WBTS: the new class of WSTS without WQOfinkel/2016-2017/slides-WBTS-workshop-etaps-2017.pdf · on...

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Introduction News on coverability Erdös and Tarski Theorem WBTS = WSTS - WQO + FAC Conclusion WBTS: the new class of WSTS without WQO Alain Finkel LSV, ENS Paris-Saclay (ex ENS Cachan) SynCoP + PV 22-23 April 2017 Blasenhus, Uppsala, Sweden Based on joint works with Michael Blondin & Pierre McKenzie, to appear in LMCS. 1 / 35

Transcript of WBTS: the new class of WSTS without WQOfinkel/2016-2017/slides-WBTS-workshop-etaps-2017.pdf · on...

Page 1: WBTS: the new class of WSTS without WQOfinkel/2016-2017/slides-WBTS-workshop-etaps-2017.pdf · on the Grand Central Dispatch (GCD) technology [13] , and to perform parametric verication

IntroductionNews on coverability

Erdös and Tarski TheoremWBTS = WSTS - WQO + FAC

Conclusion

WBTS: the new class of WSTS without WQO

Alain Finkel

LSV, ENS Paris-Saclay (ex ENS Cachan)

SynCoP + PV22-23 April 2017 Blasenhus, Uppsala, Sweden

Based on joint works with Michael Blondin & Pierre McKenzie, to appearin LMCS.

1 / 35

Page 2: WBTS: the new class of WSTS without WQOfinkel/2016-2017/slides-WBTS-workshop-etaps-2017.pdf · on the Grand Central Dispatch (GCD) technology [13] , and to perform parametric verication

2 Geeraerts, Heußner, Praveen, Raskin /ω-Petri nets

1 one_task(int k) {

2 // some work ...

3 }

4 main(int P ) {

5 for i := 1 to P step 1

6 fork(one_task(i))

7 }

fork K

p1

p2

one task

(a)fork •

p1

p2

one task

(b)fork •

p1

p2

one task

(c)

ω

Figure 1. An example of a parametric system with three possible models

particularly well suited for modeling parametric concurrent systems — see for instance our recent workon the Grand Central Dispatch (GCD) technology [13] —, and to perform parametric verification [16]on those systems, as we illustrate now by means of the example in Fig. 1. The example present askeleton of a distributed program, in which a main function forks P parallel threads (where P is aparameter of the program), each executing the one task function. Many distributed programs followthis abstract skeleton that allows us to perform calculations in parallel, and being able to model preciselysuch concurrent behaviors is an important issue. In particular, we would like that the model captures thefact that P is a parameter, so that we can, for instance, check that the execution of the program alwaysterminates (assuming each individual execution of one task does), for all possible values of P . Clearly,the Petri net (a) in Fig. 1 does not capture the parametric nature of the example, as place p1 contains afixed number K of tokens. The PN (b), on the other hand captures the fact that the program can fork

an unbounded number of threads, but does not preserve termination: (fork)ω is an infinite execution ofPN (b), while the programme terminates (assuming each one task thread terminates) for all values ofP , because the for loop in line 5 executes exactly P times. Finally, observe that the ωPN (c) has thedesired properties: firing transition fork creates non-deterministically an unbounded albeit finite numberof tokens in p2 (to model all the possible executions of the for loop in line 5), and all possible executionsof this ωPN terminate, because the number of tokens produced in p2 remains finite and no further tokencreation in p2 is allowed after the firing of the fork transition.

While close to Petri nets, ωPN are sufficiently different that a thorough and careful study of theircomputational properties is required. This is the main contribution of the paper. A first example ofdiscrepancy is that the semantics of ωPN is an infinite transition system which is infinitely branching.This is not the case for plain PN: their transition systems can be infinite but they are finitely branching.As a consequence, some of the classical techniques for the analysis of Petri nets cannot be applied.Consider for example the finite unfolding of the transition system [12] that stops the development of abranch of the reachability tree whenever a node with a smaller ancestor is found. This tree is finite (andcan be built effectively) for any plain Petri net and any initial marking because the set of markings Nkis well-quasi ordered, and finite branching of plain Petri nets allows for the use of Konig’s lemma1.However, this technique cannot be applied to ωPN, as they are infinitely branching. Such peculiaritiesof ωPN motivate our study of three different tools for analysing them. First, we consider, in Section 3, avariant of the Karp and Miller tree [18] that applies to ωPN. In order to cope with the infinite branchingof the semantics of ωPN, we need to introduce in the Karp and Miller tree ω’s that are not the result ofaccelerations but the result of ω-output arcs. Our variant of the Karp and Miller construction is recursive,

1In fact, this construction is applicable to any well-structured transition system which is finitely branching and permits to decidethe termination problem for example.

Page 3: WBTS: the new class of WSTS without WQOfinkel/2016-2017/slides-WBTS-workshop-etaps-2017.pdf · on the Grand Central Dispatch (GCD) technology [13] , and to perform parametric verication

IntroductionNews on coverability

Erdös and Tarski TheoremWBTS = WSTS - WQO + FAC

Conclusion

WSTSReachability problemsA quick story of WSTSA quick story of coverability in WSTS

From parameterized systems

Parameterized protocols: mutual exclusion, broadcastprotocolsParametric concurrent systems with dynamic thread creation(ex: Geeraerts & al. FI, 2015)Parameterized systems in BIP (ex: Konnov & al.Concur’2016)

To infinite-state models

Counter (abstraction) machines with reset-transfer-affineextensionsω-Petri nets (infinite branching)WSTS 2 / 35

Page 4: WBTS: the new class of WSTS without WQOfinkel/2016-2017/slides-WBTS-workshop-etaps-2017.pdf · on the Grand Central Dispatch (GCD) technology [13] , and to perform parametric verication

IntroductionNews on coverability

Erdös and Tarski TheoremWBTS = WSTS - WQO + FAC

Conclusion

WSTSReachability problemsA quick story of WSTSA quick story of coverability in WSTS

Well Structured Transition Systems (WSTS) encompass a largenumber of infinite state systems (PN and reset-transfer-affine-ω extensions, lossy fifo

systems, broadcast protocols, CFG, graph rewriting, depth bounded processes, fragments of the π-calculus,....)

Example of WSTS:

ω–

Petri nets

(Geeraerts, Heußner, Praveen & Raskin PN’13, FU’15)

Post( ) =

3 / 35

Page 5: WBTS: the new class of WSTS without WQOfinkel/2016-2017/slides-WBTS-workshop-etaps-2017.pdf · on the Grand Central Dispatch (GCD) technology [13] , and to perform parametric verication

IntroductionNews on coverability

Erdös and Tarski TheoremWBTS = WSTS - WQO + FAC

Conclusion

WSTSReachability problemsA quick story of WSTSA quick story of coverability in WSTS

Well Structured Transition Systems (WSTS) encompass a largenumber of infinite state systems (PN and reset-transfer-affine-ω extensions, lossy fifo

systems, broadcast protocols, CFG, graph rewriting, depth bounded processes, fragments of the π-calculus,....)

Example of WSTS:

ω–

Petri nets

(Geeraerts, Heußner, Praveen & Raskin PN’13, FU’15)

Post( ) =

3 / 35

Page 6: WBTS: the new class of WSTS without WQOfinkel/2016-2017/slides-WBTS-workshop-etaps-2017.pdf · on the Grand Central Dispatch (GCD) technology [13] , and to perform parametric verication

IntroductionNews on coverability

Erdös and Tarski TheoremWBTS = WSTS - WQO + FAC

Conclusion

WSTSReachability problemsA quick story of WSTSA quick story of coverability in WSTS

Well Structured Transition Systems (WSTS) encompass a largenumber of infinite state systems (PN and reset-transfer-affine-ω extensions, lossy fifo

systems, broadcast protocols, CFG, graph rewriting, depth bounded processes, fragments of the π-calculus,....)

Example of WSTS:

ω–

Petri nets

(Geeraerts, Heußner, Praveen & Raskin PN’13, FU’15)

Post( ) =

3 / 35

Page 7: WBTS: the new class of WSTS without WQOfinkel/2016-2017/slides-WBTS-workshop-etaps-2017.pdf · on the Grand Central Dispatch (GCD) technology [13] , and to perform parametric verication

IntroductionNews on coverability

Erdös and Tarski TheoremWBTS = WSTS - WQO + FAC

Conclusion

WSTSReachability problemsA quick story of WSTSA quick story of coverability in WSTS

Multiple decidability results are known for (finitely branching)WSTS.

Example of WSTS:

ω–

Petri nets

(Geeraerts, Heußner, Praveen & Raskin PN’13, FU’15)

Post( ) =

3 / 35

Page 8: WBTS: the new class of WSTS without WQOfinkel/2016-2017/slides-WBTS-workshop-etaps-2017.pdf · on the Grand Central Dispatch (GCD) technology [13] , and to perform parametric verication

IntroductionNews on coverability

Erdös and Tarski TheoremWBTS = WSTS - WQO + FAC

Conclusion

WSTSReachability problemsA quick story of WSTSA quick story of coverability in WSTS

And also for (infinitely branching) WSTS such as systems withinfinitely many initial states and parametric systems

Example of WSTS: ω–Petri nets (Geeraerts, Heußner, Praveen & Raskin PN’13, FU’15)

ω

Post( ) =

3 / 35

Page 9: WBTS: the new class of WSTS without WQOfinkel/2016-2017/slides-WBTS-workshop-etaps-2017.pdf · on the Grand Central Dispatch (GCD) technology [13] , and to perform parametric verication

IntroductionNews on coverability

Erdös and Tarski TheoremWBTS = WSTS - WQO + FAC

Conclusion

WSTSReachability problemsA quick story of WSTSA quick story of coverability in WSTS

And also for (infinitely branching) WSTS such as systems withinfinitely many initial states and parametric systems

Example of WSTS: ω–Petri nets (Geeraerts, Heußner, Praveen & Raskin PN’13, FU’15)

ω

Post( ) =

3 / 35

Page 10: WBTS: the new class of WSTS without WQOfinkel/2016-2017/slides-WBTS-workshop-etaps-2017.pdf · on the Grand Central Dispatch (GCD) technology [13] , and to perform parametric verication

IntroductionNews on coverability

Erdös and Tarski TheoremWBTS = WSTS - WQO + FAC

Conclusion

WSTSReachability problemsA quick story of WSTSA quick story of coverability in WSTS

And also for (infinitely branching) WSTS such as systems withinfinitely many initial states and parametric systems

Example of WSTS: ω–Petri nets (Geeraerts, Heußner, Praveen & Raskin PN’13, FU’15)

ω

Post( ) =

3 / 35

Page 11: WBTS: the new class of WSTS without WQOfinkel/2016-2017/slides-WBTS-workshop-etaps-2017.pdf · on the Grand Central Dispatch (GCD) technology [13] , and to perform parametric verication

IntroductionNews on coverability

Erdös and Tarski TheoremWBTS = WSTS - WQO + FAC

Conclusion

WSTSReachability problemsA quick story of WSTSA quick story of coverability in WSTS

And also for (infinitely branching) WSTS such as systems withinfinitely many initial states and parametric systems

Example of WSTS: ω–Petri nets (Geeraerts, Heußner, Praveen & Raskin PN’13, FU’15)

ω

Post( ) =

3 / 35

Page 12: WBTS: the new class of WSTS without WQOfinkel/2016-2017/slides-WBTS-workshop-etaps-2017.pdf · on the Grand Central Dispatch (GCD) technology [13] , and to perform parametric verication

IntroductionNews on coverability

Erdös and Tarski TheoremWBTS = WSTS - WQO + FAC

Conclusion

WSTSReachability problemsA quick story of WSTSA quick story of coverability in WSTS

And also for (infinitely branching) WSTS such as systems withinfinitely many initial states and parametric systems

Example of WSTS: ω–Petri nets (Geeraerts, Heußner, Praveen & Raskin PN’13, FU’15)

ω

Post( ) =

3 / 35

Page 13: WBTS: the new class of WSTS without WQOfinkel/2016-2017/slides-WBTS-workshop-etaps-2017.pdf · on the Grand Central Dispatch (GCD) technology [13] , and to perform parametric verication

IntroductionNews on coverability

Erdös and Tarski TheoremWBTS = WSTS - WQO + FAC

Conclusion

WSTSReachability problemsA quick story of WSTSA quick story of coverability in WSTS

And also for (infinitely branching) WSTS such as systems withinfinitely many initial states and parametric systems

Example of WSTS: ω–Petri nets (Geeraerts, Heußner, Praveen & Raskin PN’13, FU’15)

ω

Post( ) = , , , . . .

