A Knowledge Compilation Map for Conditional Preference Statements-based Languages
aa r X i v : . [ c s . A I] F e b A Knowledge Compilation Map for Conditional PreferenceStatements-based Languages
Research Report IRIT/RR–2021–02–FRInstitut de Recherche en Informatique de Toulouse
Hélène Fargier
IRIT-CNRS, Université de ToulouseToulouse, [email protected]
Jérôme Mengin
IRIT-CNRS, Université de ToulouseToulouse, [email protected]
ABSTRACT
Conditional preference statements have been used to compactlyrepresent preferences over combinatorial domains. They are at thecore of CP-nets and their generalizations, and lexicographic pref-erence trees. Several works have addressed the complexity of somequeries (optimization, dominance in particular). We extend in thispaper some of these results, and study other queries which havenot been addressed so far, like equivalence, thereby contributingto a knowledge compilation map for languages based on condi-tional preference statements. We also introduce a new parame-terised family of languages, which enables to balance expressive-ness against the complexity of some queries.
Preference handling is a key component in several areas of Artifi-cial Intelligence, notably for decision-aid systems. Research in Arti-ficial Intelligence has led to the development of several languagesthat enable compact representation of preferences over complex,combinatorial domains. Some preference models rank alternativesaccording to their values given by some multivariate function; thisis the case for instance with valued constraints [30], additive utili-ties and their generalizations [9, 25]. Ordinal models like CP netsand their generalisations [4, 8, 33], or lexicographic preferencesand their generalisations [3, 10, 18, 22, 31, 34] use sets of condi-tional preference statements to represent a pre-order over the setof alternatives.Many problems of interest, like comparing alternatives or find-ing optimal alternatives, are NP-hard for many of these models,even PSPACE hard for some models, which makes these repre-sentations difficult to use in some decision-aid systems like con-figurators, where real-time interaction with a decision maker isneeded. One approach to tackle this problem is Knowledge Compi-lation, whereby a model, or a part of it, is compiled , off-line, into an-other representation which enables fast query answering, even ifthe compiled representation has a much bigger size. This approachhas first been studied in propositional logic: [13, 14] compare howvarious subsets of propositional logic can succinctly, or not, ex-press some propositional knowledge bases, and the complexity ofqueries of interest. [12] follow a similar approach to compare exten-sions of propositional logic which associate real values to modelsof a knowledge base; [19] provide such a map for value function-based models. The aim of this paper is to initiate such a compilation map formodels of preferences based on the language of conditional pref-erence statements. We compare the expressiveness and succinct-ness of various languages on these conditional preference state-ments, and the complexity of several queries of interest for theselanguages.The next section recalls some basic definitions about combinato-rial domains and pre-orders, and introduces notations that will beused throughout. Section 3 gives an overview of various languagesbased on conditional preference statements that have been studiedin the literature. We also introduce a new parameterised familyof languages, which enables to balance expressiveness against thecomplexity of some queries. Section 4 and 5 respectively study ex-pressiveness and succinctness for languages based on conditionalpreference statements. Sections 6 study the complexity of queriesfor these languages. Proofs can be found in [20].
We consider languages that can be used to represent the prefer-ences of a decision maker over a combinatorial space X : here X isa set of attributes that characterise the possible alternatives, eachattribute 𝑋 ∈ X having a finite set of possible values 𝑋 ; we assumethat | 𝑋 | ≥ 𝑋 ∈ X ; then X denotes the cartesian prod-uct of the domains of the attributes in X , its elements are calledalternatives. For a binary attribute 𝑋 , we will often denote by 𝑥, ¯ 𝑥 its two possible values. In the sequel, 𝑛 is the number of attributesin X .For a subset 𝑈 of X , we will denote by 𝑈 the cartesian productof the domains of the attributes in 𝑈 , called instantiations of 𝑈 , orpartial instantiations (of X ). If 𝑣 is an instantiation of some 𝑉 ⊆ X , 𝑣 [ 𝑈 ] denotes the restriction of 𝑣 to the attributes in 𝑉 ∩ 𝑈 ; wesay that instantiation 𝑢 ∈ 𝑈 and 𝑣 are compatible if 𝑣 [ 𝑈 ∩ 𝑉 ] = 𝑢 [ 𝑈 ∩ 𝑉 ] ; if 𝑈 ⊆ 𝑉 and 𝑣 [ 𝑈 ] = 𝑢 , we say that 𝑣 extends 𝑢 .Sets of partial instantiations can often be conveniently, and com-pactly, specified with propositional formulas: the atoms are 𝑋 = 𝑥 for every 𝑋 ∈ X and 𝑥 ∈ 𝑋 , and we use the standard connectives ∧ (conjunction), ∨ (disjunction), → (implication), ↔ (equivalence)and ¬ (negation); we denote by ⊤ (resp. ⊥ ) the formula always true(resp. false). Implicitly, this propositional logic is equipped with atheory that enforces that every attribute has precisely one valuefrom its domain; so, for two distinct values 𝑥, 𝑥 ′ of attribute 𝑋 , theormula 𝑋 = 𝑥 ∧ 𝑋 = 𝑥 ′ is a contradiction; also, the interpreta-tions are thus in one-to-one correspondence with X . If 𝛼 is sucha propositional formula over X and 𝑜 ∈ X , we will write 𝑜 | = 𝛼 when 𝑜 satisfies 𝛼 , that is when, assigning to every literal 𝑋 = 𝑥 that appears in 𝛼 the value true if 𝑜 [ 𝑋 ] = 𝑥 , and the value false otherwise, makes 𝛼 true.Given a formula 𝛼 , or a partial instantiation 𝑢 , Var ( 𝛼 ) and Var ( 𝑢 ) denote the set of attributes, the values of which appear in 𝛼 and 𝑢 respectively.When it is not ambiguous, we will use 𝑥 as a shorthand for theliteral 𝑋 = 𝑥 ; also, for a conjunction of such literals, we will omitthe ∧ symbol, thus 𝑋 = 𝑥 ∧ 𝑌 = ¯ 𝑦 for instance will be denoted 𝑥 ¯ 𝑦 . Depending on the knowledge that we have about a decision maker’spreferences, given any pair of distinct alternatives 𝑜, 𝑜 ′ ∈ X , one ofthe following situations must hold: one may be strictly preferredover the other, or 𝑜 and 𝑜 ′ may be equally preferred, or 𝑜 and 𝑜 ′ may be incomparable.Assuming that preferences are transitive, such a state of knowl-edge about the DM’s preferences can be characterised by a pre-order (cid:23) over X : (cid:23) is a binary, reflexive and transitive relation; foralternatives 𝑜, 𝑜 ′ , we then write 𝑜 (cid:23) 𝑜 ′ when ( 𝑜, 𝑜 ′ ) ∈ (cid:23) ; 𝑜 ≻ 𝑜 ′ when ( 𝑜, 𝑜 ′ ) ∈ (cid:23) and ( 𝑜 ′ , 𝑜 ) ∉ (cid:23) ; 𝑜 ∼ 𝑜 ′ when ( 𝑜, 𝑜 ′ ) ∈ (cid:23) and ( 𝑜 ′ , 𝑜 ) ∈ (cid:23) ; 𝑜 ⊲⊳ 𝑜 ′ when ( 𝑜, 𝑜 ′ ) ∉ (cid:23) and ( 𝑜 ′ , 𝑜 ) ∉ (cid:23) . Note that forany pair of alternatives 𝑜, 𝑜 ′ ∈ X either 𝑜 ≻ 𝑜 ′ , or 𝑜 ′ ≻ 𝑜 , or 𝑜 ∼ 𝑜 ′ or 𝑜 ⊲⊳ 𝑜 ′ .The relation ∼ defined in this way is the symmetric part of (cid:23) , itis reflexive and transitive, ⊲⊳ is irreflexive, they are both symmetric.The relation ≻ is the irreflexive part of (cid:23) , it is what is usually calleda strict partial order: it is irreflexive and transitive. Terminology and notations.
We say that alternative 𝑜 dominates alternative 𝑜 ′ (w.r.t. (cid:23) ) if and only if 𝑜 (cid:23) 𝑜 ′ ; if 𝑜 ≻ 𝑜 ′ , then wesay that 𝑜 strictly dominates 𝑜 ′ . We use standard notations for thecomplements of ≻ and (cid:23) : we write 𝑜 (cid:15) 𝑜 ′ when it is not the casethat 𝑜 (cid:23) 𝑜 ′ , and 𝑜 ⊁ 𝑜 ′ when it is not the case that 𝑜 ≻ 𝑜 ′ . A conditional preference statement (aka., CP statement) over X isan expression of the form 𝛼 | 𝑉 : 𝑤 ≥ 𝑤 ′ , where 𝛼 is a propositionalformula over 𝑈 ⊆ X , 𝑤, 𝑤 ′ ∈ 𝑊 are such that 𝑤 [ 𝑋 ] ≠ 𝑤 ′ [ 𝑋 ] forevery 𝑋 ∈ 𝑊 , and 𝑈 ,𝑉 ,𝑊 are disjoint subsets of X , not necessarilyforming a partition of X . Informally, such a statement representsthe piece of knowledge that, when comparing alternatives 𝑜, 𝑜 ′ thatboth satisfy 𝛼 , the one that has values 𝑤 for 𝑊 is preferred to theone that has values 𝑤 ′ for 𝑊 , irrespective of the values of the at-tributes in 𝑉 , every attribute in X \ ( 𝑉 ∪ 𝑊 ) being fixed. We call 𝛼 the conditioning part of the statement; we call 𝑊 the swappedattributes, and 𝑉 the free part. Example . Consider plan-ning a holiday, with three choices / attributes: wait til next month( 𝑊 = 𝑤 ) or leave now ( 𝑊 = ¯ 𝑤 ), going to city 1, 2 or 3 ( 𝐶 = 𝑐 , 𝐶 = 𝑐 or 𝐶 = 𝑐 ), travelling by plane ( 𝑃 = 𝑝 ) or by car ( 𝑃 = ¯ 𝑝 ). I wouldrather go now, irrespective of the other attributes: ⊤ |{ 𝐶𝑃 } : ¯ 𝑤 ≥ 𝑤 . All else being equal, I prefer to go to city 3, city 1 being my sec-ond best choice: ⊤ | ∅ : 𝑐 ≥ 𝑐 ≥ 𝑐 . Also, if I go now, I prefer to fly:¯ 𝑤 | ∅ : 𝑝 ≥ ¯ 𝑝 . Together, the last two statements imply that if I gonow, I prefer to go to city 3 by plane than go to city 1 by car; how-ever these statements do not say what I prefer between flying tocity 1 or driving to city 3. In fact, I prefer the former, this tradeoff can be expressed with the statement ¯ 𝑤 | ∅ : 𝑐 𝑝 ≥ 𝑐 ¯ 𝑝 . Finally, if Igo later, I prefer to drive, irrespective of the city: 𝑤 |{ 𝐶 } : ¯ 𝑝 ≥ 𝑝 .Conditional preference statements have been studied in manyworks, under various language restrictions. They are the basis forCP-nets [4, 6] and their extensions, and have been studied in amore logic-based fashion by e.g. [24] and [32, 33, 35]. They areclosely related to
CI-statements by [7]For the semantics sets of CP statements, we use the definitionsof [35]. Given a statement 𝛼 | 𝑉 : 𝑤 ≥ 𝑤 ′ , let 𝑈 = Var ( 𝛼 ) and 𝑊 = Var ( 𝑤 ) = Var ( 𝑤 ′ ) : a worsening swap is any pair of alternatives ( 𝑜, 𝑜 ′ ) such that 𝑜 [ 𝑈 ] = 𝑜 ′ [ 𝑈 ] | = 𝛼 , 𝑜 [ 𝑊 ] = 𝑤 and 𝑜 ′ [ 𝑊 ] = 𝑤 ′ ,and such that for every attribute 𝑌 ∉ 𝑈 ∪ 𝑉 ∪ 𝑊 it holds that 𝑜 [ 𝑌 ] = 𝑜 ′ [ 𝑌 ] ; we say that 𝛼 | 𝑉 : 𝑤 ≥ 𝑤 ′ sanctions ( 𝑜, 𝑜 ′ ) . For a set of CP-statements 𝜑 , let 𝜑 ∗ be the set of all worsening swaps sanctionedby statements of 𝜑 , and define (cid:23) 𝜑 to be the reflexive and transitiveclosure of 𝜑 ∗ . [35] proves that 𝑜 (cid:23) 𝜑 𝑜 ′ if and only if 𝑜 = 𝑜 ′ or 𝜑 ∗ contains a finite sequence of worsening swaps ( 𝑜 𝑖 , 𝑜 𝑖 + ) ≤ 𝑖 ≤ 𝑘 − with 𝑜 = 𝑜 and 𝑜 𝑘 = 𝑜 ′ . Example . Let 𝜑 = {⊤ |{ 𝐶𝑃 } : ¯ 𝑤 ≥ 𝑤, ⊤ | ∅ : 𝑐 ≥ 𝑐 ≥ 𝑐 , 𝑛 | ∅ : 𝑝 ≥ ¯ 𝑝, ¯ 𝑤 | ∅ : 𝑐 𝑝 ≥ 𝑐 ¯ 𝑝, 𝑤 |{ 𝐶 } : ¯ 𝑝 ≥ 𝑝 } . Then ⊤ |{ 𝐶𝑃 } : ¯ 𝑤 ≥ 𝑤 sanctions for instance ( ¯ 𝑤𝑐 𝑝, 𝑤𝑐 ¯ 𝑝 ) , so ¯ 𝑤𝑐 𝑝 (cid:23) 𝜑 𝑤𝑐 ¯ 𝑝 . Also, ⊤ | ∅ : 𝑐 ≥ 𝑐 ≥ 𝑐 sanctions ( ¯ 𝑤𝑐 𝑝, ¯ 𝑤𝑐 𝑝 ) , ¯ 𝑤 | ∅ : 𝑝 ≥ ¯ 𝑝 sanctions ( ¯ 𝑤𝑐 𝑝, ¯ 𝑤𝑐 ¯ 𝑝 ) , so, by transitivity, ¯ 𝑤𝑐 𝑝 (cid:23) 𝜑 ¯ 𝑤𝑐 ¯ 𝑝 . It is notdifficult to check that ¯ 𝑤𝑐 𝑝 ⊲⊳ 𝜑 ¯ 𝑤𝑐 ¯ 𝑝 .Let us call CP the language where formulas are sets of state-ments of the general form 𝛼 | 𝑉 : 𝑤 ≥ 𝑤 ′ . This language is very ex-pressive: it is possible to represent any preorder “in extension”with preference statements of the form 𝑜 ≥ 𝑜 ′ – they all have 𝑊 = X as set of swapped attributes, 𝛼 = ⊤ as condition, and no freeattribute.This expressiveness has a cost: we will see that many queriesabout pre-orders represented by CP -statements are PSPACE -hardfor the language CP . Several restrictions / sub-languages have beenstudied in the literature, we review them below. Linearisability.
