Advanced Automata Theory Exercise Sheet 6
Prof Roland Meyer TU Braunschweig
Dr Prakash Saivasan Summer term 2017
Out: May 16 Due: May 22, 12:00
Exercise 1: Naive Interpretation of NFAs as NBAs
LetA= (Σ, Q, q0,→, QF) be an NFA with ∅ 6=L(A)⊆Σ+ and, for any two states q, q0 ∈Q, define L6=q,q0 := {w ∈ Σ+ | q −→w q0 in A}. If Lω(A) is the ω-regular language accepted by A (interpreted as an NBA), one can wrongly believe that Lω(A) =L(A)ω.
a) Find a counterexample to Lω(A) =L(A)ω when ∅ 6=L6=q,q ⊆L(A) for all q ∈QF. b) Given an NFA A, provide a construction for an NBA Aω such that L(Aω) =L(A)ω.
Exercise 2: NBA languages = ω-regular Languages
a) Prove thatω-regular languages are NBA definable.
b) Show that if there exists an NBA that acceptsL⊆Σω then L isω-regular.
c) Construct an NBA that accepts L= (ab+c)∗((aa+b)c)ω+ (a∗c)ω.
Exercise 3: Shuffle ω-regular Languages
Given an infinite set of positionsI ⊆ {0,1, . . .} withI ={i1, i2, . . .} and i1 < i2 < . . ., and anω-word w, we writew|I for the ω-word w(i1)w(i2). . ., i.e. the sub-word of w obtained by selecting the letters in the positions ofI.
The fair shuffle of two ω-languages L1, L2 is defined as
L1L2 :={w| ∃partition I, J of positions {0,1, . . .} such that w|I ∈L1 and w|J ∈L2}
Note in particular, that sinceI and J form a partition of the positions, I 6=∅ 6=J. Show thatω-regular languages are closed under fair shuffle.
Exercise 4: Variation of Ramsey’s Theorem
Let (V, E) be an infinite graph such that for every infinite set of vertices X ⊆V there are v, v0 ∈X with (v, v0)∈E. Prove that (V, E) contains an infinite complete subgraph.
