Exercise 2 9 Points Evaluate the following ML-formulae on the given Kripke-structure by constructing the model- checking game and computing the winning regions of the respective players

Lehr- und Forschungsgebiet

Mathematische Grundlagen der Informatik RWTH Aachen

Prof. Dr. E. Grädel, R. Wilke

WS 2018/19

Logic and Games — Assignment 1 Due : Tuesday the 16th at 12:00 in the lecture or at our chair.

Exercise 1 4 Points

Consider the game graphs G_i = (Vⁱ, V₀ⁱ, V₁ⁱ, Eⁱ), where j denotes a position of player 0 and

k one of player 1.

1

2 3

4 5 6

7 8 9 10

1 2 3

4 5 6

7

G₁ G₂

Compute the winning regionsW₀ andW₁ in both games. Infinite plays are considered as ties.

Exercise 2 9 Points

Evaluate the following ML-formulae on the given Kripke-structure by constructing the model- checking game and computing the winning regions of the respective players.

(a) ϕ_a=hbihaiP;

(b) ϕ_b = [a]hai(Q∨ hbiP);

(c) ϕc= [b] ([b]P∧ hai¬P).

1 2

Q

3 P

4 b

a a, b

a a

b

Exercise 3 8 Points

Let G = (V, V₀, V₁, E) be a reachability game. Consider the two inductive definitions of the winning regions:

(i) W_σ⁰ :={v∈V1−σ : vE=∅}

W_σⁿ⁺¹ :={v∈Vσ : vE∩W_σⁿ6=∅} ∪ {v∈V1−σ : vE⊆W_σⁿ} (ii) Wf_σ⁰ :={v∈V1−σ : vE=∅}

Wf_σⁿ⁺¹ :=Wf_σⁿ∪ⁿv∈Vσ : vE∩Wf_σⁿ6=∅

o∪ⁿv∈V1−σ : vE⊆Wf_σ^no (a) Show that W_σⁿ=Wf_σⁿ holds for all n∈N.

http://logic.rwth-aachen.de/Teaching/LS-WS19/

(b) IfG is afinite reachability game (that means, |V| ∈N), then Wσ = ^[

n∈N

W_σⁿ.

In infinitereachability games only “⊇” holds. Provide a counterexample showing that ine- quality is possible!

Exercise 4 9 Points

The (n, k)-Nim-game is played withnmatches, given (n≥k≥1). Two players alternate their turns, where the current one removes at least 1 and at most k matches from the board. If the last match is removed, the current player loses. Otherwise it is the opponents turn.

For which choices of nand k does the starting player win the (n, k)-Nim-game?

http://logic.rwth-aachen.de/Teaching/LS-WS19/