Handling Infinitely Branching WSTS Michael Blondin 1 2 , Alain Finkel - - PowerPoint PPT Presentation

Introduction WSTS completion Applications Conclusion

Handling Infinitely Branching WSTS

Michael Blondin 1 2, Alain Finkel1 & Pierre McKenzie 1 2

1LSV, ENS Cachan 2DIRO, Université de Montréal

PV 2015, Madrid, September 4, 2015

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Introduction WSTS completion Applications Conclusion Overview WSTS Reachability problems

Well-structured transition systems (WSTS) encompass a large number of infinite state systems. Example of WSTS: Petri nets (Geeraerts, Heußner, Praveen & Raskin PN’13)

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Introduction WSTS completion Applications Conclusion Overview WSTS Reachability problems

Well-structured transition systems (WSTS) encompass a large number of infinite state systems. Example of WSTS: Petri nets (Geeraerts, Heußner, Praveen & Raskin PN’13)

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Introduction WSTS completion Applications Conclusion Overview WSTS Reachability problems

Well-structured transition systems (WSTS) encompass a large number of infinite state systems. Example of WSTS: Petri nets (Geeraerts, Heußner, Praveen & Raskin PN’13)

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Introduction WSTS completion Applications Conclusion Overview WSTS Reachability problems

Multiple decidability results are known for finitely branching WSTS. Example of WSTS: Petri nets (Geeraerts, Heußner, Praveen & Raskin PN’13)

Post( ) =

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Introduction WSTS completion Applications Conclusion Overview WSTS Reachability problems

How to handle infinitely branching WSTS such as systems with infinitely many initial states, and parametric systems? Example of WSTS: ω–Petri nets (Geeraerts, Heußner, Praveen & Raskin PN’13) ω

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Introduction WSTS completion Applications Conclusion Overview WSTS Reachability problems

How to handle infinitely branching WSTS such as systems with infinitely many initial states, and parametric systems? Example of WSTS: ω–Petri nets (Geeraerts, Heußner, Praveen & Raskin PN’13) ω

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Introduction WSTS completion Applications Conclusion Overview WSTS Reachability problems

How to handle infinitely branching WSTS such as systems with infinitely many initial states, and parametric systems? Example of WSTS: ω–Petri nets (Geeraerts, Heußner, Praveen & Raskin PN’13) ω

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Introduction WSTS completion Applications Conclusion Overview WSTS Reachability problems

How to handle infinitely branching WSTS such as systems with infinitely many initial states, and parametric systems? Example of WSTS: ω–Petri nets (Geeraerts, Heußner, Praveen & Raskin PN’13) ω

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Introduction WSTS completion Applications Conclusion Overview WSTS Reachability problems

How to handle infinitely branching WSTS such as systems with infinitely many initial states, and parametric systems? Example of WSTS: ω–Petri nets (Geeraerts, Heußner, Praveen & Raskin PN’13) ω

2 / 24

Introduction WSTS completion Applications Conclusion Overview WSTS Reachability problems

How to handle infinitely branching WSTS such as systems with infinitely many initial states, and parametric systems? Example of WSTS: ω–Petri nets (Geeraerts, Heußner, Praveen & Raskin PN’13) ω

Post( ) =

, , , . . .

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Introduction WSTS completion Applications Conclusion Overview WSTS Reachability problems

Well-structured transition system (Finkel ICALP’87, Finkel & Schnoebelen TCS’01) S = (X, − →, ≤) where X set, − → ⊆X × X, monotony, well-quasi-ordered.

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Introduction WSTS completion Applications Conclusion Overview WSTS Reachability problems

Well-structured transition system (Finkel ICALP’87, Finkel & Schnoebelen TCS’01) S = (X, − →, ≤) where N3, − → ⊆X × X, monotony, well-quasi-ordered.

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Introduction WSTS completion Applications Conclusion Overview WSTS Reachability problems

Well-structured transition system (Finkel ICALP’87, Finkel & Schnoebelen TCS’01) S = (X, − →, ≤) where X set, − → ⊆N3 × N3, monotony, well-quasi-ordered.

