DS201005: Kontinen, Jarmo (2010) Coherence and Complexity in Fragments of Dependence Logic. Doctoral thesis, University of Amsterdam.
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Abstract
We study the properties of fragments of dependence logic (D) over finite
structures.
One essential notion used to distinguish between Dformulas is socalled
kcoherence of a formula. Satisfaction of a kcoherent formula in all teams
can be reduced to the satisfaction in the kelement subteams. We will
characterize the coherence of quantifierfree Dformulas and give an example
of a formula which is not kcoherent for any natural number. We show that
all coherent formulas are equivalent to firstorder sentences when there is
an extra predicate interpreting the team.
We also seek to characterize the computational complexity of model checking
of Dformulas. A classic example in the field of descriptive complexity
theory is the Fagin's theorem, which establishes a perfect match between
existential second order (ESO) formulas and languages in NP. Dformulas are
known to have a definition in ESO and vice versa. When we combine this with
Fagin's result we get that the properties definable in D over finite
structures are exactly the ones recognized in NP.
We use the notion of coherence to give a characterization for the
computational complexity of the model checking for Dformulas.
We establish three thresholds in the computational complexity of the model
checking, namely when the model checking can be done in logarithmic space
(L), in nondeterministic logarithmic space (NL) and when the checking
becomes complete for nondeterministic polynomial time (NP). We give
complete instances for NL and NP.
Another criterion we use to find structure inside dependence logic is
asymptotic probability and the 01law. We show that the 01law holds for
universal and existential Dsentences as well as for all the quantifierfree
formulas in the case of atomic probability 1/2.
In the second part of the thesis we give a characterization for the 01law
for proportional quantifiers over uniform distribution of finite graphs. We
will give a precise threshold when the 01 law holds for finite variable
infinitary logic extended with a proportional quantifier and when it does
not.
Item Type:  Thesis (Doctoral) 

Report Nr:  DS201005 
Series Name:  ILLC Dissertation (DS) Series 
Year:  2010 
Subjects:  Computation Language Logic 
Depositing User:  Dr Marco Vervoort 
Date Deposited:  14 Jun 2022 15:16 
Last Modified:  14 Jun 2022 15:16 
URI:  https://eprints.illc.uva.nl/id/eprint/2088 
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