EXPSPACE-Complete Variant of Countdown Games, and Simulation on - - PowerPoint PPT Presentation

EXPSPACE-Complete Variant of Countdown Games, and Simulation on Succinct One-Counter Nets

Petr Janˇ car1, Petr Osiˇ cka1, Zdenˇ ek Sawa2

1Palack´

y University Olomouc, Czechia

• 2Techn. Univ. Ostrava, Czechia

12th Int. Conf. on Reachability Problems, Marseille, 25 Sept 2018

Janˇ car, Osiˇ cka, Sawa EXPSPACE-Complete Countdown Games 25 Sept 2018 1 / 8

Hofman, P., Lasota, S., Mayr, R., Totzke, P.: Simulation problems

• ver one-counter nets. Logical Methods in Comp. Sci. 12(1) (2016)

Open: simulation on concise OCN (PSPACE – EXPSPACE) Here: EXPSPACE-complete Hunter, P.: Reachability in succinct one-counter games. In: RP 2015

G¨

• ller, S., Haase, C., Ouaknine, J., Worrell, J.: Model checking

succinct and parametric one-counter automata. In: ICALP 2010. G¨

• ller, S., Lohrey, M.: Branching-time model checking of one-counter
• processes. In: STACS 2010.

Here: Reachability game reduces to (bi)simulation relations Jurdzinski, M., Sproston, J., Laroussinie, F.: Model checking probabilistic timed automata with one or two clocks. Logical Methods in Comp. Sci. 4(3) (2008) Countdown games ... EXPTIME-complete Here: “Existential” countdown games EXPSPACE-complete

Chandra, A.K., Kozen, D., Stockmeyer, L.J.: Alternation. J. ACM 28(1), 114–133 (1981)

Janˇ car, Osiˇ cka, Sawa EXPSPACE-Complete Countdown Games 25 Sept 2018 2 / 8

G = (QE, QA, δ), δ being a finite set of rules p

x

− → p′, x ∈ Z, x < 0. From p0(23) Eve has a strategy to reach E-win(0), from p0(14) no. Can Eve force E-win(0) from p(k) ? ... ExpTime-complete. Can Eve force E-win(0) from p(k) for some k ? ... ExpSpace-complete.

Janˇ car, Osiˇ cka, Sawa EXPSPACE-Complete Countdown Games 25 Sept 2018 3 / 8

➣ ➣ ➣ ➣ $ $ $ $ ➣ ➣ $ $

β3 β2 a2 a3 an

• β

β1

q0

a1

q+

x

1 2 3 n 1 2 3 n+1 m−1 j i t−1 t

C w

3

C w

2

C w

1

C w C w

i

C w

t

C w

t−1 Janˇ car, Osiˇ cka, Sawa EXPSPACE-Complete Countdown Games 25 Sept 2018 4 / 8

+1

. . . . . .

−(m−2) −(m−1) −m −(m−2) −3 −1 −2 −1 −1 −(m−n)

. . .

−3 −n −(m−2) −(m−2)

. . . . . . . . .

2 3 1 n+2 n n+1 m−2 m−1

E E E E E E E E E A A E E E E A

(if c = 0)

β1 β2 β3 $ E-win ➣

q+

• a3

β ➣

x

a2 a3

• $

an

• β1β2β3

β′

1β′ 2β′ 3 q0 a1

• Janˇ

car, Osiˇ cka, Sawa EXPSPACE-Complete Countdown Games 25 Sept 2018 5 / 8

We assume a labelled OCN N = (Q, A, δ), δ being a finite set of rules p

a,x

− → p′, a ∈ A, x ∈ Z. Configurations p(m) ∈ Q × N. Transitions: if p

a,x

− → p′ is a rule then p(m)

a,x

− → p′(m+x) if m+x ≥ 0. Simulation game, In a pair

• p(m), q(n)
• 1 Attacker makes a move p(m)

a,x

− → p′(m+x)

2 Defender responds by q(n)

a,y

− → q′(n+y) The play continues from

• p′(m+x), q′(n+y)
• .

If a player has no possible move, (s)he loses. An infinite play is Defender’s win. p(m) q(n) iff Defender has a winning strategy (from

• p(m), q(n)
• )

Janˇ car, Osiˇ cka, Sawa EXPSPACE-Complete Countdown Games 25 Sept 2018 6 / 8

s2 s3 s′

2

s′

3

s1

a1

2(x)

a1

3(y)

s′

1

a1

2(x)

a1

3(y)

s2 s3 s1

(x) (y)

E s2 s3 s′

2

s′

3

s1

23

a1

2(x)

a1

3(y)

s1

2

a1

2(x′′)

a1

3(y−x′)

s1

3

a1

3(y ′′)

a1

2(x−y ′)

s1

ac(0) ac(x′) ac(y ′)

s′

1

ac(x′) ac(y ′)

s2 s3 s1

(x) (y)

A

x′ = min {x, 0}, x′′ = max {x, 0}, and y′ = min {y, 0}, y′′ = max {y, 0}

Janˇ car, Osiˇ cka, Sawa EXPSPACE-Complete Countdown Games 25 Sept 2018 7 / 8

P<p,q>(m, n) = black iff p(m) q(n) Exponential thickness of belts, double-exponential period

Janˇ car, Osiˇ cka, Sawa EXPSPACE-Complete Countdown Games 25 Sept 2018 8 / 8