Although the original definition of CP-nets by[6] does not impose it, many works on CP-nets, especially follow-ing [4], consider that they are intended to represent a strict partialorder, that is, that (cid:23) 𝜑 should be antisymmetric; equivalently, thismeans that the irreflexive part ≻ 𝜑 of (cid:23) 𝜑 can be extended to a linearorder. We say that a set 𝜑 of CP-statements is linearisable in thiscase. The formula 𝑢 | 𝑉 : 𝑥 ≥ 𝑥 ′ is written 𝑢 : 𝑥 > 𝑥 ′ [ 𝑉 ] by [35]. Actually, [35] proves that ( 𝑜, 𝑜 ′ ) is in the transitive closure of 𝜑 ∗ if and only thereis such a worsening sequence from 𝑜 to 𝑜 ′ , but adding the reflexive closure to thistransitive closure does not change the result, since we can add any pair ( 𝑜, 𝑜 ) to, orremove it from, any sequence of worsening swaps without changing the validity ofthe sequence. Such sets of CP-statements are often called consistent in the standard terminology onCP-nets, but we prefer to depart from this definition which only makes sense whenone asserts that 𝜑 should indeed represent a strict partial order. otations. We write 𝛼 : 𝑤 ≥ 𝑤 ′ when 𝑉 is empty, and 𝑤 ≥ 𝑤 ′ when 𝑉 is empty and 𝛼 = ⊤ . Note that we reserve the symbol ≥ for conditional preference statements, whereas “curly” symbols ≻ , ⊁ , (cid:23) , (cid:15) are used to represent relations over the set of alternatives.In the remainder of this section, we present various sublanguagesof CP . Some are defined by imposing various simple syntactical re-strictions on the formulas, two are languages which have been wellstudied (CP-nets and lexicographic preference trees); we close thesection by introducing a new, parameterised class of sublanguagesof CP which have interesting properties, as will be shown in sub-sequent subsections. Some restrictions are on the syntactical form of statements allowed;they bear on the size of the set of free attributes, or on the size ofthe set of swapped attributes, or on the type of conditioning formu-las allowed. Given some language
L ⊆ CP , we define the followingrestrictions: L ⋫ = only formulas with empty free parts ( 𝑉 = ∅ ) for every state-ment; L∧ = only formulas where the condition 𝛼 of every statement isa conjunction of literals; k- L = only formulas where the set of swapped attributes containsno more than 𝑘 attributes ( | 𝑊 | ≤ 𝑘 ) for every statement; inparticular, we call elements of unary statements.In particular, ∧ is the language studied by [35], and ⋫ isthe language of generalized CP-nets as defined by [24]. Given 𝜑 ∈ CP over set of attributes X , we define 𝐷 𝜑 as the graphwith sets of vertices X , and such that there is an edge ( 𝑋, 𝑌 ) if thereis 𝛼 | 𝑉 : 𝑤 ≥ 𝑤 ′ ∈ 𝜑 such that 𝑋 ∈ Var ( 𝛼 ) and 𝑌 ∈ Var ( 𝑤 ) , or 𝑋 ∈ Var ( 𝑤 ) and 𝑌 ∈ 𝑉 . We call 𝐷 𝜑 the dependency graph of 𝜑 . Note that 𝐷 𝜑 can be computed in polynomial time. This definition, inspiredby [35, Def. 15], generalises that of [4], which is restricted to thecase where all CP statements are unary and have no free attributes,and that of [8], who study statements with free attributes. Manytractability results on sets of CP statements have been obtainedwhen 𝐷 𝜑 has good properties. Given some language L ⊆ CP , wedefine: L6 (cid:8) = the restriction of L to acyclic formulas, which are those 𝜑 such that 𝐷 𝜑 is acyclic; L6 (cid:8) poly = the restriction of L to formulas where the dependencygraph is a polytree.[35] also defines a weaker graphical restriction, called “context-uniform conditional acyclicity”, but it turns out that it does givesrise to the same complexities as another, weaker restriction called“conditional acyclicity” by [35], which we generalize in section 3.6. In the literature, the symbol ⊲ is sometimes used to represent an importance relationbetween attributes; and, as explained by [35], statement 𝛼 | 𝑉 : 𝑤 ≥ 𝑤 ′ is a way toexpress that attributes in Var ( 𝑤 ) are more important than those in 𝑉 (when 𝛼 istrue). This is full acyclicity in [35]. 𝑊 ¯ 𝑤 ≥ 𝑤𝐶𝑃𝑐 𝑝 ≥ 𝑐 𝑝 ≥ 𝑐 ¯ 𝑝 ≥ 𝑐 ¯ 𝑝 ≥ 𝑐 ¯ 𝑝𝑐 𝑝 ≥ 𝑐 𝑝 ≥ 𝑐 ¯ 𝑝 𝑃 ¯ 𝑝 ≥ 𝑝𝐶 𝑐 ≥ 𝑐 ≥ 𝑐 ¯ 𝑤 𝑤 Figure 1: An LP-tree equivalent to the set of CP-statementsof Example 2. CP -nets In their seminal work, [4] define a CP-net over a set of attributes X to be composed of two elements:(1) a directed graph over X , which should represent preferentialdependencies between attributes; (2) a set of conditional preference tables, one for every attribute 𝑋 : if 𝑈 is the set of parents of 𝑋 in the graph, the conditionalpreference table for 𝑋 contains exactly | 𝑈 | rules 𝑢 : ≥ , forevery 𝑢 ∈ 𝑈 , where the ≥ ’s are linear orders over 𝑋 .Therefore, as shown by [35], CP-nets can be seen as sets of unary CP statements in conjunctive form with no free attribute. Specifi-cally, given a CP-net N over X , define 𝜑 N to be the set of all CPstatements 𝑢 : 𝑥 ≥ 𝑥 ′ , for every attribute 𝑋 , every 𝑢 ∈ 𝑈 where 𝑈 is the set of parents of 𝑋 in the graph, every 𝑥, 𝑥 ′ ∈ 𝑋 such that 𝑥, 𝑥 ′ are consecutive values in the linear order ≥ specified by therule 𝑢 : ≥ of N . Then the dependency graph of 𝜑 N , as defined inSection 3.3, coincides with the graph of N . We call CPnet = the language that contains all 𝜑 N , for every CP-net N .Note that CPnet ⊆ ∧ ⋫ . For a given 𝜑 ∈ ∧ ⋫ , beinga CP-net necessitates a very strong form of local consistency andcompleteness: for every attribute 𝑋 with parents 𝑈 in 𝐷 𝜑 , for every 𝑢 ∈ 𝑈 , for every 𝑥, 𝑥 ′ ∈ 𝑋 , 𝜑 must explicitly, and uniquely, order 𝑢𝑥 and 𝑢𝑥 ′ .[8] define TCP-nets as an extension of CP-nets where it is pos-sible to represent tradeoffs, by stating that, under some conditions,some attributes are more important than other ones. [35] describeshow TCP-nets can be transformed, in polynomial time, into equiv-alent sets of ∧ statements. LP-trees generalise lexicographic orders, which have been widelystudied in decision making – see e.g. [21]. As an inference mech-anism, they are equivalent to search trees used by [5], and for-malised by [32, 35]. As a preference representation, and elicitation,language, slightly different definitions for LP-trees have been pro-posed by [3, 10, 18]. We use here a definition which subsumes theothers.An LP-tree that is equivalent to the set of CP-statements of Ex-ample 2 is depicted on Figure 1. More generally, an LP-tree over X is a rooted tree with labelled nodes and edges, and a set of pref-erence tables; specifically • every node 𝑁 is labelled with a set of attributes, denoted Var ( 𝑁 ) ; Given some pre-order (cid:23) over X , attribute 𝑋 is said to be preferentially dependenton attribute 𝑌 if there exist 𝑥, 𝑥 ′ ∈ 𝑋 , 𝑦, 𝑦 ′ ∈ 𝑌 , 𝑧 ∈ X \ ({ 𝑋,𝑌 }) such that 𝑥𝑦𝑧 (cid:23) 𝜑 𝑥 ′ 𝑦𝑧 but 𝑥𝑦 ′ 𝑧 (cid:15) 𝜑 𝑥 ′ 𝑦 ′ 𝑧 . if 𝑁 is not a leaf, it can have one child, or | Var ( 𝑁 ) | children; • in the latter case, the edges that connect 𝑁 to its childrenare labelled with the instantiations in Var ( 𝑁 ) ; • if 𝑁 has one child only, the edge that connects 𝑁 to its childis not labelled: all instantiations in Var ( 𝑁 ) lead to the samesubtree; • we denote by Anc ( 𝑁 ) the set of attributes that appear in thenodes between the root and 𝑁 (excluding those at 𝑁 ), and by Inst ( 𝑁 ) (resp. NonInst ( 𝑁 ) ) the set of attributes that appearin the nodes above 𝑁 that have more than one children (resp.only one child); • a conditional preference table CPT ( 𝑁 ) is associated with 𝑁 :it contains local preference rules of the form 𝛼 : ≥ , where ≥ is a preorder over Var ( 𝑁 ) , and 𝛼 is a propositional formulaover some attributes in NonInst ( 𝑁 ) .We assume that the rules in CPT ( 𝑁 ) define their preorder over Var ( 𝑁 ) in extension. Additionally, two constraints guarantee thatan LP-tree 𝜑 defines a unique preorder over X : • no attribute can appear at more than one node on any branchof 𝜑 ; and, • at every node 𝑁 of 𝜑 , for every 𝑢 ∈ NonInst ( 𝑁 ) , CPT ( 𝑁 ) must contain exactly one rule 𝛼 : ≥ such that 𝑢 | = 𝛼 .Given an LP-tree 𝜑 and an alternative 𝑜 ∈ X , there is a unique wayto traverse the tree, starting at the root, and along edges that areeither not labelled, or labelled with instantiations that agree with 𝑜 , until a leaf is reached. Now, given two distinct alternatives 𝑜, 𝑜 ′ ,it is possible to traverse the tree along the same edges as long as 𝑜 and 𝑜 ′ agree, until a node 𝑁 is reached which is labelled with some 𝑊 such that 𝑜 [ 𝑊 ] ≠ 𝑜 ′ [ 𝑊 ] : we say that 𝑁 decides { 𝑜, 𝑜 ′ } .In order to define (cid:23) 𝜑 for some LP-tree 𝜑 , let 𝜑 ∗ be the set ofall pairs of distinct alternatives ( 𝑜, 𝑜 ′ ) such that there is a node 𝑁 that decides { 𝑜, 𝑜 ′ } and the only rule 𝛼 : ≥ ∈ CPT ( 𝑁 ) with 𝑜 [ NonInst ( 𝑁 )] = 𝑜 ′ [ NonInst ( 𝑁 )] | = 𝛼 is such that 𝑜 [ 𝑊 ] ≥ 𝑜 ′ [ 𝑊 ] .Then (cid:23) 𝜑 is the reflexive closure of 𝜑 ∗ . Proposition 1.
Let 𝜑 be an LP-tree over X , then (cid:23) 𝜑 as definedabove is a preorder. Furthermore, (cid:23) 𝜑 is a linear order if and only if 1)every attribute appears on every branch and 2) every preference rulespecifies a linear order. An LP-tree 𝜑 is said to be complete if the two conditions in Propo-sition 1 hold, that is, if (cid:23) 𝜑 is a linear order.From a semantic point of view, an LP-tree 𝜑 is equivalent to theset that contains, for every node 𝑁 of 𝜑 labelled with 𝑊 = Var ( 𝑁 ) ,and every rule 𝛼 : ≥ 𝛼𝑁 in CPT ( 𝑁 ) , all CP statements of the form 𝛼 ∧ 𝑢 | 𝑉 : 𝑤 ♯ ≥ 𝑤 ′ ♯ , where • 𝑢 is the combination of values given to the attributes in Inst ( 𝑁 ) along the edges between the root and 𝑁 , and • 𝑤, 𝑤 ′ ∈ 𝑊 such that 𝑤 ≥ 𝛼𝑁 𝑤 ′ , and 𝑊 ♯ is the set of attributeson which 𝑤 and 𝑤 ′ have distinct values, and 𝑤 ♯ = 𝑤 [ 𝑊 ♯ ] ,and 𝑤 ′ ♯ = 𝑤 ′ [ 𝑊 ♯ ] ; and • 𝑉 = [X − ( Anc ( 𝑁 ) ∪ 𝑊 )] .This set of statements indicate that outcomes that agree on Anc ( 𝑁 ) and satisfy 𝑢 ∧ 𝛼 , but have different values for Var ( 𝑁 ) , should beordered according to ≥ 𝛼𝑁 , whatever their values for attributes in 𝑉 . LPT = the language of LP-trees as defined above; we consider that
LPT is a subset of CP . Note that, using the notations defined above, k-LPT = LPT ∩ 𝑘 -CP is the restriction of LPT where every node has at most 𝑘 at-tributes, for every 𝑘 ∈ N ; in particular, is the language ofLP-trees with one attribute at each node; and LPT ∧ = LPT ∩ CP ∧ is the restriction of LPT where the condition 𝛼 in every rule atevery node is a conjunction of literals. Search trees of [32, 35] andLP-trees as defined by [3, 27] are sublanguages of ∧ ; LP-treesof [18] and [10] are sublanguages of LPT ∧ . Many graphical restrictions that have been proposed in order to en-able polytime answers to some queries are in fact particular casesof a more general property which we study now. We define a new,parameterised family of languages. Given some language
L ⊆ CP and 𝑘 ∈ N , we define: L6 (cid:8) lex 𝑘 = the restriction of L to formulas 𝜑 such that there existssome complete LP-tree 𝜓 ∈ 𝑘 -LPT such that (cid:23) 𝜓 extends (cid:23) 𝜑 .We say that formulas of CP (cid:8) lex 𝑘 are 𝑘 -lexico-compatible . [35] proves that acyclic formulas of are 1-lexico-compatiblewhen they enjoy some local consistency property; it illustrates that 𝑘 -lexico-compatibility is indeed a weak form of acyclicity. We willsee that k-lexico-compatibility makes some queries tractable.The next result shows that proving that some 𝜑 ∈ CP is 𝑘 -lexico-compatible, for a fixed 𝑘 , is not always easy: Proposition 2.