3 / 24

Introduction WSTS completion Applications Conclusion Overview WSTS Reachability problems

Well-structured transition system (Finkel ICALP’87, Finkel & Schnoebelen TCS’01) S = (X, − →, ≤) where X set, − → ⊆X × X, monotony, well-quasi-ordered.

• 3 / 24

Introduction WSTS completion Applications Conclusion Overview WSTS Reachability problems

Well-structured transition system (Finkel ICALP’87, Finkel & Schnoebelen TCS’01) S = (X, − →, ≤) where X set, − → ⊆X × X, monotony, well-quasi-ordered.

• 3 / 24

Introduction WSTS completion Applications Conclusion Overview WSTS Reachability problems

Well-structured transition system (Finkel ICALP’87, Finkel & Schnoebelen TCS’01) S = (X, − →, ≤) where X set, − → ⊆X × X, monotony, well-quasi-ordered.

• 3 / 24

Introduction WSTS completion Applications Conclusion Overview WSTS Reachability problems

Well-structured transition system (Finkel ICALP’87, Finkel & Schnoebelen TCS’01) S = (X, − →, ≤) where X set, − → ⊆X × X, monotony, well-quasi-ordered. ∀ x − → y

• x′

∗

− → y′ ∃

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Introduction WSTS completion Applications Conclusion Overview WSTS Reachability problems

Well-structured transition system (Finkel ICALP’87, Finkel & Schnoebelen TCS’01) S = (X, − →, ≤) where X set, − → ⊆X × X, transitive monotony, well-quasi-ordered. ∀ x − → y

• x′

+

− → y′ ∃

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Introduction WSTS completion Applications Conclusion Overview WSTS Reachability problems

Well-structured transition system (Finkel ICALP’87, Finkel & Schnoebelen TCS’01) S = (X, − →, ≤) where X set, − → ⊆X × X, strong monotony, well-quasi-ordered. ∀ x − → y

• x′

− → y′ ∃

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Introduction WSTS completion Applications Conclusion Overview WSTS Reachability problems

Well-structured transition system (Finkel ICALP’87, Finkel & Schnoebelen TCS’01) S = (X, − →, ≤) where X set, − → ⊆X × X, monotony, well-quasi-ordered: ∀x0, x1, . . . ∃i < j s.t. xi ≤ xj.

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Introduction WSTS completion Applications Conclusion Overview WSTS Reachability problems

Branching A WSTS (X, − →, ≤) is finitely branching if Post(x) is finite for every x ∈ X.

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Introduction WSTS completion Applications Conclusion Overview WSTS Reachability problems

Branching A WSTS (X, − →, ≤) is finitely branching if Post(x) is finite for every x ∈ X. Some finitely branching WSTS Petri nets, vector addition systems,

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Introduction WSTS completion Applications Conclusion Overview WSTS Reachability problems

Branching A WSTS (X, − →, ≤) is finitely branching if Post(x) is finite for every x ∈ X. Some finitely branching WSTS Petri nets, vector addition systems, Counter machines with affine updates,

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Introduction WSTS completion Applications Conclusion Overview WSTS Reachability problems

Branching A WSTS (X, − →, ≤) is finitely branching if Post(x) is finite for every x ∈ X. Some finitely branching WSTS Petri nets, vector addition systems, Counter machines with affine updates, Lossy channel systems (Abdulla, Cerans, Jonsson & Tsay LICS’96),

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Introduction WSTS completion Applications Conclusion Overview WSTS Reachability problems

Branching A WSTS (X, − →, ≤) is finitely branching if Post(x) is finite for every x ∈ X. Some finitely branching WSTS Petri nets, vector addition systems, Counter machines with affine updates, Lossy channel systems (Abdulla, Cerans, Jonsson & Tsay LICS’96), Much more.