3 / 35

Page 14: WBTS: the new class of WSTS without WQOfinkel/2016-2017/slides-WBTS-workshop-etaps-2017.pdf · on the Grand Central Dispatch (GCD) technology [13] , and to perform parametric verication

IntroductionNews on coverability

Erdös and Tarski TheoremWBTS = WSTS - WQO + FAC

Conclusion

WSTSReachability problemsA quick story of WSTSA quick story of coverability in WSTS

Well structured transition system (F, ICALP’87)

S = (X ,−→,≤) where

X set,−→ ⊆X × X ,monotony,well-quasi-ordered.

:∀x0, x1, . . . ∃i < j s.t. xi ≤ xj .

∀ x −→ y

> >

x ′ y ′ ∃

4 / 35

Page 15: WBTS: the new class of WSTS without WQOfinkel/2016-2017/slides-WBTS-workshop-etaps-2017.pdf · on the Grand Central Dispatch (GCD) technology [13] , and to perform parametric verication

IntroductionNews on coverability

Erdös and Tarski TheoremWBTS = WSTS - WQO + FAC

Conclusion

WSTSReachability problemsA quick story of WSTSA quick story of coverability in WSTS

Well structured transition system (F, ICALP’87)

S = (X ,−→,≤) where

N3,−→ ⊆X × X ,monotony,well-quasi-ordered.

:∀x0, x1, . . . ∃i < j s.t. xi ≤ xj .

∀ x −→ y

> >

x ′ y ′ ∃

4 / 35

Page 16: WBTS: the new class of WSTS without WQOfinkel/2016-2017/slides-WBTS-workshop-etaps-2017.pdf · on the Grand Central Dispatch (GCD) technology [13] , and to perform parametric verication

IntroductionNews on coverability

Erdös and Tarski TheoremWBTS = WSTS - WQO + FAC

Conclusion

WSTSReachability problemsA quick story of WSTSA quick story of coverability in WSTS

Well structured transition system (F, ICALP’87)

S = (X ,−→,≤) where

X set,−→ ⊆N3 × N3,monotony,well-quasi-ordered.

:∀x0, x1, . . . ∃i < j s.t. xi ≤ xj .

∀ x −→ y

> >

x ′ y ′ ∃

4 / 35

Page 17: WBTS: the new class of WSTS without WQOfinkel/2016-2017/slides-WBTS-workshop-etaps-2017.pdf · on the Grand Central Dispatch (GCD) technology [13] , and to perform parametric verication

IntroductionNews on coverability

Erdös and Tarski TheoremWBTS = WSTS - WQO + FAC

Conclusion

WSTSReachability problemsA quick story of WSTSA quick story of coverability in WSTS

Well structured transition system (F, ICALP’87)

S = (X ,−→,≤) where

X set,−→ ⊆X × X ,monotony,well-quasi-ordered.

:∀x0, x1, . . . ∃i < j s.t. xi ≤ xj .

∀ x −→ y

> >

x ′ y ′ ∃

>

4 / 35

Page 18: WBTS: the new class of WSTS without WQOfinkel/2016-2017/slides-WBTS-workshop-etaps-2017.pdf · on the Grand Central Dispatch (GCD) technology [13] , and to perform parametric verication

IntroductionNews on coverability

Erdös and Tarski TheoremWBTS = WSTS - WQO + FAC

Conclusion

WSTSReachability problemsA quick story of WSTSA quick story of coverability in WSTS

Well structured transition system (F, ICALP’87)

S = (X ,−→,≤) where

X set,−→ ⊆X × X ,monotony,well-quasi-ordered.

:∀x0, x1, . . . ∃i < j s.t. xi ≤ xj .

∀ x −→ y

> >

x ′ y ′ ∃

>

4 / 35

Page 19: WBTS: the new class of WSTS without WQOfinkel/2016-2017/slides-WBTS-workshop-etaps-2017.pdf · on the Grand Central Dispatch (GCD) technology [13] , and to perform parametric verication

IntroductionNews on coverability

Erdös and Tarski TheoremWBTS = WSTS - WQO + FAC

Conclusion

WSTSReachability problemsA quick story of WSTSA quick story of coverability in WSTS

Well structured transition system (F, ICALP’87)

S = (X ,−→,≤) where

X set,−→ ⊆X × X ,monotony,well-quasi-ordered.

:∀x0, x1, . . . ∃i < j s.t. xi ≤ xj .

∀ x −→ y

> >

x ′ y ′ ∃

>

4 / 35

Page 20: WBTS: the new class of WSTS without WQOfinkel/2016-2017/slides-WBTS-workshop-etaps-2017.pdf · on the Grand Central Dispatch (GCD) technology [13] , and to perform parametric verication

IntroductionNews on coverability

Erdös and Tarski TheoremWBTS = WSTS - WQO + FAC

Conclusion

WSTSReachability problemsA quick story of WSTSA quick story of coverability in WSTS

Well structured transition system (F, ICALP’87)

S = (X ,−→,≤) where

X set,−→ ⊆X × X ,monotony,well-quasi-ordered.

:∀x0, x1, . . . ∃i < j s.t. xi ≤ xj .

∀ x −→ y

> >

x ′ ∗−→ y ′ ∃

4 / 35

Page 21: WBTS: the new class of WSTS without WQOfinkel/2016-2017/slides-WBTS-workshop-etaps-2017.pdf · on the Grand Central Dispatch (GCD) technology [13] , and to perform parametric verication

IntroductionNews on coverability

Erdös and Tarski TheoremWBTS = WSTS - WQO + FAC

Conclusion

WSTSReachability problemsA quick story of WSTSA quick story of coverability in WSTS

Well structured transition system (F, ICALP’87)

S = (X ,−→,≤) where

X set,−→ ⊆X × X ,transitive monotony,well-quasi-ordered.

:∀x0, x1, . . . ∃i < j s.t. xi ≤ xj .

∀ x −→ y

> >

x ′ +−→ y ′ ∃

4 / 35

Page 22: WBTS: the new class of WSTS without WQOfinkel/2016-2017/slides-WBTS-workshop-etaps-2017.pdf · on the Grand Central Dispatch (GCD) technology [13] , and to perform parametric verication

IntroductionNews on coverability

Erdös and Tarski TheoremWBTS = WSTS - WQO + FAC

Conclusion

WSTSReachability problemsA quick story of WSTSA quick story of coverability in WSTS

Well structured transition system (F, ICALP’87)

S = (X ,−→,≤) where

X set,−→ ⊆X × X ,strong monotony,well-quasi-ordered.

:∀x0, x1, . . . ∃i < j s.t. xi ≤ xj .

∀ x −→ y

> >

x ′ −→ y ′ ∃

4 / 35

Page 23: WBTS: the new class of WSTS without WQOfinkel/2016-2017/slides-WBTS-workshop-etaps-2017.pdf · on the Grand Central Dispatch (GCD) technology [13] , and to perform parametric verication

IntroductionNews on coverability

Erdös and Tarski TheoremWBTS = WSTS - WQO + FAC

Conclusion

WSTSReachability problemsA quick story of WSTSA quick story of coverability in WSTS

Well structured transition system (F, ICALP’87)

S = (X ,−→,≤) where

X set,−→ ⊆X × X ,monotony,well-quasi-ordered:∀x0, x1, . . . ∃i < j s.t. xi ≤ xj .

∀ x −→ y

> >

x ′ y ′ ∃

4 / 35

Page 24: WBTS: the new class of WSTS without WQOfinkel/2016-2017/slides-WBTS-workshop-etaps-2017.pdf · on the Grand Central Dispatch (GCD) technology [13] , and to perform parametric verication

IntroductionNews on coverability

Erdös and Tarski TheoremWBTS = WSTS - WQO + FAC

Conclusion

WSTSReachability problemsA quick story of WSTSA quick story of coverability in WSTS

WSTS Everywhere! (F, Schnoebelen LATIN’98, TCS’01)

T (w) = length of a longest computation starting fromw ∈ Σ∗.

T (w) ∈ Nω.

w≤Tw ′ if T (w) ≤ T (w ′).

≤T is a wqo on Σ∗.

TheoremTuring machines are WSTS with strict and strong monotony wrt≤T .

5 / 35

Page 25: WBTS: the new class of WSTS without WQOfinkel/2016-2017/slides-WBTS-workshop-etaps-2017.pdf · on the Grand Central Dispatch (GCD) technology [13] , and to perform parametric verication

IntroductionNews on coverability

Erdös and Tarski TheoremWBTS = WSTS - WQO + FAC

Conclusion

WSTSReachability problemsA quick story of WSTSA quick story of coverability in WSTS

WSTS Everywhere! (F, Schnoebelen LATIN’98, TCS’01)

T (w) = length of a longest computation starting fromw ∈ Σ∗.

T (w) ∈ Nω.

w≤Tw ′ if T (w) ≤ T (w ′).

≤T is a wqo on Σ∗.

TheoremTuring machines are WSTS with strict and strong monotony wrt≤T .

5 / 35

Page 26: WBTS: the new class of WSTS without WQOfinkel/2016-2017/slides-WBTS-workshop-etaps-2017.pdf · on the Grand Central Dispatch (GCD) technology [13] , and to perform parametric verication

IntroductionNews on coverability

Erdös and Tarski TheoremWBTS = WSTS - WQO + FAC

Conclusion

WSTSReachability problemsA quick story of WSTSA quick story of coverability in WSTS

WSTS Everywhere!

≤T is not decidable.

Hence TM are non-effective WSTS.

This also proves that there is no (non-trivial) decidabilityresult for non-effective WSTS (not surprising !).