For a fixed 𝑘 ∈ N , checking if a formula 𝜑 ∈ CP is 𝑘 -lexico-compatible is coNP -complete. Algorithm 1 checks if a given formula is k-lexico-compatible.Given 𝜑 ∈ CP , it builds, in a top-down fashion, a complete 𝜓 ∈ k-LPT that is compatible with 𝜑 . The algorithm is similar to the al-gorithm proposed by [3] to learn an LP-tree that sanctions a givenset of pairs ( 𝑜, 𝑜 ′ ) . It starts with an empty root node at step 1; then,while there is some empty node, it picks one of them, call it 𝑁 , andcalls at step 2b the function chooseAttribute to get a pair ( 𝑇 , ≥) to label 𝑁 , where 𝑇 is a set of at most 𝑘 attributes, none of whichappear above 𝑁 , and ≥ is a linear order over 𝑇 ; if no such pair iscompatible with 𝜑 , in a sense that will be defined shortly, chooseAt-tribute returns failure and the algorithm stops at step 2c; otherwise,if there remain some attributes that do not appear in 𝑇 nor at anynode above 𝑁 , then the algorithm expands the tree below 𝑁 atstep 2e by creating a branch and a new node for every instantia-tion 𝑡 ∈ 𝑇 , and loops.Note that all edges of the tree built by the algorithm are labelled,so that, at every node 𝑁 , NonInst ( 𝑁 ) = ∅ , so CPT ( 𝑁 ) must containonly one rule of the form ⊤ : ≥ , where ⊤ is the formula always true.This is why chooseAttribute needs to return one linear order over 𝑇 only, we do not need to specify the trivial condition ⊤ here. Theremay be a more compact 𝑘 -LP-tree compatible with 𝜑 than the onereturned by the above algorithm when it does not fail, but we areonly interested here in checking if 𝜑 is 𝑘 -lexico-compatible, and we Strictly speaking, for
LPT ⊆ CP to hold, we can add the possibility to augment everyformula in CP with a tree structure. This definition generalises conditionally acyclic formulas of [35], which are the for-mulas of CP (cid:8) lex . lgorithm 1: Build complete LP tree Input: 𝜑 ∈ CP ; 𝑘 ∈ N ;Output: 𝜓 ∈ k-LPT , 𝜓 complete, s.t. (cid:23) 𝜓 ⊇ (cid:23) 𝜑 , or FAILURE ;(1) 𝜓 ← { an unlabelled root node } ;(2) while 𝜓 contains some unlabelled node:(a) choose unlabelled node 𝑁 of 𝜓 ;(b) ( 𝑇 , ≥) ← chooseAttribute ( 𝑁 , 𝑘, 𝜑 ) ;(c) if 𝑇 = FAILURE then STOP and return
FAILURE ;(d) label 𝑁 with ( 𝑇 , ≥) ;(e) if Anc ( 𝑁 ) ∪ 𝑇 ≠ X , for each 𝑡 ∈ 𝑇 : add new unlabellednode to 𝜓 , attached to 𝑁 with edge labelled with 𝑡 ;(3) return 𝜓 .have seen that the problem is coNP -complete, so it seems difficultto avoid exploring a tree with size exponential in the size of 𝜑 in theworst case. We now specifiy some condition that chooseAttribute must verify in order for the algorithm to be correct and complete.Given any yet unlabelled node 𝑁 of the tree being build, let 𝜑 ( 𝑁 ) = { 𝛼 | 𝑉 : 𝑤 ≥ 𝑤 ′ ∈ 𝜑 | 𝛼 ∧ inst ( 𝑁 ) 6| = ⊥ ,𝑊 ∩ Anc ( 𝑁 ) = ∅} . Definition . We say that chooseAttribute is 𝜑 -compatible if thepair ( 𝑇 , ≥) that chooseAttribute returns at some yet unlabellednode 𝑁 is such that for every 𝛼 | 𝑉 : 𝑤 ≥ 𝑤 ′ ∈ 𝜑 ( 𝑁 ) : (1) if Var ( 𝑤 ) ∩ 𝑇 = ∅ , then 𝑉 ∩ 𝑇 = ∅ ; (2) if Var ( 𝑤 ) ∩ 𝑇 ≠ ∅ , then 𝑡 > 𝑁 𝑡 ′ for every 𝑡, 𝑡 ′ ∈ 𝑇 such that 𝑡 ∧ 𝑤 = ⊥ , 𝑡 ′ ∧ 𝑤 ′ = ⊥ , 𝑡 [X \ ( 𝑉 ∪ 𝑊 )] = 𝑡 ′ [X \ ( 𝑉 ∪ 𝑊 )] and 𝑡 ∧ 𝛼 = ⊥ . If no such pair ( 𝑇 , ≥) can be found,then chooseAttribute must return failure.Condition (2) guarantees that 𝑁 will correctly decide every pairof alternatives that is sanctionned by 𝛼 | 𝑉 : 𝑤 ≥ 𝑤 ′ and that will bedecided at 𝑁 . When the entire tree is built in this way, condition ?? guarantees that at every node 𝑁 , if 𝛼 | 𝑉 : 𝑤 ≥ 𝑤 ′ ∈ 𝜑 ( 𝑁 ) then 𝑉 ∩ Anc ( 𝑁 ) = ∅ . Proposition 3.
Given 𝜑 ∈ CP and some 𝑘 ∈ N , suppose that chooseAttribute is 𝜑 -compatible, then 𝜑 ∈ CP (cid:8) lex 𝑘 if and only ifthe algorithm above returns some 𝜓 ∈ k-LPT such that (cid:23) 𝜓 ⊇ (cid:23) 𝜑 ;otherwise, it returns FAILURE . Note that chooseAttribute can be implemented to run in polyno-mial time, for fixed 𝑘 : there are no more than Í 𝑘𝑖 = (cid:0) 𝑛𝑖 (cid:1) ≤ 𝑘𝑛 𝑘 possi-bilities for the 𝑇 it can return, and the number of pairs 𝑡, 𝑡 ′ that itmust check against every statement in 𝜑 ( 𝑁 ) is bounded by | 𝑇 | ,and | 𝑇 | is bounded by 𝑑 𝑘 , where 𝑑 is the size of the largest domainof the attributes in X . Also, each branch of the tree returned by thealgorithm, when it succeeds, can have at most 𝑛 nodes, but the treecan have up to 𝑑 𝑛 leaves. We detail our results about expressiveness of the various languagesstudied here in this section, the results about succinctness are inthe next section. These results are summarised on Figure 2.
Definition . Let L and L ′ be two languages for representing pre-orders. We say that L is at least as expressive as L ′ , written L ⊒ L ′ ,if every preorder that can be represented with a formula of L ′ can also be represented with a formula of L ; we write L = L ′ if L ⊒ L ′ but it is not the case that L ′ ⊒ L , and say in this case that L is strictly more expressive than L ′ . We write L ⊑⊒ L ′ when thetwo languages are equally expressive.We reserve the usual “rounded” symbols ⊂ and ⊆ for (strict) setinclusion, and ⊃ and ⊇ for the reverse inclusions. Note that ⊒ is apreorder, and obviously L ⊇ L ′ implies L ⊒ L ′ .Clearly, CP ⋫ ⊂ CP and CP ∧ ⊂ CP ; however, these three lan-guages have the same expressiveness, because of the following: Property 4.
Given some preorder (cid:23) , define 𝜑 ∈ { 𝑜 [ Δ ( 𝑜, 𝑜 ′ )] ≥ 𝑜 ′ [ Δ ( 𝑜, 𝑜 ′ )] | 𝑜 (cid:23) 𝑜 ′ , 𝑜 ≠ 𝑜 ′ } , where Δ ( 𝑜, 𝑜 ′ ) is the set of attributesthat have different values in 𝑜 and 𝑜 ′ , then 𝜑 ∈ CP ⋫ ∩ CP ∧ , and (cid:23) 𝜑 = (cid:23) . A large body of works on CP-statements since the seminal paperby [5] concentrate on various subsets of . With this strongrestriction on the number of swapped attributes, CP-statementshave a reduced expressiveness.
Example . Consider two binary attributes 𝐴 and 𝐵 , with respec-tive domains { 𝑎, ¯ 𝑎 } and { 𝑏, ¯ 𝑏 } . Define preorder (cid:23) such that 𝑎𝑏 ≻ ¯ 𝑎 ¯ 𝑏 ≻ 𝑎 ¯ 𝑏 ≻ ¯ 𝑎𝑏 . This can be represented in CP with 𝜑 = { 𝑎𝑏 ≥ ¯ 𝑎 ¯ 𝑏, ¯ 𝑎 ¯ 𝑏 ≥ 𝑎 ¯ 𝑏, 𝑎 ¯ 𝑏 ≥ ¯ 𝑎𝑏 } . But it cannot be represented in : { 𝑏 : 𝑎 ≥ ¯ 𝑎, ¯ 𝑏 : ¯ 𝑎 ≥ 𝑎, 𝑎 : 𝑏 ≥ ¯ 𝑏, ¯ 𝑎 : ¯ 𝑏 ≥ 𝑏 } ∗ ⊆ 𝜑 ∗ , but this is not sufficient to com-pare 𝑎 ¯ 𝑏 with ¯ 𝑎𝑏 . The four remaining formulas of over thesetwo attributes are 𝐵 : 𝑎 ≥ ¯ 𝑎 , 𝐵 : ¯ 𝑎 ≥ 𝑎 , 𝐴 : 𝑏 ≥ ¯ 𝑏 , 𝐴 : ¯ 𝑏 ≥ 𝑏 , adding anyof them to 𝜑 yields a preorder which would not be antisymmetric.Forbidding free parts incurs an additional loss in expressiveness: Example . Consider two binary attributes 𝐴 and 𝐵 , with respec-tive domains { 𝑎, ¯ 𝑎 } and { 𝑏, ¯ 𝑏 } . Define preorder (cid:23) such that 𝑎𝑏 ≻ 𝑎 ¯ 𝑏 ≻ ¯ 𝑎𝑏 ≻ ¯ 𝑎 ¯ 𝑏 . This can be represented in with 𝜑 = { 𝐵 : 𝑎 ≥ ¯ 𝑎, 𝑏 ≥ ¯ 𝑏 } .But the “tradeoff” 𝑎 ¯ 𝑏 ≻ ¯ 𝑎𝑏 cannot be represented in ⋫ , anyformula of ⋫ that implies it will put some intermediate alter-native between 𝑎 ¯ 𝑏 and ¯ 𝑎𝑏 However, restricting to conjunctive statements does not incur aloss in expressiveness.
Proposition 5. CP = Ð 𝑘 ∈ N k-CP and, for every 𝑘 ∈ N : CP ∧ ⊑⊒ CP ⋫ ⊑⊒ CP = k-CP ⊑⊒ k-CP ∧ = k-CP ⋫ ⊑⊒ k-CP ∧ ⋫ k-CP = (k-1)-CP . Because an LP-tree can be a single node labelled with X , anda single preference rule ⊤ : ≥ where ≥ can be any preorder, LPT can represent any preorder. Limiting to conjunctive conditions inthe rules is not restrictive. However, restricting to reducesexpressiveness, even if one considers formulas of that repre-sent total, linear orders:
Example . Let 𝜑 = { 𝑎 ≥ ¯ 𝑎, ¯ 𝑐 | 𝐴 : ¯ 𝑏 ≥ 𝑏, ¯ 𝑎𝑐 : ¯ 𝑏 ≥ 𝑏, 𝑎𝑐 : 𝑏 ≥ ¯ 𝑏, 𝑎 : 𝑐 ≥ ¯ 𝑐, ¯ 𝑎 | 𝐵 : ¯ 𝑐 ≥ 𝑐 } . This yields the following linear order: 𝑎𝑏𝑐 (cid:23) 𝜑 𝑎 ¯ 𝑏𝑐 (cid:23) 𝜑 𝑎 ¯ 𝑏 ¯ 𝑐 (cid:23) 𝜑 ¯ 𝑎 ¯ 𝑏 ¯ 𝑐 (cid:23) 𝜑 𝑎𝑏 ¯ 𝑐 (cid:23) 𝜑 ¯ 𝑎𝑏 ¯ 𝑐 (cid:23) 𝜑 ¯ 𝑎 ¯ 𝑏𝑐 (cid:23) 𝜑 ¯ 𝑎𝑏𝑐. No 𝜓 ∈ can represent it: 𝐴 could not be at the root of such a tree becausefor instance 𝑎 ¯ 𝑏 ¯ 𝑐 (cid:23) 𝜑 ¯ 𝑎 ¯ 𝑏 ¯ 𝑐 and ¯ 𝑎 ¯ 𝑏 ¯ 𝑐 (cid:23) 𝜑 𝑎𝑏 ¯ 𝑐 ; neither could 𝐶 , since 𝑎 ¯ 𝑏𝑐 (cid:23) 𝜑 𝑎 ¯ 𝑏 ¯ 𝑐 and ¯ 𝑎𝑏 ¯ 𝑐 (cid:23) 𝜑 ¯ 𝑎 ¯ 𝑏𝑐 ; and finally 𝐵 could not be at the rooteither, because 𝑎𝑏𝑐 (cid:23) 𝜑 𝑎 ¯ 𝑏𝑐 and ¯ 𝑎 ¯ 𝑏 ¯ 𝑐 (cid:23) 𝜑 𝑎𝑏 ¯ 𝑐 . Proposition 6.