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Introduction WSTS completion Applications Conclusion Overview WSTS Reachability problems

Branching A WSTS (X, − →, ≤) is finitely branching if Post(x) is finite for every x ∈ X. Some infinitely branching WSTS Inserting FIFO automata (Cécé, Finkel, Iyer IC’96),

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Introduction WSTS completion Applications Conclusion Overview WSTS Reachability problems

Branching A WSTS (X, − →, ≤) is finitely branching if Post(x) is finite for every x ∈ X. Some infinitely branching WSTS Inserting FIFO automata (Cécé, Finkel, Iyer IC’96), Inserting automata (Bouyer, Markey, Ouaknine, Schnoebelen, Worrell

FAC’12),

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Introduction WSTS completion Applications Conclusion Overview WSTS Reachability problems

Branching A WSTS (X, − →, ≤) is finitely branching if Post(x) is finite for every x ∈ X. Some infinitely branching WSTS Inserting FIFO automata (Cécé, Finkel, Iyer IC’96), Inserting automata (Bouyer, Markey, Ouaknine, Schnoebelen, Worrell

FAC’12),

ω-Petri nets (Geeraerts, Heussner, Praveen & Raskin PN’13),

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Introduction WSTS completion Applications Conclusion Overview WSTS Reachability problems

Branching A WSTS (X, − →, ≤) is finitely branching if Post(x) is finite for every x ∈ X. Some infinitely branching WSTS Inserting FIFO automata (Cécé, Finkel, Iyer IC’96), Inserting automata (Bouyer, Markey, Ouaknine, Schnoebelen, Worrell

FAC’12),

ω-Petri nets (Geeraerts, Heussner, Praveen & Raskin PN’13), Parametric WSTS.

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Introduction WSTS completion Applications Conclusion Overview WSTS Reachability problems

Proposition Finite branching is undecidable for post-effective WSTS.

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Introduction WSTS completion Applications Conclusion Overview WSTS Reachability problems

Proposition Finite branching is undecidable for post-effective WSTS. Proof Let Si = (N, − →Si, ≤) be the WSTS such that:

x − →Si x + 1 if TMi does not halt within ≤ x steps, x − →Si 0, 1, 2, . . . otherwise.

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Introduction WSTS completion Applications Conclusion Overview WSTS Reachability problems

Proposition Finite branching is undecidable for post-effective WSTS. Proof Let Si = (N, − →Si, ≤) be the WSTS such that:

x − →Si x + 1 if TMi does not halt within ≤ x steps, x − →Si 0, 1, 2, . . . otherwise.

Si is post-effective (the cardinal of PostSi(x) is computable). Si has strong and strict monotony since x − →Si x + 1 for every x ∈ N.

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Introduction WSTS completion Applications Conclusion Overview WSTS Reachability problems

Proposition Finite branching is undecidable for post-effective WSTS. Proof Let Si = (N, − →Si, ≤) be the WSTS such that:

x − →Si x + 1 if TMi does not halt within ≤ x steps, x − →Si 0, 1, 2, . . . otherwise.

Si is post-effective (the cardinal of PostSi(x) is computable). Si has strong and strict monotony since x − →Si x + 1 for every x ∈ N. TMi halts iff there exist x ∈ N and an execution 0 ∗ − →Si x such that PostSi(x) is infinite. The halting problem thus Turing-reduces to the infinite branching problem.

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Introduction WSTS completion Applications Conclusion Overview WSTS Reachability problems

Objective We want to study the usual reachability problems for these infinitely branching systems, e.g.,

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Introduction WSTS completion Applications Conclusion Overview WSTS Reachability problems

Objective We want to study the usual reachability problems for these infinitely branching systems, e.g., Termination,

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Introduction WSTS completion Applications Conclusion Overview WSTS Reachability problems

Objective We want to study the usual reachability problems for these infinitely branching systems, e.g., Termination, Coverability,

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Introduction WSTS completion Applications Conclusion Overview WSTS Reachability problems

Objective We want to study the usual reachability problems for these infinitely branching systems, e.g., Termination, Coverability, Boundedness.

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Introduction WSTS completion Applications Conclusion Overview WSTS Reachability problems

Termination Input: (X, − →, ≤) a WSTS, x0 ∈ X. Question: ∃x0 − → x1 − → x2 − → . . .?

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Introduction WSTS completion Applications Conclusion Overview WSTS Reachability problems

Termination Input: (X, − →, ≤) a WSTS, x0 ∈ X. Question: ∃x0 − → x1 − → x2 − → . . .? Theorem (Finkel ICALP’87) Termination is decidable for finitely branching WSTS with transitive monotony.