6 / 35

Page 27: WBTS: the new class of WSTS without WQOfinkel/2016-2017/slides-WBTS-workshop-etaps-2017.pdf · on the Grand Central Dispatch (GCD) technology [13] , and to perform parametric verication

IntroductionNews on coverability

Erdös and Tarski TheoremWBTS = WSTS - WQO + FAC

Conclusion

WSTSReachability problemsA quick story of WSTSA quick story of coverability in WSTS

Effective WSTSA WSTS S = (X ,−→,≤), given as a tuple (MX ,M→,M≤) ofTuring machines, is effective if:(1) MX decides X(2) M→ decides →.(3) M≤ decides ≤.

7 / 35

Page 28: WBTS: the new class of WSTS without WQOfinkel/2016-2017/slides-WBTS-workshop-etaps-2017.pdf · on the Grand Central Dispatch (GCD) technology [13] , and to perform parametric verication

IntroductionNews on coverability

Erdös and Tarski TheoremWBTS = WSTS - WQO + FAC

Conclusion

WSTSReachability problemsA quick story of WSTSA quick story of coverability in WSTS

(Effective) WSTS Everywhere!

S = (Nk ,≤).Petri nets: WSTS with strict and strong monotony.Positive Affine nets, Reset/Transfer Petri nets: WSTS withstrong (but not strict) monotony.

S = (Q × Σ∗k ,= × vk).LCS: WSTS with non-strict monotony.

8 / 35

Page 29: WBTS: the new class of WSTS without WQOfinkel/2016-2017/slides-WBTS-workshop-etaps-2017.pdf · on the Grand Central Dispatch (GCD) technology [13] , and to perform parametric verication

IntroductionNews on coverability

Erdös and Tarski TheoremWBTS = WSTS - WQO + FAC

Conclusion

WSTSReachability problemsA quick story of WSTSA quick story of coverability in WSTS

(Effective) WSTS Everywhere!

S = (Nk ,≤).Petri nets: WSTS with strict and strong monotony.Positive Affine nets, Reset/Transfer Petri nets: WSTS withstrong (but not strict) monotony.

S = (Q × Σ∗k ,= × vk).LCS: WSTS with non-strict monotony.

8 / 35

Page 30: WBTS: the new class of WSTS without WQOfinkel/2016-2017/slides-WBTS-workshop-etaps-2017.pdf · on the Grand Central Dispatch (GCD) technology [13] , and to perform parametric verication

IntroductionNews on coverability

Erdös and Tarski TheoremWBTS = WSTS - WQO + FAC

Conclusion

WSTSReachability problemsA quick story of WSTSA quick story of coverability in WSTS

WSTS still verywhere!Data nets: S = (Q × Nk)∗

Lazic, Newcomb, Ouaknine, Roscoe, Worrell (PN’07)Hofman, Lasota, Lazić, Leroux, Schmitz, Totzke(FOSSACS’16).Lasota (PN’16)

ν-Petri nets: S = (Q × Nk)⊕.Rosa-Velardo, de Frutos-Escrig (PN’07)Lazić and Schmitz (LICS’16).

Pi-calculus: Depth-Bounded Processes (trees).Wies, Zufferey, Henzinger (FOSSACS’10, VMCAI’12).

Timed Petri nets: Regions = ((Q × Nk)⊕)∗Bonnet, F, Haddad, Rosa-Velardo (FOSSACS’10)Haddad, Schmitz, Schnoebelen (LICS’12).

Process algebra (BPP,...).9 / 35

Page 31: WBTS: the new class of WSTS without WQOfinkel/2016-2017/slides-WBTS-workshop-etaps-2017.pdf · on the Grand Central Dispatch (GCD) technology [13] , and to perform parametric verication

IntroductionNews on coverability

Erdös and Tarski TheoremWBTS = WSTS - WQO + FAC

Conclusion

WSTSReachability problemsA quick story of WSTSA quick story of coverability in WSTS

ObjectiveWe want to study the usual reachability problems, e.g.,

Reachability...but it is undecidable for general WSTS :((

TerminationCoverability (the most used property)BoundednessAnd other properties like eventuality, simulation by finiteautomaton...

10 / 35

Page 32: WBTS: the new class of WSTS without WQOfinkel/2016-2017/slides-WBTS-workshop-etaps-2017.pdf · on the Grand Central Dispatch (GCD) technology [13] , and to perform parametric verication

IntroductionNews on coverability

Erdös and Tarski TheoremWBTS = WSTS - WQO + FAC

Conclusion

WSTSReachability problemsA quick story of WSTSA quick story of coverability in WSTS

ObjectiveWe want to study the usual reachability problems, e.g.,

Reachability...but it is undecidable for general WSTS :((Termination

Coverability (the most used property)BoundednessAnd other properties like eventuality, simulation by finiteautomaton...

10 / 35

Page 33: WBTS: the new class of WSTS without WQOfinkel/2016-2017/slides-WBTS-workshop-etaps-2017.pdf · on the Grand Central Dispatch (GCD) technology [13] , and to perform parametric verication

IntroductionNews on coverability

Erdös and Tarski TheoremWBTS = WSTS - WQO + FAC

Conclusion

WSTSReachability problemsA quick story of WSTSA quick story of coverability in WSTS

ObjectiveWe want to study the usual reachability problems, e.g.,

Reachability...but it is undecidable for general WSTS :((TerminationCoverability (the most used property)

BoundednessAnd other properties like eventuality, simulation by finiteautomaton...

10 / 35

Page 34: WBTS: the new class of WSTS without WQOfinkel/2016-2017/slides-WBTS-workshop-etaps-2017.pdf · on the Grand Central Dispatch (GCD) technology [13] , and to perform parametric verication

IntroductionNews on coverability

Erdös and Tarski TheoremWBTS = WSTS - WQO + FAC

Conclusion

WSTSReachability problemsA quick story of WSTSA quick story of coverability in WSTS

ObjectiveWe want to study the usual reachability problems, e.g.,

Reachability...but it is undecidable for general WSTS :((TerminationCoverability (the most used property)Boundedness

And other properties like eventuality, simulation by finiteautomaton...

10 / 35

Page 35: WBTS: the new class of WSTS without WQOfinkel/2016-2017/slides-WBTS-workshop-etaps-2017.pdf · on the Grand Central Dispatch (GCD) technology [13] , and to perform parametric verication

IntroductionNews on coverability

Erdös and Tarski TheoremWBTS = WSTS - WQO + FAC

Conclusion

WSTSReachability problemsA quick story of WSTSA quick story of coverability in WSTS

ObjectiveWe want to study the usual reachability problems, e.g.,

Reachability...but it is undecidable for general WSTS :((TerminationCoverability (the most used property)BoundednessAnd other properties like eventuality, simulation by finiteautomaton...

10 / 35

Page 36: WBTS: the new class of WSTS without WQOfinkel/2016-2017/slides-WBTS-workshop-etaps-2017.pdf · on the Grand Central Dispatch (GCD) technology [13] , and to perform parametric verication

IntroductionNews on coverability

Erdös and Tarski TheoremWBTS = WSTS - WQO + FAC

Conclusion

WSTSReachability problemsA quick story of WSTSA quick story of coverability in WSTS

ICALP’87 (F)WSTS definitionsdecidability of terminationdecidability of boundednesscomputation of the coverability set hence decidability ofcoverability (under stronger hyp.)

LICS’96 (Abdulla, Cerans, Jonsson, Tsay)decidability of coverability with a backward algorithmdecidability of simulation with finite-state systemsundecidability of repeated control-state (for LCS).

LICS’98 (Emerson, Namjoshi), LICS’99 (Esparza, F, Mayr)broadcast protocols are WSTSmodel checking of WSTS (with procedures)

WSTS everywhere, TCS’01 (F, Schnoebelen)

11 / 35

Page 37: WBTS: the new class of WSTS without WQOfinkel/2016-2017/slides-WBTS-workshop-etaps-2017.pdf · on the Grand Central Dispatch (GCD) technology [13] , and to perform parametric verication

IntroductionNews on coverability

Erdös and Tarski TheoremWBTS = WSTS - WQO + FAC

Conclusion

WSTSReachability problemsA quick story of WSTSA quick story of coverability in WSTS

ICALP’87 (F)WSTS definitionsdecidability of terminationdecidability of boundednesscomputation of the coverability set hence decidability ofcoverability (under stronger hyp.)

LICS’96 (Abdulla, Cerans, Jonsson, Tsay)decidability of coverability with a backward algorithmdecidability of simulation with finite-state systemsundecidability of repeated control-state (for LCS).

LICS’98 (Emerson, Namjoshi), LICS’99 (Esparza, F, Mayr)broadcast protocols are WSTSmodel checking of WSTS (with procedures)

WSTS everywhere, TCS’01 (F, Schnoebelen)

11 / 35

Page 38: WBTS: the new class of WSTS without WQOfinkel/2016-2017/slides-WBTS-workshop-etaps-2017.pdf · on the Grand Central Dispatch (GCD) technology [13] , and to perform parametric verication

IntroductionNews on coverability

Erdös and Tarski TheoremWBTS = WSTS - WQO + FAC

Conclusion

WSTSReachability problemsA quick story of WSTSA quick story of coverability in WSTS

ICALP’87 (F)WSTS definitionsdecidability of terminationdecidability of boundednesscomputation of the coverability set hence decidability ofcoverability (under stronger hyp.)

LICS’96 (Abdulla, Cerans, Jonsson, Tsay)decidability of coverability with a backward algorithmdecidability of simulation with finite-state systemsundecidability of repeated control-state (for LCS).

LICS’98 (Emerson, Namjoshi), LICS’99 (Esparza, F, Mayr)broadcast protocols are WSTSmodel checking of WSTS (with procedures)

WSTS everywhere, TCS’01 (F, Schnoebelen)11 / 35

Page 39: WBTS: the new class of WSTS without WQOfinkel/2016-2017/slides-WBTS-workshop-etaps-2017.pdf · on the Grand Central Dispatch (GCD) technology [13] , and to perform parametric verication

IntroductionNews on coverability

Erdös and Tarski TheoremWBTS = WSTS - WQO + FAC

Conclusion

WSTSReachability problemsA quick story of WSTSA quick story of coverability in WSTS

FSTTCS’04 (Geeraerts, Raskin and Van Begin):The first forward coverability algorithm for WSTS (with ADL).

STACS’09, ICALP’09 (F, Goubault-Larrecq), ICALP’14(Blondin, F, McKenzie)

ADL is not an hypothesis.Ideal completion of any WSTSComputation of the clover for flattable WSTSω2-WSTS are completable and robust....

2015-2016: Use of ideals decomposition in:RP’15: The Ideal View on Rackoff’s Coverability Technique(Lazić, Schmitz)LICS’15: Demystifying Reachability in Vector AdditionSystems (Leroux, Schmitz).FOSSACS’16: Coverability Trees for Petri Nets with UnorderedData (Schmitz and a lot of authors...)LICS’16: ν-Petri nets (Lazić, Schmitz)....

12 / 35

Page 40: WBTS: the new class of WSTS without WQOfinkel/2016-2017/slides-WBTS-workshop-etaps-2017.pdf · on the Grand Central Dispatch (GCD) technology [13] , and to perform parametric verication

IntroductionNews on coverability

Erdös and Tarski TheoremWBTS = WSTS - WQO + FAC

Conclusion

WSTSReachability problemsA quick story of WSTSA quick story of coverability in WSTS

FSTTCS’04 (Geeraerts, Raskin and Van Begin):The first forward coverability algorithm for WSTS (with ADL).