LPT = Ð 𝑘 ∈ N k-LPT and, for every 𝑘 ∈ N : CP ⊑⊒ LPT ⊑⊒ LPT ∧ = k-LPT ⊑⊒ k-LPT ∧ = (k-1)-LPT . inally, note that k-lexico-compatibility is a weaker restrictionthan being a 𝑘 -LP-tree. Proposition 7.
For every 𝑘 ∈ N : CP (cid:8) lex 𝑘 = CP (cid:8) lex 𝑘 − , and CP (cid:8) lex 𝑘 = k-LPT . [35] proves that (cid:8) ⊆ CP (cid:8) lex . Whether this property canbe generalised, with an appropriate definition of 𝑘 -acyclicity , is leftfor future work. Another criterion is the relative sizes of formulas that can repre-sent the same preorder in different languages. [11] study the spaceefficiency of various propositional knowledge representation for-malisms. An often used definition of succinctness [14, 23] makes ita particular case of expressiveness, which is not a problem whencomparing languages of same expressiveness. However, we studyhere languages with very different expressiveness, so we need amore fine grained definition:
Definition . Let L and L ′ be two languages for representing pre-orders. We say that L is at least as succinct as L ′ , written L ≦ L ′ ,if there exists a polynomial 𝑝 such that for every 𝜑 ′ ∈ L ′ , thereexists 𝜑 ∈ L that represent the same preorder as 𝜑 ′ and such that | 𝜑 | < 𝑝 (| 𝜑 ′ |) . Moreover, we say that L is strictly more succinctthan L ′ , written L ≪ L ′ , if L ≦ L ′ and for every polynomial 𝑝 ,there exists 𝜑 ∈ L such that: • there exists 𝜑 ′ ∈ L ′ such that (cid:23) 𝜑 = (cid:23) 𝜑 ′ , but • for every 𝜑 ′ ∈ L ′ such that (cid:23) 𝜑 = (cid:23) 𝜑 ′ , | 𝜑 ′ | > 𝑝 (| 𝜑 |) .With this definition, L≪L ′ if every formula of L ′ has an equiv-alent formula in L which is “no bigger” (up to some polynomialtransformation of the size of 𝜑 ), and there is at least one sequenceof formulas (one formula for every polynomial 𝑝 ) in L that haveequivalent formulas in L ′ but necessarily “exponentially bigger”. Proposition 8.
The following hold, for languages L , L ′ , L ′′ : • if L ⊇ L ′ then L ≦ L ′ ; and if L ≦ L ′ , then L ⊒ L ′ ; • if L ≪ L ′ then L ≦ L ′ and L ′ ≦ L ; • if L ⊑⊒ L ′ , the reverse implication holds:if L ≦ L ′ and L ′ ≦ L then L ≪ L ′ (otherwise, it might be that L ′ ≦ L because L ′
6⊒ L ); • if L ⊇ L ′ and L ′ ≪ L ′′ , then L ≪ L ′′ . Restricting the conditioning part of the statements to be con-junctions of literals does induce a loss in succinctness.
Example . Consider 2 𝑛 + 𝑋 , 𝑋 , . . . , 𝑋 𝑛 , 𝑌 , 𝑌 ,. . . , 𝑌 𝑛 , 𝑍 , and let 𝜑 contain 2 𝑛 + ( 𝑥 ∨ 𝑦 ) ∧ ( 𝑥 ∨ 𝑦 ) ∧ . . . ∧ ( 𝑥 𝑛 ∨ 𝑦 𝑛 ) : 𝑧 ≥ ¯ 𝑧 , ¬[( 𝑥 ∨ 𝑦 ) ∧( 𝑥 ∨ 𝑦 ) ∧ . . . ∧ ( 𝑥 𝑛 ∨ 𝑦 𝑛 )] : ¯ 𝑧 ≥ 𝑧 and ¯ 𝑥 𝑖 ≥ 𝑥 𝑖 and ¯ 𝑦 𝑖 ≥ 𝑦 𝑖 for every 𝑖 ∈ { , . . . , 𝑛 } . Then 𝜑 ∈ ⋫ , but 𝜑 is not in conjunctive form. Aset of conjunctive CP-statements equivalent to 𝜑 has to contain all2 𝑛 statements of the form 𝜇 𝜇 . . . 𝜇 𝑛 : 𝑧 ≥ ¯ 𝑧 with 𝜇 𝑖 = 𝑥 𝑖 or 𝜇 𝑖 = 𝑦 𝑖 for every 𝑖 . Where | 𝜑 | = Í 𝛼 | 𝑉 : 𝑤 ≥ 𝑤 ′∈ 𝜑 ( | 𝛼 | + | 𝑉 | + | Var ( 𝑤 ) |) , with | 𝛼 | = the numberof connectives plus the number of atoms of 𝛼 . When ≪ is defined as the strict counterpart of ≦ , it can happen that L≪L ′ evenif there is no real difference in representation size in the two languages, but L = L ′ . Also, free attributes enable succinct representation of relativeimportance of some attributes over others; disabling free attributesthus incurs a loss in succinctness:
Example . Consider 𝑛 + 𝑋 , 𝑋 , . . . , 𝑋 𝑛 , 𝑌 , let 𝑈 = { 𝑋 , 𝑋 , . . . , 𝑋 𝑛 } , and let 𝜑 = { 𝑈 | 𝑦 ≥ ¯ 𝑦 } . Then 𝜑 ∗ = {( 𝑢𝑦,𝑢 ′ ¯ 𝑦 ) | 𝑢, 𝑢 ′ ∈ 𝑈 } , and 𝜑 ∗ is equal to its transitive closure, so, if 𝑜 ≠ 𝑜 ′ , then 𝑜 (cid:23) 𝜑 𝑜 ′ if and only if 𝑜 [ 𝑌 ] = 𝑦 and 𝑜 ′ [ 𝑌 ] = ¯ 𝑦 . This can be rep-resented, without free attribute, with formula 𝜓 that contains, forevery 𝑉 ⊆ 𝑈 and every 𝑣 ∈ 𝑉 , the statement 𝑣𝑦 ≥ ¯ 𝑣 ¯ 𝑦 , where ¯ 𝑣 denotes the tuple obtained by inverting all values of 𝑣 . For every0 ≤ 𝑖 ≤ 𝑛 there are (cid:0) 𝑛𝑖 (cid:1) subsets of 𝑉 of size 𝑖 , with 2 𝑖 ways to choose 𝑣 ∈ 𝑉 , thus 𝜓 contains Í 𝑛 (cid:0) 𝑛𝑖 (cid:1) 𝑖 = 𝑛 statements.Restricting to CP-nets induces a further loss in succinctness, asthe next example shows: Example . Consider 𝑛 + 𝑋 , 𝑋 , . . . , 𝑋 𝑛 , 𝑌 , andlet 𝜑 be the ⋫ ∧ formula that contains the following state-ments: 𝑥 𝑖 ≥ ¯ 𝑥 𝑖 for 𝑖 = , . . . , 𝑛 ; 𝑥 𝑥 . . . 𝑥 𝑛 : 𝑦 ≥ ¯ 𝑦 ; ¯ 𝑥 𝑖 : ¯ 𝑦 ≥ 𝑦 for 𝑖 = , . . . , 𝑛 . The size of 𝜑 is linear in 𝑛 . Because preferences for 𝑌 de-pend on all 𝑋 𝑖 ’s, a CP-net equivalent to 𝜑 will contain, in the tablefor 𝑌 , 2 𝑛 CP statements.
Proposition 9.
The following hold: • L ≪ L∧ for every L such that ⋫ ⊆ L ⊆ CP ; • L ≪ L ⋫ for every L such that ∧ ⊆ L ⊆ CP ; • ⋫ ∧ ≪ CPnet . Table 1 gives an overview of the tractability of the queries that westudy in this section. We begin this section with the two queriesthat have generated most interest in the literature on CP state-ments.
Linearisability.
Knowing that a given 𝜑 ∈ CP is linearisable (thatis, that (cid:23) 𝜑 is antisymmetric) is valuable, as it makes several otherqueries easier. It also gives some interesting insights into the se-mantics of 𝜑 . The following query has been addressed in manyworks on CP statements: linearisability Given 𝜑 , is 𝜑 linearisable?[4] prove that when its dependency graph 𝐷 𝜑 is acyclic, then aCP-net 𝜑 is linearisable. This result has been extended by [8, 15, 35],who give weaker, sufficient syntactical conditions that guaranteethat a locally consistent set of unary, conjunctive CP statements islinearisable; more generally, by definition of 𝑘 -lexico-compatibility,every formula of CP (cid:8) lex 𝑘 is linearisable (since it is compatible witha complete LP-tree). [24, Theorem 3 and 4] prove that linearis-ability is PSPACE -complete for ⋫ ∧ . Proposition 10. linearisability can be checked in polynomialtime for
LPT . Comparing alternatives.
A basic question, given a formula 𝜑 and two alternatives 𝑜, 𝑜 ′ is: how do 𝑜 and 𝑜 ′ compare, accordingto 𝜑 ? Is it the case that 𝑜 ≻ 𝜑 𝑜 ′ , or 𝑜 ′ ≻ 𝜑 𝑜 , or 𝑜 ⊲⊳ 𝜑 𝑜 ′ , or 𝑜 ∼ 𝜑 𝑜 ′ ?We define the following query, for any relation 𝑅 ∈ {≻ , (cid:23) , ∼ , ⊲⊳ } : This query is often called consistency . PCP ∧ CP ⋫ CP ∧ ⋫ ≪ ≪≪ ≪ k-CP k-CP ∧≪ k-LPT k-LPT ∧≪ k-CP ⋫ k-CP ∧ ⋫ ≪ ≪ ≪ CP (cid:8) lex 𝑘 (k-1)-CP (k-1)-CP ∧≪ (k-1)-CP ⋫ (k-1)-CP ∧ ⋫ ≪ ≪ ≪ (k-1)-LPT (k-1)-LPT ∧≪ CP (cid:8) lex 𝑘 − CPnet ≪ CPnet (cid:8) L L ′′ : L is strictly more expressive than L ′ L ≪ L ′′ : L is strictly more succinct than L ′ For 𝑘 >
2. Boxes contain languages that areequally expressive.
Figure 2: Rel. expressiveness and succinctness C P - C P ⋫ - C P ⋫ ∧ C P n e t C P (cid:8) l e x 𝑘 C P n e t (cid:8) C P n e t (cid:8) p o l y L P T L T P w . c a n . t a b l e s linearisability ✘✘ ✘✘ ✘✘ ⊤ ⊤ ⊤ ✓ ✓ R-comparison , 𝑅 ∈ {(cid:23) , ≻ , ⊲⊳ } ✘✘ ✘✘ ✘✘ ✘ ◦ ✘ ◦ ✘ ✓ ✓ ✓ ∼ -comparison ✘✘ ✘✘ ✘✘ ⊥ ⊥ ⊥ ✓ ✓ eqivalence ✘✘ ✘ ◦ ✘ ◦ ✓ ✘ ◦ ✓ ✓ ✘ ◦ ✓ top- 𝑝 ✓ ✓ ✓ ✓ ✓ w. undominated ∃ ⊤ ⊤ ⊤ ⊤ ⊤ ⊤ ⊤ ⊤ ⊤ undominated ∃ ✘✘ ✘✘ ✘✘ ⊤ ⊤ ⊤ ⊤ ⊤ s. dom. ∃ , dom. ∃ ✘✘ ✘✘ ⊤ ⊤ ✓ ✓ undom. check , (cid:23) -cut extract. ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ s. dom., dom., w. undom. check ✘✘ ✘✘ ✘✘ ✓ ✓ ✓ ✓ ≻ -cut extraction ✘✘ ✘✘ ✘✘ ✓ ✓ ✓ ✓ ✓ ≻ -cut counting ✘✘ ✘✘ ✘✘ ✓ ✓ Each column corresponds to one sub-language of CP . They are sorted in order ofdecreasing expressiveness from left to right, except when columns are separatedby double lines. For each query and sub-language: ⊤ = always true for thelanguage ; ⊥ = always false for the language; ✓ = polytime answer; ✘ = NP -complete query; ✘ ◦ = NP / coNP -hard query; ✘✘ = PSPACE -complete query.