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Introduction WSTS completion Applications Conclusion Overview WSTS Reachability problems

Termination Input: (X, − →, ≤) a WSTS, x0 ∈ X. Question: ∃x0 − → x1 − → x2 − → . . .? Theorem (deduced from Dufourd, Jančar & Schnoebelen ICALP’99) Termination is undecidable for infinitely branching WSTS.

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Introduction WSTS completion Applications Conclusion Overview WSTS Reachability problems

Strong termination Input: (X, − →, ≤) a WSTS, x0 ∈ X. Question: ∃k bounding length of executions from x0?

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Introduction WSTS completion Applications Conclusion Overview WSTS Reachability problems

Strong termination Input: (X, − →, ≤) a WSTS, x0 ∈ X. Question: ∃k bounding length of executions from x0? Remark Strong termination and termination are the same in finitely branching WSTS.

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Introduction WSTS completion Applications Conclusion Overview WSTS Reachability problems

Strong termination Input: (X, − →, ≤) a WSTS, x0 ∈ X. Question: ∃k bounding length of executions from x0? Theorem Strong termination is decidable for infinitely branching WSTS under some assumptions.

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Introduction WSTS completion Applications Conclusion Ideals Completion

Issues with finite branching techniques Some techniques for WSTS based on finite reachability trees; impossible for infinite branching. Some rely on upward closed sets; what about downward closed, in particular with infinite branching?

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Introduction WSTS completion Applications Conclusion Ideals Completion

Issues with finite branching techniques Some techniques for WSTS based on finite reachability trees; impossible for infinite branching. Some rely on upward closed sets; what about downward closed, in particular with infinite branching? A tool Develop from the WSTS completion introduced by Finkel & Goubault-Larrecq in STACS’09 and ICALP’09.

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Introduction WSTS completion Applications Conclusion Ideals Completion

Ideals I ⊆ X is an ideal if downward closed: I = ↓ I,

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Introduction WSTS completion Applications Conclusion Ideals Completion

Ideals I ⊆ X is an ideal if downward closed: I = ↓ I, directed: a, b ∈ I = ⇒ ∃c ∈ I s.t. a ≤ c and b ≤ c.

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Introduction WSTS completion Applications Conclusion Ideals Completion

Ideals I ⊆ X is an ideal if downward closed: I = ↓ I, directed: a, b ∈ I = ⇒ ∃c ∈ I s.t. a ≤ c and b ≤ c.

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Introduction WSTS completion Applications Conclusion Ideals Completion

Ideals I ⊆ X is an ideal if downward closed: I = ↓ I, directed: a, b ∈ I = ⇒ ∃c ∈ I s.t. a ≤ c and b ≤ c.

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Introduction WSTS completion Applications Conclusion Ideals Completion

Theorem (Finkel & Goubault-Larrecq ICALP’09; Goubault-Larrecq ’14) D downward closed = ⇒ D =

• finite

Ideals

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Introduction WSTS completion Applications Conclusion Ideals Completion

Theorem (Finkel & Goubault-Larrecq ICALP’09; Goubault-Larrecq ’14) D downward closed = ⇒ D =

• finite

Ideals Corollary Every downward closed set decomposes canonically as the union of its maximal ideals.

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Introduction WSTS completion Applications Conclusion Ideals Completion

Completion The completion of S = (X, − →S, ≤) is S = ( X, − →

S, ⊆) such that

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Introduction WSTS completion Applications Conclusion Ideals Completion

Completion The completion of S = (X, − →S, ≤) is S = ( X, − →

S, ⊆) such that

• X = Ideals(X),

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Introduction WSTS completion Applications Conclusion Ideals Completion

Completion The completion of S = (X, − →S, ≤) is S = ( X, − →

S, ⊆) such that

• X = Ideals(X),

I − →

S J if ↓ Post(I) =

. . . ∪ J ∪ . . .

• canonical decomposition

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Introduction WSTS completion Applications Conclusion Ideals Completion

Theorem Let S = (X, − →S, ≤) be a WSTS, then S = ( X, − →

S, ⊆) is such

that:

• S is finitely branching,

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Introduction WSTS completion Applications Conclusion Ideals Completion

Theorem Let S = (X, − →S, ≤) be a WSTS, then S = ( X, − →

S, ⊆) is such

that:

• S is finitely branching,
• S has (strong) monotony,

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Introduction WSTS completion Applications Conclusion Ideals Completion

Theorem Let S = (X, − →S, ≤) be a WSTS, then S = ( X, − →

S, ⊆) is such

that:

• S is finitely branching,
• S has (strong) monotony,
• S is not always a WSTS (Jančar IPL’99).

