[PPT] - Reachability In Parametric Timed Automata With Two Parametric Clocks PowerPoint Presentation

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Introduction Upper Bound Lower Bound

Reachability In Parametric Timed Automata With Two Parametric Clocks And One Parameter Is EXPSPACE-complete

Mathieu Hilaire Paris-Saclay University, LSV, France joint work with Stefan G¨

ller

MOVEP 2020 June 23, 2020

Mathieu Hilaire Paris-Saclay University, LSV, France joint work with Stefan G¨

ller

MOVEP 2020 Reachability In Parametric Timed Automata With Two Parametric

SLIDE 2

Introduction Upper Bound Lower Bound

Outline

1

Introduction Definition Reachability Problem State of the Art

2

Upper Bound Parametric One-Counter Automata Difficulties Unpumping N-runs Depumping Lemma The 5/6-Lemma

3

Lower Bound

Mathieu Hilaire Paris-Saclay University, LSV, France joint work with Stefan G¨

ller

MOVEP 2020 Reachability In Parametric Timed Automata With Two Parametric

SLIDE 3

Introduction Upper Bound Lower Bound

Definition An automaton is a tuple (Q, R, q0, F) where Q finite set of states R finite set of transitions q0 initial state F finite set of final states

Mathieu Hilaire Paris-Saclay University, LSV, France joint work with Stefan G¨

ller

MOVEP 2020 Reachability In Parametric Timed Automata With Two Parametric

SLIDE 4

Introduction Upper Bound Lower Bound

Parametric Timed Automata

Automaton A : q1 q0 start qf x := 0 x > 3 y = p Definition PTA A parametric timed automaton (PTA) is an automaton extended with finite set of clocks X = {x, y, z, . . .} over N finite set of parameters P = {p, p1, p2, . . .} transitions with

guards x ⊲ ⊳ c or x ⊲ ⊳ p, where ⊲ ⊳ ∈ {<, =, >} resets of the form x := 0

Mathieu Hilaire Paris-Saclay University, LSV, France joint work with Stefan G¨

ller

MOVEP 2020 Reachability In Parametric Timed Automata With Two Parametric

SLIDE 5

Introduction Upper Bound Lower Bound

Run in PTAs

PTA A : q1 q0 start qf x := 0 x > 3 y = p 10-run i.e. run with µ(p) = 10 configurations: state × clock valuations q0(0, 0) x>3 − − → q1(4, 4) x:=0 − − − → q0(0, 5) x>3 − − → q1(4, 9)

y=p

− − → qf (5, 10)

Mathieu Hilaire Paris-Saclay University, LSV, France joint work with Stefan G¨

ller

MOVEP 2020 Reachability In Parametric Timed Automata With Two Parametric

SLIDE 6

Introduction Upper Bound Lower Bound

PTA-Reachability

Reachability in Parametric Timed Automata Input: parametric timed automata A with parameters P and two parametric clocks, state qf . Output: Does there exists a valuation µ : P → N of P such that there is a run ending in qf ?

Mathieu Hilaire Paris-Saclay University, LSV, France joint work with Stefan G¨

ller

MOVEP 2020 Reachability In Parametric Timed Automata With Two Parametric

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Introduction Upper Bound Lower Bound

PTA-Reachability : State of the Art

Central open problem: Is reachability in PTA with two parametric clocks decidable ? parametric clocks lower bound upper bound 1 clock NEXP[BBLS15] NEXP[BBLS15] 2 clocks PSPACENEXP[BO17]

pen

3+ clocks undecidable[AHV93] undecidable[AHV93]

Mathieu Hilaire Paris-Saclay University, LSV, France joint work with Stefan G¨

ller

MOVEP 2020 Reachability In Parametric Timed Automata With Two Parametric

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Introduction Upper Bound Lower Bound

Our Contribution

Main Theorem PTA-Reachability is EXPSPACE-complete for two parametric clocks and one parameter. Improves best previously known bounds from Bundala & Ouaknine [BO17] in 2017 PSPACENEXP lower bound decidabilty upper bound

Mathieu Hilaire Paris-Saclay University, LSV, France joint work with Stefan G¨

ller

MOVEP 2020 Reachability In Parametric Timed Automata With Two Parametric

SLIDE 9

Introduction Upper Bound Lower Bound The 5/6-Lemma

Outline

1

Introduction Definition Reachability Problem State of the Art

2

Upper Bound Parametric One-Counter Automata Difficulties Unpumping N-runs Depumping Lemma The 5/6-Lemma