Table 1: Complexity of queries. 𝑅 -comparison Given formula 𝜑 , alternatives 𝑜 ≠ 𝑜 ′ , is it the casethat 𝑜𝑅 𝜑 𝑜 ′ ?For LP-trees, in order to compare alternatives 𝑜 and 𝑜 ′ , one onlyhas to traverse the tree from the root downwards until a node thatdecides the pair is reached, or up to a leaf if no such node is encoun-tered: in this case 𝑜 and 𝑜 ′ are incomparable. Note that checking ifa node decides the pair, and checking if a rule at that nodes appliesto order them, can both be done in polynomial time. Proposition 11. 𝑅 -comparison is in P for LPT for 𝑅 ∈ {≻ , (cid:23) , ∼ , ⊲⊳ } . Tractability of comparisons, except in some trivial cases, comesat a heavy price in terms of expressiveness: (cid:23) -comparison is tractablefor CP-nets when the dependency graph is a polytree [4, Theo-rem 14], but [4, Theorems 15, 16] prove that (cid:23) -comparison is al-ready NP -hard for the quite restrictive language of binary-valued,directed-path singly connected CP-nets, which are acyclic. [24, Prop.7, Corrolary 1] prove that (cid:23) -comparison , ≻ -comparison , ⊲⊳ -comparison and ∼ -comparison are PSPACE complete for ⋫ ∧ and for lin-earisable, locally complete formulas of ⋫ . More precise hard-ness results for acyclic CP-nets are also proved by [28]. Proposi-tion 12 completes the picture. Proposition 12. ≻ -comparison and ⊲⊳ -comparison are NP -hardfor the language of fully acyclic CP-nets, and tractable for polytreeCP-nets. ∼ -comparison is easy for (cid:8) lex . Comparing theories.
Checking if two theories yield the samepreorder can be useful during the compilation process. We say thattwo formulas 𝜑 and 𝜑 ′ are equivalent if they represent the samepreorder, that is, if (cid:23) 𝜑 and (cid:23) 𝜑 ′ are identical; we then write 𝜑 ≡ 𝜑 ′ . eqivalence Given two formulas 𝜑 and 𝜑 ′ , are they equivalent?Consider a formula 𝜑 ∈ CP , two alternatives 𝑜, 𝑜 ′ , and let 𝜑 ′ = 𝜑 ∪ { 𝑜 ≥ 𝑜 ′ } : clearly 𝑜 (cid:23) 𝜑 ′ 𝑜 ′ , thus 𝜑 ≡ 𝜑 ′ if and only if 𝑜 (cid:23) 𝜑 𝑜 ′ .Therefore, if language L is such that adding CP statement 𝑜 ≥ 𝑜 ′ toany of its formulas yields a formula that is still in L , then eqiv-alence has to be at least as hard as (cid:23) -comparison for L . Thisis the case of CP . The problem remains hard for ⋫ , becauseit is hard to check the equivalence, in propositional logic, of theconditions of statements that entail a particular swap 𝑥 ≥ 𝑥 ′ . Example . Consider three attributes 𝐴 , 𝐵 and 𝐶 with respectivedomains { 𝑎, ¯ 𝑎 } , { 𝑏, ¯ 𝑏 } and { 𝑐 , 𝑐 , 𝑐 } . Consider two CP statements 𝑠 = ¯ 𝑎 : 𝑐 ≥ 𝑐 and 𝑠 ′ = 𝑏 : 𝑐 ≥ 𝑐 , and let 𝜑 = { 𝑠, 𝑠 ′ , 𝑎 : 𝑐 ≥ 𝑐 } .Because of statements 𝑠 and 𝑠 ′ we have ¯ 𝑎𝑏𝑐 ≥ 𝜑 ¯ 𝑎𝑏𝑐 ≥ 𝜑 ¯ 𝑎𝑏𝑐 ; also, 𝑎𝑏𝑐 ≥ 𝜑 𝑎𝑏𝑐 because of statement 𝑎 : 𝑐 ≥ 𝑐 . Hence, for any 𝑢 ∈ 𝐴 × 𝐵 , if 𝑢 | = 𝑎 ∨ ( ¯ 𝑎𝑏 ) then 𝑢𝑐 ≥ 𝑢𝑐 . Thus 𝜑 ≡ { 𝑠, 𝑠 ′ } ∪ { 𝑎 ∨( ¯ 𝑎𝑏 ) : 𝑐 ≥ 𝑐 } ≡ 𝜑 ∪ { 𝑏 : 𝑐 ≥ 𝑐 } . Proposition 13. equivalence is coNP -hard for ⋫ ∧6 (cid:8) , andfor ∧ , both restricted to binary attributes. As usual, comparing two formulas is easier for languages wherethere exists a canonical form. This is the case of CP-nets, as shownby [26, Lemma 2]; their proof makes it clear that the canonical formof any CP-net 𝜑 can be computed in polynomial time. Hence: Proposition 14. equivalence is in P for CP-net . eqivalence also becomes tractable if some form of canonicityis imposed for the conditions of the rules in an LP-tree; this is be-cause, as with CP-net, it is possible to define a canonical form forthe structure, by imposing that the labels of the node be as small aspossible – which may lead, in some cases, to splitting some nodes. op 𝑝 alternatives. Given a set of alternatives 𝑆 and some integer 𝑝 , we may be interested in finding a subset 𝑆 ′ of 𝑆 that contains 𝑝 “best” alternatives of 𝑆 , in the sense that for every 𝑜 ∈ 𝑆 ′ , for every 𝑜 ′ ∈ 𝑆 \ 𝑆 ′ it is not the case that 𝑜 ′ ≻ 𝜑 𝑜 . Note that such a set mustexist, because ≻ 𝜑 is acyclic. The Top- 𝑝 query is usually defined fortotally ordered sets; a definition suited to partial orders is given in[35] (where it is called ordering ), we adopt this definition here: Top- 𝑝 Given 𝑆 ⊆ X , 𝑝 < | 𝑆 | , and 𝜑 , find 𝑜 , 𝑜 , . . . , 𝑜 𝑝 ∈ 𝑆 suchthat for every 𝑖 ∈ , . . . , 𝑝 , for every 𝑜 ′ ∈ 𝑆 , if 𝑜 ′ ≻ 𝜑 𝑜 𝑖 then 𝑜 ′ ∈{ 𝑜 , . . . , 𝑜 𝑖 − } .Note that if 𝑜 , 𝑜 , . . . , 𝑜 𝑝 is the answer to such query, if 1 ≤ 𝑖 < 𝑗 ≤ 𝑝 , then it can be the case that 𝑜 𝑖 ⊲⊳ 𝑜 𝑗 , but it is guaranteedthat 𝑜 𝑗 ⊁ 𝑜 𝑖 : in the context of a recommender system for instance,where one would expect alternatives to be presented in order ofnon-increasing preference, 𝑜 𝑖 could be safely presented before 𝑜 𝑗 .[4] prove that top- 𝑝 is tractable for acyclic CP-nets for the spe-cific case where | 𝑆 | =
2. More generally, ≻ -comparison queriescan be used to compute an answer to a top- 𝑝 query (by asking ≻ -comparison queries for every pair of elements of 𝑆 , the num-ber of such pairs being in Θ (| 𝑆 | ) ). However, [35] shows that anupper approximation of ≻ is sufficient, and proves that such anapproximation can be obtained in time polynomial in | 𝜑 | for arestricted class of lexico-compatible formulas of ∧ [35, Th. 5].We prove that this result does indeed hold for the full class of lexico-compatible formulas of ∧ . The top- 𝑝 query is also tractablefor LPT . Proposition 15. top- 𝑝 can be answered in time which is polyno-mial in the size of 𝜑 and the size of 𝑆 for k-lexico-compatible formulas(for fixed 𝑘 ); and for LPT . Optimization.
Instead of ordering a given set, we may want tofind a globally optimal alternative. Following [24], given 𝜑 , we saythat alternative 𝑜 is: • weakly undominated if there is no 𝑜 ′ ∈X such that 𝑜 ′ ≻ 𝜑 𝑜 ; • undominated if there is no 𝑜 ′ ∈ X , 𝑜 ′ ≠ 𝑜 , such that 𝑜 ′ (cid:23) 𝜑 𝑜 ; • dominating if for every 𝑜 ′ ∈ X , 𝑜 (cid:23) 𝜑 𝑜 ′ ; • strongly dominating if for every 𝑜 ′ ∈ X with 𝑜 ′ ≠ 𝑜 , 𝑜 ≻ 𝜑 𝑜 ′ .Note that 𝑜 is strongly dominating if and only if it is dominat-ing and undominated; and that if 𝑜 is dominating or undominated,then it is weakly undominated. This gives rise to several types ofqueries: [w | s] (undominated | dominating) existence Given 𝜑 , is therea [weakly | strongly] (undominated | dominating) alternative? [w | s] (undominated | dominating) checking Given 𝜑 , 𝑜 , is 𝑜 a [weakly | strongly] (undominated | dominating) alternative?All these queries are easily shown to be tractable for LPT . Theproblem undominated check has been shown to be tractable forCP-nets [4] and for ⋫ [24]. This can be generalized: Proposition 16. undominated check is in P for CP . The existence of a weakly undominated alternative is triviallytrue for CP (in any finite directed graph, at least one vertex hasno "strict" predecessor). Linearisability also ensure that there is atleast one undominated alternative. For CP -nets, [4] give a polytime algorithm that computes theonly dominating alternative when the dependency graph is acyclic;in this case, this alternative is also the only strongly dominatingone and the only undominated one, since the CP -net is linearis-able: this implies that dominating ∃ , s. dominating ∃ , undomi-nated ∃ , s. dominating check , dominating check and w. un-dominated check are tractable for acyclic CP-nets.[24, Prop. 8, 9 and 11] prove that w. undominated check , dom-inating check , s. dominating check , dominating ∃ and s. dom-inating ∃ are PSPACE -complete for ⋫ , and their reductionsfor proving hardness of w. undominated check , dominating check , s. dominating check indeed yield formulas of ⋫ ∧ . NP -hardnessof undominated ∃ for ⋫ ∧ is proved by [16], Cuts.
Cuts are sets of alternatives that are at the same “level” withrespect to (cid:23) . For rankings defined with real-valued functions, cutsare defined with respect to possible real values. In the case of pre-orders, we define cuts with respect to some alternative 𝑜 : given 𝜑 ∈ CP , for any 𝑅 ∈ {≻ , (cid:23)} , for every alternative 𝑜 , we define • CUT
𝑅,𝑜 ( 𝜑 ) = { 𝑜 ′ ∈ X | 𝑜 ′ ≠ 𝑜, 𝑜 ′ 𝑅 𝜑 𝑜 } .Following [19], we define two families of queries: 𝑅 -cut counting Given 𝜑, 𝑜 , count the elements of
CUT
𝑅,𝑜 ( 𝜑 ) 𝑅 -cut extraction Given 𝜑, 𝑜 , return an element of
CUT
𝑅,𝑜 ( 𝜑 ) (or that it is empty) Proposition 17. (cid:23) -cut extraction is tractable for CP . ≻ -cutcounting and ≻ -cut extraction are PSPACE -hard for ⋫ ∧ .For CP (cid:8) lex 𝑘 , ≻ -cut extraction is equivalent to (cid:23) -cut extraction and is tractable. ≻ -cut counting is tractable for LP-trees. The literature on languages on CP statements has long focused onstatements with unary swaps. Several examples in Section 4 showthat this strongly degrades expressiveness. We have introduced anew parameterised family of languages, CP (cid:8) lex 𝑘 , which permits tobalance expressiveness against query complexity: the lower 𝑘 is,the less expressive the language is, but the faster answering mostqueries will be. Table 1 shows that comparison queries seem to re-sist tractability, even for CP (cid:8) lex 𝑘 , but queries like the top- 𝑝 querymay be sufficient in many applications. Tractability of the eqiva-lence query relies on the existence of canonical form: it is the casewhen the language enforces a structure like a dependency graphor a tree, and when the conditions of the statements are restrictedto some propositional language with a canonical form.We have not studied here transformations, like conditioning orother forms of projection for instance. Some initial results on pro-jections can be found in [1]. This is an important direction for fu-ture work, as well as properties of the various languages studiedhere with respect to machine learning. ACKNOWLEDGMENTS
We thank anonymous referees for their valuable comments on pre-vious versions of this paper. This work has benefited from the AIInterdisciplinary Institute ANITI. ANITI is funded by the French"Investing for the Future – PIA3" program under grant agreementANR-19-PI3A-0004. This work has also been supported by the PING/ACKroject of the French National Agency for Research, grant agree-ment ANR-18-CE40-0011.
REFERENCES [1] Philippe Besnard, Jérôme Lang, and Pierre Marquis. 2005. Variable forgettingin preference relations over combinatorial domains. In
Proceedings of the IJCAIMultidisciplinary Workshop on Advances in Preference Handling (MPREF’05) .[2] Richard Booth, Yann Chevaleyre, Jérôme Lang, Jérôme Mengin, and ChattrakulSombattheera. 2009.
Learning various classes of models of lexicographic order-ings
Proceedings of the 19th European Conference on Artificial Intelligence (ECAI2010) (Frontiers in Artificial Intelligence and Applications, Vol. 215) , Helder Coelho,Rudi Studer, and Michael Wooldridge (Eds.). IOS Press, 269–274.[4] Craig Boutilier, Romen I. Brafman, Carmel Domshlak, Holger H. Hoos, andDavid Poole. 2004. CP-nets: a tool for representing and reasoning with con-ditional ceteris paribus preference statements.
Journal of Artificial IntelligenceResearch
21 (2004), 135–191.[5] Craig Boutilier, Ronen I. Brafman, Carmel Domshlak, Holger H. Hoos, and DavidPoole. 2004. Preference-Based Constrained Optimization with CP-Nets.
Compu-tational Intelligence
20, 2 (2004), 137–157.[6] Craig Boutilier, Ronen I. Brafman, Holger H. Hoos, and David Poole. 1999.Reasoning With Conditional Ceteris Paribus Preference Statements. In
Pro-ceedings of the 15th Annual Conference on Uncertainty in Artificial Intelligence(UAI-99) , Kathryn B. Laskey and Henri Prade (Eds.). Morgan Kaufmann, 71–80.https://dslpitt.org/uai/displayArticleDetails.jsp?mmnu=1&smnu=2&article_id=155&proceeding_id=15[7] Sylvain Bouveret, Ulle Endriss, and Jérôme Lang. 2009. Conditional Impor-tance Networks: A Graphical Language for Representing Ordinal, MonotonicPreferences over Sets of Goods. In
Proceedings of the 21st International JointConference on Artificial Intelligence (IJCAI’09) , Craig Boutilier (Ed.). 67–72.http://ijcai.org/Proceedings/09/Papers/022.pdf[8] Ronen I. Brafman, Carmel Domshlak, and Solomon E. Shimony. 2006. On graph-ical modeling of preference and importance.
