward Closure

Computing the Upward Closure Given: Petri net (N, M0, Mf). Compute: FSA A with L(A) = L ( N, M0, Mf ) ↑.

5

Computing the Upward Closure

Computing the Upward Closure Given: Petri net (N, M0, Mf). Compute: FSA A with L(A) = L ( N, M0, Mf ) ↑. Theorem Upper bound: One can compute an FSA of doubly exponential size representing the upward closure in doubly exponential time. Lower bound: This is optimal.

5

Computing the Upward Closure - Upper Bound

Lemma (Upper Bound) One can compute an FSA of doubly exponential size for the upward closure in doubly exponential time.

6

Computing the Upward Closure - Upper Bound

Lemma (Upper Bound) One can compute an FSA of doubly exponential size for the upward closure in doubly exponential time. Theorem (Rackoff 1978) Petri net coverability can be solved using exponential space (and doubly exponential time).

6

Computing the Upward Closure - Upper Bound

Theorem (Rackoff 1978) Petri net coverability can be solved using exponential space (and doubly exponential time).

7

Computing the Upward Closure - Upper Bound

Theorem (Rackoff 1978) Petri net coverability can be solved using exponential space (and doubly exponential time). Assume places are ordered, P = [1..ℓ].

7

Computing the Upward Closure - Upper Bound

Theorem (Rackoff 1978) Petri net coverability can be solved using exponential space (and doubly exponential time). Assume places are ordered, P = [1..ℓ]. Consider i-bounded computations: Allow negative values on the places [i + 1..ℓ]

7

Computing the Upward Closure - Upper Bound

Theorem (Rackoff 1978) Petri net coverability can be solved using exponential space (and doubly exponential time). Assume places are ordered, P = [1..ℓ]. Consider i-bounded computations: Allow negative values on the places [i + 1..ℓ] Define f(i) upper bound on the length of an i-bounded, i-covering computation from an arbitrary initial marking

7

Computing the Upward Closure - Upper Bound

Theorem (Rackoff 1978) Petri net coverability can be solved using exponential space (and doubly exponential time). Assume places are ordered, P = [1..ℓ]. Consider i-bounded computations: Allow negative values on the places [i + 1..ℓ] Define f(i) upper bound on the length of an i-bounded, i-covering computation from an arbitrary initial marking Prove f(ℓ) ⩽ 22O(n·log n)

7

Computing the Upward Closure - Upper Bound

Theorem (Rackoff 1978) Petri net coverability can be solved using exponential space (and doubly exponential time). Assume places are ordered, P = [1..ℓ]. Consider i-bounded computations: Allow negative values on the places [i + 1..ℓ] Define f(i) upper bound on the length of an i-bounded, i-covering computation from an arbitrary initial marking Prove f(ℓ) ⩽ 22O(n·log n) Show f(i + 1) ⩽ (2nf(i))i+1 + f(i)

7

Computing the Upward Closure - Upper Bound

f(i + 1) ⩽ (2nf(i))i+1 + f(i) Take an arbitrary (i + 1)-bounded, (i + 1)-covering computation

8

Computing the Upward Closure - Upper Bound

f(i + 1) ⩽ (2nf(i))i+1 + f(i) Take an arbitrary (i + 1)-bounded, (i + 1)-covering computation 1st case: Values on all places [1..i + 1] bounded by 2n · f(i): Identify repetitions Delete loops Obtain new computation of length at most 2nf i

i 1

2nf(i)

8

Computing the Upward Closure - Upper Bound

f(i + 1) ⩽ (2nf(i))i+1 + f(i) Take an arbitrary (i + 1)-bounded, (i + 1)-covering computation 1st case: Values on all places [1..i + 1] bounded by 2n · f(i): Identify repetitions Delete loops Obtain new computation of length at most 2nf i

i 1

2nf(i)