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Introduction WSTS completion Applications Conclusion Ideals Completion

Theorem Let S = (X, − →S, ≤) be a WSTS, then S = ( X, − →

S, ⊆) is such

that:

• S is finitely branching,
• S has (strong) monotony,
• S is not always a WSTS (Jančar IPL’99).

Jančar IPL’99 A wqo ≤ is a ω2-wqo iff ≤# is a wqo, where ≤# is the Hoare

• rdering defined by A ≤# B iff ↑ B ⊆↑ A.

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Introduction WSTS completion Applications Conclusion Ideals Completion

Theorem Let S = (X, − →S, ≤) be a WSTS, then S = ( X, − →

S, ⊆) is such

that:

• S is finitely branching,
• S has (strong) monotony,
• S is not always a WSTS (Jančar IPL’99).

Jančar IPL’99 A wqo ≤ is a ω2-wqo iff ≤# is a wqo, where ≤# is the Hoare

• rdering defined by A ≤# B iff ↑ B ⊆↑ A.

Theorem Let S be a WSTS, then S is a WSTS iff S is a ω2-WSTS.

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Introduction WSTS completion Applications Conclusion Termination Coverability

Relating executions of S and S Let S = (X, − →S, ≤) be a WSTS, then if x

k

− →S y,

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Introduction WSTS completion Applications Conclusion Termination Coverability

Relating executions of S and S Let S = (X, − →S, ≤) be a WSTS, then if x

k

− →S y, then for every ideal I ⊇ ↓ x

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Introduction WSTS completion Applications Conclusion Termination Coverability

Relating executions of S and S Let S = (X, − →S, ≤) be a WSTS, then if x

k

− →S y, then for every ideal I ⊇ ↓ x there exists an ideal J ⊇ ↓ y

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Introduction WSTS completion Applications Conclusion Termination Coverability

Relating executions of S and S Let S = (X, − →S, ≤) be a WSTS, then if x

k

− →S y, then for every ideal I ⊇ ↓ x there exists an ideal J ⊇ ↓ y such that I

k

− →

S J,

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Introduction WSTS completion Applications Conclusion Termination Coverability

Relating executions of S and S Let S = (X, − →S, ≤) be a WSTS, then if x

k

− →S y, then for every ideal I ⊇ ↓ x there exists an ideal J ⊇ ↓ y such that I

k

− →

S J,

if I

k

− →

S J,

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Introduction WSTS completion Applications Conclusion Termination Coverability

Relating executions of S and S Let S = (X, − →S, ≤) be a WSTS, then if x

k

− →S y, then for every ideal I ⊇ ↓ x there exists an ideal J ⊇ ↓ y such that I

k

− →

S J,

if I

k

− →

S J, then for every y ∈ J

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Introduction WSTS completion Applications Conclusion Termination Coverability

Relating executions of S and S Let S = (X, − →S, ≤) be a WSTS, then if x

k

− →S y, then for every ideal I ⊇ ↓ x there exists an ideal J ⊇ ↓ y such that I

k

− →

S J,

if I

k

− →

S J, then for every y ∈ J there exists x ∈ I

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Introduction WSTS completion Applications Conclusion Termination Coverability

Relating executions of S and S Let S = (X, − →S, ≤) be a WSTS, then if x

k

− →S y, then for every ideal I ⊇ ↓ x there exists an ideal J ⊇ ↓ y such that I

k

− →

S J,

if I

k

− →

S J, then for every y ∈ J there exists x ∈ I such that

x

∗

− →S y′ ≥ y.

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Introduction WSTS completion Applications Conclusion Termination Coverability

Relating executions of S and S Let S = (X, − →S, ≤) be a WSTS with transitive monotony, then if x

k

− →S y, then for every ideal I ⊇ ↓ x there exists an ideal J ⊇ ↓ y such that I

k

− →

S J,

if I

k

− →

S J, then for every y ∈ J there exists x ∈ I such that

x

≥k

− − →S y′ ≥ y.

