n}) i) FO over the class of finite linear orders Lin ={([n], <) :n∈N, &lt

Lehr- und Forschungsgebiet

Mathematische Grundlagen der Informatik RWTH Aachen

Prof. Dr. E. Grädel, F. Abu Zaid, W. Pakusa, F. Reinhardt

WS 2013/14

Algorithmic Model Theory — Assignment 12

Due: Monday, 27 January, 12:00

Exercise 1

Calculate the asymptotic probabilities of the following graph properties with respect to the uniform distribution on the class G of undirected graphs:

i) K₁={G∈ G :Ghas no isolated node}

ii) K₂={G∈ G :Gis bipartite}

iii) K₃={G∈ G :Gis a tree}

iv) K₄={G∈ G :G= (V, E) contains a clique of size ≥log(|V|)}

Exercise 2

Prove or disprove that the following logics have the zero-one law with respect to the uniform probability distribution on the respective classes. ([n] :={1,2, . . . , n})

i) FO over the class of finite linear orders Lin ={([n], <) :n∈N, < linear order on [n]}

ii) FO over the class of finite binary words W ={([n], <, P) : ([n], <)∈Lin, P ⊆[n]}

iii) FO over the class of bipartite graphs Bip ={([n]× {0,1}, E) :E ⊆([n]×0)×([n]×1)}

iv) C^ω_∞,ω over the class of all graphs v) SO over the class of all graphs

http://logic.rwth-aachen.de/Teaching/AMT-WS13/