Problem 1: Hardest Language

Concurrency Theory(SS 2015) Out: Wed, 1 Jul Due: Tue, 7 Jul

Exercise Sheet 11

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

Problem 1: Hardest Language

A languageL₀is ahardest languagefor a language classC if for everyL∈ Cthere exists an homomorphismαsuch thatL=α⁻¹(L₀).

For any regular language L, letkLk be the minimum number of states of an automaton generatingL, i.e.kLk= min{|Q| |A= (Q, E, q₀, F), L=L(A)}. Show that:

a) L=α⁻¹(L⁰)implieskLk ≤ kL⁰k.

b) For eachn ∈N, there exists a regular language withkLk> n.

c) Use 1.a and 1.b to show that there is no hardest language for the class of regular languages.

Problem 2: Concatenation

a) Show that ifL⁰ ⊆X^∗ is closed under concatenation andL =α⁻¹(L⁰), thenLis closed under concatenation.

b) Show thatD⁰₂is not a hardest language for CFL.

Problem 3: Kleene Iteration

LetL⊆X^∗. Show that the full trio generated by(L#)^∗is closed under Kleene iteration.

Problem 4: Generating languages

LetM ={aⁿbⁿ|n ∈N}andC_M be the full trio generated byM.

a) LetR_n={ww|w∈X_n^∗}wherewis the wordwwith each occurrence of each letterx_i replaced by the letterx_i ∈X_n.

Show that for eachn∈N, the complement ofRnis inCM.

b) Sketch how you can adapt the solution of the previous problem to prove that for each homomorphism α: X^∗ → Y^∗, with X ∩Y = ∅, the complement of the language {wα(w)|w∈X^∗}is inCM.