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Introduction WSTS completion Applications Conclusion Termination Coverability

Relating executions of S and S Let S = (X, − →S, ≤) be a WSTS with strong monotony, then if x

k

− →S y, then for every ideal I ⊇ ↓ x there exists an ideal J ⊇ ↓ y such that I

k

− →

S J,

if I

k

− →

S J, then for every y ∈ J there exists x ∈ I such that

x

k

− →S y′ ≥ y.

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Introduction WSTS completion Applications Conclusion Termination Coverability

Relations between S and S

A generality The completion S = ( X, − →

S, ⊆) computes exactly the downward

closure of the reachability set of its original system S = (X, − →S, ≤). An equality We have: Post∗

• S(↓ x)= ↓ Post∗

S(x).

In fact, it is more exactly: Theorem If Post∗

• S(↓ x) = {J1, . . . , Jn} then ↓ Post∗

S(x) = J1 ∪ . . . ∪ Jn.

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Introduction WSTS completion Applications Conclusion Termination Coverability

Theorem Strong termination is decidable for infinitely branching WSTS with transitive monotony and such that S is a post-effective WSTS.

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Introduction WSTS completion Applications Conclusion Termination Coverability

Theorem Strong termination is decidable for infinitely branching WSTS with transitive monotony and such that S is a post-effective WSTS. Post-effectiveness Possible to compute cardinality of

Post( ) =

, , , . . .

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Introduction WSTS completion Applications Conclusion Termination Coverability

Theorem Strong termination is decidable for infinitely branching WSTS with transitive monotony and such that S is a post-effective WSTS. Proof Executions bounded in S iff bounded in S.

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Introduction WSTS completion Applications Conclusion Termination Coverability

Theorem Strong termination is decidable for infinitely branching WSTS with transitive monotony and such that S is a post-effective WSTS. Proof Executions bounded in S iff bounded in S.

• S finitely branching, can decide termination in

S by Finkel ICALP’87, Finkel & Schnoebelen TCS’01.

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Introduction WSTS completion Applications Conclusion Termination Coverability

Coverability Input: (X, − →, ≤) a WSTS, x0, x ∈ X. Question: x0

∗

− → x′ ≥ x?

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Introduction WSTS completion Applications Conclusion Termination Coverability

Coverability Input: (X, − →, ≤) a WSTS, x0, x ∈ X. Question: x0 ∈ ↑ Pre∗(↑ x)? Backward method (Abdulla, Cerans, Jonsson & Tsay IC’00) Compute ↑ Pre∗(↑ x) iteratively assuming ↑ Pre(↑ x) computable.

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Introduction WSTS completion Applications Conclusion Termination Coverability

Coverability Input: (X, − →, ≤) a WSTS, x0, x ∈ X. Question: x0 ∈ ↑ Pre∗(↑ x)? Backward method (Abdulla, Cerans, Jonsson & Tsay IC’00) Compute ↑ Pre∗(↑ x) iteratively assuming ↑ Pre(↑ x) computable.

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Introduction WSTS completion Applications Conclusion Termination Coverability

Coverability Input: (X, − →, ≤) a WSTS, x0, x ∈ X. Question: x ∈ ↓ Post∗(x0)?

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Introduction WSTS completion Applications Conclusion Termination Coverability

Coverability Input: (X, − →, ≤) a WSTS, x0, x ∈ X. Question: x ∈ ↓ Post∗(x0)? Forward method Coverability: Enumerate executions ↓ x0

∗

− →

S I,

Accept if x ∈ I.

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Introduction WSTS completion Applications Conclusion Termination Coverability

Coverability Input: (X, − →, ≤) a WSTS, x0, x ∈ X. Question: x ∈ ↓ Post∗(x0)? Forward method Coverability: Enumerate executions ↓ x0

∗

− →

S I,

Accept if x ∈ I. Non coverability: Enumerate D ⊆ X downward closed, x0 ∈ D and ↓ PostS(D) ⊆ D Reject if x ∈ D.

