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Show that this extension of SO-HORN has the full power of second-order logic

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Lehr- und Forschungsgebiet

Mathematische Grundlagen der Informatik RWTH Aachen

Prof. Dr. E. Grädel, W. Pakusa, F. Reinhardt, M. Voit

SS 2016

Algorithmic Model Theory — Assignment 6 Due: Friday, 3 June, 13:00

Exercise 1

Construct an SO-HORN-sentenceψwhich defines the class of (undirected) graphsG= (V, E, c, d) (with two constant symbolsc andd) in which there is no path from ctod.

Exercise 2

To justify the definition of SO-HORN, show that the admission of arbitrary first-order prefixes would make the restriction to Horn clauses pointless. Show that this extension of SO-HORN has the full power of second-order logic.

Exercise 3

weak-SO-HORN is the subclasss of SO-HORN consisting of all sentences of the form QR1. . . QRk∀x1, . . .∀xl ^

1≤i≤r

Ci,

where the clauses Ci are of the form β1. . .βnH and where the βi are either atoms or negated atoms with the restriction that the relations R1, . . . , Rk only occur positively. In other words, weak-SO-HORN differs from SO-HORN in the fact that only atomic or negated atomic first-order formulas are allowed in the clauses (instead of arbitrary first-order formulas which do not contain R1, . . . , Rk).

(a) Show that on ordered structures weak-SO-HORN is strictly less expressive than SO-HORN.

Hint: Show that for every weak-SO-HORN sentence ψ the class {A : A |= ψ} is closed under substructures.

(b) Show that, however, on ordered structures with the additional successor relation and constants 0, efor the first and last element in the order weak-SO-HORN and SO-HORN are equally expressive.

Hint: Show that on this domain weak-SO-HORN captures PTIME.

http://logic.rwth-aachen.de/Teaching/AMT-SS16/

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