8

Computing the Upward Closure - Upper Bound

f(i + 1) ⩽ (2nf(i))i+1 + f(i) Take an arbitrary (i + 1)-bounded, (i + 1)-covering computation 1st case: Values on all places [1..i + 1] bounded by 2n · f(i): Identify repetitions Delete loops Obtain new computation of length at most 2nf i

i 1

2nf(i)

8

Computing the Upward Closure - Upper Bound

f(i + 1) ⩽ (2nf(i))i+1 + f(i) Take an arbitrary (i + 1)-bounded, (i + 1)-covering computation 1st case: Values on all places [1..i + 1] bounded by 2n · f(i): Identify repetitions Delete loops ↰ Obtain new computation of length at most (2nf(i))i+1 2nf(i) ⩽ (2nf(i))i+1

8

Computing the Upward Closure - Upper Bound

2nd case: Some place, say i + 1, exceeds 2n · f(i): Treat first part as in 1st case Replace second part by i-covering, i-bounded computation of length f i 2nf(i)

9

Computing the Upward Closure - Upper Bound

2nd case: Some place, say i + 1, exceeds 2n · f(i): Treat first part as in 1st case Replace second part by i-covering, i-bounded computation of length f i 2nf(i) ⩽ (2nf(i))i+1

9

Computing the Upward Closure - Upper Bound

2nd case: Some place, say i + 1, exceeds 2n · f(i): Treat first part as in 1st case Replace second part by i-covering, i-bounded computation of length ⩽ f(i) 2nf(i) ⩽ (2nf(i))i+1

9

Computing the Upward Closure - Upper Bound

2nd case: Some place, say i + 1, exceeds 2n · f(i): Treat first part as in 1st case Replace second part by i-covering, i-bounded computation of length ⩽ f(i) 2nf(i) ⩽ (2nf(i))i+1 ⩽ f(i)

9

Computing the Upward Closure - Upper Bound

What do we need to change? Definition of f i : upper bound on the length of a i-bounded, i-covering computations from an arbitrary initial marking that generate all minimal words 1st case: Deleting loops creates a subword 2nd case: Replacing second part of the computation handle with care

10

Computing the Upward Closure - Upper Bound

What do we need to change? Definition of f(i): upper bound on the length of a i-bounded, i-covering computations from an arbitrary initial marking that generate all minimal words 1st case: Deleting loops creates a subword 2nd case: Replacing second part of the computation handle with care

10

Computing the Upward Closure - Upper Bound

What do we need to change? Definition of f(i): upper bound on the length of a i-bounded, i-covering computations from an arbitrary initial marking that generate all minimal words 1st case: Deleting loops creates a subword ✓ 2nd case: Replacing second part of the computation handle with care

10

Computing the Upward Closure - Upper Bound

What do we need to change? Definition of f(i): upper bound on the length of a i-bounded, i-covering computations from an arbitrary initial marking that generate all minimal words 1st case: Deleting loops creates a subword ✓ 2nd case: Replacing second part of the computation ✗ handle with care

10

Computing the Upward Closure - Upper Bound

What do we need to change? Definition of f(i): upper bound on the length of a i-bounded, i-covering computations from an arbitrary initial marking that generate all minimal words 1st case: Deleting loops creates a subword ✓ 2nd case: Replacing second part of the computation ✗ ↰ handle with care

10

Computing the Upward Closure - Upper Bound

Finally: The minimal words of L ( N, M0, Mf ) ↑ have a computation of length ⩽ f(ℓ) ⩽ 22O(n·log n).

11

Computing the Upward Closure - Upper Bound

Finally: The minimal words of L ( N, M0, Mf ) ↑ have a computation of length ⩽ f(ℓ) ⩽ 22O(n·log n). FSA can simulate the net for f(ℓ) steps to accept them (and their upward-closure)

11

Computing the Upward Closure - Lower Bound

Lemma (Lower Bound) There is a family of Petri nets such that the upward closure cannot be represented by an FSA of less than doubly exponential size.

12

Computing the Upward Closure - Lower Bound

Lemma (Lower Bound) There is a family of Petri nets such that the upward closure cannot be represented by an FSA of less than doubly exponential size. Theorem (Lipton 1976) Petri net reachability is EXPSPACE-hard.

