
## 4.1 Typical Atoms

S is supposed to be formulated in a first order language, so using the terminology of Predicate Calculus let the base of S **be a set of all ground instances of**
all atomic formulas corresponding to all predicates occurring in S:

$$B a s e(S)=\{P^{(k)}(t_{1},\ldots,t_{k})\},$$
$$\left({4}\right)$$

such that P
(k)(x1, . . . , xk) occurs in S, Diis the domain of xi**, and** ti ∈ Di for 1 ≤ i ≤ k.

Definition 4.1 For each ground atomic formula a ∈ Base(S) let ˆa **denote the**
typical atom corresponding to a such that the evidence of ˆa **is at least as large**
as that of ¬ˆa**. So**2

$${\hat{\mathbf{a}}}={\left\{\begin{array}{l l}{\mathbf{a}}&{{\mathrm{if~}}E(S,\mathbf{a})\geq0.5}\\ {\neg\mathbf{a}}&{{\mathrm{otherwise}}}\end{array}\right.}$$
$$\left(5\right)$$
$\widehat{\mathbb{E}}_{\mathbb{E}^{n}}$
For a formula F we define its typical value Fb by substituting F for a **in expression (5).**
Evidence E(S, a**) is introduced in order to be used as an approximation of**
p(a). The larger the difference |E(S, a) − 0.5|**, the better this approximation.**
However, if E(S, a) is close to 0.5, it may diverge from p(a**) even qualitatively**
such as E(S, a) > 0.5 but p(a) < 0.**5. But in the absence of a sufficient statistics** regarding p(w) one has to rely on E(S, a**) and believe that any typical atom is**
not less likely to be true than false in the present state of the world. On the other hand, the need to avoid inconsistency may force a non-oblivious reasoner to adopt beliefs in negation of some typical atoms. So questions arising in any reasoning system intended for answering multiple queries are:
Given a set of queries {F1, . . . , Fk}**, is there a state of the world in**
which beliefs in the truth of typical values {Fb1, . . . , Fbk} **hold for all**
1 ≤ i ≤ k, i.e. is there a model m of S **such that** m |=Vk i=1 Fbi?

What is the value of evidence E(S, Vk i=1 Fbi)?

The answer to the first question is positive if the latter evidence is larger than zero.

Let A(k) be a set of k literals l such that l ∈ {a,¬a}, a ∈ Base(S**), and**
A
(k)
∧ =Vl∈A(k) l**. The following theorem estimates the value of evidence of** A
(k)
∧ .

Theorem 4.1 max(0, α, β) ≤ E(**S, A**(k)

$$\beta)\leq E(S,A_{\land}^{(k)})\leq\operatorname*{min}(1,\gamma),\,\,\,w h e r e$$
$$\alpha=\sum_{l\in A^{(k)}}E(S,l)\quad-\quad k+1,$$  $$\beta=1-\frac{2^{|B a s e(S)|-k}(2^{k}-1)}{|M O D(S)|},\quad\gamma\quad=\quad\frac{2^{|B a s e(S)|-k}}{|M O D(S)|}.$$

. (7)
2If E(S, a) = 0.5 then E(S, a) = E(S,¬a), and so any one of a or ¬a can be considered a typical atom. However, in practice a proper choice of one of a or ¬a should be made based on relevant knowledge of the real world.