4. Representation of databases with incomplete information and sentential data manipulation languages

Let Dt be the metadabase. Then database with incomplete information (for short, DBI or, if it is necessary to underline "set of strings" DBI, then SDBI), denoted Xt, is the finite set of the so-called incomplete facts (N-facts) being sentential forms (SF) of CF grammar Gt:

$$X\_t = \{\mathfrak{x}\_1, \dots, \mathfrak{x}\_m\} \subseteq \mathrm{SF}(\mathcal{G}\_t), \tag{45}$$

where

$$SF(G\_t) = \left\{ \mathbf{x} | \mathfrak{a}\_0 \stackrel{\*}{\underset{G\_t}{\rightleftharpoons}} \mathbf{x} \right\} \tag{46}$$

is the set of all sentential forms of grammar Gt. Obviously, L Gð Þ<sup>t</sup> ⊂SF Gð Þ<sup>t</sup> : Example 3. Consider MDB from Example 2 and corresponding DBI

> Xt ¼ fAREA LONELY TREES IS NORMAL AT 12:31, AREA LONELY TREES<state>AT 12:< minutes>, AREA <name of area>IS SMOKED AT 15:30g:

The first N-fact of the three, entering this DBI, does not contain nonterminals, so it is fact in the sense of S-semantics of DML. The second N-fact contains nonterminals <state> and <minutes>, which correspond to the uncertainty of the state of the area Lonely Trees and time moment, when this state occurs; however, the aforementioned moment enters interval from 12.01 to 12.59. The last N-fact contains information about the same area, which was detected as smoked at 15.30. ∎

Before consideration of equations, defining M-semantics of operations on DBI, we shall introduce interpretation of relation ∗ ¼) Gt of the mutual derivability of sentential forms of context-free grammar as relation of mutual informativity of N-facts. Let Gt be acyclic and unambiguous CF grammar [49]. If so, ∗ ¼) Gt is the relation of

partial order on the set SF Gð Þ<sup>t</sup> [38, 39]. There is maximal element of the set SF Gð Þ<sup>t</sup> —it is axiom α<sup>0</sup> ("fact") because

$$\text{for every } \mathbf{x} \in \text{SF}(\mathbf{G}\_l), \ a\_0 \xrightarrow[\mathbf{G}\_l]{\ast} \text{x. For every subset } \mathbf{X} \subseteq \text{SF}(\mathbf{G}\_l), \text{ there exists set of its } l$$

upper bounds, e.g., sentential forms ("N-facts") y∈SF Gð Þ<sup>t</sup> , such that y ∗ ¼) Gt x for all x∈X, and minimal (least) upper bound, sup X, such that for every other upper bound y from the mentioned set, the relation y ∗ ¼) Gt sup X is true. For some X ⊆SF Gð Þ<sup>t</sup> , there may exist set of lower bounds, e.g., sentential forms ("N-facts") y∈SF Gð Þ<sup>t</sup> , such that x ∗ ¼) Gt y for all x∈X, and maximal lower bound inf X such that

for every other lower bound y, inf X ∗ ¼) Gt y is true.

Algorithms of sup and inf generation are described in detail in [38, 39]. Example 4. For DBI from Example 3, inf Xt does not exist, but

sup Xt ¼ AREA <name of area>IS <state>AT 1<0 to 9>:< minutes>:

At the same time,
