Coherence and Complexity in Fragments of Dependence Logic
Jarmo Kontinen
Abstract:
We study the properties of fragments of dependence logic (D) over finite
structures.
One essential notion used to distinguish between D-formulas is so-called
k-coherence of a formula. Satisfaction of a k-coherent formula in all teams
can be reduced to the satisfaction in the k-element sub-teams. We will
characterize the coherence of quantifier-free D-formulas and give an example
of a formula which is not k-coherent for any natural number. We show that
all coherent formulas are equivalent to first-order sentences when there is
an extra predicate interpreting the team.
We also seek to characterize the computational complexity of model checking
of D-formulas. 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. D-formulas 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 D-formulas.
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 non-deterministic logarithmic space (NL) and when the checking
becomes complete for non-deterministic 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 0-1-law. We show that the 0-1-law holds for
universal and existential D-sentences as well as for all the quantifier-free
formulas in the case of atomic probability 1/2.
In the second part of the thesis we give a characterization for the 0-1-law
for proportional quantifiers over uniform distribution of finite graphs. We
will give a precise threshold when the 0-1- law holds for finite variable
infinitary logic extended with a proportional quantifier and when it does
not.