Journal of Artificial IntelligenceResearch
25 (2006), 389–424.[9] Darius Braziunas and Craig Boutilier. 2005. Local Utility Elicitation in GAI Mod-els. In
Proceedings of the 21st Conference on Uncertainty in Artificial Intelligence(UAI’05) , Fahiem Bacchus and Tommi Jaakkola (Eds.). AUAI Press, 42–49.[10] Michael Bräuning and Eyke Hüllermeyer. 2012. Learning Conditional Lexico-graphic Preference Trees. In
Preference Learning: Problems and Applications inAI. Proceedings of the ECAI 2012 workshop , Johannes Fürnkranz and Eyke Hüller-meyer (Eds.). 11–15.[11] Marco Cadoli, Francesco M. Donini, Paolo Liberatore, and Marco Schaerf.2000. Space Efficiency of Propositional Knowledge Representation For-malisms.
Journal of Artificial Intelligence Research
Proceedings of the Sixteenth International Joint Conference onArtificial Intelligence (IJCAI 99) , Thomas Dean (Ed.). Morgan Kaufmann, 284–289.http://ijcai.org/Proceedings/99-1/Papers/042.pdf[14] Adnan Darwiche and Pierre Marquis. 2002. A Knowledge Compila-tion Map.
Journal of Artificial Intelligence Research
17 (2002), 229–264.https://doi.org/10.1613/jair.989[15] Carmel Domshlak and Ronen I. Brafman. 2002. CP-nets: Reasoning and Con-sistency Testing. In
Proceedings of the Eights International Conference on Princi-ples of Knowledge Representation and Reasoning (KR-02) , Dieter Fensel, FaustoGiunchiglia, Deborah L. McGuinness, and Mary-Anne Williams (Eds.). MorganKaufmann, 121–132.[16] Carmel Domshlak, Steven David Prestwich, Francesca Rossi, Kristen Brent Ven-able, and Toby Walsh. 2006. Hard and soft constraints for reasoning about qual-itative conditional preferences.
J. Heuristics
12, 4-5 (2006), 263–285.[17] Didier Dubois, Christopher A. Welty, and Mary-Anne Williams (Eds.). 2004.
Pro-ceedings of the Ninth International Conference on the Principles of Knowledge Rep-resentation and Reasoning . AAAI Press.[18] Hélène Fargier, Pierre Francois Gimenez, and Jérôme Mengin. 2018. Learn-ing Lexicographic Preference Trees From Positive Examples. In
Proceedingsof the Thirty-Second AAAI Conference on Artificial Intelligence (AAAI 2018)
Proceedings of the Twenty-Eighth AAAI Confer-ence on Artificial Intelligence, July 27 -31, 2014, Québec City, Québec,Canada
A Knowledge Compilation Mapfor Conditional Preference Statements-based Languages . Research ReportIRIT/RR–2021–02–FR. IRIT - Institut de recherche en informatique de Toulouse.[21] Peter C. Fishburn. 1974. Lexicographic Orders, Utilities and DecisionRules: A Survey.
Management Science
Psychological Review
Jour-nal of Artificial Intelligence Research
33 (2008), 403–432.[25] Christophe Gonzales and Patrice Perny. 2004. GAI Networks for Utility Elicita-tion, See [17], 224–233.[26] Frédéric Koriche and Bruno Zanuttini. 2010. Learning conditionalpreference networks.
Artificial Intelligence
Artificial Intelligence
Artificial Intelligence
272 (2019), 101–142. https://doi.org/10.1016/j.artint.2018.12.010[29] Chris S. Mellish (Ed.). 1995.
Proceedings of the Fourteenth InternationalJoint Conference on Artificial Intelligence (IJCAI 95) . Morgan Kaufmann.http://ijcai.org/proceedings/1995-1[30] Thomas Schiex, Hélène Fargier, and Gérard Verfaillie. 1995. Valued Con-straint Satisfaction Problems: Hard and Easy Problems, See [29], 631–639.http://ijcai.org/Proceedings/95-1/Papers/083.pdf[31] Michael Schmitt and Laura Martignon. 2006. On the Complexity of LearningLexicographic Strategies.
Journal of Machine Learning Research
Proceedings of the 16th Eureopean Conference on Artificial Intel-ligence (ECAI 2004) , Ramón López de Mántaras and Lorenza Saitta (Eds.). IOSPress, 888–892.[33] Nic Wilson. 2004. Extending CP-Nets with Stronger Conditional PreferenceStatements. In
Proceedings of the Nineteenth National Conference on Artificial In-telligence (AAAI’04) , Deborah L. McGuinness and George Ferguson (Eds.). AAAIPress / The MIT Press, 735–741.[34] Nic Wilson. 2006. An Effcient Upper Approximation for Conditional Preference.In
Proceedings of the 17th European Conference on Artificial Intelligence (ECAI2006) (Frontiers in Artificial Intelligence and Applications) , Gerhard Brewka, SilviaCoradeschi, Anna Perini, and Paolo Traverso (Eds.). IOS Press.[35] Nic Wilson. 2011. Computational techniques for a simple theory ofconditional preferences.
Artificial Intelligence
175 (2011), 1053–1091.https://doi.org/10.1016/j.artint.2010.11.018
A PROOFS
Proposition 1.
Let 𝜑 be an LP-tree over X , then (cid:23) 𝜑 as definedabove is a preorder. Furthermore, (cid:23) 𝜑 is a linear order if and only if 1)every attribute appears on every branch and 2) every preference rulespecifies a linear order. Proof.
By definition, (cid:23) 𝜑 is reflexive. For transitivity, the proofgiven by [2] is for a restricted family of LP-trees, so we recast ithere for our more general family of LP-trees. Suppose that 𝑜 (cid:23) 𝜑 𝑜 ′ (cid:23) 𝜑 𝑜 ′′ and 𝑜 , 𝑜 ′ , 𝑜 ′′ are distinct. There must be a node 𝑁 atwhich { 𝑜, 𝑜 ′ } is decided, let 𝑊 be the set of attributes that labels 𝑁 , then 𝑜 [ Anc ( 𝑁 )] = 𝑜 ′ [ Anc ( 𝑁 )] , and there is one rule 𝛼 : ≥ suchthat 𝑜 [ NonInst ( 𝑁 )] = 𝑜 ′ [ NonInst ( 𝑁 )] | = 𝛼 and 𝑜 [ 𝑊 ] ≥ 𝑜 ′ [ 𝑊 ] .Similarly, let 𝑁 ′ be the node at which { 𝑜 ′ , 𝑜 ′′ } is decided, let 𝑊 ′ bethe set of attributes that labels 𝑁 ′ , then 𝑜 [ Anc ( 𝑁 ′ )] = 𝑜 ′ [ Anc ( 𝑁 ′ )] ,nd there is one rule 𝛼 ′ : ≥ ′ s.t. 𝑜 ′ [ NonInst ( 𝑁 ′ )] = 𝑜 ′′ [ NonInst ( 𝑁 ′ )] | = 𝛼 ′ and 𝑜 ′ [ 𝑊 ′ ] ≥ ′ 𝑜 ′′ [ 𝑊 ′ ] . If 𝑁 = 𝑁 ′ , then 𝑜 [ Anc ( 𝑁 )] = 𝑜 ′ [ Anc ( 𝑁 )] = 𝑜 ′′ [ Anc ( 𝑁 )] and ≥ = ≥ ′ is transitive (it is a preorder) thus 𝑜 [ 𝑊 ] ≥ 𝑜 ′′ [ 𝑊 ] hence 𝑜 (cid:23) 𝜑 𝑜 ′′ . If 𝑁 ≠ 𝑁 ′ , note that both nodesare in the unique branch in 𝜑 that corresponds to 𝑜 ′ , so one of 𝑁 , 𝑁 ′ must be above the other. Suppose that 𝑁 is above 𝑁 ′ , then, itmust be the case that 𝑜 ′ [ 𝑊 ] = 𝑜 ′′ [ 𝑊 ] , and 𝑜 [ 𝑊 ] ≠ 𝑜 ′ [ 𝑊 ] , thus 𝑁 decides { 𝑜, 𝑜 ′′ } ; moreover, since NonInst ( 𝑁 ) ⊆ NonInst ( 𝑁 ′ ) , 𝑜 ′ [ NonInst ( 𝑁 )] = 𝑜 ′′ [ NonInst ( 𝑁 )] | = 𝛼 , and 𝑜 [ 𝑊 ] ≥ 𝑜 ′ [ 𝑊 ] = 𝑜 ′′ [ 𝑊 ] ; hence 𝑜 (cid:23) 𝜑 𝑜 ′′ . The case where 𝑁 ′ is above 𝑁 is similar.For the second part of the proposition, suppose first that everyattribute appears on every branch and that every preference rulespecifies a linear order: we will show that (cid:23) 𝜑 is antisymmetric andconnex. For antisymmetry, consider distinct alternatives 𝑜, 𝑜 ′ ∈ X :because every attribute appears on every branch, there must bea node 𝑁 , labelled with some 𝑊 ⊆ X , that decides { 𝑜, 𝑜 ′ } , and aunique rule 𝛼 : ≥ at 𝑁 such that 𝑜 [ NonInst ( 𝑁 )] = 𝑜 ′ [ NonInst ( 𝑁 )] | = 𝛼 ; ≥ must be a linear order over 𝑊 , so either 𝑜 [ 𝑊 ] > 𝑜 ′ [ 𝑊 ] and 𝑜 ≻ 𝜑 𝑜 ′ , or 𝑜 ′ [ 𝑊 ] > 𝑜 [ 𝑊 ] and 𝑜 ′ ≻ 𝜑 𝑜 : (cid:23) 𝜑 is connex and antisym-metric. For the converse, assuming that either there is some branchwhere some attribute does not appear, or that there is some rule atsome node that does not define a linear order, it is not difficultto define two distinct alternatives that cannot be compared with (cid:23) 𝜑 . (cid:3) Proposition 2.
For a fixed 𝑘 ∈ N , checking if a formula 𝜑 ∈ CP is 𝑘 -lexico-compatible is coNP -complete. Proof.
For membership in coNP : a certificate that some given 𝜑 is not lexico-compatible is a branch of a tree built using the al-gorithm above where failure occurs. coNP -completeness can beproved using the same reduction of used by [35, Prop. 24]to prove that checking cuc-acyclicity is coNP -hard. Consider 𝑚 clauses 𝐶 , . . . , 𝐶 𝑚 over 𝑛 binary attributes 𝑋 , . . . , 𝑋 𝑛 . Let X = { 𝑋 , . . . , 𝑋 𝑛 , 𝑌 , 𝑌 , . . . , 𝑌 𝑚 } , where the 𝑌 𝑖 s are new binary attributes.Define 𝜑 = { 𝑙 | 𝑌 𝑘 : 𝑦 𝑘 − ≥ ¯ 𝑦 𝑘 − | 𝑙 ∈ 𝐶 𝑘 , ≥ 𝑘 ≥ 𝑚 } ∪ {| 𝑌 : 𝑦 𝑚 ≥ ¯ 𝑦 𝑚 } . and consider some complete LP tree 𝜓 over set of attributes X . Ev-ery attribute 𝑌 𝑘 appears in the free part of at least one rule of 𝜑 ,thus cannot be at the root of any complete LP tree 𝜓 that is compat-ible with 𝜑 . On the other hand, any of the 𝑋 𝑖 ’s can be at the root,or at any level, in any branch of 𝜓 . Suppose now that 𝐶 ∧ . . . ∧ 𝐶 𝑚 is satisfiable: let 𝑢 be an instantiation of 𝑋 , . . . , 𝑋 𝑛 that satisfiesthis CNF, and consider a branch of 𝜓 where all the 𝑋 𝑖 s have thesame value as in 𝑢 : at every node 𝑁 in such a branch, for every 𝑌 𝑘 , inst ( 𝑁 ) is consistent with the condition of at least one rulewhich has 𝑌 𝑘 as free part; therefore, no ordering of the 𝑌 𝑘 s in sucha branch can be compatible with condition 1 in Proposition 18. Onthe other hand, if 𝐶 ∧ . . . ∧ 𝐶 𝑚 is unsatisfiable, then it is not dif-ficult to see that it is possible to build 𝜓 in such a way that theconditions of Proposition 18 are satisfied w.r.t. 𝜑 , by taking, for in-stance, the 𝑋 𝑖 s for the nodes at the 𝑛 first levels of 𝜓 : then, since 𝐶 ∧ . . . ∧ 𝐶 𝑚 is unsatisfiable, in every branch of the tree there mustbe one clause 𝐶 𝑘 that is not satisfied by the corresponding instan-tiation of the 𝑋 𝑖 s, so none of the conditions of the correspondingrules 𝑙 | 𝑌 𝑘 : 𝑦 𝑘 − ≥ ¯ 𝑦 𝑘 − is satisfied; then attribute 𝑌 𝑘 − can be cho-sen for the label of the node at the next level, then 𝑌 𝑘 − , and soon . . . (cid:3) Proposition 3.