STACS’09, ICALP’09 (F, Goubault-Larrecq), ICALP’14(Blondin, F, McKenzie)

ADL is not an hypothesis.Ideal completion of any WSTSComputation of the clover for flattable WSTSω2-WSTS are completable and robust....

2015-2016: Use of ideals decomposition in:RP’15: The Ideal View on Rackoff’s Coverability Technique(Lazić, Schmitz)LICS’15: Demystifying Reachability in Vector AdditionSystems (Leroux, Schmitz).FOSSACS’16: Coverability Trees for Petri Nets with UnorderedData (Schmitz and a lot of authors...)LICS’16: ν-Petri nets (Lazić, Schmitz).... 12 / 35

Page 41: WBTS: the new class of WSTS without WQOfinkel/2016-2017/slides-WBTS-workshop-etaps-2017.pdf · on the Grand Central Dispatch (GCD) technology [13] , and to perform parametric verication

IntroductionNews on coverability

Erdös and Tarski TheoremWBTS = WSTS - WQO + FAC

Conclusion

WSTSReachability problemsA quick story of WSTSA quick story of coverability in WSTS

The survey/story of coverability for WSTSYear Authors Mathematical hyp. Effectivity hyp. back/forward1978 Arnold & Latteux reset VAS reset VAS are effective backward1987 F. (very) WSTS+strong+strict+ω2-wqo+... effective very WSTS forward1996 Abdulla & CJT strong monotony PreS (↑ x) comp. backward1998 F. Schnoebelen monotony ↑ PreS (↑ x) comp. backward2004 Geeraerts & RV strong monotony, ADL effective ADL forward2006 Geeraerts & RV monotony, ADL effective ADL forward2009 F. & Goubault-Larrecq strong monotony, weak ADL, flattable effective WADL forward2009 F. & Goubault-Larrecq strong monotony, flattable ideally effective forward2014 Blondin & FM monotony, ideally effective forward2016 Blondin & FM monotony, no wqo but FAC ideally effective forward

13 / 35

Page 42: WBTS: the new class of WSTS without WQOfinkel/2016-2017/slides-WBTS-workshop-etaps-2017.pdf · on the Grand Central Dispatch (GCD) technology [13] , and to perform parametric verication

IntroductionNews on coverability

Erdös and Tarski TheoremWBTS = WSTS - WQO + FAC

Conclusion

CoverabilityThe backward coverability algorithmA conceptual algorithmThe TheoremA conceptual coverability algorithm based on downward closed setsProcedure 1: coverability certificateProcedure 2: non coverability certificate

CoverabilityFor monotone transition systems, y is coverable from x if

∃x ′ | x ∗−→ x ′ ≥ y ( this is the definition !) iff

x ∈ Pre∗(↑ y) ( this could be the definition !) iffy ∈ ↓Post∗(x) ( this could be the definition !).

Remark

Pre∗(↑ y) = ↑Pre∗(↑ y)↓Post∗(x) = ↓Post∗(↓ x).

ConsequenceCompute ↑Pre∗(↑ y) or ↓Post∗(x).Compute ↑ or ↓ over-approximations invariants.

14 / 35

Page 43: WBTS: the new class of WSTS without WQOfinkel/2016-2017/slides-WBTS-workshop-etaps-2017.pdf · on the Grand Central Dispatch (GCD) technology [13] , and to perform parametric verication

IntroductionNews on coverability

Erdös and Tarski TheoremWBTS = WSTS - WQO + FAC

Conclusion

CoverabilityThe backward coverability algorithmA conceptual algorithmThe TheoremA conceptual coverability algorithm based on downward closed setsProcedure 1: coverability certificateProcedure 2: non coverability certificate

CoverabilityFor monotone transition systems, y is coverable from x if

∃x ′ | x ∗−→ x ′ ≥ y ( this is the definition !) iffx ∈ Pre∗(↑ y) ( this could be the definition !) iff

y ∈ ↓Post∗(x) ( this could be the definition !).

Remark

Pre∗(↑ y) = ↑Pre∗(↑ y)↓Post∗(x) = ↓Post∗(↓ x).

ConsequenceCompute ↑Pre∗(↑ y) or ↓Post∗(x).Compute ↑ or ↓ over-approximations invariants.

14 / 35

Page 44: WBTS: the new class of WSTS without WQOfinkel/2016-2017/slides-WBTS-workshop-etaps-2017.pdf · on the Grand Central Dispatch (GCD) technology [13] , and to perform parametric verication

IntroductionNews on coverability

Erdös and Tarski TheoremWBTS = WSTS - WQO + FAC

Conclusion

CoverabilityThe backward coverability algorithmA conceptual algorithmThe TheoremA conceptual coverability algorithm based on downward closed setsProcedure 1: coverability certificateProcedure 2: non coverability certificate

CoverabilityFor monotone transition systems, y is coverable from x if

∃x ′ | x ∗−→ x ′ ≥ y ( this is the definition !) iffx ∈ Pre∗(↑ y) ( this could be the definition !) iffy ∈ ↓Post∗(x) ( this could be the definition !).

Remark

Pre∗(↑ y) = ↑Pre∗(↑ y)

↓Post∗(x) = ↓Post∗(↓ x).

ConsequenceCompute ↑Pre∗(↑ y) or ↓Post∗(x).Compute ↑ or ↓ over-approximations invariants.

14 / 35

Page 45: WBTS: the new class of WSTS without WQOfinkel/2016-2017/slides-WBTS-workshop-etaps-2017.pdf · on the Grand Central Dispatch (GCD) technology [13] , and to perform parametric verication

IntroductionNews on coverability

Erdös and Tarski TheoremWBTS = WSTS - WQO + FAC

Conclusion

CoverabilityThe backward coverability algorithmA conceptual algorithmThe TheoremA conceptual coverability algorithm based on downward closed setsProcedure 1: coverability certificateProcedure 2: non coverability certificate

CoverabilityFor monotone transition systems, y is coverable from x if

∃x ′ | x ∗−→ x ′ ≥ y ( this is the definition !) iffx ∈ Pre∗(↑ y) ( this could be the definition !) iffy ∈ ↓Post∗(x) ( this could be the definition !).

Remark

Pre∗(↑ y) = ↑Pre∗(↑ y)↓Post∗(x) = ↓Post∗(↓ x).

ConsequenceCompute ↑Pre∗(↑ y) or ↓Post∗(x).Compute ↑ or ↓ over-approximations invariants. 14 / 35

Page 46: WBTS: the new class of WSTS without WQOfinkel/2016-2017/slides-WBTS-workshop-etaps-2017.pdf · on the Grand Central Dispatch (GCD) technology [13] , and to perform parametric verication

IntroductionNews on coverability

Erdös and Tarski TheoremWBTS = WSTS - WQO + FAC

Conclusion

CoverabilityThe backward coverability algorithmA conceptual algorithmThe TheoremA conceptual coverability algorithm based on downward closed setsProcedure 1: coverability certificateProcedure 2: non coverability certificate

The backward coverability algorithm (based on ↑-sets)

1978: coverability for reset VAS is decidable (Arnold andLatteux published in French in CALCOLO’78). Theiralgorithm is an instance of the backward algorithm (LICS’96).

1993: decidability of coverability for LCS (Abdulla, Cerans,Jonsson, Tsay, LICS’93)

1996: decidability of coverability for strong WSTS assumingPre(↑ x) is computable (Abdulla, Cerans, Jonsson, Tsay,LICS’96)1998: decidability of coverability for WSTS assuming↑Pre(↑ x) is computable (F., Schnoebelen LATIN’98)

15 / 35

Page 47: WBTS: the new class of WSTS without WQOfinkel/2016-2017/slides-WBTS-workshop-etaps-2017.pdf · on the Grand Central Dispatch (GCD) technology [13] , and to perform parametric verication

IntroductionNews on coverability

Erdös and Tarski TheoremWBTS = WSTS - WQO + FAC

Conclusion

CoverabilityThe backward coverability algorithmA conceptual algorithmThe TheoremA conceptual coverability algorithm based on downward closed setsProcedure 1: coverability certificateProcedure 2: non coverability certificate

Remarks on the backward coverability algorithm

It computes Pre∗(↑ y) that is more than solving coverability.

It is often but not always computable, ex: depth-boundedprocesses (Wies, Zufferey, Henzinger, FOSSACS’10)

Backward algorithms are often less efficient than forwardalgorithms.

16 / 35

Page 48: WBTS: the new class of WSTS without WQOfinkel/2016-2017/slides-WBTS-workshop-etaps-2017.pdf · on the Grand Central Dispatch (GCD) technology [13] , and to perform parametric verication

IntroductionNews on coverability

Erdös and Tarski TheoremWBTS = WSTS - WQO + FAC

Conclusion

CoverabilityThe backward coverability algorithmA conceptual algorithmThe TheoremA conceptual coverability algorithm based on downward closed setsProcedure 1: coverability certificateProcedure 2: non coverability certificate

Coverability: a conceptual algorithm (still based on ↑-sets)

Execute two procedures in parallel, one looking for a coverabilitycertificate and one looking for a non coverability certificate.

Coverability is semi-decidable:if ∃x ′ ≥ y , x ∗−→ x ′, one finally will find x ′.

Non-coverability is also semi-decidable:¬(∃x ′ ≥ y , x ∗−→ x ′) iff x 6∈ Pre∗(↑ y) =↑ Jm for some m.One enumerates all the finite sets J ⊆ X such that y ∈↑ J andPre(↑ J) ⊆↑ J and x 6∈↑ J , hence ↑ Jm ⊆ Pre∗(↑ J) =↑ J .Enumeration of upward closed sets by a finite set of minimal elements is aconsequence of (X ,≤) is WQO.One finally will find such an invariant J . May be we find a large Jp

s.t. ↑ Jm = Pre∗(↑ y) (↑ Jp but x 6∈↑ Jp =⇒ x 6∈↑ Jm = Pre∗(↑ y).

17 / 35

Page 49: WBTS: the new class of WSTS without WQOfinkel/2016-2017/slides-WBTS-workshop-etaps-2017.pdf · on the Grand Central Dispatch (GCD) technology [13] , and to perform parametric verication

IntroductionNews on coverability

Erdös and Tarski TheoremWBTS = WSTS - WQO + FAC

Conclusion

CoverabilityThe backward coverability algorithmA conceptual algorithmThe TheoremA conceptual coverability algorithm based on downward closed setsProcedure 1: coverability certificateProcedure 2: non coverability certificate

Coverability: a conceptual algorithm (still based on ↑-sets)

Execute two procedures in parallel, one looking for a coverabilitycertificate and one looking for a non coverability certificate.

Coverability is semi-decidable:if ∃x ′ ≥ y , x ∗−→ x ′, one finally will find x ′.