3

Lower Bound

Mathieu Hilaire Paris-Saclay University, LSV, France joint work with Stefan G¨

ller

MOVEP 2020 Reachability In Parametric Timed Automata With Two Parametric

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Introduction Upper Bound Lower Bound The 5/6-Lemma

Parametric One-Counter Automata

Automata C : q1 q0 start qf +1 < p mod 3 Definition A parametric one-counter automaton (POCA) is an automaton extended with transitions extended with +1/−1/+p/−p ⊲ ⊳ c / ⊲ ⊳ p where ⊲ ⊳∈ {<, =, >} and c ∈ N mod c where c ∈ N

Mathieu Hilaire Paris-Saclay University, LSV, France joint work with Stefan G¨

ller

MOVEP 2020 Reachability In Parametric Timed Automata With Two Parametric

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Introduction Upper Bound Lower Bound The 5/6-Lemma

Runs in POCAs

POCA C : q1 q0 start qf +1 < p mod 3 5-run i.e. run with µ(p) = 5 q0(0)

x<p

− − → q1(0) +1 − − → q0(1) x<5 − − → q1(1) +1 − − → q0(2) x<5 − − → q1(2) +1 − − → q0(3)

x<p

− − → q1(3) mod 3 − − − − → qf (3)

Mathieu Hilaire Paris-Saclay University, LSV, France joint work with Stefan G¨

ller

MOVEP 2020 Reachability In Parametric Timed Automata With Two Parametric

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Introduction Upper Bound Lower Bound The 5/6-Lemma

From PTA to POCA revisited

Lemma The following is computable in exponential time: INPUT: PTA A over one parameter p and two parametric clocks. OUTPUT: A POCA C such that:

1 all accepting N-runs have values in [0, 6 · max(N, |C|)] 2 reachability holds for A if, and only if, reachability holds for C

Inspired by a construction due to Bundala/Ouaknine [BO17]

Mathieu Hilaire Paris-Saclay University, LSV, France joint work with Stefan G¨

ller

MOVEP 2020 Reachability In Parametric Timed Automata With Two Parametric

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Introduction Upper Bound Lower Bound The 5/6-Lemma

Main Theorem

Theorem For a POCA C with parameter p given an accepting N-run with N > 2poly(|C|) and countervalues in [0, 6 · N] there exists an accepting N′-run for N′ smaller with countervalues in [0, 6 · N′]. Corollary PTA-Reachability is in EXPSPACE for two parametric clocks and one parameter.

Mathieu Hilaire Paris-Saclay University, LSV, France joint work with Stefan G¨

ller

MOVEP 2020 Reachability In Parametric Timed Automata With Two Parametric

SLIDE 14

Introduction Upper Bound Lower Bound The 5/6-Lemma

Outline

1

Introduction Definition Reachability Problem State of the Art

2

Upper Bound Parametric One-Counter Automata Difficulties Unpumping N-runs Depumping Lemma The 5/6-Lemma

3

Lower Bound

Mathieu Hilaire Paris-Saclay University, LSV, France joint work with Stefan G¨

ller

MOVEP 2020 Reachability In Parametric Timed Automata With Two Parametric

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Introduction Upper Bound Lower Bound The 5/6-Lemma

Difficulties unpumping N-runs

Figure: Parametric update followed by some arbitrary run

Mathieu Hilaire Paris-Saclay University, LSV, France joint work with Stefan G¨

ller

MOVEP 2020 Reachability In Parametric Timed Automata With Two Parametric

SLIDE 16

Introduction Upper Bound Lower Bound The 5/6-Lemma

Difficulties unpumping N-runs

Figure: Parametric update with a lower value followed by same arbitrary run

Mathieu Hilaire Paris-Saclay University, LSV, France joint work with Stefan G¨

ller

MOVEP 2020 Reachability In Parametric Timed Automata With Two Parametric

SLIDE 17

Introduction Upper Bound Lower Bound The 5/6-Lemma

Difficulties unpumping N-runs

We need to change the run following the parametric update!

