the size of S, so the value of r is a convenient measure for estimating p(mtm).

There is another syntactic (and so easily computable) measure of S **that controls**
features of S in a way similar to that of r. This is impurity **studied in Lozinskii**
(2006).

Let pos(v), neg(v**) stand, respectively, for the number of unnegated and**
negated occurrences of a variable v in a set of clauses S. If v occurs in S **either** only unnegated or only negated (neg(v) = 0 or pos(v) = 0) then v is a **pure** variable in S, otherwise v is an impure **one. Denote**

$$m a x(v)=\operatorname*{max}(p o s(v),n e g(v)),\quad m i n(v)=\operatorname*{min}(p o s(v),n e g(v)).$$

Let imp(v) = min(v)/max(v) be called the impurity of v, and imp(S**) stand**
for the impurity of S**, that is the average impurity of its variable:**

$$\begin{array}{c}{{i m p(S)=\frac{1}{B}\sum_{i=1}^{B}m i n(v_{i})/m a x(v_{i})}}\\ {{0\leq i m p(S)\leq1.}}\end{array}$$
$$(23)$$
$$s(v),n e g(v)).$$
$$(24)$$
$$\left(25\right)$$
$$0\leq i m p(S)\leq1.$$

It has been shown in Lozinskii (2006) that while the impurity **of a set of**
clauses S growth from 0 to 1, the probability that S **is satisfiable decreases**
and undergoes a phase transition in the vicinity of a certain **value of impurity**
depending on r. The number of models of S **is a monotone decreasing function** of imp(S) like it is as a function of r. Figure 3 presents p(mtm) and ER(mtm)
of a set of clauses as functions of its impurity (averaged over 10000 random sets with B = 30, 100, r = 4.26, and 0 ≤ imp(S) ≤ 0.**92). The patterns are similar** to those of Figure 2. So given S, both r(S) and imp(S**) can be used for a quick**
estimation of the probability that S **has a most typical model.**

## 8 Conclusion

In general, a knowledge system S **describing a real world does not contain**
complete information about it. Reasoning with incomplete information is prone to errors since any belief derived from S **may turn out to be false in the present** state of the world. The smaller the expected number of false beliefs produced by an approach to reasoning with incomplete information, the more reliable the approach.

In regard to the main goal - choosing a model that would represent the reality most faithfully - this work is close to the previous research on reasoning with incomplete information, but presents a completely **different approach**
introducing typical models and showing that any knowledge system has a typical model that is the most trustworthy one since it minimizes the number of false beliefs. So if minimization of reasoning errors is important, the semantics of typical models is the best one among all approaches to **reasoning with**
incomplete information.