Non-coverability is also semi-decidable:¬(∃x ′ ≥ y , x ∗−→ x ′) iff x 6∈ Pre∗(↑ y) =↑ Jm for some m.One enumerates all the finite sets J ⊆ X such that y ∈↑ J andPre(↑ J) ⊆↑ J and x 6∈↑ J , hence ↑ Jm ⊆ Pre∗(↑ J) =↑ J .Enumeration of upward closed sets by a finite set of minimal elements is aconsequence of (X ,≤) is WQO.One finally will find such an invariant J . May be we find a large Jp

s.t. ↑ Jm = Pre∗(↑ y) (↑ Jp but x 6∈↑ Jp =⇒ x 6∈↑ Jm = Pre∗(↑ y).

17 / 35

Page 50: WBTS: the new class of WSTS without WQOfinkel/2016-2017/slides-WBTS-workshop-etaps-2017.pdf · on the Grand Central Dispatch (GCD) technology [13] , and to perform parametric verication

IntroductionNews on coverability

Erdös and Tarski TheoremWBTS = WSTS - WQO + FAC

Conclusion

CoverabilityThe backward coverability algorithmA conceptual algorithmThe TheoremA conceptual coverability algorithm based on downward closed setsProcedure 1: coverability certificateProcedure 2: non coverability certificate

Coverability: a conceptual algorithm (still based on ↑-sets)

Execute two procedures in parallel, one looking for a coverabilitycertificate and one looking for a non coverability certificate.

Coverability is semi-decidable:if ∃x ′ ≥ y , x ∗−→ x ′, one finally will find x ′.

Non-coverability is also semi-decidable:¬(∃x ′ ≥ y , x ∗−→ x ′) iff x 6∈ Pre∗(↑ y) =↑ Jm for some m.

One enumerates all the finite sets J ⊆ X such that y ∈↑ J andPre(↑ J) ⊆↑ J and x 6∈↑ J , hence ↑ Jm ⊆ Pre∗(↑ J) =↑ J .Enumeration of upward closed sets by a finite set of minimal elements is aconsequence of (X ,≤) is WQO.One finally will find such an invariant J . May be we find a large Jp

s.t. ↑ Jm = Pre∗(↑ y) (↑ Jp but x 6∈↑ Jp =⇒ x 6∈↑ Jm = Pre∗(↑ y).

17 / 35

Page 51: WBTS: the new class of WSTS without WQOfinkel/2016-2017/slides-WBTS-workshop-etaps-2017.pdf · on the Grand Central Dispatch (GCD) technology [13] , and to perform parametric verication

IntroductionNews on coverability

Erdös and Tarski TheoremWBTS = WSTS - WQO + FAC

Conclusion

CoverabilityThe backward coverability algorithmA conceptual algorithmThe TheoremA conceptual coverability algorithm based on downward closed setsProcedure 1: coverability certificateProcedure 2: non coverability certificate

Coverability: a conceptual algorithm (still based on ↑-sets)

Execute two procedures in parallel, one looking for a coverabilitycertificate and one looking for a non coverability certificate.

Coverability is semi-decidable:if ∃x ′ ≥ y , x ∗−→ x ′, one finally will find x ′.

Non-coverability is also semi-decidable:¬(∃x ′ ≥ y , x ∗−→ x ′) iff x 6∈ Pre∗(↑ y) =↑ Jm for some m.One enumerates all the finite sets J ⊆ X such that y ∈↑ J andPre(↑ J) ⊆↑ J and x 6∈↑ J , hence ↑ Jm ⊆ Pre∗(↑ J) =↑ J .Enumeration of upward closed sets by a finite set of minimal elements is aconsequence of (X ,≤) is WQO.

One finally will find such an invariant J . May be we find a large Jp

s.t. ↑ Jm = Pre∗(↑ y) (↑ Jp but x 6∈↑ Jp =⇒ x 6∈↑ Jm = Pre∗(↑ y).

17 / 35

Page 52: WBTS: the new class of WSTS without WQOfinkel/2016-2017/slides-WBTS-workshop-etaps-2017.pdf · on the Grand Central Dispatch (GCD) technology [13] , and to perform parametric verication

IntroductionNews on coverability

Erdös and Tarski TheoremWBTS = WSTS - WQO + FAC

Conclusion

CoverabilityThe backward coverability algorithmA conceptual algorithmThe TheoremA conceptual coverability algorithm based on downward closed setsProcedure 1: coverability certificateProcedure 2: non coverability certificate

Coverability: a conceptual algorithm (still based on ↑-sets)

Execute two procedures in parallel, one looking for a coverabilitycertificate and one looking for a non coverability certificate.

Coverability is semi-decidable:if ∃x ′ ≥ y , x ∗−→ x ′, one finally will find x ′.

Non-coverability is also semi-decidable:¬(∃x ′ ≥ y , x ∗−→ x ′) iff x 6∈ Pre∗(↑ y) =↑ Jm for some m.One enumerates all the finite sets J ⊆ X such that y ∈↑ J andPre(↑ J) ⊆↑ J and x 6∈↑ J , hence ↑ Jm ⊆ Pre∗(↑ J) =↑ J .Enumeration of upward closed sets by a finite set of minimal elements is aconsequence of (X ,≤) is WQO.One finally will find such an invariant J . May be we find a large Jp

s.t. ↑ Jm = Pre∗(↑ y) (↑ Jp but x 6∈↑ Jp =⇒ x 6∈↑ Jm = Pre∗(↑ y).

17 / 35

Page 53: WBTS: the new class of WSTS without WQOfinkel/2016-2017/slides-WBTS-workshop-etaps-2017.pdf · on the Grand Central Dispatch (GCD) technology [13] , and to perform parametric verication

IntroductionNews on coverability

Erdös and Tarski TheoremWBTS = WSTS - WQO + FAC

Conclusion

CoverabilityThe backward coverability algorithmA conceptual algorithmThe TheoremA conceptual coverability algorithm based on downward closed setsProcedure 1: coverability certificateProcedure 2: non coverability certificate

A very interesting characterization of WQO (Finite Basis Property):

Theorem (Higman’52, Nash-Williams’ 63-64, Fraïsse’86,...)

(X ,≤) WQO ⇐⇒ for all U = ↑U ⊆ X we have: U =⋃finite↑ x

18 / 35

Page 54: WBTS: the new class of WSTS without WQOfinkel/2016-2017/slides-WBTS-workshop-etaps-2017.pdf · on the Grand Central Dispatch (GCD) technology [13] , and to perform parametric verication

IntroductionNews on coverability

Erdös and Tarski TheoremWBTS = WSTS - WQO + FAC

Conclusion

CoverabilityThe backward coverability algorithmA conceptual algorithmThe TheoremA conceptual coverability algorithm based on downward closed setsProcedure 1: coverability certificateProcedure 2: non coverability certificate

The downward approach for coverability

Initially presented by Geeraerts, Raskin, and Van Begin(FSTTCS’04) for strongly monotone WSTS with AdequateDomain of Limits (ADL).

Simplified and extended with Goubault-Larrecq (STACS’09):ADL is not an hypothesis, it always exists.Still simplified and extended with Blondin, McKenzie(ICALP’14): ideal completion for infinitely branching.Still simplified and extended with Blondin, McKenzie: WQO isnot necessary. (arxiv, august 2016, to appear in LMCS’2017).

19 / 35

Page 55: WBTS: the new class of WSTS without WQOfinkel/2016-2017/slides-WBTS-workshop-etaps-2017.pdf · on the Grand Central Dispatch (GCD) technology [13] , and to perform parametric verication

IntroductionNews on coverability

Erdös and Tarski TheoremWBTS = WSTS - WQO + FAC

Conclusion

CoverabilityThe backward coverability algorithmA conceptual algorithmThe TheoremA conceptual coverability algorithm based on downward closed setsProcedure 1: coverability certificateProcedure 2: non coverability certificate

The downward approach for coverability

Initially presented by Geeraerts, Raskin, and Van Begin(FSTTCS’04) for strongly monotone WSTS with AdequateDomain of Limits (ADL).Simplified and extended with Goubault-Larrecq (STACS’09):ADL is not an hypothesis, it always exists.

Still simplified and extended with Blondin, McKenzie(ICALP’14): ideal completion for infinitely branching.Still simplified and extended with Blondin, McKenzie: WQO isnot necessary. (arxiv, august 2016, to appear in LMCS’2017).

19 / 35

Page 56: WBTS: the new class of WSTS without WQOfinkel/2016-2017/slides-WBTS-workshop-etaps-2017.pdf · on the Grand Central Dispatch (GCD) technology [13] , and to perform parametric verication

IntroductionNews on coverability

Erdös and Tarski TheoremWBTS = WSTS - WQO + FAC

Conclusion

CoverabilityThe backward coverability algorithmA conceptual algorithmThe TheoremA conceptual coverability algorithm based on downward closed setsProcedure 1: coverability certificateProcedure 2: non coverability certificate

The downward approach for coverability

Initially presented by Geeraerts, Raskin, and Van Begin(FSTTCS’04) for strongly monotone WSTS with AdequateDomain of Limits (ADL).Simplified and extended with Goubault-Larrecq (STACS’09):ADL is not an hypothesis, it always exists.Still simplified and extended with Blondin, McKenzie(ICALP’14): ideal completion for infinitely branching.

Still simplified and extended with Blondin, McKenzie: WQO isnot necessary. (arxiv, august 2016, to appear in LMCS’2017).

19 / 35

Page 57: WBTS: the new class of WSTS without WQOfinkel/2016-2017/slides-WBTS-workshop-etaps-2017.pdf · on the Grand Central Dispatch (GCD) technology [13] , and to perform parametric verication

IntroductionNews on coverability

Erdös and Tarski TheoremWBTS = WSTS - WQO + FAC

Conclusion

CoverabilityThe backward coverability algorithmA conceptual algorithmThe TheoremA conceptual coverability algorithm based on downward closed setsProcedure 1: coverability certificateProcedure 2: non coverability certificate

The downward approach for coverability

Initially presented by Geeraerts, Raskin, and Van Begin(FSTTCS’04) for strongly monotone WSTS with AdequateDomain of Limits (ADL).Simplified and extended with Goubault-Larrecq (STACS’09):ADL is not an hypothesis, it always exists.Still simplified and extended with Blondin, McKenzie(ICALP’14): ideal completion for infinitely branching.Still simplified and extended with Blondin, McKenzie: WQO isnot necessary. (arxiv, august 2016, to appear in LMCS’2017).

19 / 35

Page 58: WBTS: the new class of WSTS without WQOfinkel/2016-2017/slides-WBTS-workshop-etaps-2017.pdf · on the Grand Central Dispatch (GCD) technology [13] , and to perform parametric verication

IntroductionNews on coverability

Erdös and Tarski TheoremWBTS = WSTS - WQO + FAC

Conclusion

CoverabilityThe backward coverability algorithmA conceptual algorithmThe TheoremA conceptual coverability algorithm based on downward closed setsProcedure 1: coverability certificateProcedure 2: non coverability certificate

In order to decide whether y is coverable from x , Procedure 1iteratively computes

Post(x),Post(Post(x)),Post3(x), . . .

until it finds y in ↓Postn(x).

procedure 1: searches for a coverability certificate of y from xD ← x ;while y 6∈ ↓D do

D ← D ∪ Post(D)return true

20 / 35

Page 59: WBTS: the new class of WSTS without WQOfinkel/2016-2017/slides-WBTS-workshop-etaps-2017.pdf · on the Grand Central Dispatch (GCD) technology [13] , and to perform parametric verication

IntroductionNews on coverability

Erdös and Tarski TheoremWBTS = WSTS - WQO + FAC

Conclusion

CoverabilityThe backward coverability algorithmA conceptual algorithmThe TheoremA conceptual coverability algorithm based on downward closed setsProcedure 1: coverability certificateProcedure 2: non coverability certificate

y is not coverable from x iff y 6∈ ↓Post∗(x).

