Preference Decomposition and the Expressiveness of Preference Query Languages

Abstract

Preferences in the scope of relational databases allow modeling user wishes by queries with soft constraints. There are different frameworks for database preferences including commercially available systems. They slightly vary in semantics and expressiveness but have in common that preferences induce strict partial orders on a given data set. In the present paper we study the expressiveness of preference operators in the available implementations. Particularly, we search for decompositions of strict partial orders into fundamental preference constructs. We study which preference operators and operands are necessary to express any strict partial order. Finally, we present two decomposition algorithms and show their correctness.

A Unique Tuple Decompositions

Given the data set $r = x_1 + ... + x_4$ and the preference

$$ a = ((\mathsf {t}(x_1) \otimes \mathsf {t}(x_3)) \mathbin { \& }\mathsf {t}(x_2)) \otimes (\mathsf {t}(x_3) \mathbin { \& }\mathsf {t}(x_4)), $$

there is no decomposition into an $r$-equivalent preference within $\mathsf {un}_\mathsf {\{\mathbin { \& },\otimes \}}(r)$. We will show this in the following R-Script, which is a snippet from [11].

The model finder function

extends the temporary preference term

by $...\,\otimes \,\mathsf {t}(x_i)$ or $...\,\mathbin { \& }\,\mathsf {t}(x_i)$ in the recursive step. The term extension at the end is sufficient to get all possible terms, as $\mathbin { \& }$ and $\otimes$ are associative operators. The variable

stores those $x \in r$ which can still be used for $\mathsf {t}(x)$ without violating the uniqueness of the $x_i$. Hence

is comparable to the parameter $s$ in $\mathsf {un}_\mathsf {\mathsf {op}}(s)$, Definition 3.3.

The comparison w.r.t. $r$-equivalency of two preferences (cf. Definition 2.5) is done by comparing the adjacency lists of their Hasse diagrams. Note that we can rely on a predefined sorting (lexicographic) of the adjacency list of a Hasse diagram in the result of

. Hence the equivalency of these adjacency lists imply the $r$-equivalency of the corresponding preferences.

Note that this is some kind of a brute-force search. It could be optimized by e.g. exploiting the commutativity of $\otimes$ and a more efficient $r$-equivalence check. We omitted such optimizations to keep the code as simple as possible. The script generates and checks 633 possible terms. On our off-the-shelf computer the execution time of this program is about 8 s. Finally, it returns

, i.e., no decomposition is found.