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Introduction WSTS completion Applications Conclusion Termination Coverability

Coverability Input: (X, − →, ≤) a WSTS, x0, x ∈ X. Question: x ∈ ↓ Post∗(x0)? Forward method Coverability: Enumerate executions ↓ x0

∗

− →

S I,

Accept if x ∈ I. Non coverability: Enumerate D = I1 ∪ . . . ∪ Ik Reject if x ∈ D.

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Introduction WSTS completion Applications Conclusion Termination Coverability

Coverability Input: (X, − →, ≤) a WSTS, x0, x ∈ X. Question: x ∈ ↓ Post∗(x0)? Forward method Coverability: Enumerate executions ↓ x0

∗

− →

S I,

Accept if x ∈ I. Non coverability: Enumerate D ⊆ X downward closed Reject if x ∈ D.

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Introduction WSTS completion Applications Conclusion Termination Coverability

Coverability Input: (X, − →, ≤) a WSTS, x0, x ∈ X. Question: x ∈ ↓ Post∗(x0)? Forward method Coverability: Enumerate executions ↓ x0

∗

− →

S I,

Accept if x ∈ I. Non coverability: Enumerate D ⊆ X downward closed, x0 ∈ D Reject if x ∈ D.

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Introduction WSTS completion Applications Conclusion Termination Coverability

Coverability Input: (X, − →, ≤) a WSTS, x0, x ∈ X. Question: x ∈ ↓ Post∗(x0)? Forward method Coverability: Enumerate executions ↓ x0

∗

− →

S I,

Accept if x ∈ I. Non coverability: Enumerate D ⊆ X downward closed, ↓ x0 ⊆ I1 ∪ . . . ∪ Ik Reject if x ∈ D.

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Introduction WSTS completion Applications Conclusion Termination Coverability

Coverability Input: (X, − →, ≤) a WSTS, x0, x ∈ X. Question: x ∈ ↓ Post∗(x0)? Forward method Coverability: Enumerate executions ↓ x0

∗

− →

S I,

Accept if x ∈ I. Non coverability: Enumerate D ⊆ X downward closed, ∃j s.t. ↓ x0 ⊆ Ij Reject if x ∈ D.

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Introduction WSTS completion Applications Conclusion Termination Coverability

Coverability Input: (X, − →, ≤) a WSTS, x0, x ∈ X. Question: x ∈ ↓ Post∗(x0)? Forward method Coverability: Enumerate executions ↓ x0

∗

− →

S I,

Accept if x ∈ I. Non coverability: Enumerate D ⊆ X downward closed, x0 ∈ D and ↓ PostS(D) ⊆ D Reject if x ∈ D.

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Introduction WSTS completion Applications Conclusion Termination Coverability

Prebasis computability Prebasis computability is sufficient, but not necessary, to ensure decidability of coverability.

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Introduction WSTS completion Applications Conclusion Termination Coverability

Prebasis computability Prebasis computability is sufficient, but not necessary, to ensure decidability of coverability. Coverability is decidable in F1 The algorithm consists to enumerate strictly increasing reachable sequences until finding an y ≥ x.

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Introduction WSTS completion Applications Conclusion Termination Coverability

Prebasis computability Prebasis computability is sufficient, but not necessary, to ensure decidability of coverability. Coverability is decidable in F1 The algorithm consists to enumerate strictly increasing reachable sequences until finding an y ≥ x. Prebasis is not computable for F1 Let Si = (N, − →Si, ≤) be the WSTS such that: x − →Si 0 if TMi does not halt on its encoding in ≤ x steps, x − →Si 1 otherwise.

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Introduction WSTS completion Applications Conclusion Termination Coverability

Prebasis computability Prebasis computability is sufficient, but not necessary, to ensure decidability of coverability. Coverability is decidable in F1 The algorithm consists to enumerate strictly increasing reachable sequences until finding an y ≥ x. Prebasis is not computable for F1 Let Si = (N, − →Si, ≤) be the WSTS such that: x − →Si 0 if TMi does not halt on its encoding in ≤ x steps, x − →Si 1 otherwise. Then Si ∈ F1 and Si is effective.

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Introduction WSTS completion Applications Conclusion Termination Coverability

Three Pre sets PreSi(0) = {x ∈ N : TMi does not halt in ≤ x steps }, PreSi(1) = {x ∈ N : TMi halts in ≤ x steps }, PreSi(x) = ∅ for x ≥ 2.