Let (Di )i be an enumeration of dc sets, hence ↓Post∗(x) = Dm,for some m.procedure 2: enumerates dcs to find non coverability certificate ofy from xi ← 0;while ¬(↓Post(Di ) ⊆ Di and x ∈ Di and y 6∈ Di ) do

i ← i + 1return false

21 / 35

Page 60: WBTS: the new class of WSTS without WQOfinkel/2016-2017/slides-WBTS-workshop-etaps-2017.pdf · on the Grand Central Dispatch (GCD) technology [13] , and to perform parametric verication

IntroductionNews on coverability

Erdös and Tarski TheoremWBTS = WSTS - WQO + FAC

Conclusion

CoverabilityThe backward coverability algorithmA conceptual algorithmThe TheoremA conceptual coverability algorithm based on downward closed setsProcedure 1: coverability certificateProcedure 2: non coverability certificate

procedure 1: searches for a coverability certificate of y from xD ← x ;while y 6∈ ↓D do

D ← D ∪ Post(D)return true

procedure 2: enumerates dcs to find non coverability certificatei ← 0;while ¬(↓Post(Di ) ⊆ Di and x ∈ Di and y 6∈ Di ) do

i ← i + 1return false

Effective hypothesesdcs are recursive.Union of dcs is computable↓ Post(D) is computable.Inclusion between dcs is decidable. 22 / 35

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IntroductionNews on coverability

Erdös and Tarski TheoremWBTS = WSTS - WQO + FAC

Conclusion

CoverabilityThe backward coverability algorithmA conceptual algorithmThe TheoremA conceptual coverability algorithm based on downward closed setsProcedure 1: coverability certificateProcedure 2: non coverability certificate

TheoremLet S = (X ,−→,≤) be a monotone transition system + there existsan enumeration of downward closed sets of X , and let x , y ∈ X .

1 y is coverable from x iff Procedure 1 terminates.2 y is not coverable from x iff Procedure 2 terminates.

This theorem does not provide an algorithm.

RemarkWSTS, hence WQO implies possible enumeration of downwardclosed sets (by minimal elements of upward closed sets) but theconverse is false: (Z,≤) is not WQO but one may enumerate theDi as follows: Di = ↓ xi for xi ∈ Z or Di = Z.

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IntroductionNews on coverability

Erdös and Tarski TheoremWBTS = WSTS - WQO + FAC

Conclusion

CoverabilityThe backward coverability algorithmA conceptual algorithmThe TheoremA conceptual coverability algorithm based on downward closed setsProcedure 1: coverability certificateProcedure 2: non coverability certificate

QuestionHow to enumerate downward closed sets ?

AnswerBy enumerating ideals ! (Erdös & Tarski)

24 / 35

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IntroductionNews on coverability

Erdös and Tarski TheoremWBTS = WSTS - WQO + FAC

Conclusion

DefinitionsIdealsThe TheoremErdös & Tarski Theorem for WQO

Definition

Now (X ,≤) is a qo (shortly written X or ≤)

A ⊆ X is an antichain if all couples of A are uncomparable.X is Finite AntiChain (FAC) if all antichains in X are finite.X is Well Founded (WF) if all strictly decreasing sequences inX are finite.X is WQO if X is FAC & X is WF (can be a Theorem).

Every WQO is FACThe converse is false since Z,Q,R are FAC but not WF:13/7, π, 3, 2, 1, 0,−1/2,−1,−2,−3,−41,−78695/12, ...Z2 contains infinite antichains, A = {(n,−n) | n ∈ N}, hencethe cartesian product of two FAC’s is not necessarly a FAC.

25 / 35

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IntroductionNews on coverability

Erdös and Tarski TheoremWBTS = WSTS - WQO + FAC

Conclusion

DefinitionsIdealsThe TheoremErdös & Tarski Theorem for WQO

Definition

Now (X ,≤) is a qo (shortly written X or ≤)A ⊆ X is an antichain if all couples of A are uncomparable.

X is Finite AntiChain (FAC) if all antichains in X are finite.X is Well Founded (WF) if all strictly decreasing sequences inX are finite.X is WQO if X is FAC & X is WF (can be a Theorem).

Every WQO is FACThe converse is false since Z,Q,R are FAC but not WF:13/7, π, 3, 2, 1, 0,−1/2,−1,−2,−3,−41,−78695/12, ...Z2 contains infinite antichains, A = {(n,−n) | n ∈ N}, hencethe cartesian product of two FAC’s is not necessarly a FAC.

25 / 35

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IntroductionNews on coverability

Erdös and Tarski TheoremWBTS = WSTS - WQO + FAC

Conclusion

DefinitionsIdealsThe TheoremErdös & Tarski Theorem for WQO

Definition

Now (X ,≤) is a qo (shortly written X or ≤)A ⊆ X is an antichain if all couples of A are uncomparable.X is Finite AntiChain (FAC) if all antichains in X are finite.

X is Well Founded (WF) if all strictly decreasing sequences inX are finite.X is WQO if X is FAC & X is WF (can be a Theorem).

Every WQO is FACThe converse is false since Z,Q,R are FAC but not WF:13/7, π, 3, 2, 1, 0,−1/2,−1,−2,−3,−41,−78695/12, ...Z2 contains infinite antichains, A = {(n,−n) | n ∈ N}, hencethe cartesian product of two FAC’s is not necessarly a FAC.

25 / 35

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IntroductionNews on coverability

Erdös and Tarski TheoremWBTS = WSTS - WQO + FAC

Conclusion

DefinitionsIdealsThe TheoremErdös & Tarski Theorem for WQO

Definition

Now (X ,≤) is a qo (shortly written X or ≤)A ⊆ X is an antichain if all couples of A are uncomparable.X is Finite AntiChain (FAC) if all antichains in X are finite.X is Well Founded (WF) if all strictly decreasing sequences inX are finite.

X is WQO if X is FAC & X is WF (can be a Theorem).

Every WQO is FACThe converse is false since Z,Q,R are FAC but not WF:13/7, π, 3, 2, 1, 0,−1/2,−1,−2,−3,−41,−78695/12, ...Z2 contains infinite antichains, A = {(n,−n) | n ∈ N}, hencethe cartesian product of two FAC’s is not necessarly a FAC.

25 / 35

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IntroductionNews on coverability

Erdös and Tarski TheoremWBTS = WSTS - WQO + FAC

Conclusion

DefinitionsIdealsThe TheoremErdös & Tarski Theorem for WQO

Definition

Now (X ,≤) is a qo (shortly written X or ≤)A ⊆ X is an antichain if all couples of A are uncomparable.X is Finite AntiChain (FAC) if all antichains in X are finite.X is Well Founded (WF) if all strictly decreasing sequences inX are finite.X is WQO if X is FAC & X is WF (can be a Theorem).

Every WQO is FACThe converse is false since Z,Q,R are FAC but not WF:13/7, π, 3, 2, 1, 0,−1/2,−1,−2,−3,−41,−78695/12, ...Z2 contains infinite antichains, A = {(n,−n) | n ∈ N}, hencethe cartesian product of two FAC’s is not necessarly a FAC.

25 / 35

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IntroductionNews on coverability

Erdös and Tarski TheoremWBTS = WSTS - WQO + FAC

Conclusion

DefinitionsIdealsThe TheoremErdös & Tarski Theorem for WQO

Definition

Now (X ,≤) is a qo (shortly written X or ≤)A ⊆ X is an antichain if all couples of A are uncomparable.X is Finite AntiChain (FAC) if all antichains in X are finite.X is Well Founded (WF) if all strictly decreasing sequences inX are finite.X is WQO if X is FAC & X is WF (can be a Theorem).

Every WQO is FAC

The converse is false since Z,Q,R are FAC but not WF:13/7, π, 3, 2, 1, 0,−1/2,−1,−2,−3,−41,−78695/12, ...Z2 contains infinite antichains, A = {(n,−n) | n ∈ N}, hencethe cartesian product of two FAC’s is not necessarly a FAC.

25 / 35

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IntroductionNews on coverability

Erdös and Tarski TheoremWBTS = WSTS - WQO + FAC

Conclusion

DefinitionsIdealsThe TheoremErdös & Tarski Theorem for WQO

Definition

Now (X ,≤) is a qo (shortly written X or ≤)A ⊆ X is an antichain if all couples of A are uncomparable.X is Finite AntiChain (FAC) if all antichains in X are finite.X is Well Founded (WF) if all strictly decreasing sequences inX are finite.X is WQO if X is FAC & X is WF (can be a Theorem).

Every WQO is FACThe converse is false since Z,Q,R are FAC but not WF:13/7, π, 3, 2, 1, 0,−1/2,−1,−2,−3,−41,−78695/12, ...

Z2 contains infinite antichains, A = {(n,−n) | n ∈ N}, hencethe cartesian product of two FAC’s is not necessarly a FAC.

25 / 35

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IntroductionNews on coverability

Erdös and Tarski TheoremWBTS = WSTS - WQO + FAC

Conclusion

DefinitionsIdealsThe TheoremErdös & Tarski Theorem for WQO

Definition

Now (X ,≤) is a qo (shortly written X or ≤)A ⊆ X is an antichain if all couples of A are uncomparable.X is Finite AntiChain (FAC) if all antichains in X are finite.X is Well Founded (WF) if all strictly decreasing sequences inX are finite.X is WQO if X is FAC & X is WF (can be a Theorem).

Every WQO is FACThe converse is false since Z,Q,R are FAC but not WF:13/7, π, 3, 2, 1, 0,−1/2,−1,−2,−3,−41,−78695/12, ...

Z2 contains infinite antichains, A = {(n,−n) | n ∈ N}, hencethe cartesian product of two FAC’s is not necessarly a FAC.

25 / 35

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IntroductionNews on coverability

Erdös and Tarski TheoremWBTS = WSTS - WQO + FAC

Conclusion

DefinitionsIdealsThe TheoremErdös & Tarski Theorem for WQO

Definition

Now (X ,≤) is a qo (shortly written X or ≤)A ⊆ X is an antichain if all couples of A are uncomparable.X is Finite AntiChain (FAC) if all antichains in X are finite.X is Well Founded (WF) if all strictly decreasing sequences inX are finite.X is WQO if X is FAC & X is WF (can be a Theorem).

Every WQO is FACThe converse is false since Z,Q,R are FAC but not WF:13/7, π, 3, 2, 1, 0,−1/2,−1,−2,−3,−41,−78695/12, ...Z2 contains infinite antichains, A = {(n,−n) | n ∈ N}, hencethe cartesian product of two FAC’s is not necessarly a FAC.

