

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

allow the distribution to be different for each modal depth from the top level to d  1. We also allow direct specification of the probability distribution for the number of literals in a clause at each modal depth. Thus, the probability distribution for the number of propositional atoms depends on both the modal depth and the number of literals in the clause.

## 5.1 Generalization: Shaping The Probability Distributions.

The generator has two parameters to control the shape of formulae. The first parameter, C, is a list of lists (e.g., [[0,0,1]]) telling it how many disjuncts to put in each disjunction at each modal level. Each internal list represents a finite discrete probability distribution. For instance, the
"[0,0,1]" says "0=1 of the disjunctions have 1 disjunct, 0=1 have 2 disjuncts, and 1=1 have 3 disjunctions" (fixed length 3). Because there is only one element of the list, this frequency is used at each modal depth, until the last. Other possibilities are, e.g., [[1,1,1,1]] (maximum length 4 with uniform distribution), [[16,8,4,2,1]] (maximum length 5 with exponential distribution),
and so on.

The second parameter, p, is a list of lists of lists (e.g., [[[],[],[0,3,3,0]]]) that controls the propositional/modal rate. The top-level elements are for each modal depth (here all the same). The second-level elements are for disjunctions with 1,2,3,... disjunctions (here only the third matters as all disjunctions have three disjuncts). For instance, the "[0,3,3,0]" says "0=6 of the disjunctions have 0 propositional atoms, 3=6 have 1 propositional atom, 3=6 have 2 propositional atoms, and 0=6 have 3 propositional atoms" (that is, our new scheme discussed in the paper with