Given 𝜑 ∈ CP and some 𝑘 ∈ N , suppose that chooseAttribute is 𝜑 -compatible, then 𝜑 ∈ CP (cid:8) lex 𝑘 if and only ifthe algorithm above returns some 𝜓 ∈ k-LPT such that (cid:23) 𝜓 ⊇ (cid:23) 𝜑 ;otherwise, it returns FAILURE . In order to prove Proposition 3, we first state and prove the fol-lowing result, similar to that of [35, Prop. 13], which shows that itis possible to check in polynomial time, given 𝜑 ∈ CP and somecomplete 𝜓 ∈ LPT , if (cid:23) 𝜓 extends (cid:23) 𝜑 . For ease of presentation, weintroduce the following definition: Definition . Statement 𝛼 | 𝑉 : 𝑤 ≥ 𝑤 ′ is relevant at node 𝑁 of someLP-tree 𝜓 if Var ( 𝑤 ) ∩ Anc ( 𝑁 ) = ∅ , and 𝛼 ∧ inst ( 𝑁 ) 6| = ⊥ , and Var ( 𝑤 ) ∩ Var ( 𝑁 ) ≠ ∅ .Informally, 𝛼 | 𝑉 : 𝑤 ≥ 𝑤 ′ is relevant at node 𝑁 if it sanctionssome swap ( 𝑢𝑟𝑣𝑤, 𝑢𝑟𝑣𝑤 ′ ) , with 𝑢 ∈ Var ( 𝛼 ) such that 𝑢 | = 𝛼 and 𝑣 ∈ 𝑉 , which may be decided at 𝑁 . Proposition 18 (Generalisation of Prop. 13 by [35]).
Let 𝜑 ∈ CP and let 𝜓 be some complete LP-tree. Then (cid:23) 𝜓 ⊇ (cid:23) 𝜑 if and only iffor every node 𝑁 ∈ 𝜓 , for every 𝛼 | 𝑉 : 𝑤 ≥ 𝑤 ′ ∈ 𝜑 that is relevant at 𝑁 , the following two conditions hold:(1) 𝑉 ∩ Anc ( 𝑁 ) = ∅ ; and(2) for every rule 𝛽 : ≥ 𝛽 ∈ CPT ( 𝑁 ) such that 𝛼 ∧ 𝛽 = ⊥ , forevery 𝑢 ∈ Var ( 𝛼 ) such that 𝑢 ∧ 𝛼 ∧ 𝛽 ∧ inst ( 𝑁 ) 6| = ⊥ , for every 𝑠 ∈ Var ( 𝑁 ) \ ( Var ( 𝛼 ) ∪ 𝑉 ∪ Var ( 𝑤 )) , for every 𝑣 , 𝑣 ∈ 𝑉 , itmust be the case that 𝑢𝑠𝑣 𝑤 [ Var ( 𝑁 )] > 𝛽 𝑢𝑠𝑣 𝑤 ′ [ Var ( 𝑁 )] . Proof.
Let us first prove that conditions 1 and 2 are necessaryfor every node 𝑁 of 𝜓 labelled with 𝑇 and every statement 𝛼 | 𝑉 : 𝑤 ≥ 𝑤 ′ ∈ 𝜑 that is relevant at 𝑁 , assuming that (cid:23) 𝜓 ⊇ (cid:23) 𝜑 . Let 𝑅 = X \(
Var ( 𝛼 ) ∪ 𝑉 ∪ Var ( 𝑤 )) . Suppose first that there is some 𝑌 ∈ 𝑉 ∩ Anc ( 𝑁 ) : there must be some node 𝑁 between the root and 𝑁 in 𝜓 such that 𝑌 is in the label 𝑆 of 𝑁 . Choose some 𝑢 ∈ Var ( 𝛼 ) ,˜ 𝑣 ∈ 𝑉 \ { 𝑌 } , 𝑟 ∈ 𝑅 such that 𝑢 | = 𝛼 and 𝑢 ˜ 𝑣𝑟 ∧ inst ( 𝑁 ) 6| = ⊥ (thisis possible because 𝛼 ∧ inst ( 𝑁 ) 6| = ⊥ ); pick 𝑦 , 𝑦 ∈ 𝑌 with 𝑦 ≠ 𝑦 , then 𝜑 sanctions ( 𝑢𝑟𝑦 ˜ 𝑣𝑤, 𝑢𝑟𝑦 ˜ 𝑣𝑤 ′ ) and ( 𝑢𝑟𝑦 ˜ 𝑣𝑤, 𝑢𝑟𝑦 ˜ 𝑣𝑤 ′ ) , so,since (cid:23) 𝜓 ⊇ (cid:23) 𝜑 , 𝑢𝑟𝑦 ˜ 𝑣𝑤 (cid:23) 𝜓 𝑢𝑟𝑦 ˜ 𝑣𝑤 ′ and 𝑢𝑟𝑦 ˜ 𝑣𝑤 (cid:23) 𝜓 𝑢𝑟𝑦 ˜ 𝑣𝑤 ′ .Moreover, { 𝑢𝑟𝑦 ˜ 𝑣𝑤, 𝑢𝑟𝑦 ˜ 𝑣𝑤 ′ } and { 𝑢𝑟𝑦 ˜ 𝑣𝑤, 𝑢𝑟𝑦 ˜ 𝑣𝑤 ′ } are both de-cided in 𝜓 at 𝑁 because 𝑊 ∩ Anc ( 𝑁 ) ⊆ 𝑊 ∩ Anc ( 𝑁 ) = ∅ ; let 𝛽 : ≥ 𝛽 be the unique rule in CPT ( 𝑁 ) such that 𝑢𝑟 ˜ 𝑣 [ Anc ( 𝑁 )] | = 𝛽 : then 𝑢𝑟𝑦 ˜ 𝑣 [ 𝑆 ] ≥ 𝛽 𝑢𝑟𝑦 ˜ 𝑣 [ 𝑆 ] because 𝑢𝑟𝑦 ˜ 𝑣𝑤 (cid:23) 𝜓 𝑢𝑟𝑦 ˜ 𝑣𝑤 ′ , and 𝑢𝑟𝑦 ˜ 𝑣 [ 𝑆 ] ≥ 𝛽 𝑢𝑟𝑦 ˜ 𝑣 [ 𝑆 ] because 𝑢𝑟𝑦 ˜ 𝑣𝑤 (cid:23) 𝜓 𝑢𝑟𝑦 ˜ 𝑣𝑤 ′ : this would im-ply 𝑢𝑟𝑦 ˜ 𝑣 [ 𝑆 ] ∼ 𝛽 𝑢𝑟𝑦 ˜ 𝑣 [ 𝑆 ] , which is impossible since 𝜓 is a com-plete LP-tree, and so, by definition, every rule at every node as-signs a linear order. This proves that 𝑉 ∩ Anc ( 𝑁 ) = ∅ . Now let 𝛽 : ≥ 𝛽 ∈ CPT ( 𝑁 ) such that 𝛼 ∧ 𝛽 = ⊥ : let • 𝑢 ∈ Var ( 𝛼 ) such that 𝑢 ∧ 𝛼 ∧ 𝛽 ∧ inst ( 𝑁 ) 6| = ⊥ , • 𝑠 ∈ Var ( 𝑁 ) \ ( Var ( 𝛼 ) ∪ 𝑉 ∪ Var ( 𝑤 )) , • 𝑣 , 𝑣 ∈ 𝑉 , • 𝑟 ∈ X \ Var ( 𝑁 ) ∪ Var ( 𝛼 ) ∪ 𝑉 ∪ Var ( 𝑤 ) such that 𝑟 ∧ 𝛽 = ⊥ ,then 𝛼 | 𝑉 : 𝑤 ≥ 𝑤 ′ ∈ 𝜑 sanctions ( 𝑢𝑣 𝑠𝑟𝑤, 𝑢𝑣 𝑠𝑟𝑤 ′ ) , so 𝑢𝑣 𝑠𝑟𝑤 (cid:23) 𝜓 𝑢𝑣 𝑠𝑟𝑤 ′ , and the pair { 𝑢𝑣 𝑠𝑟𝑤, 𝑢𝑣 𝑠𝑟𝑤 ′ } is decided in 𝜓 at 𝑁 withthe rule 𝛽 : ≥ 𝛽 , so it must be the case that 𝑢𝑣 𝑠𝑤 [ Var ( 𝑁 )] > 𝛽 𝑢𝑣 𝑠𝑤 ′ [ Var ( 𝑁 )] .et us now prove that the condition is sufficient: it is in fact suf-ficient to prove that, if the condition above holds, then for everystatement 𝛼 | 𝑉 : 𝑤 ≥ 𝑤 ′ ∈ 𝜑 , for every pair of alternatives { 𝑜, 𝑜 ′ } sanctioned by the statement, it is the case that 𝑜 (cid:23) 𝜓 𝑜 ′ . We musthave that 𝑜 = 𝑢𝑟𝑣 𝑤 and 𝑜 ′ = 𝑢𝑟𝑣 𝑤 ′ for some 𝑢 ∈ Var ( 𝛼 ) suchthat 𝑢 | = 𝛼 , some 𝑣 , 𝑣 ∈ 𝑉 , some 𝑟 ∈ X \ ( Var ( 𝛼 ) ∪ 𝑉 ∪ Var ( 𝑤 )) .Let 𝑁 be the node that decides { 𝑜, 𝑜 ′ } in 𝜓 , let us prove that 𝛼 | 𝑉 : 𝑤 ≥ 𝑤 ′ is relevant at 𝑁 . That Var ( 𝑤 ) ∩ Anc ( 𝑁 ) = ∅ follows from thefact that { 𝑜, 𝑜 ′ } is not decided at some other node between theroot and 𝑁 ; also, the common part of 𝑜 and 𝑜 ′ must be consis-tent with inst ( 𝑁 ) (otherwise, the pair would have taken anotherbranch on the tree), so 𝑢 ∧ inst ( 𝑁 ) 6| = ⊥ , and 𝑢 is a model of 𝛼 , so 𝛼 ∧ inst ( 𝑁 ) 6| = ⊥ ; there remains to prove that Var ( 𝑤 ) ∩ Var ( 𝑁 ) ≠ ∅ . Suppose first that 𝑣 = 𝑣 , then it must be the casethat 𝑤 [ Var ( 𝑁 )] ≠ 𝑤 ′ [ Var ( 𝑁 )] (since { 𝑜, 𝑜 ′ } is decided at 𝑁 , so Var ( 𝑤 ) ∩ Var ( 𝑁 ) ≠ ∅ : 𝛼 | 𝑉 : 𝑤 ≥ 𝑤 ′ is indeed relevant at 𝑁 , andcondition 1 implies that 𝑉 ∩ Anc ( 𝑁 ) = ∅ . Suppose now that 𝑣 ≠ 𝑣 ,let 𝑜 ′′ = 𝑢𝑟𝑣 𝑤 ′ , let 𝑁 ′ be the node that decides { 𝑜, 𝑜 ′′ } , fromwhat we have just seen, in particular that 𝑉 ∩ Anc ( 𝑁 ′ ) = ∅ wecan conclude that 𝑁 ′ also decides { 𝑜, 𝑜 ′ } , thus 𝑁 = 𝑁 ′ : it followsthat 𝑉 ∩ Anc ( 𝑁 ) = ∅ , thus, since 𝑁 decides { 𝑜, 𝑜 ′ } , it must be thecase that Var ( 𝑤 ) ∩ Var ( 𝑁 ) ≠ ∅ : statement 𝛼 | 𝑉 : 𝑤 ≥ 𝑤 ′ is againrelevant at 𝑁 . Consider now the unique rule 𝛽 : ≥ 𝛽 ∈ CPT ( 𝑁 ) such that 𝑢𝑟 [ Anc ( 𝑁 )] | = 𝛽 , then 𝑢 ∧ 𝛼 ∧ 𝛽 = ⊥ since 𝑢 | = 𝛼 ;let 𝑠 = 𝑟 [ Var ( 𝑁 )] , then 𝑠 ∈ Var ( 𝑁 ) \ ( Var ( 𝛼 ) ∪ 𝑉 ∪ Var ( 𝑤 )) , thuscondition 2 implies that 𝑢𝑠𝑣 𝑤 [ Var ( 𝑁 )] > 𝛽 𝑢𝑠𝑣 𝑤 ′ [ Var ( 𝑁 )] ; since { 𝑜, 𝑜 ′ } is decided at 𝑁 , it follows that 𝑜 (cid:23) 𝜓 𝑜 ′ . (cid:3) Lemma 1.
Let 𝜑 ∈ CP , suppose that chooseAttribute is 𝜑 -compatible,that the algorithm above terminates without failure on 𝜑 , 𝑘 , and re-turns some complete 𝑘 -LP-tree 𝜓 , then (cid:23) 𝜑 ⊆ (cid:23) 𝜓 . Proof.
