

![0_image_0.png](0_image_0.png)

As an illustration of our general method, we present a set of tests with m = 1, d = 3; 4, N = 3; 4;, C =[[1,8,1]], and p =[[[1,0],[0,1,0],[0,1,1,0]]]. This set of tests introduces a small fraction of single-literal clauses that contain a modal literal (except at the greatest modal depth, where they contain, of course, a single propositional literal). The results of tests are given in Figure 14. Again, trivial instances occur only out the interesting zone. Here we can generate interesting test sets even with modal depth 4.

## 5.2 Varying The Probability Distributions With The Depth

Our new method provides the ability to fine-tune the distribution of both the size and the propositional/modal rate of the clauses *at every depth*. This fine tuning results in a very large number of parameters, and so far in this paper we have only investigated distributions that conform to the scheme described above or ones that correspond to the 3CNF2m generation method previously used.

To give an example of how to vary the probability distributions with the nesting depth of the clauses, we consider the case with d = 4; 5, m = 1, N = 3; 4; 5, C =[[1,8,1],[1,2]],
p =[[[1,0],[0,1,0],[0,1,1,0]],[[1,0],[0,1,0]]]. The results of the tests are given in Figure 15.

The C parameter says that the probability distributions of the length of the clauses occurring at nesting depth 0 and  1 are [1,8,1] and [1,2] respectively. (When not explicitly specified, it is