Mathieu Hilaire Paris-Saclay University, LSV, France joint work with Stefan G¨

ller

MOVEP 2020 Reachability In Parametric Timed Automata With Two Parametric

SLIDE 18

Introduction Upper Bound Lower Bound The 5/6-Lemma

Outline

1

Introduction Definition Reachability Problem State of the Art

2

Upper Bound Parametric One-Counter Automata Difficulties Unpumping N-runs Depumping Lemma The 5/6-Lemma

3

Lower Bound

Mathieu Hilaire Paris-Saclay University, LSV, France joint work with Stefan G¨

ller

MOVEP 2020 Reachability In Parametric Timed Automata With Two Parametric

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Introduction Upper Bound Lower Bound The 5/6-Lemma

Depumping Lemma

Lemma For every N-run π of a POCA C with N > 2poly(|C|) counter effect larger than 2poly(|C|) same number of +p and −p never more than 12 pending +p / −p there exists an N-run with smaller absolute counter effect.

Mathieu Hilaire Paris-Saclay University, LSV, France joint work with Stefan G¨

ller

MOVEP 2020 Reachability In Parametric Timed Automata With Two Parametric

SLIDE 20

Introduction Upper Bound Lower Bound The 5/6-Lemma

Application of the Depumping Lemma

Figure: Parametric update followed by a run with the right properties

Mathieu Hilaire Paris-Saclay University, LSV, France joint work with Stefan G¨

ller

MOVEP 2020 Reachability In Parametric Timed Automata With Two Parametric

SLIDE 21

Introduction Upper Bound Lower Bound The 5/6-Lemma

Application of the Depumping Lemma

Figure: Parametric update with a smaller value followed by same run

Mathieu Hilaire Paris-Saclay University, LSV, France joint work with Stefan G¨

ller

MOVEP 2020 Reachability In Parametric Timed Automata With Two Parametric

SLIDE 22

Introduction Upper Bound Lower Bound The 5/6-Lemma

Application of the Depumping Lemma

Figure: Section where to cut subruns from the well-bracketed subrun

Mathieu Hilaire Paris-Saclay University, LSV, France joint work with Stefan G¨

ller

MOVEP 2020 Reachability In Parametric Timed Automata With Two Parametric

SLIDE 23

Introduction Upper Bound Lower Bound The 5/6-Lemma

Application of the Depumping Lemma

Figure: Parametric update with a different value followed by de-pumped run

Mathieu Hilaire Paris-Saclay University, LSV, France joint work with Stefan G¨

ller

MOVEP 2020 Reachability In Parametric Timed Automata With Two Parametric

SLIDE 24

Introduction Upper Bound Lower Bound The 5/6-Lemma

The 5/6-Lemma

Lemma Given an N-run π from q0(z0) to qn(zn) with N > 2poly(|C|) and countervalues in [0, 6 · N], max(z0, zn, N) − min(z0, zn, N) ≤ 5/6N there exists an (N − ∆)-run from q0(z0 − ∆) to qn(zn − ∆).

Mathieu Hilaire Paris-Saclay University, LSV, France joint work with Stefan G¨

ller

MOVEP 2020 Reachability In Parametric Timed Automata With Two Parametric

SLIDE 25

Introduction Upper Bound Lower Bound

Outline

1

Introduction Definition Reachability Problem State of the Art

2

Upper Bound Parametric One-Counter Automata Difficulties Unpumping N-runs Depumping Lemma The 5/6-Lemma

3

Lower Bound

Mathieu Hilaire Paris-Saclay University, LSV, France joint work with Stefan G¨

ller

MOVEP 2020 Reachability In Parametric Timed Automata With Two Parametric

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Introduction Upper Bound Lower Bound

Serializability

Definition Let C be a complexity class. We say a language L is exponentially C-serializable if there is a regular language R, and there is a U ∈ C such that for all x ∈ {0, 1}n : x ∈ L ⇐ ⇒ χU(x, 02p(n)) . . . χU(x, 12p(n)) ∈ R

Mathieu Hilaire Paris-Saclay University, LSV, France joint work with Stefan G¨

ller

MOVEP 2020 Reachability In Parametric Timed Automata With Two Parametric

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Introduction Upper Bound Lower Bound

Theorem For every language L ∈ EXPSPACE there is some regular language R such that L is exponentially LOGSPACE-serializable via R.