25 / 35

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IntroductionNews on coverability

Erdös and Tarski TheoremWBTS = WSTS - WQO + FAC

Conclusion

DefinitionsIdealsThe TheoremErdös & Tarski Theorem for WQO

IdealsI ⊆ X is an ideal if

downward closed: I = ↓ I,

directed: a, b ∈ I =⇒ ∃c ∈ I s.t. a ≤ c and b ≤ c.

26 / 35

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IntroductionNews on coverability

Erdös and Tarski TheoremWBTS = WSTS - WQO + FAC

Conclusion

DefinitionsIdealsThe TheoremErdös & Tarski Theorem for WQO

IdealsI ⊆ X is an ideal if

downward closed: I = ↓ I,directed: a, b ∈ I =⇒ ∃c ∈ I s.t. a ≤ c and b ≤ c.

26 / 35

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IntroductionNews on coverability

Erdös and Tarski TheoremWBTS = WSTS - WQO + FAC

Conclusion

DefinitionsIdealsThe TheoremErdös & Tarski Theorem for WQO

IdealsI ⊆ X is an ideal if

downward closed: I = ↓ I,directed: a, b ∈ I =⇒ ∃c ∈ I s.t. a ≤ c and b ≤ c.

26 / 35

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IntroductionNews on coverability

Erdös and Tarski TheoremWBTS = WSTS - WQO + FAC

Conclusion

DefinitionsIdealsThe TheoremErdös & Tarski Theorem for WQO

IdealsI ⊆ X is an ideal if

downward closed: I = ↓ I,directed: a, b ∈ I =⇒ ∃c ∈ I s.t. a ≤ c and b ≤ c.

26 / 35

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IntroductionNews on coverability

Erdös and Tarski TheoremWBTS = WSTS - WQO + FAC

Conclusion

DefinitionsIdealsThe TheoremErdös & Tarski Theorem for WQO

0 2 4 6 8 10 12 140

5

10

15

x1

x 2

Figure: Decomposition ofX = {(x1, x2) ∈ N2 : (x1 ≤ 4) ∨ (x1 ≤ 8 ∧ x2 ≤ 10) ∨ (x2 ≤ 5)} intofinitely many ideals. The three ideals ↓ 4× N, ↓ 8× ↓ 10 and N× ↓ 5appear respectively in blue, orange and green.

27 / 35

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IntroductionNews on coverability

Erdös and Tarski TheoremWBTS = WSTS - WQO + FAC

Conclusion

DefinitionsIdealsThe TheoremErdös & Tarski Theorem for WQO

A very interesting but unknown theorem in the verificationcommunity like was the Higman theorem before WSTS’87 - frontlossy channel systems’89 - lossy channel systems’93.Theorem (Erdös & Tarski’43, Bonnet’75, Fraïsse’86,...)

(X ,≤) FAC ⇐⇒ for all D = ↓D ⊆ X we have: D =⋃finite

Ideals

CorollaryEvery downward closed set decomposes canonically as the union ofits ⊆-maximal ideals.

28 / 35

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IntroductionNews on coverability

Erdös and Tarski TheoremWBTS = WSTS - WQO + FAC

Conclusion

DefinitionsIdealsThe TheoremErdös & Tarski Theorem for WQO

A very interesting but unknown theorem in the verificationcommunity like was the Higman theorem before WSTS’87 - frontlossy channel systems’89 - lossy channel systems’93.Theorem (Erdös & Tarski’43, Bonnet’75, Fraïsse’86,...)

(X ,≤) FAC ⇐⇒ for all D = ↓D ⊆ X we have: D =⋃finite

Ideals

CorollaryEvery downward closed set decomposes canonically as the union ofits ⊆-maximal ideals.

28 / 35

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This content downloaded from 130.223.2.20 on Thu, 23 May 2013 17:27:26 PMAll use subject to JSTOR Terms and Conditions

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IntroductionNews on coverability

Erdös and Tarski TheoremWBTS = WSTS - WQO + FAC

Conclusion

DefinitionsIdealsThe TheoremErdös & Tarski Theorem for WQO

TheoremLet D = ↓D ⊆ X and X a WQO. Then D = I1 ∪ I2 ∪ · · · ∪ Im forsome I1, I2, . . . , Im ∈ Ideals(X ).

Assume that a bad D (bad = dc set that does not admit a finite(may be empty) decomposition in ideals) exists.

∃D bad and minimal for inclusion among bad subsets (strictlydecreasing subsequences of dc subsets are finite in a WQO).

D 6= ∅ since ∅ is equal to an empty union (∅ is not an ideal).D 6= {d} since {d} is an ideal.Let x1 6= x2 ∈ D.

Since D \ ↑ x1 and D \ ↑ x2 are dc and strictly included in D, theyare not bad (by minimality of D).Thus, D \ ↑ x1 =

⋃nj=1 Ij and D \ ↑ x2 =

⋃mj=n+1 Ij for some ideals

I1, I2, . . . , Im ⊆ X .

29 / 35

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IntroductionNews on coverability

Erdös and Tarski TheoremWBTS = WSTS - WQO + FAC

Conclusion

DefinitionsIdealsThe TheoremErdös & Tarski Theorem for WQO

TheoremLet D = ↓D ⊆ X and X a WQO. Then D = I1 ∪ I2 ∪ · · · ∪ Im forsome I1, I2, . . . , Im ∈ Ideals(X ).

Assume that a bad D (bad = dc set that does not admit a finite(may be empty) decomposition in ideals) exists.

∃D bad and minimal for inclusion among bad subsets (strictlydecreasing subsequences of dc subsets are finite in a WQO).

D 6= ∅ since ∅ is equal to an empty union (∅ is not an ideal).D 6= {d} since {d} is an ideal.

Let x1 6= x2 ∈ D.Since D \ ↑ x1 and D \ ↑ x2 are dc and strictly included in D, theyare not bad (by minimality of D).Thus, D \ ↑ x1 =

⋃nj=1 Ij and D \ ↑ x2 =

⋃mj=n+1 Ij for some ideals

I1, I2, . . . , Im ⊆ X .

29 / 35

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IntroductionNews on coverability

Erdös and Tarski TheoremWBTS = WSTS - WQO + FAC

Conclusion

DefinitionsIdealsThe TheoremErdös & Tarski Theorem for WQO

TheoremLet D = ↓D ⊆ X and X a WQO. Then D = I1 ∪ I2 ∪ · · · ∪ Im forsome I1, I2, . . . , Im ∈ Ideals(X ).

Assume that a bad D (bad = dc set that does not admit a finite(may be empty) decomposition in ideals) exists.

∃D bad and minimal for inclusion among bad subsets (strictlydecreasing subsequences of dc subsets are finite in a WQO).

D 6= ∅ since ∅ is equal to an empty union (∅ is not an ideal).D 6= {d} since {d} is an ideal.Let x1 6= x2 ∈ D.

Since D \ ↑ x1 and D \ ↑ x2 are dc and strictly included in D, theyare not bad (by minimality of D).

Thus, D \ ↑ x1 =⋃n

j=1 Ij and D \ ↑ x2 =⋃m

j=n+1 Ij for some idealsI1, I2, . . . , Im ⊆ X .

29 / 35

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IntroductionNews on coverability

Erdös and Tarski TheoremWBTS = WSTS - WQO + FAC

Conclusion

DefinitionsIdealsThe TheoremErdös & Tarski Theorem for WQO

TheoremLet D = ↓D ⊆ X and X a WQO. Then D = I1 ∪ I2 ∪ · · · ∪ Im forsome I1, I2, . . . , Im ∈ Ideals(X ).

Assume that a bad D (bad = dc set that does not admit a finite(may be empty) decomposition in ideals) exists.

∃D bad and minimal for inclusion among bad subsets (strictlydecreasing subsequences of dc subsets are finite in a WQO).

D 6= ∅ since ∅ is equal to an empty union (∅ is not an ideal).D 6= {d} since {d} is an ideal.Let x1 6= x2 ∈ D.

Since D \ ↑ x1 and D \ ↑ x2 are dc and strictly included in D, theyare not bad (by minimality of D).Thus, D \ ↑ x1 =

⋃nj=1 Ij and D \ ↑ x2 =

⋃mj=n+1 Ij for some ideals

I1, I2, . . . , Im ⊆ X .29 / 35

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IntroductionNews on coverability

Erdös and Tarski TheoremWBTS = WSTS - WQO + FAC

Conclusion

DefinitionsIdealsThe TheoremErdös & Tarski Theorem for WQO

HenceD′ = (D \ ↑ x1) ∪ (D \ ↑ x2) =

m⋃j=1

Ij

We have:D′ = D \ (↑ x1 ∩ ↑ x2)

As D′ 6= D, therefore, D ∩ (↑ x1 ∩ ↑ x2) 6= ∅

Thus:

∃d ∈ D ∩ (↑ x1 ∩ ↑ x2) s.t. x1 ≤ d and x2 ≤ d .

Hence D is directed and therefore D is an ideal, contradicting ourassumption. Thus, D is equal to a finite union of ideals.

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IntroductionNews on coverability

Erdös and Tarski TheoremWBTS = WSTS - WQO + FAC

Conclusion

DefinitionsIdealsThe TheoremErdös & Tarski Theorem for WQO

HenceD′ = (D \ ↑ x1) ∪ (D \ ↑ x2) =

m⋃j=1

Ij

We have:D′ = D \ (↑ x1 ∩ ↑ x2)

As D′ 6= D, therefore, D ∩ (↑ x1 ∩ ↑ x2) 6= ∅

Thus:

∃d ∈ D ∩ (↑ x1 ∩ ↑ x2) s.t. x1 ≤ d and x2 ≤ d .

Hence D is directed and therefore D is an ideal, contradicting ourassumption. Thus, D is equal to a finite union of ideals.

30 / 35

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IntroductionNews on coverability

Erdös and Tarski TheoremWBTS = WSTS - WQO + FAC

Conclusion

DefinitionsIdealsThe TheoremErdös & Tarski Theorem for WQO

HenceD′ = (D \ ↑ x1) ∪ (D \ ↑ x2) =

m⋃j=1

Ij

We have:D′ = D \ (↑ x1 ∩ ↑ x2)

As D′ 6= D,

therefore, D ∩ (↑ x1 ∩ ↑ x2) 6= ∅

Thus:

∃d ∈ D ∩ (↑ x1 ∩ ↑ x2) s.t. x1 ≤ d and x2 ≤ d .

Hence D is directed and therefore D is an ideal, contradicting ourassumption. Thus, D is equal to a finite union of ideals.

30 / 35

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IntroductionNews on coverability

Erdös and Tarski TheoremWBTS = WSTS - WQO + FAC

Conclusion

DefinitionsIdealsThe TheoremErdös & Tarski Theorem for WQO

HenceD′ = (D \ ↑ x1) ∪ (D \ ↑ x2) =

m⋃j=1

Ij

We have:D′ = D \ (↑ x1 ∩ ↑ x2)

As D′ 6= D, therefore, D ∩ (↑ x1 ∩ ↑ x2) 6= ∅

Thus:

∃d ∈ D ∩ (↑ x1 ∩ ↑ x2) s.t. x1 ≤ d and x2 ≤ d .

Hence D is directed and therefore D is an ideal, contradicting ourassumption. Thus, D is equal to a finite union of ideals.