12

Computing the Upward Closure - Lower Bound

Theorem (Lipton 1976) Petri net reachability is EXPSPACE-hard.

13

Computing the Upward Closure - Lower Bound

Theorem (Lipton 1976) Petri net reachability is EXPSPACE-hard. In the proof, a Petri net of size polynomial in n simulates a counter machine with counter values bounded by 22n (including zero tests!)

13

Computing the Upward Closure - Lower Bound

Theorem (Lipton 1976) Petri net reachability is EXPSPACE-hard. In the proof, a Petri net of size polynomial in n simulates a counter machine with counter values bounded by 22n (including zero tests!) Using this idea, we construct for each n ∈ N a Petri net with L ( N(n), M0, Mf ) = { a22n} .

13

Computing the Upward Closure - Lower Bound

Theorem (Lipton 1976) Petri net reachability is EXPSPACE-hard. In the proof, a Petri net of size polynomial in n simulates a counter machine with counter values bounded by 22n (including zero tests!) Using this idea, we construct for each n ∈ N a Petri net with L ( N(n), M0, Mf ) = { a22n} . We obtain L ( N(n), M0, Mf ) ↑ = { ak

• k ⩾ 22n}

.

13

Computing the Downward Closure

Computing the Downward Closure

Computing the Downward Closure Given: Petri net (N, M0, Mf). Compute: FSA A with L(A) = L ( N, M0, Mf ) ↓.

14

Computing the Downward Closure

Computing the Downward Closure Given: Petri net (N, M0, Mf). Compute: FSA A with L(A) = L ( N, M0, Mf ) ↓. Theorem Upper bound: One can compute an FSA of non-primitive recursive size representing the downward closure (in non-primitive recursive time). Lower bound: This is optimal.

14

Computing the Downward Closure - Upper Bound

Lemma (Upper Bound) One can compute an FSA of non-primitive recursive size representing the downward closure.

15

Computing the Downward Closure - Upper Bound

Lemma (Upper Bound) One can compute an FSA of non-primitive recursive size representing the downward closure. Proof Sketch. The Karp-Miller tree (coverability graph) of the Petri net can be seen as finite automaton KMT Its language is a subset of the downward closure, N M0 Mf KMT N M0 Mf KMT N M0 Mf Its size might be non-primitive recursive

15

Computing the Downward Closure - Upper Bound

Lemma (Upper Bound) One can compute an FSA of non-primitive recursive size representing the downward closure. Proof Sketch. The Karp-Miller tree (coverability graph) of the Petri net can be seen as finite automaton KMT Its language is a subset of the downward closure, N M0 Mf KMT N M0 Mf KMT N M0 Mf Its size might be non-primitive recursive

15

Computing the Downward Closure - Upper Bound

Lemma (Upper Bound) One can compute an FSA of non-primitive recursive size representing the downward closure. Proof Sketch. The Karp-Miller tree (coverability graph) of the Petri net can be seen as finite automaton KMT Its language is a subset of the downward closure, L ( N, M0, Mf ) ⊆ L(KMT) ⊆ L ( N, M0, Mf ) ↓ KMT N M0 Mf Its size might be non-primitive recursive

15

Computing the Downward Closure - Upper Bound

Lemma (Upper Bound) One can compute an FSA of non-primitive recursive size representing the downward closure. Proof Sketch. The Karp-Miller tree (coverability graph) of the Petri net can be seen as finite automaton KMT Its language is a subset of the downward closure, L ( N, M0, Mf ) ⊆ L(KMT) ⊆ L ( N, M0, Mf ) ↓ L(KMT)↓= L ( N, M0, Mf ) ↓ Its size might be non-primitive recursive

15

Computing the Downward Closure - Upper Bound

Lemma (Upper Bound) One can compute an FSA of non-primitive recursive size representing the downward closure. Proof Sketch. The Karp-Miller tree (coverability graph) of the Petri net can be seen as finite automaton KMT Its language is a subset of the downward closure, L ( N, M0, Mf ) ⊆ L(KMT) ⊆ L ( N, M0, Mf ) ↓ L(KMT)↓= L ( N, M0, Mf ) ↓ Its size might be non-primitive recursive

15

Computing the Downward Closure - Lower Bound

Lemma (Lower Bound) There is a family of Petri nets such that the downward closure cannot be represented by an FSA of primitive recursive size.

