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Concurrency Theory(SS 2015) Out: Thu, 23 Jul Due: Whenever

Exercise Sheet 14

Zetzsche, Furbach, D’Osualdo Technische Universit¨at Kaiserslautern

Problem 1: Parikh Images

Show the following: If the Parikh images of a full trioC are effectively semilinear, then the simultaneous unboundedness problem is decidable forC.

Problem 2: C-Grammars

a) Show how the algorithm for eliminating ε-productions can be applied to C-grammars.

Assume emptiness is decidable forC and thatC is a full trio..

b) Show that emptiness is decidable for a full trioC if and only if emptiness is decidable forAlg(C).

Problem 3: Petri net languages

a) Show that for a full trioC,Alg(C)is closed under Kleene iteration.

b) Show that for the full trio of Petri net languagesP, it holdsP (Alg(P).

Problem 4: Algebraic Extensions

a) Show that ifC is a full trio then so isAlg(C).

[Hint: Use Problem 2 on Sheet 12 and for regular intersections, apply the triple construc- tion.]

b) Show that if emptiness is decidable for a full trio C, then Alg(C)is not closed under intersection.

[Hint:Use 2b) and the fact that(D01#1)∗is context-free.]

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