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Prompt and Parametric LTL Games Martin Zimmermann RWTH Aachen University September 17th, 2009 Games Workshop 2009 Udine, Italy Martin Zimmermann RWTH Aachen University Prompt and Parametric LTL Games 1/12 Outline 1. Introduction 2.

SLIDE 1

Prompt and Parametric LTL Games

Martin Zimmermann

RWTH Aachen University

September 17th, 2009

Games Workshop 2009 Udine, Italy

Martin Zimmermann RWTH Aachen University Prompt and Parametric LTL Games 1/12

SLIDE 2

Outline

1. Introduction
2. Parametric LTL
3. Conclusion

Martin Zimmermann RWTH Aachen University Prompt and Parametric LTL Games 2/12

SLIDE 3

Infinite Games

Played in finite arena A = (V , V0, V1, E, v0, l) with labeling l : V → 2P. Winning conditions are expressed in extensions of LTL over P.

v0

p, q

v1

p

v2

∅

v3

q, r

v4

r

Martin Zimmermann RWTH Aachen University Prompt and Parametric LTL Games 3/12

SLIDE 4

Infinite Games

Played in finite arena A = (V , V0, V1, E, v0, l) with labeling l : V → 2P. Winning conditions are expressed in extensions of LTL over P.

v0

p, q

v1

p

v2

∅

v3

q, r

v4

r

Theorem (Pnueli, Rosner ’89)

Determining the winner of an LTL game is 2EXPTIME-complete. Finite-state strategies suffice to win an LTL game.

Martin Zimmermann RWTH Aachen University Prompt and Parametric LTL Games 3/12

SLIDE 5

Infinite Games

Played in finite arena A = (V , V0, V1, E, v0, l) with labeling l : V → 2P. Winning conditions are expressed in extensions of LTL over P.

v0

p, q

v1

p

v2

∅

v3

q, r

v4

r

Theorem (Pnueli, Rosner ’89)

Determining the winner of an LTL game is 2EXPTIME-complete. Finite-state strategies suffice to win an LTL game. However, LTL lacks capabilities to express timing constraints. There are many extensions of LTL to overcome this. Here, we consider two of them: PLTL: Parametric LTL (Alur et. al., ’99) PROMPT − LTL (Kupferman et. al., ’07)

Martin Zimmermann RWTH Aachen University Prompt and Parametric LTL Games 3/12

SLIDE 6

Outline

1. Introduction
2. Parametric LTL
3. Conclusion

Martin Zimmermann RWTH Aachen University Prompt and Parametric LTL Games 4/12

SLIDE 7

PLTL

Let X and Y two disjoint sets of variables. Add F≤x for x ∈ X and G≤y for y ∈ Y to LTL. Semantics defined w.r.t. variable valuation α: X ∪ Y → N. (ρ, i, α) | = G≤yϕ: ρ i i + α(y) ϕ ϕ ϕ ϕ ϕ (ρ, i, α) | = F≤xϕ: ρ i i + α(x) ϕ

Martin Zimmermann RWTH Aachen University Prompt and Parametric LTL Games 5/12

SLIDE 8

PLTL

Let X and Y two disjoint sets of variables. Add F≤x for x ∈ X and G≤y for y ∈ Y to LTL. Semantics defined w.r.t. variable valuation α: X ∪ Y → N. (ρ, i, α) | = G≤yϕ: ρ i i + α(y) ϕ ϕ ϕ ϕ ϕ (ρ, i, α) | = F≤xϕ: ρ i i + α(x) ϕ PLTL game (A, ϕ): σ winning strategy for Player 0 w.r.t. α iff for all plays ρ consistent with σ: (ρ, 0, α) | = ϕ. τ winning strategy for Player 1 w.r.t. α iff for all plays ρ consistent with τ: (ρ, 0, α) | = ϕ. Wi

G = {α | Player i has winning strategy for G w.r.t. α}.

Martin Zimmermann RWTH Aachen University Prompt and Parametric LTL Games 5/12

SLIDE 9

PLTL Games: Examples

Winning condition FG≤yp: Player 0’s goal: eventually satisfy p for at least α(y) steps. p p p p

≥ α(y)

Martin Zimmermann RWTH Aachen University Prompt and Parametric LTL Games 6/12

SLIDE 10

PLTL Games: Examples

Winning condition FG≤yp: Player 0’s goal: eventually satisfy p for at least α(y) steps. p p p p

≥ α(y)

Player 1’s goal: reach vertex with ¬p at least every α(y) steps. ¬p ¬p ¬p ¬p

≤ α(y) ≤ α(y) ≤ α(y) ≤ α(y)

Martin Zimmermann RWTH Aachen University Prompt and Parametric LTL Games 6/12

SLIDE 11

PLTL Games: Examples

Winning condition G(q → F≤xp): “Every request q is eventually responded by p”. Player 0’s goal: uniformly bound the waiting times between requests q and responses p by α(x). q p q p

≤ α(x) ≤ α(x)

Martin Zimmermann RWTH Aachen University Prompt and Parametric LTL Games 7/12

SLIDE 12

PLTL Games: Examples

Winning condition G(q → F≤xp): “Every request q is eventually responded by p”. Player 0’s goal: uniformly bound the waiting times between requests q and responses p by α(x). q p q p

≤ α(x) ≤ α(x)

Player 1’s goal: either request q and prevent response p or enforce waiting time greater than α(x). q, ¬p ¬p ¬p p

≥ α(x)

Martin Zimmermann RWTH Aachen University Prompt and Parametric LTL Games 7/12

SLIDE 13

PLTL Games: Examples

Winning condition G(q → F≤xp): “Every request q is eventually responded by p”. Player 0’s goal: uniformly bound the waiting times between requests q and responses p by α(x). q p q p

≤ α(x) ≤ α(x)

Player 1’s goal: either request q and prevent response p or enforce waiting time greater than α(x). q, ¬p ¬p ¬p p

≥ α(x)

Note: both winning conditions induce an optimization problem: maximize α(y) resp. minimize α(x).