16

Computing the Downward Closure - Lower Bound

Lemma (Lower Bound) There is a family of Petri nets such that the downward closure cannot be represented by an FSA of primitive recursive size. Inductive construction from [Mayr, Meyer 1981], adapted to labeled Petri nets: ∀n, x ∈ N ∃ ( N(n), M(x)

0 , Mf

) polynomial in (n + x) such that L ( N(n), M(x)

0 , Mf

) = { ak

• k ⩽ Acker(n, x)

} = L ( N(n), M(x)

0 , Mf

) ↓

16

Computing the Downward Closure - Lower Bound

Lemma (Lower Bound) There is a family of Petri nets such that the downward closure cannot be represented by an FSA of primitive recursive size. Inductive construction from [Mayr, Meyer 1981], adapted to labeled Petri nets: ∀n, x ∈ N ∃ ( N(n), M(x)

0 , Mf

) polynomial in (n + x) such that L ( N(n), M(x)

0 , Mf

) = { ak

• k ⩽ Acker(n, x)

} = L ( N(n), M(x)

0 , Mf

) ↓

in0 tcp0

• ut0

strt0 tstrt0 cp0 tstp0 stp0

inn+1 strtn+1 ANn inn strtn

• utn

stpn tstrtn+1 tswpn+1 swpn+1 trestartn+1 tinn+1 tcpn+1 ttmpn+1 tmpn+1 tstpn+1 stpn+1

• utn+1

16

SRE in Downward Closure

Simple Regular Expression

Simple regular expression sre ::= p sre + sre p ::= a (a + ε) Γ∗ p.p where Γ ⊆ Σ Known: Downward and upward closures can be described by SREs

17

SRE in Downward Closure

SRE in Downward Closure Given: SRE sre, Petri net (N, M0, Mf). Decide: L(sre) ⊆ L ( N, M0, Mf ) ↓ ?

18

SRE in Downward Closure

SRE in Downward Closure Given: SRE sre, Petri net (N, M0, Mf). Decide: L(sre) ⊆ L ( N, M0, Mf ) ↓ ? Theorem SRE in Downward Closure is EXPSPACE-complete.

18

SRE in Downward Closure - Upper Bound

Lemma (Upper Bound) SRE in Downward Closure can be solved in EXPSPACE.

19

SRE in Downward Closure - Upper Bound

Lemma (Upper Bound) SRE in Downward Closure can be solved in EXPSPACE. SRE is a choice among products sre ::= p sre + sre Show inclusion for each product separately p a a p p Problem: Need to enforce that for each word in , a covering computation exists

19

SRE in Downward Closure - Upper Bound

Lemma (Upper Bound) SRE in Downward Closure can be solved in EXPSPACE. SRE is a choice among products sre ::= p sre + sre Show inclusion for each product separately p ::= a (a + ε) Γ∗ p.p Problem: Need to enforce that for each word in , a covering computation exists

19

SRE in Downward Closure - Upper Bound

Lemma (Upper Bound) SRE in Downward Closure can be solved in EXPSPACE. SRE is a choice among products sre ::= p sre + sre Show inclusion for each product separately p ::= a (a + ε) Γ∗ p.p Problem: Need to enforce that for each word in Γ∗, a covering computation exists

19

SRE in Downward Closure - Upper Bound

[Zetzsche 2015]: Downward closures computable iff a certain unboundedness problem decidable

20

SRE in Downward Closure - Upper Bound

[Zetzsche 2015]: Downward closures computable iff a certain unboundedness problem decidable Theorem (Demri 2013) The Simultaneous Unboundedness Problem for Petri Nets is EXPSPACE-complete.

