  We consider extensions of the two-variable guarded fragment, GF2, where
distinguished binary predicates that occur only in guards are required to be
interpreted in a special way (as transitive relations, equivalence relations,
pre-orders or partial orders). We prove that the only fragment that retains the
finite (exponential) model property is GF2 with equivalence guards without
equality. For remaining fragments we show that the size of a minimal finite
model is at most doubly exponential. To obtain the result we invent a strategy
of building finite models that are formed from a number of multidimensional
grids placed over a cylindrical surface. The construction yields a
2NExpTime-upper bound on the complexity of the finite satisfiability problem
for these fragments. We improve the bounds and obtain optimal ones for all the
fragments considered, in particular NExpTime for GF2 with equivalence guards,
and 2ExpTime for GF2 with transitive guards. To obtain our results we
essentially use some results from integer programming.