Mathieu Hilaire Paris-Saclay University, LSV, France joint work with Stefan G¨

ller

MOVEP 2020 Reachability In Parametric Timed Automata With Two Parametric

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Introduction Upper Bound Lower Bound

Conclusion

Our contribution : EXPSPACE upper bound introducing intricate pumping techniques EXPSPACE lower bound using serializability techniques To go further : Extend technique to more than one parameter

Mathieu Hilaire Paris-Saclay University, LSV, France joint work with Stefan G¨

ller

MOVEP 2020 Reachability In Parametric Timed Automata With Two Parametric

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Introduction Upper Bound Lower Bound

Bibliography

Rajeev Alur, Thomas A Henzinger, and Moshe Y Vardi. Parametric real-time reasoning. In Proc. STOC’93, pages 592–601. ACM, 1993. Nikola Beneˇ s, Peter Bezdˇ ek, Kim G Larsen, and Jiˇ r´ ı Srba. Language emptiness of continuous-time parametric timed automata. In International Colloquium on Automata, Languages, and Programming, pages 69–81. Springer, 2015. Daniel Bundala and Joel Ouaknine. On parametric timed automata and one-counter machines. Information and Computation, 253:272–303, 2017. Christoph Haase. On the complexity of model checking counter automata. 2012.

Mathieu Hilaire Paris-Saclay University, LSV, France joint work with Stefan G¨

ller

MOVEP 2020 Reachability In Parametric Timed Automata With Two Parametric

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Introduction Upper Bound Lower Bound

Appendix : the programming language

Programing langage with sets of variables X, Y and B. Input: Some word w ∈ {0, 1}n

θ := B := B ± X | Xi := f1(Xj) | Xi := f2(B, Xj) | Y := g1(X) | Y := g2(B, X) π = θ | if Y then π else π | π; π | while Y do π

f1 (resp. f2) : {0, 1}n → {0, 1}n PSPACE g1 (resp. g2) : {0, 1}n → {0, 1}2n LOGSPACE

Mathieu Hilaire Paris-Saclay University, LSV, France joint work with Stefan G¨

ller

MOVEP 2020 Reachability In Parametric Timed Automata With Two Parametric

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Introduction Upper Bound Lower Bound

Appendix : Semantic

Definition An n-assignment σ is a tuple (α, β, z) where α : X → {0, 1}n β : Y → {0, 1} z ∈ {0, 1}2n

Mathieu Hilaire Paris-Saclay University, LSV, France joint work with Stefan G¨

ller

MOVEP 2020 Reachability In Parametric Timed Automata With Two Parametric

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Introduction Upper Bound Lower Bound

Appendix : Semantic

[B := B ± X](α, β, z) = (α, β, z ± α(X)) [Xi := f1(Xj)](α, β, z) = (α[Xi ← f1(α(Xj))], β, z) [Xi := f2(B, Xj)](α, β, z) = (α[Xi ← f2(z, α(Xj))], β, z) [Y := g1(X)](α, β, z) = (α, β[Y ← g1(α(X))], z) [Y := g2(B, X)](α, β, z) = (α, β[Y ← g2(z, α(X))], z).

Mathieu Hilaire Paris-Saclay University, LSV, France joint work with Stefan G¨

ller

MOVEP 2020 Reachability In Parametric Timed Automata With Two Parametric

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Introduction Upper Bound Lower Bound

Appenndix: State of the Art

Theorem ([Haa12]) Reachability in two-clock timed automata is polynomial-time inter-reducible with reachability in one-counter automata. Theorem ([BO17]) Reachability in parametric two-clock timed automata is inter-reducible with reachability in parametric one-counter automata where we allow the counter to be incremented by +[0, p] transitions.

Mathieu Hilaire Paris-Saclay University, LSV, France joint work with Stefan G¨

ller

MOVEP 2020 Reachability In Parametric Timed Automata With Two Parametric

SLIDE 34

Introduction Upper Bound Lower Bound

Appendix

Theorem The following is computable in exponential time: INPUT: A PTA A over one parameter p with two parametric clocks. OUTPUT: A POCA C such that:

1 all accepting N-runs π in C have countervalues in

[0, 6 · max(N, |C|)]

2 reachability holds for A if, and only if, reachability holds for C Mathieu Hilaire Paris-Saclay University, LSV, France joint work with Stefan G¨

ller

MOVEP 2020 Reachability In Parametric Timed Automata With Two Parametric

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Introduction Upper Bound Lower Bound

Appendix

Theorem Let C be a POCA with one parameter p. Then there exist constants MC and ΓC bounded by 2poly(|C|) such that if there exists an accepting N-run in C with values in [0, 6 · N] for some N > MC, then there exists an accepting (N − ΓC)-run in C.

