behind a choice of quality measure should come from the hidden user's preference model. Given a Hasse diagram induced from the user's preference model, a k-plan set that will be presented to the user can be considered to be randomly selected from the diagram. The probability of having one plan in the set classified in a class at the optimal layer would increase when the individual plans are more likely to be at different layers, and this chance in turn will increase if they are less likely to be equally prefered by the user.5 On the other hand, the effect of base level elements of a plan on high level features relevant to the user suggests that plans similar with respect to base level features are more likely to be close to each other on the *high level feature space determining* user's preference model.

In order to define a quality measure using base level features of plans, we proceed with the following assumption: plans that are different from each other with respect to the base level features are less likely to be equally prefered by the user, in other words they are more likely to be at different layers of the Hasse diagram. With the purpose of increasing the chance of having a plan that the user prefers, we propose the quality measure of plan sets as its *diversity* measure, defined using the distance between two plans in the set with respect to a base level element. More formally, the quality measure ζ : 2S → R of a plan set P can be defined as either the minimal, maximal, or average distance between plans:
- Minimal distance:

$$\zeta_{m i n}({\mathcal{P}})=\operatorname*{min}_{p,p^{\prime}\in{\mathcal{P}}}\delta(p,p^{\prime})$$
$$(1)$$
δ(**p, p**′) (1)
 $\bullet$  Maximal distance: . 
$$\zeta_{m a x}({\mathcal{P}})=\operatorname*{max}_{p,p^{\prime}\in{\mathcal{P}}}\delta(p,p^{\prime})$$
$$(2)$$
$$(3)$$
δ(**p, p**′) (2)
- Average distance:

$$\zeta_{a v g}({\mathcal{P}})={\binom{|{\mathcal{P}}|}{2}}^{-1}\times\sum_{p,p^{\prime}\in{\mathcal{P}}}\delta(p,p^{\prime})$$

where δ : **S × S →** [0, 1] is the distance measures between two plans.

3.1.1. Distance measures between plans There are various choices on how to define the distance measure δ(**p, p**′) between two plans using plan actions, sequence of states or causal links, and each way can have different impact on the diversity of plan set on the Hasse diagram. In the following, we propose distance measures in which a plan is considered as (i) a set of actions and causal links, or (ii) sequence of states the agent goes through, which could be used independently of plan representation (e.g. total order, partial order plans).