special case of speciousness occurs when two sets of attributes, $Q$ and $X$, cover the same set of elements, i.e., when $cov(X)=cov(Q)$. In this case, $X$ and $Q$ are called [*equivalent*]{} and they have exactly the same dependency rules. In this special case, speciousness of $Q\rightarrow C{=}c$ by $X\rightarrow C{=}c$ is of type a or c in Fig \[spectypes\]. On the other hand, if $X$ and $\neg Q$ are equivalent, then $X$ and $Q$ have complement rules of the form $X\rightarrow C{=}c$ and $Q\rightarrow C{\neq}c$. This corresponds to cases b and d in Fig \[spectypes\].

For a pair of equivalent rules, all conditional leverages (of $Q\rightarrow C{=}c$ given $X\rightarrow C{=}c$ or $X\rightarrow C{\neq}c$ and vice versa) are zero. This matches the definition of speciousness, even if one cannot consider them specious per se. Still, equivalent rules capture the same information and there is no need to report both, once the equivalence rule $X\rightarrow Q$ is reported. This type of speciousness has potentially important computational implications, because it enables radical pruning of the search space.

Equivalent rules are also related to the concepts of [*closed item sets*]{} and [*minimal generators*]{} [@pasquier99]. An attribute set $X$ is called closed, if for all 