

![0_image_0.png](0_image_0.png)

![0_image_1.png](0_image_1.png)

or incomparable with ones at high-value layers.3 Figure 2 show examples of Hasse diagrams representing a total and partial preference ordering between plans.

When the preference model is explicitly specified, answering queries such as comparing two plans, finding a most preferred (optimal) plan becomes an easy task. This is possible, however, only if the set of plans is small and known upfront. Many preference *languages*, therefore, have been proposed to represent the relation  in a more compact way, and serve as starting points for algorithms to answer queries. Most preference languages fall into the following two categories:

- Quantitative languages define a *value function* V : S → R which assigns a real number to each plan, with a precise interpretation that p  p
′ ⇐⇒ V (p) ≤ V (p
′). Although this function is defined differently in many languages, at a high level it combines the user's preferences on various aspects of plan that can be measured quantitatively. For instance, in the context of decision-theoretic planning (Boutilier et al., 1999), the value function of a policy is defined as the expected rewards of states that are visited when the policy executes. In partial satisfaction (over-subcription) planning (PSP) (Smith, 2004; Van Den Briel et al., 2004), the quality of plans is defined as its total rewards of soft goals achieved minus its total action costs. In PDDL2.1 (Fox and Long, 2003), the value function is an arithmetic function of numerical fluents such as plan makespans, fuel used etc.,
and in PDDL3 (Gerevini et al., 2009) it is enhanced with individual preference specification defined as formulae over state trajectory using linear temporal logic (LTL) (Pnueli, 1977).

- Qualitative languages provide qualitative statements that are more intuitive for lay users to specify. A commonly used language of this type is *CP-networks* (Boutilier et al., 2004), where the user can specify her preference statements on values of plan attributes, possibly given specification of others (for instance, "Among tickets with the same prices, I prefer airline A to airline B."). Another example is LPP (Bienvenu et al., 2006) in which the