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Introduction WSTS completion Applications Conclusion Termination Coverability

Three Pre sets PreSi(0) = {x ∈ N : TMi does not halt in ≤ x steps }, PreSi(1) = {x ∈ N : TMi halts in ≤ x steps }, PreSi(x) = ∅ for x ≥ 2. Conclusion: prebasis is not computable for F1 Therefore, ↑ PreSi(↑ 1) = ↑ PreSi(1) = PreSi(1). If an algorithm outputting a finite basis of ↑ PreSi(↑ 1) existed, then it would be possible to decide whether PreSi(1) = ∅.

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Introduction WSTS completion Applications Conclusion Termination Coverability

Three Pre sets PreSi(0) = {x ∈ N : TMi does not halt in ≤ x steps }, PreSi(1) = {x ∈ N : TMi halts in ≤ x steps }, PreSi(x) = ∅ for x ≥ 2. Conclusion: prebasis is not computable for F1 Therefore, ↑ PreSi(↑ 1) = ↑ PreSi(1) = PreSi(1). If an algorithm outputting a finite basis of ↑ PreSi(↑ 1) existed, then it would be possible to decide whether PreSi(1) = ∅. But PreSi(1) = ∅ iff TMi does not halt. The halting problem thus Turing-reduces to the prebasis computation.

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Introduction WSTS completion Applications Conclusion Termination Coverability

Boundedness for infinitely branching WSTS Boundedness is decidable for post-effective WSTS with strict monotony and a wpo.

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Introduction WSTS completion Applications Conclusion Termination Coverability

Boundedness for infinitely branching WSTS Boundedness is decidable for post-effective WSTS with strict monotony and a wpo. Proof We build a reachability tree T with root c0 labelled x0. If PostS(x0) is infinite, then we return “unbounded”,

• therwise we mark c0 and for every x ∈ PostS(x0) we add a

child labelled x to c0.

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Introduction WSTS completion Applications Conclusion Termination Coverability

Boundedness for infinitely branching WSTS Boundedness is decidable for post-effective WSTS with strict monotony and a wpo. Proof We build a reachability tree T with root c0 labelled x0. If PostS(x0) is infinite, then we return “unbounded”,

• therwise we mark c0 and for every x ∈ PostS(x0) we add a

child labelled x to c0. If c has an ancestor c′ labelled x′ such that x′ < x, we return “unbounded”. Otherwise,

if c has an ancestor c′ labelled x′ such that x′ = x, we mark c. Otherwise, if PostS(x) is infinite, then we return “unbounded”. Otherwise we mark c and for every y ∈ PostS(x) we add a child labelled y to c.

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Introduction WSTS completion Applications Conclusion Termination Coverability

Boundedness for infinitely branching WSTS Boundedness is decidable for post-effective WSTS with strict monotony and a wpo. Proof We build a reachability tree T with root c0 labelled x0. If PostS(x0) is infinite, then we return “unbounded”,

• therwise we mark c0 and for every x ∈ PostS(x0) we add a

child labelled x to c0. If c has an ancestor c′ labelled x′ such that x′ < x, we return “unbounded”. Otherwise,

if c has an ancestor c′ labelled x′ such that x′ = x, we mark c. Otherwise, if PostS(x) is infinite, then we return “unbounded”. Otherwise we mark c and for every y ∈ PostS(x) we add a child labelled y to c.

T is finite and correct.

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Introduction WSTS completion Applications Conclusion

Further result for infinitely branching WSTS Strong maintainability is decidable for WSTS with strong monotony and such that S is a post-effective WSTS.

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Introduction WSTS completion Applications Conclusion

Further work ∃ general class of infinitely branching WSTS with a Karp-Miller procedure?

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Introduction WSTS completion Applications Conclusion

Further work ∃ general class of infinitely branching WSTS with a Karp-Miller procedure? Toward the algorithmics of complete WSTS.

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Introduction WSTS completion Applications Conclusion

Further work ∃ general class of infinitely branching WSTS with a Karp-Miller procedure? Toward the algorithmics of complete WSTS. What else can we do with the WSTS completion?

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Introduction WSTS completion Applications Conclusion

Thank you!

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