30 / 35

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IntroductionNews on coverability

Erdös and Tarski TheoremWBTS = WSTS - WQO + FAC

Conclusion

DefinitionsIdealsThe TheoremErdös & Tarski Theorem for WQO

HenceD′ = (D \ ↑ x1) ∪ (D \ ↑ x2) =

m⋃j=1

Ij

We have:D′ = D \ (↑ x1 ∩ ↑ x2)

As D′ 6= D, therefore, D ∩ (↑ x1 ∩ ↑ x2) 6= ∅

Thus:

∃d ∈ D ∩ (↑ x1 ∩ ↑ x2) s.t. x1 ≤ d and x2 ≤ d .

Hence D is directed and therefore D is an ideal, contradicting ourassumption. Thus, D is equal to a finite union of ideals.

30 / 35

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IntroductionNews on coverability

Erdös and Tarski TheoremWBTS = WSTS - WQO + FAC

Conclusion

DefinitionsIdealsThe TheoremErdös & Tarski Theorem for WQO

HenceD′ = (D \ ↑ x1) ∪ (D \ ↑ x2) =

m⋃j=1

Ij

We have:D′ = D \ (↑ x1 ∩ ↑ x2)

As D′ 6= D, therefore, D ∩ (↑ x1 ∩ ↑ x2) 6= ∅

Thus:

∃d ∈ D ∩ (↑ x1 ∩ ↑ x2) s.t. x1 ≤ d and x2 ≤ d .

Hence D is directed and therefore D is an ideal, contradicting ourassumption. Thus, D is equal to a finite union of ideals.

30 / 35

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IntroductionNews on coverability

Erdös and Tarski TheoremWBTS = WSTS - WQO + FAC

Conclusion

Definition (new)

A Well Behaved Transition System (WBTS) is a monotonetransition system S = (X ,−→,≤) such that (X ,≤) is FAC.

WSTS and WBTS

WSTS are WBTS but the converse is false: Z-VASS areWBTS but are not WSTS.Weighted VASS are WBTS for ≤ (but are not WSTS).Multi-weighted VASS are WBTS for ≤lex (but are not WSTS).

Corollary (new)

Coverability is decidable for any ideally effective class of WBTS.31 / 35

Page 91: WBTS: the new class of WSTS without WQOfinkel/2016-2017/slides-WBTS-workshop-etaps-2017.pdf · on the Grand Central Dispatch (GCD) technology [13] , and to perform parametric verication

IntroductionNews on coverability

Erdös and Tarski TheoremWBTS = WSTS - WQO + FAC

Conclusion

RemarkThe backward coverability procedure does not terminate onZ-VASS, weighted VASS and multi-weighted VASS.

32 / 35

Page 92: WBTS: the new class of WSTS without WQOfinkel/2016-2017/slides-WBTS-workshop-etaps-2017.pdf · on the Grand Central Dispatch (GCD) technology [13] , and to perform parametric verication

IntroductionNews on coverability

Erdös and Tarski TheoremWBTS = WSTS - WQO + FAC

Conclusion

The survey/story of coverability for WSTSYear Authors Mathematical hyp. Effectivity hyp. back/forward1978 Arnold & Latteux reset VAS YES backward1987 F. very WSTS (strong+strict, ω2-wqo,...) effective very WSTS forward1996 Abdulla & CJT strong monotony PreS (↑ x) comp. backward1998 F. Schnoebelen monotony ↑ PreS (↑ x) comp. backward2004 Geeraerts & RV strong monotony, ADL effective ADL forward2006 Geeraerts & RV monotony, ADL effective ADL forward2009 F. & Goubault-Larrecq strong monotony, weak ADL, flattable effective WADL forward2009 F. & Goubault-Larrecq strong monotony, flattable ideally effective forward2014 Blondin & FM monotony, ideally effective forward2016 Blondin & FM monotony, no wqo but FAC ideally effective forward2017 Trivial no monotony, wqo (Minsky machines) ideally effective Undec.2017 New question monotony, no wqo but WF ideally effective Undec ?

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Page 93: WBTS: the new class of WSTS without WQOfinkel/2016-2017/slides-WBTS-workshop-etaps-2017.pdf · on the Grand Central Dispatch (GCD) technology [13] , and to perform parametric verication

IntroductionNews on coverability

Erdös and Tarski TheoremWBTS = WSTS - WQO + FAC

Conclusion

And now ?

Further work

Explore more in details WBTS.Find applications of WBTS.Computing efficiently with ideals (no brut force enumeration).Design Karp and Miller tree algorithm for ω2-WSTS.

Interships available at the ENS Paris-Saclay

Different topics: theoretical and/or applied subjects.Developping the WSTS theory or a prototype for finding bugsin web services and choreographies.Different levels: Bachelor, Master, PhD, post-PhD.

34 / 35

Page 94: WBTS: the new class of WSTS without WQOfinkel/2016-2017/slides-WBTS-workshop-etaps-2017.pdf · on the Grand Central Dispatch (GCD) technology [13] , and to perform parametric verication

IntroductionNews on coverability

Erdös and Tarski TheoremWBTS = WSTS - WQO + FAC

Conclusion

And now ?

Further work

Explore more in details WBTS.

Find applications of WBTS.Computing efficiently with ideals (no brut force enumeration).Design Karp and Miller tree algorithm for ω2-WSTS.

Interships available at the ENS Paris-Saclay

Different topics: theoretical and/or applied subjects.Developping the WSTS theory or a prototype for finding bugsin web services and choreographies.Different levels: Bachelor, Master, PhD, post-PhD.

34 / 35

Page 95: WBTS: the new class of WSTS without WQOfinkel/2016-2017/slides-WBTS-workshop-etaps-2017.pdf · on the Grand Central Dispatch (GCD) technology [13] , and to perform parametric verication

IntroductionNews on coverability

Erdös and Tarski TheoremWBTS = WSTS - WQO + FAC

Conclusion

And now ?

Further work

Explore more in details WBTS.Find applications of WBTS.

Computing efficiently with ideals (no brut force enumeration).Design Karp and Miller tree algorithm for ω2-WSTS.

Interships available at the ENS Paris-Saclay

Different topics: theoretical and/or applied subjects.Developping the WSTS theory or a prototype for finding bugsin web services and choreographies.Different levels: Bachelor, Master, PhD, post-PhD.

34 / 35

Page 96: WBTS: the new class of WSTS without WQOfinkel/2016-2017/slides-WBTS-workshop-etaps-2017.pdf · on the Grand Central Dispatch (GCD) technology [13] , and to perform parametric verication

IntroductionNews on coverability

Erdös and Tarski TheoremWBTS = WSTS - WQO + FAC

Conclusion

And now ?

Further work

Explore more in details WBTS.Find applications of WBTS.Computing efficiently with ideals (no brut force enumeration).

Design Karp and Miller tree algorithm for ω2-WSTS.

Interships available at the ENS Paris-Saclay

Different topics: theoretical and/or applied subjects.Developping the WSTS theory or a prototype for finding bugsin web services and choreographies.Different levels: Bachelor, Master, PhD, post-PhD.

34 / 35

Page 97: WBTS: the new class of WSTS without WQOfinkel/2016-2017/slides-WBTS-workshop-etaps-2017.pdf · on the Grand Central Dispatch (GCD) technology [13] , and to perform parametric verication

IntroductionNews on coverability

Erdös and Tarski TheoremWBTS = WSTS - WQO + FAC

Conclusion

And now ?

Further work

Explore more in details WBTS.Find applications of WBTS.Computing efficiently with ideals (no brut force enumeration).Design Karp and Miller tree algorithm for ω2-WSTS.

Interships available at the ENS Paris-Saclay

Different topics: theoretical and/or applied subjects.Developping the WSTS theory or a prototype for finding bugsin web services and choreographies.Different levels: Bachelor, Master, PhD, post-PhD.

34 / 35

Page 98: WBTS: the new class of WSTS without WQOfinkel/2016-2017/slides-WBTS-workshop-etaps-2017.pdf · on the Grand Central Dispatch (GCD) technology [13] , and to perform parametric verication

IntroductionNews on coverability

Erdös and Tarski TheoremWBTS = WSTS - WQO + FAC

Conclusion

And now ?

Further work

Explore more in details WBTS.Find applications of WBTS.Computing efficiently with ideals (no brut force enumeration).Design Karp and Miller tree algorithm for ω2-WSTS.

Interships available at the ENS Paris-Saclay

Different topics: theoretical and/or applied subjects.Developping the WSTS theory or a prototype for finding bugsin web services and choreographies.Different levels: Bachelor, Master, PhD, post-PhD.

34 / 35

Page 99: WBTS: the new class of WSTS without WQOfinkel/2016-2017/slides-WBTS-workshop-etaps-2017.pdf · on the Grand Central Dispatch (GCD) technology [13] , and to perform parametric verication

IntroductionNews on coverability

Erdös and Tarski TheoremWBTS = WSTS - WQO + FAC

Conclusion

And now ?

Further work

Explore more in details WBTS.Find applications of WBTS.Computing efficiently with ideals (no brut force enumeration).Design Karp and Miller tree algorithm for ω2-WSTS.

Interships available at the ENS Paris-Saclay

Different topics: theoretical and/or applied subjects.

Developping the WSTS theory or a prototype for finding bugsin web services and choreographies.Different levels: Bachelor, Master, PhD, post-PhD.

34 / 35

Page 100: WBTS: the new class of WSTS without WQOfinkel/2016-2017/slides-WBTS-workshop-etaps-2017.pdf · on the Grand Central Dispatch (GCD) technology [13] , and to perform parametric verication

IntroductionNews on coverability

Erdös and Tarski TheoremWBTS = WSTS - WQO + FAC

Conclusion

And now ?

Further work

Explore more in details WBTS.Find applications of WBTS.Computing efficiently with ideals (no brut force enumeration).Design Karp and Miller tree algorithm for ω2-WSTS.

Interships available at the ENS Paris-Saclay

Different topics: theoretical and/or applied subjects.Developping the WSTS theory or a prototype for finding bugsin web services and choreographies.

Different levels: Bachelor, Master, PhD, post-PhD.

34 / 35

Page 101: WBTS: the new class of WSTS without WQOfinkel/2016-2017/slides-WBTS-workshop-etaps-2017.pdf · on the Grand Central Dispatch (GCD) technology [13] , and to perform parametric verication

IntroductionNews on coverability

Erdös and Tarski TheoremWBTS = WSTS - WQO + FAC

Conclusion

And now ?

Further work

Explore more in details WBTS.Find applications of WBTS.Computing efficiently with ideals (no brut force enumeration).Design Karp and Miller tree algorithm for ω2-WSTS.

Interships available at the ENS Paris-Saclay

Different topics: theoretical and/or applied subjects.Developping the WSTS theory or a prototype for finding bugsin web services and choreographies.Different levels: Bachelor, Master, PhD, post-PhD.

34 / 35

Page 102: WBTS: the new class of WSTS without WQOfinkel/2016-2017/slides-WBTS-workshop-etaps-2017.pdf · on the Grand Central Dispatch (GCD) technology [13] , and to perform parametric verication

IntroductionNews on coverability

Erdös and Tarski TheoremWBTS = WSTS - WQO + FAC

Conclusion

And now ?

Thank you!

35 / 35