  We study the detection problem of finding planted solutions in random
instances of flat satisfiability problems, a generalization of boolean
satisfiability formulas. We describe the properties of random instances of flat
satisfiability, as well of the optimal rates of detection of the associated
hypothesis testing problem. We also study the performance of an algorithmically
efficient testing procedure. We introduce a modification of our model, the
light planting of solutions, and show that it is as hard as the problem of
learning parity with noise. This hints strongly at the difficulty of detecting
planted flat satisfiability for a wide class of tests.