20

SRE in Downward Closure - Upper Bound

[Zetzsche 2015]: Downward closures computable iff a certain unboundedness problem decidable Theorem (Demri 2013) The Simultaneous Unboundedness Problem for Petri Nets is EXPSPACE-complete. Simultaneous Unboundedness Problem for Petri Nets Given: Petri net N, marking M0, set of places X ⊆ P Decide: ∀n ∈ N ∃M0[σ⟩M with M(p) ⩾ n ∀p ∈ X?

20

SRE in Downward Closure - Upper Bound

Lemma (Upper Bound) SRE in Downward Closure can be solved in EXPSPACE. Handle each product p separately For each expression in p, add a place that tracks

• ccurrence of all symbols in

(also track the rest of p) Check whether the places for the are simultaneously unbounded

21

SRE in Downward Closure - Upper Bound

Lemma (Upper Bound) SRE in Downward Closure can be solved in EXPSPACE. Handle each product p separately For each expression Γ∗ in p, add a place that tracks

• ccurrence of all symbols in Γ

(also track the rest of p) Check whether the places for the are simultaneously unbounded

21

SRE in Downward Closure - Upper Bound

Lemma (Upper Bound) SRE in Downward Closure can be solved in EXPSPACE. Handle each product p separately For each expression Γ∗ in p, add a place that tracks

• ccurrence of all symbols in Γ

(also track the rest of p) Check whether the places for the Γ∗ are simultaneously unbounded

21

SRE in Downward Closure - Lower Bound

Lemma (Lower Bound) SRE in Downward Closure is EXPSPACE-hard.

22

SRE in Downward Closure - Lower Bound

Lemma (Lower Bound) SRE in Downward Closure is EXPSPACE-hard. Proof. Coverability for (unlabeled) Petri nets is EXPSPACE-hard Label all transitions by Note: N M0 Mf iff Mf coverable, else N M0 Mf iff Mf coverable

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SRE in Downward Closure - Lower Bound

Lemma (Lower Bound) SRE in Downward Closure is EXPSPACE-hard. Proof. Coverability for (unlabeled) Petri nets is EXPSPACE-hard Label all transitions by Note: N M0 Mf iff Mf coverable, else N M0 Mf iff Mf coverable

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SRE in Downward Closure - Lower Bound

Lemma (Lower Bound) SRE in Downward Closure is EXPSPACE-hard. Proof. Coverability for (unlabeled) Petri nets is EXPSPACE-hard Label all transitions by ε Note: N M0 Mf iff Mf coverable, else N M0 Mf iff Mf coverable

22

SRE in Downward Closure - Lower Bound

Lemma (Lower Bound) SRE in Downward Closure is EXPSPACE-hard. Proof. Coverability for (unlabeled) Petri nets is EXPSPACE-hard Label all transitions by ε Note: L ( N, M0, Mf ) = {ε} iff Mf coverable, ∅ else N M0 Mf iff Mf coverable

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SRE in Downward Closure - Lower Bound

Lemma (Lower Bound) SRE in Downward Closure is EXPSPACE-hard. Proof. Coverability for (unlabeled) Petri nets is EXPSPACE-hard Label all transitions by ε Note: L ( N, M0, Mf ) = {ε} iff Mf coverable, ∅ else L(∅∗) = {ε} ⊆ L ( N, M0, Mf ) ↓ iff Mf coverable

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SRE in Upward Closure

SRE in Upward Closure

SRE in Upward Closure Given: SRE sre, Petri net (N, M0, Mf). Decide: L(sre) ⊆ L ( N, M0, Mf ) ↑ ?

23

SRE in Upward Closure

SRE in Upward Closure Given: SRE sre, Petri net (N, M0, Mf). Decide: L(sre) ⊆ L ( N, M0, Mf ) ↑ ? Theorem SRE in Upward Closure is EXPSPACE-complete.