Mathieu Hilaire Paris-Saclay University, LSV, France joint work with Stefan G¨

ller

MOVEP 2020 Reachability In Parametric Timed Automata With Two Parametric

SLIDE 36

Introduction Upper Bound Lower Bound

Figure: A well-bracketed run π from q0(z0) to qn(zn)

Mathieu Hilaire Paris-Saclay University, LSV, France joint work with Stefan G¨

ller

MOVEP 2020 Reachability In Parametric Timed Automata With Two Parametric

SLIDE 37

Introduction Upper Bound Lower Bound

Appendix

Figure: Only considering the non parametric updates

Mathieu Hilaire Paris-Saclay University, LSV, France joint work with Stefan G¨

ller

MOVEP 2020 Reachability In Parametric Timed Automata With Two Parametric

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Introduction Upper Bound Lower Bound

Appendix

Claim (by pigeonhole principle) ∀ 0 ≤ a < b ≤ n s.t. pot(b) − pot(a)> 25 · |Q| · LCM(Const(C)) there exist positions a ≤ s < t ≤ b, with same states same number of pending +p/-p zt − zs = d · LCM(Const(C)) for some d ∈ [1, 25 · |Q|]

Mathieu Hilaire Paris-Saclay University, LSV, France joint work with Stefan G¨

ller

MOVEP 2020 Reachability In Parametric Timed Automata With Two Parametric

SLIDE 39

Introduction Upper Bound Lower Bound

Appendix : Bracketing properties

Mapping φ(τ) =      [ if opτ = +p ] if opτ = −p ε

therwise.

Mathieu Hilaire Paris-Saclay University, LSV, France joint work with Stefan G¨

ller

MOVEP 2020 Reachability In Parametric Timed Automata With Two Parametric

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Introduction Upper Bound Lower Bound

Potential

λ(i) = |φ(π[0, i])|[−|φ(π[0, i])|] and pot(i) = zi−z0−λ(i)·N By asumption λ(i) ∈ [−12, 12] and

1 |pot(i − 1) − pot(i)| ≤ 1 for all i ∈ [1, n], 2 if λ(i) = λ(j), then pot(j) − pot(i) = zj − zi for all

i, j ∈ [0, n], and

3 pot(n) = zn − z0 = ∆(π) and pot(0) = 0 Mathieu Hilaire Paris-Saclay University, LSV, France joint work with Stefan G¨

ller

MOVEP 2020 Reachability In Parametric Timed Automata With Two Parametric

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Introduction Upper Bound Lower Bound

Since the counter effect is large enough, find several a, b with pot(b) − pot(a)> 25 · |Q| · LCM(Const(C)) Apply the above Claim to all pairings a, b Since the counter effect is large enough, there exists some d ∈ [1, 25 · |Q|] appearing LCM(25 · |Q|) times. Cut away LCM(25 · |Q|)/d such intervals

Mathieu Hilaire Paris-Saclay University, LSV, France joint work with Stefan G¨

ller

MOVEP 2020 Reachability In Parametric Timed Automata With Two Parametric

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Introduction Upper Bound Lower Bound

Appendix : Details of the upper bound

Figure: What we do now

Mathieu Hilaire Paris-Saclay University, LSV, France joint work with Stefan G¨

ller

MOVEP 2020 Reachability In Parametric Timed Automata With Two Parametric

SLIDE 43

Introduction Upper Bound Lower Bound

Detailed semantic

[B := B ± X](α, β, z) = (α, β, z ± α(X)) [Xi := f1(Xj)](α, β, z) = (α[Xi ← f1(α(Xj))], β, z) [Xi := f2(B, Xj)](α, β, z) = (α[Xi ← f2(z, α(Xj))], β, z) [Y := g1(X)](α, β, z) = (α, β[Y ← g1(α(X))], z) [Y := g2(B, X)](α, β, z) = (α, β[Y ← g2(z, α(X))], z). [ if Y then π1 else π2](σ) = [π1](α, β, z) if β(Y ) = 1 [π2](α, β, z)

therwise

[ while αi do π](σ) = ρ iff exists λ1, λ2, ... λt such that σ = λ1, λt = ρ, avec ρA(αi) = 0 (i.e. False) and for all 0 < j < t[π](λj)λj+1 and λj

A(αi) = 1 (i.e. True)

[ π1 ; π2](σ) = [π2]([π1](σ))

