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Lehr- und Forschungsgebiet

Mathematische Grundlagen der Informatik RWTH Aachen

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

WS 2011/12

Algorithmic Model Theory — Assignment 13 Due: Monday, 30 January, 12:00

Exercise 1

For any directed graphG= (V, E) we define its double graph 2G= (V0, E0) over the vertex set V0 =V × {0,1} with edge relation

E0 :={((u, i),(v, j))∈V0×V0 : (u, v)∈E}.

LetPndenote the directed path of lengthnand letK be the class of all double directed paths, i.e.K :={2Pn:n≥1}.

(a) Show that K does not admit an FP-definable linear order.Hint: Exercise 2, Sheet 11.

(b) Show that FP captures polynomial time onK by using the method of canonisation. Con- struct FP-interpretations (when necessary using equivalences) to show:

• Pn is interpretable in 2Pn,

• (Cn,0) (an undirected circle of lengthnwith a constant 0) is interpretable in Pn,

• (2Pn, <) is interpretable in (Cn,0).

Hint:Use the edge relation (in both directions) as the domain formula of your inter- pretation.

Exercise 2

In the lecture, thek-pebble bijection game was introduced which characterisesC∞ωk -equivalence of structures.

(a) Modify the rules of the game to capture equivalence inLk∞ω rather than C∞ωk . Hint: Relax the requirement for Duplicator to choose a bijection.

(b) Use this game to show that the following classes of structures are undefinable in FP:

• The class of (undirected) graphs with an Eulerian cycle.

Hint: Consider complete graphs.

• The class of (undirected) graphs with an Hamiltonian cycle.

Hint: Consider complete bipartite graphs.

Exercise 3

Show that the CFI-query is decidable in polynomial time, i.e. show that given a CFI-graph XS(G) one can decide in polynomial time, whetherS is even or odd.

http://logic.rwth-aachen.de/Teaching/AMT-WS12/

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