23

SRE in Upward Closure

SRE in Upward Closure Given: SRE sre, Petri net (N, M0, Mf). Decide: L(sre) ⊆ L ( N, M0, Mf ) ↑ ? Theorem SRE in Upward Closure is EXPSPACE-complete. Note: Lower bound (EXPSPACE-hardness) as for SRE in Downward Closure

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SRE in Upward Closure - Upper Bound

Lemma (Upper Bound) SRE in Upward Closure can be solved in EXPSPACE.

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SRE in Upward Closure - Upper Bound

Lemma (Upper Bound) SRE in Upward Closure can be solved in EXPSPACE. SRE sre is a choice among products p Check inclusion p N M0 Mf for each product For each product, compute its minimal word: min a a min p p min p min p min a min Check this using a coverability query in a modified net

24

SRE in Upward Closure - Upper Bound

Lemma (Upper Bound) SRE in Upward Closure can be solved in EXPSPACE. SRE sre is a choice among products p Check inclusion L(p) ⊆ L ( N, M0, Mf ) ↑ for each product For each product, compute its minimal word: min a a min p p min p min p min a min Check this using a coverability query in a modified net

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SRE in Upward Closure - Upper Bound

Lemma (Upper Bound) SRE in Upward Closure can be solved in EXPSPACE. SRE sre is a choice among products p Check inclusion L(p) ⊆ L ( N, M0, Mf ) ↑ for each product For each product, compute its minimal word: min(a) = a min(p.p′) = min(p).min(p′) min(a + ε) = ε min(Γ∗) = ε Check this using a coverability query in a modified net

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SRE in Upward Closure - Upper Bound

Lemma (Upper Bound) SRE in Upward Closure can be solved in EXPSPACE. SRE sre is a choice among products p Check inclusion L(p) ⊆ L ( N, M0, Mf ) ↑ for each product For each product, compute its minimal word: min(a) = a min(p.p′) = min(p).min(p′) min(a + ε) = ε min(Γ∗) = ε We have L(p) ⊆ L ( N, M0, Mf ) ↑ iff min(p) ∈ L ( N, M0, Mf ) ↑ Check this using a coverability query in a modified net

24

SRE in Upward Closure - Upper Bound

Lemma (Upper Bound) SRE in Upward Closure can be solved in EXPSPACE. SRE sre is a choice among products p Check inclusion L(p) ⊆ L ( N, M0, Mf ) ↑ for each product For each product, compute its minimal word: min(a) = a min(p.p′) = min(p).min(p′) min(a + ε) = ε min(Γ∗) = ε We have L(p) ⊆ L ( N, M0, Mf ) ↑ iff min(p) ∈ L ( N, M0, Mf ) ↑ Check this using a coverability query in a modified net

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Being Downward/Upward closed

Being DC/UC

Being Downward/Upward Closed Given: Petri net (N, M0, Mf). Decide: L ( N, M0, Mf ) = L ( N, M0, Mf ) ↓ /↑ ?

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Being DC/UC

Being Downward/Upward Closed Given: Petri net (N, M0, Mf). Decide: L ( N, M0, Mf ) = L ( N, M0, Mf ) ↓ /↑ ? Theorem Being DC and Being UC are decidable.

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Being DC/UC

Being Downward/Upward Closed Given: Petri net (N, M0, Mf). Decide: L ( N, M0, Mf ) = L ( N, M0, Mf ) ↓ /↑ ? Theorem Being DC and Being UC are decidable. Note: L ⊆ L↑ and L ⊆ L↓ always hold and are effectively regular

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Being DC/UC

Being Downward/Upward Closed Given: Petri net (N, M0, Mf). Decide: L ( N, M0, Mf ) = L ( N, M0, Mf ) ↓ /↑ ? Theorem Being DC and Being UC are decidable. Note: L ⊆ L↑ and L ⊆ L↓ always hold L↑ and L↓ are effectively regular

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REG-IN-PNCOV

Regular lang. included in PN coverability lang. Given: Petri net (N, M0, Mf), FSA A. Decide: L(A) ⊆ L ( N, M0, Mf ) ?

