A Knowledge Compilation Map for Conditional Preference Statements-based Languages

Abstract

Conditional preference statements have been used to compactly represent preferences over combinatorial domains. They are at the core of CP-nets and their generalizations, and lexicographic preference trees. Several works have addressed the complexity of some queries (optimization, dominance in particular). We extend in this paper some of these results, and study other queries which have not been addressed so far, like equivalence, thereby contributing to a knowledge compilation map for languages based on conditional preference statements. We also introduce a new parameterised family of languages, which enables to balance expressiveness against the complexity of some queries. formula = ∧ = ′ is a contradiction; also, the interpretations are thus in one-to-one correspondence with X. If is such a propositional formula over X and ∈ X, we will write |= when satisfies , that is when, assigning to every literal = that appears in the value true if [ ] = , and the value false otherwise, makes true.Given a formula , or a partial instantiation , Var( ) and Var( ) denote the set of attributes, the values of which appear in and respectively.When it is not ambiguous, we will use as a shorthand for the literal = ; also, for a conjunction of such literals, we will omit the ∧ symbol, thus = ∧ = ¯ for instance will be denoted ¯ . Preference RelationsDepending on the knowledge that we have about a decision maker's preferences, given any pair of distinct alternatives , ′ ∈ X, one of the following situations must hold: one may be strictly preferred over the other, or and ′ may be equally preferred, or and ′ may be incomparable.Assuming that preferences are transitive, such a state of knowledge about the DM's preferences can be characterised by a preorder over X: is a binary, reflexive and transitive relation; for alternatives , ′ , we then write ′ when ( , ′ ) ∈ ; ≻ ′ when ( , ′ ) ∈ and ( ′ , ) ∉ ; ∼ ′ when ( , ′ ) ∈ and ( ′ , ) ∈ ; ⊲⊳ ′ when ( , ′ ) ∉ and ( ′ , ) ∉ . Note that for any pair of alternatives , ′ ∈ X either ≻ ′ , or ′ ≻ , or ∼ ′ or ⊲⊳ ′ .The relation ∼ defined in this way is the symmetric part of , it is reflexive and transitive, ⊲⊳ is irreflexive, they are both symmetric. The relation ≻ is the irreflexive part of , it is what is usually called a strict partial order: it is irreflexive and transitive.Terminology and notations. We say that alternative dominates alternative ′ (w.r.t. ) if and only if ′ ; if ≻ ′ , then we say that strictly dominates ′ . We use standard notations for the complements of ≻ and : we write ′ when it is not the case that ′ , and ⊁ ′ when it is not the case that ≻ ′ . LANGUAGES 3.1 Conditional Preference StatementsA conditional preference statement (aka., CP statement) over X is an expression of the form | : ≥ ′ , where is a propositional formula over ⊆ X, , ′ ∈ are such that [ ] ≠ ′ [ ] for every ∈ , and , , are disjoint subsets of X, not necessarily forming a partition of X. Informally, such a statement represents the piece of knowledge that, when comparing alternatives , ′ that both satisfy , ...