Martin Zimmermann RWTH Aachen University Prompt and Parametric LTL Games 7/12

SLIDE 14

PROMPT-LTL

PROMPT − LTL: No G≤y, all F≤x parameterized by the same variable.

Martin Zimmermann RWTH Aachen University Prompt and Parametric LTL Games 8/12

SLIDE 15

PROMPT-LTL

PROMPT − LTL: No G≤y, all F≤x parameterized by the same variable.

Theorem

Let G be a PROMPT − LTL game. The emptiness problem for W0

G is 2EXPTIME complete.

Martin Zimmermann RWTH Aachen University Prompt and Parametric LTL Games 8/12

SLIDE 16

PROMPT-LTL

PROMPT − LTL: No G≤y, all F≤x parameterized by the same variable.

Theorem

Let G be a PROMPT − LTL game. The emptiness problem for W0

G is 2EXPTIME complete.

Proof 2EXPTIME algorithm: apply alternating-color technique of Kupferman et al.. Reduce G to an LTL game G′ such that a finite-state winning strategy for G′ can be transformed into a winning strategy for G that bounds the waiting times. 2EXPTIME hardness follows from 2EXPTIME hardness of solving LTL games.

Martin Zimmermann RWTH Aachen University Prompt and Parametric LTL Games 8/12

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PLTL: Results

Theorem

Let G be a PLTL game. The emptiness, finiteness, and universality problem for Wi

G are 2EXPTIME-complete.

Martin Zimmermann RWTH Aachen University Prompt and Parametric LTL Games 9/12

SLIDE 18

PLTL: Results

Theorem

Let G be a PLTL game. The emptiness, finiteness, and universality problem for Wi

G are 2EXPTIME-complete.

Proof 2EXPTIME algorithms: Emptiness for formulae with only F≤x: reduction to PROMPT − LTL games. For the full logic and the other problems use: Duality of F≤x and G≤y. Monotonicity of F≤x and G≤y. 2EXPTIME hardness follows from 2EXPTIME hardness of solving LTL games.

Martin Zimmermann RWTH Aachen University Prompt and Parametric LTL Games 9/12

SLIDE 19

PLTL: Results

If ϕ contains only F≤x respectively only G≤y, then solving games is an optimization problem: which is the best valuation in W0

G?

Martin Zimmermann RWTH Aachen University Prompt and Parametric LTL Games 10/12

SLIDE 20

PLTL: Results

If ϕ contains only F≤x respectively only G≤y, then solving games is an optimization problem: which is the best valuation in W0

G?

Theorem

Let ϕF be G≤y-free and ϕG be F≤x-free, let GF = (A, ϕF) and GG = (A, ϕG). The following problems are decidable: Determine minα∈W0

GF maxx∈var(ϕF) α(x). Martin Zimmermann RWTH Aachen University Prompt and Parametric LTL Games 10/12

SLIDE 21

PLTL: Results

If ϕ contains only F≤x respectively only G≤y, then solving games is an optimization problem: which is the best valuation in W0

G?

Theorem

Let ϕF be G≤y-free and ϕG be F≤x-free, let GF = (A, ϕF) and GG = (A, ϕG). The following problems are decidable: Determine minα∈W0

GF maxx∈var(ϕF) α(x).

Determine minα∈W0

GF minx∈var(ϕF) α(x).

Determine maxα∈W0

GG maxy∈var(ϕG) α(y).

Determine maxα∈W0

GG miny∈var(ϕG) α(y). Martin Zimmermann RWTH Aachen University Prompt and Parametric LTL Games 10/12

SLIDE 22

Outline

1. Introduction
2. Parametric LTL
3. Conclusion

Martin Zimmermann RWTH Aachen University Prompt and Parametric LTL Games 11/12

SLIDE 23

Conclusion

We considered infinite games with winning conditions in extensions

f LTL with bounded temporal operators.

Solving them is as hard as solving LTL games. Several optimization problems can be solved effectively.

Martin Zimmermann RWTH Aachen University Prompt and Parametric LTL Games 12/12

SLIDE 24

Conclusion

We considered infinite games with winning conditions in extensions

f LTL with bounded temporal operators.

Solving them is as hard as solving LTL games. Several optimization problems can be solved effectively. Further research: Better algorithms for the optimization problems. Hardness results for the optimization problems. Tradeoff between size and quality of a finite-state strategy. Time-optimal winning strategies for other winning conditions.

Martin Zimmermann RWTH Aachen University Prompt and Parametric LTL Games 12/12