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REG-IN-PNCOV

Regular lang. included in PN coverability lang. Given: Petri net (N, M0, Mf), FSA A. Decide: L(A) ⊆ L ( N, M0, Mf ) ? Theorem Regular lang. included in PN coverability lang. is decidable.

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REG-IN-PNCOV

Regular lang. included in PN coverability lang. Given: Petri net (N, M0, Mf), FSA A. Decide: L(A) ⊆ L ( N, M0, Mf ) ? Theorem Regular lang. included in PN coverability lang. is decidable. Theorem (Jancar, Esparza, Moller 1999) Given Petri net (N, M0) and FSA A. T (A) ⊆ T (N, M0) is decidable.

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Reducing to Trace Inclusion

Theorem (Jancar, Esparza, Moller 1999) Given Petri net (N′, M0) and FSA B. T (B) ⊆ T (N′, M0) is decidable

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Reducing to Trace Inclusion

Theorem (Jancar, Esparza, Moller 1999) Given Petri net (N′, M0) and FSA B. T (B) ⊆ T (N′, M0) is decidable where T (B) = { w

• q0

w

− → q for some state q } , T (N′, M0) = {w | M0[σ⟩M for some M and σ, λ(σ) = w}.

27

Reducing to Trace Inclusion

Theorem (Jancar, Esparza, Moller 1999) Given Petri net (N′, M0) and FSA B. T (B) ⊆ T (N′, M0) is decidable where T (B) = { w

• q0

w

− → q for some state q } , T (N′, M0) = {w | M0[σ⟩M for some M and σ, λ(σ) = w}. Lemma L(A) ⊆ L ( N, M0, Mf ) iff T (A.a) ⊆ T (N.a, M0).

27

Reducing to Trace Inclusion

Theorem (Jancar, Esparza, Moller 1999) Given Petri net (N′, M0) and FSA B. T (B) ⊆ T (N′, M0) is decidable where T (B) = { w

• q0

w

− → q for some state q } , T (N′, M0) = {w | M0[σ⟩M for some M and σ, λ(σ) = w}. Lemma L(A) ⊆ L ( N, M0, Mf ) iff T (A.a) ⊆ T (N.a, M0). where a fresh letter A.a reduced FSA for L(A).a N.a = N plus a-labeled transition tf consuming Mf

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BPP Nets

Results

Petri nets Compute UC Doubly exponential∗ Compute DC Non-prim. rec.∗ SRE in DC EXPSPACE-compl. SRE in UC EXPSPACE-compl. Being DC/UC Decidable

∗ : Time for construction & size of minimal FSA 28

BPP Nets - Negative Example

In a BPP net, each transition consumes at most one token. 2 2 2

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BPP Nets - Positive Example

In a BPP net, each transition consumes at most one token.

p0 t0

ε

p1 t1

ε

p2 t2

ε

p3 tf

a

pf 2 2 2

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Results

Petri nets Compute UC Doubly exponential∗ Compute DC Non-prim. rec.∗ SRE in DC EXPSPACE-compl. SRE in UC EXPSPACE-compl. Being DC/UC Decidable ∗ : Time for construction & size of minimal FSA 31

Results

Petri nets BPP nets Compute UC Doubly exponential∗ Exponential∗ Compute DC Non-prim. rec.∗ Exponential∗ SRE in DC EXPSPACE-compl. NP-compl. SRE in UC EXPSPACE-compl. NP-compl. Being DC/UC Decidable ∗ : Time for construction & size of minimal FSA 31

Results

Petri nets BPP nets Techniques for upper bound lower bound Compute UC Doubly exponential∗ Exponential∗ Unfoldings Initial ex. Compute DC Non-prim. rec.∗ Exponential∗ Unfoldings Initial ex. SRE in DC EXPSPACE-compl. NP-compl. Presburger Coverability SRE in UC EXPSPACE-compl. NP-compl. Coverability Coverability Being DC/UC Decidable ∗ : Time for construction & size of minimal FSA 31

Thank you!

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Questions?

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