Mathieu Hilaire Paris-Saclay University, LSV, France joint work with Stefan G¨

ller

MOVEP 2020 Reachability In Parametric Timed Automata With Two Parametric

SLIDE 44

Introduction Upper Bound Lower Bound

Appendix : Implementation of the lower bound

1 Variables Storage 2 B ± X 3 PSPACE operation 4 LOGSPACE operation on B Mathieu Hilaire Paris-Saclay University, LSV, France joint work with Stefan G¨

ller

MOVEP 2020 Reachability In Parametric Timed Automata With Two Parametric

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Introduction Upper Bound Lower Bound

Appendix : Variables Storage

Bit : stored by two non-parametric clocks → count modulo 2: 1 if clocks are equal 0 otherwise B : stored by two parametric clocks b+ − b− mod p

Mathieu Hilaire Paris-Saclay University, LSV, France joint work with Stefan G¨

ller

MOVEP 2020 Reachability In Parametric Timed Automata With Two Parametric

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Introduction Upper Bound Lower Bound

Appendix : B ± X operation

b+ − b− store B b+ = 3, reset b+ → remove 3 to B b− = 3, reset b− → add 3 to B

Mathieu Hilaire Paris-Saclay University, LSV, France joint work with Stefan G¨

ller

MOVEP 2020 Reachability In Parametric Timed Automata With Two Parametric

SLIDE 47

Introduction Upper Bound Lower Bound

Appendix : Implementation of the lower bound

q0 q1 q2 ... qn+1 qf w := 0 b− = 0 (X)0 = 1, w = 1, w := 0 (X)0 = 0, w = 0, w := 0 (X)1 = 1, w = 21, w := 0 (X)1 = 0, w = 0, w := 0 ... b− := 0

Mathieu Hilaire Paris-Saclay University, LSV, France joint work with Stefan G¨

ller

MOVEP 2020 Reachability In Parametric Timed Automata With Two Parametric

SLIDE 48

Introduction Upper Bound Lower Bound

Appendix : Implementation of the lower bound

f PSPACE functions in the program with working tape bound by pf (n) working tape stored bit by bit positions stored in the states → able to read and modify bits

Mathieu Hilaire Paris-Saclay University, LSV, France joint work with Stefan G¨

ller

MOVEP 2020 Reachability In Parametric Timed Automata With Two Parametric

SLIDE 49

Introduction Upper Bound Lower Bound

Appendix : Implementation of the lower bound

(q, i, h, a, b) (q′, i + δ1, h + δ2, 0, 0) (q′, i + δ1, h + δ2, 0, 1) (q′, i + δ1, h + δ2, 1, 0) (q′, i + δ1, h + δ2, 1, 1) ( Xj )i+δ

1

= , ( Xf

1

)h+δ

2

= , ( Xf

1

)h : = d ( Xj )i+δ

1

= , ( Xf

1

)h+δ

2

= 1 , ( Xf

1

)h : = d ( Xj )i+δ

1

= 1 , ( Xf

1

)h+δ

2

= , ( Xf

1

)h : = d ( Xj )i+δ

1

= 1 , ( Xf

1

)h+δ

2

= 1 , ( Xf

1

)h : = d

Mathieu Hilaire Paris-Saclay University, LSV, France joint work with Stefan G¨

ller

MOVEP 2020 Reachability In Parametric Timed Automata With Two Parametric

SLIDE 50

Introduction Upper Bound Lower Bound

Appendix : Implementation of the lower bound

Definition Chinese Remainde Representation : CRRm(M) = ((bi,c)i∈[1,m],0≤c<pi) bi,c = 1 of M mod pi = c, 0 otherwise Result Computing the Binary representation of a number from its Chinese Remainder Representation is in LOGSPACE

Mathieu Hilaire Paris-Saclay University, LSV, France joint work with Stefan G¨

ller

MOVEP 2020 Reachability In Parametric Timed Automata With Two Parametric

SLIDE 51

Introduction Upper Bound Lower Bound

Program

A = (Q, δ, I, F) recognize R q ∈ Q; q := q0; B ∈ N; B := 0; b ∈ {0, 1}; while B = 22n loop b := χU(x0, bin2N(B)); q := δ(q, b); B := B + 1; endloop return q ∈ F;

Mathieu Hilaire Paris-Saclay University, LSV, France joint work with Stefan G¨

ller

MOVEP 2020 Reachability In Parametric Timed Automata With Two Parametric