Exercise 13.1 PTA Acceptance as a Parity Game

Applied Automata Theory (WS 2013/2014) Technische Universit¨ at Kaiserslautern

Exercise Sheet 13

Jun.-Prof. Roland Meyer, Georgel Calin Due: Tue, Feb 4

Exercise 13.1 PTA Acceptance as a Parity Game

Let A = ({a/2, b/2} , {e, o, x} , e, →, Ω) be a PTA with Ω(e) = Ω(o) = 1, Ω(x) = 0, and e→

(e, o) e→

(o, e) e→

(e, e) e→

(o, o) e→

(x, x)

o→

(e, e) o→

(o, o) o→

(e, o) o→

(o, e) x→

(x, x) (a) What is L(A)?

(b) Let t / ∈ L(A) be the following tree b a

b a

B a

B B

a

B B

a

B B

B

where B = b(B, B).

Your task: construct a tree t

over Σ × ({e, o, x}

→ {0, 1}) with proj

(t

) = t and such that proj

(t

) is a positional winning strategy for the ”pathfinder” P in G(A, t).

Exercise 13.2 PTA Complementation

Let A = ({a/2, b/2} , {q

, q

} , q

, →, Ω) be a PTA with Ω(q

) = 1, Ω(q

) = 2, and q

→

(q

, q

) q

→

(q

, q

) q

→

(q

, q

) q

→

(q

, q

).

(a) What is L(A)?

(b) Argument why the complemented word language L

of all labeled paths

(a

, s

, d

)(a

, s

, d

) . . . with a

∈ {a, b} , s

∈ {q

, q

}

→ {0, 1} , d

∈ {0, 1}

from the PTA complementation proof, is precisely the language that satisfies

∀n. x

= a ∨ (∃m. x

= b ∧ ∀n < m. x

= a

∧ ∃n.(n ≤ m ∧ s

(q

, q

) 6= d

) ∨ (n > m ∧ s

(q

, q

) 6= d

)) .

(c) Construct a PTA which accepts the language L(A).