Let 𝑁 ∈ 𝜓 , let 𝛼 | 𝑉 : 𝑤 ≥ 𝑤 ′ ∈ 𝜑 that is relevant at 𝑁 ,we only need to prove that conditions 1 and 2 in Proposition 18hold.Let us first prove that 𝑉 ∩ Anc ( 𝑁 ) = ∅ . Let 𝑁 ′ be any nodebetween the root and 𝑁 , excluding 𝑁 ; it is labelled with some 𝑇 ′ ⊆ X and a linear order ≥ 𝑁 ′ over 𝑇 ′ that have been returned by chooseAttribute when called at 𝑁 ′ . We know that 𝛼 ∧ inst ( 𝑁 ) 6| = ⊥ , so, since inst ( 𝑁 ) extends inst ( 𝑁 ′ ) , 𝛼 ∧ inst ( 𝑁 ′ ) 6| = ⊥ ; more-over Var ( 𝑤 ) ∩ Anc ( 𝑁 ) = ∅ because 𝛼 | 𝑉 : 𝑤 ≥ 𝑤 ′ ∈ 𝜑 is rele-vant at 𝑁 , and Anc ( 𝑁 ′ ) ⊂ Anc ( 𝑁 ) so Var ( 𝑤 ) ∩ Anc ( 𝑁 ′ ) = ∅ ,hence 𝛼 | 𝑉 : 𝑤 ≥ 𝑤 ′ ∈ 𝜑 ( 𝑁 ′ ) ; since 𝑇 ′ ⊆ Anc ( 𝑁 ) , and Var ( 𝑤 ) ∩ Anc ( 𝑁 ) = ∅ , we also have that 𝑇 ′ ∩ Var ( 𝑤 ) = ∅ ; thus, according tocondition ?? that must be verified by chooseAttribute , it must bethe case that 𝑉 ∩ 𝑇 ′ = ∅ . This holds for every label of every nodeabove 𝑁 , hence 𝑉 ∩ Anc ( 𝑁 ) = ∅ .Let us now prove that condition 2 holds. Since the algorithmreturns LP-trees with no unlabelled edge, it can be rewritten asfollows:2’. for every 𝑢 ∈ Var ( 𝛼 ) such that 𝑢 | = 𝛼 and 𝑢 ∧ inst ( 𝑁 ) 6| = ⊥ , for every 𝑠 ∈ Var ( 𝑁 ) \ ( Var ( 𝛼 ) ∪ 𝑉 ∪ Var ( 𝑤 )) , for ev-ery 𝑣 , 𝑣 ∈ 𝑉 , it must be the case that 𝑢𝑠𝑣 𝑤 [ Var ( 𝑁 )] > 𝛽 𝑢𝑠𝑣 𝑤 ′ [ Var ( 𝑁 )] .We know that Var ( 𝑤 )∩ 𝑇 ≠ ∅ because 𝛼 | 𝑉 : 𝑤 ≥ 𝑤 ′ ∈ 𝜑 is relevantat 𝑁 . Let 𝑢 ∈ Var ( 𝛼 ) such that 𝑢 | = 𝛼 and 𝑢 ∧ inst ( 𝑁 ) 6| = ⊥ , let 𝑠 ∈ Var ( 𝑁 ) \ ( Var ( 𝛼 ) ∪ 𝑉 ∪ Var ( 𝑤 )) , let 𝑣 , 𝑣 ∈ 𝑉 , and let 𝑡 = 𝑢𝑠𝑣 𝑤 [ 𝑇 ] , 𝑡 ′ = 𝑢𝑠𝑣 𝑤 ′ [ 𝑇 ] , then clearly 𝑡 ∧ 𝑤 = ⊥ , 𝑡 ′ ∧ 𝑤 ′ = ⊥ , 𝑡 [X \ ( 𝑉 ∪ Var ( 𝑤 ))] = 𝑢𝑠 = 𝑡 ′ [X \ ( 𝑉 ∪ Var ( 𝑤 ))] , and 𝑡 ∧ 𝛼 = ⊥ (because 𝑢 | = 𝛼 ), thus, according to condition ?? in thespecification of chooseAttribute , it must be the case that 𝑡 > 𝑁 𝑡 ′ ,that is 𝑢𝑠𝑣 𝑤 [ 𝑇 ] > 𝑁 𝑢𝑠𝑣 𝑤 ′ [ 𝑇 ] . (cid:3) Lemma 2.
Let 𝜑 ∈ CP , let 𝑘 ∈ N , let 𝜓 be some complete 𝑘 -LP-tree, with no uninstantiated edge, such that (cid:23) 𝜑 ⊆ (cid:23) 𝜓 . Let 𝑁 be somenode of 𝜓 , and let 𝜓 ( 𝑁 ) be the subtree of 𝜓 rooted at 𝑁 . Suppose that,when called at 𝑁 , chooseAttribute returns a pair ( 𝑇 , ≥) different thanthe one that labels 𝑁 . Then (cid:23) 𝜑 ∈ (cid:23) 𝜓 ′ , where 𝜓 ′ is a new 𝑘 -LP-treeobtained by replacing 𝜓 ( 𝑁 ) in 𝜓 with a new subtree as follows: • 𝑁 is replaced with a new node 𝑁 ′ , labelled with ( 𝑇 , ≥) ; inthe sequel, in order to avoid confusion, we denote ≥ 𝑁 ′ thisordering associated with 𝑇 returned by chooseAttribute ; • we create 𝑇 copies of 𝜓 ( 𝑁 ) , each attached to 𝑁 ′ with an edgelabelled with some 𝑡 ∈ 𝑇 ; • for each node 𝑀 in one of these copies, in the branch below 𝑁 ′ corresponding to 𝑡 , we do as follows, where ( 𝑆, ≥ 𝑀 ) is the setof attributes and the associated linear ordering that label 𝑀 : – all subtrees below 𝑀 that correspond to paths incompatiblewith 𝑡 are deleted; – if 𝑆 ⊆ 𝑇 , then there remain only one subtree of 𝑀 , we cansafely remove node 𝑀 ; – otherwise, we replace ( 𝑆, ≥ 𝑀 ) with ( 𝑆 \ 𝑇 , ≥ 𝑀 | 𝑡 ) , where ≥ 𝑀 | 𝑡 is the linear order over 𝑆 \ 𝑇 defined as follows: for every 𝑟, 𝑟 ′ ∈ 𝑆 \ 𝑇 , 𝑟 ≥ 𝑀 | 𝑡 𝑟 ′ if and only if 𝑟𝑡 [ 𝑆 ] ≥ 𝑀 𝑟 ′ 𝑡 [ 𝑆 ] . Proof.
We prove first that 𝜓 ′ ∈ k-LPT : • the nodes not in 𝜓 ( 𝑁 ) are unchanged; • since 𝑇 is returned by chooseAttribute , | 𝑇 | ≤ 𝑘 ; • the set of attributes that labels any node below 𝑁 ′ is dimin-ished, since the attributes in 𝑇 are removed.Thus, since 𝜓 ∈ k-LPT , 𝜓 ′ ∈ k-LPT .Consider now a swap ( 𝑜, 𝑜 ′ ) sanctioned by rule 𝛼 | 𝑉 : 𝑤 ≥ 𝑤 ′ ∈ 𝜑 , and let 𝑀 be the node in 𝜓 where { 𝑜, 𝑜 ′ } is decided. If 𝑀 ∉ 𝜓 ( 𝑁 ) , then 𝑜 (cid:23) 𝜓 ′ 𝑜 ′ if and only if 𝑜 (cid:23) 𝜓 𝑜 ′ , and (cid:23) 𝜓 extends (cid:23) 𝜑 ,thus 𝑜 (cid:23) 𝜓 ′ 𝑜 ′ . Let us now consider the case where 𝑀 ∈ 𝜓 ( 𝑁 ) ,and let ( 𝑆, ≥ 𝑀 ) be the label of 𝑀 . We will consider two subcases.Suppose first that 𝑜 [ 𝑇 ] = 𝑜 ′ [ 𝑇 ] = 𝑡 , since 𝑜 [ 𝑆 ] ≠ 𝑜 ′ [ 𝑆 ] it impliesthat 𝑆 * 𝑇 , thus there is in the subtree that replaces 𝜓 ( 𝑁 ) in 𝜓 ′ a node 𝑀 ′ , with inst ( 𝑀 ′ ) compatible with 𝑡 , obtained from 𝑀 byremoving attributes in 𝑆 ∩ 𝑇 , thus getting new set of attributes 𝑅 = 𝑆 \ 𝑇 , and by conditioning ≥ 𝑀 with 𝑡 . { 𝑜, 𝑜 ′ } is not decided at 𝑁 ′ , nor at any other node above 𝑀 ′ , it is now decided at 𝑀 ′ , and 𝑜 [ 𝑆 ] and 𝑜 ′ [ 𝑆 ] are of the form ˜ 𝑡𝑟 and ˜ 𝑡𝑟 ′ , where ˜ 𝑡 = 𝑡 [ 𝑆 ] ; since 𝑜 (cid:23) 𝜓 𝑜 ′ , it must be the case that ˜ 𝑡𝑟 > 𝑀 ˜ 𝑡𝑟 ′ , thus 𝑟 > 𝑀 | 𝑡 𝑟 ′ . Therefore 𝑜 (cid:23) 𝜓 ′ 𝑜 ′ . Let us finally consider the case where 𝑜 [ 𝑇 ] ≠ 𝑜 ′ [ 𝑇 ] : { 𝑜, 𝑜 ′ } is now decided at 𝑁 ′ . Since { 𝑜, 𝑜 ′ } is decided at 𝑀 in 𝜓 , weknow that Var ( 𝑤 ) ∩ Anc ( 𝑀 ) = ∅ and that 𝛼 ∧ inst ( 𝑀 ) 6| = ⊥ , thus,since 𝑁 is above 𝑀 in 𝜓 , and since inst ( 𝑁 ) = inst ( 𝑁 ′ ) , we havethat 𝛼 ∧ inst ( 𝑁 ′ ) 6| = ⊥ . Therefore, 𝛼 | 𝑉 : 𝑤 ≥ 𝑤 ∈ 𝜑 ( 𝑁 ′ ) . Recallthat 𝑇 has been returned by chooseAttribute called at 𝑁 ′ : thus, ifwe had Var ( 𝑤 ) ∩ 𝑇 = ∅ , we would also have 𝑉 ∩ 𝑇 = ∅ , whichis impossible because, since ( 𝑜, 𝑜 ′ ) is sanctioned by 𝛼 | 𝑉 : 𝑤 ≥ 𝑤 , 𝑜 [ 𝑇 \ ( 𝑉 ∪ Var ( 𝑤 )] = 𝑜 ′ [ 𝑇 \ ( 𝑉 ∪ Var ( 𝑤 )] , thus the pair would note decided at 𝑁 ′ . But, since Var ( 𝑤 )∩ 𝑇 ≠ ∅ , let 𝑡 = 𝑜 [ 𝑇 ] , 𝑡 ′ = 𝑜 ′ [ 𝑇 ] ,they are compatible with respectively 𝑤 and 𝑤 ′ (since 𝑜 and 𝑜 ′ respectively extend 𝑤 and 𝑤 ′ ), and 𝑡 [ 𝑇 \ ( 𝑉 ∪ Var ( 𝑤 )] = 𝑡 ′ [ 𝑇 \ ( 𝑉 ∪ Var ( 𝑤 )] , thus it must be the case, according to condition ?? in thespecification of chooseAttribute , that 𝑡 > 𝑁 ′ 𝑡 ′ , thus 𝑜 ≻ 𝜓 ′ 𝑜 ′ . (cid:3) Proof of Proposition 3.
Suppose first that the algorithm suc-ceeds and returns a 𝑘 -LP-tree: lemma 1 proves that (cid:23) 𝜑 ⊆ (cid:23) 𝜓 ,and we have mentioned that all labels returned by chooseAttribute have no more than 𝑘 attributes, so it is in k-LPT , so 𝜑 ∈ CP (cid:8) lex 𝑘 .Now, suppose that 𝜑 ∈ CP (cid:8) lex 𝑘 : there is some 𝑘 -LP-tree 𝜓 suchthat (cid:23) 𝜓 ⊇ (cid:23) 𝜑 . Consider the execution of the algorithm with 𝜑 and 𝑘 as inputs. Let 𝑁 be the first node such that chooseAttribute re-turns a pair ( 𝑇 , ≥) that is not in 𝜓 at 𝑁 : then lemma 2 proves thatthere is some 𝜓 ′ ∈ k-LPT that is equal to 𝜓 except for the subtreerooted at 𝑁 , and that is equal to 𝜓 for the part built before 𝑁 , andthat has ( 𝑇 , ≥) at 𝑁 : this shows that the algorithm cannot run intoa dead end. (cid:3) Proposition 5. CP = Ð 𝑘 ∈ N k-CP and, for every 𝑘 ∈ N : CP ∧ ⊑⊒ CP ⋫ ⊑⊒ CP = k-CP ⊑⊒ k-CP ∧ = k-CP ⋫ ⊑⊒ k-CP ∧ ⋫ k-CP = (k-1)-CP . Proof.
That CP ⊑⊒ CP ∧ = CP ∧ ⋫ follows from property 4.By definition CP ⊃ ⊃ ⋫ ⊃ ∧ ⋫ and ⊃ ∧ ⊃ ∧ ⋫ , thus CP ⊒ ⊒ ⋫ ⊒ ∧ ⋫ and ⊒ ∧ ⊒ ∧ ⋫ . Restricting to conjunction of literalsdoes not induce a loss in expressiveness because, given a statement 𝛼 | 𝑉 : 𝑥 ≥ 𝑥 ′ , it is possible to compute a DNF logically equivalentto 𝛼 , and then consider a set of statements, each statement havingone disjunct of the DNF as conditioning part. Example 3 prove that CP = . Example 4 proves that = ⋫ . (cid:3) Proposition 10. linearisability can be checked in polynomialtime for
LPT . Proof.
This is because 𝜑 ∈ LPT is linearisable if and only forevery rule 𝛼 : ≥ at every node, ≥ is antisymmetric. (cid:3) Proposition 12. ≻ -comparison and ⊲⊳ -comparison are NP -hardfor the language of fully acyclic CP-nets, and tractable for polytreeCP-nets. ∼ -comparison is easy for (cid:8) lex . Proposition 12 is proved using these simple observations:
Lemma 3.
Let
L ⊆ CP .(1) If (cid:23) -comparison is tractable for L , then so are ≻ -comparison , ∼ -comparison and ⊲⊳ -comparison .(2) If ordering is tractable for L , then ⊲⊳ -comparison is at leastas hard as (cid:23) -comparison for L .Furthermore, if L has the property that all its formulas are known tobe linearisable, then (cid:23) 𝜑 is antisymmetric and:(3) (cid:23) -comparison and ≻ -comparison have the same complexityfor L , because ≻ 𝜑 is the irreflexive part of (cid:23) 𝜑 ;(4) ∼ -comparison is easy for L , because 𝑜 ∼ 𝜑 𝑜 ′ holds if andonly if 𝑜 = 𝑜 ′ . Proof. (1) ≻ -comparison , ∼ -comparison and ⊲⊳ -compari-son queries can be answered with 2 (cid:23) -comparison queries.(2) In order to check if 𝑜 (cid:23) 𝜑 𝑜 ′ , one can ask if 𝑜 ⊲⊳ 𝜑 𝑜 ′ : if theanswer is yes , then 𝑜 (cid:15) 𝜑 𝑜 ′ ; if the answer is no , then onecan ask to order { 𝑜, 𝑜 ′ } : if the answer is that 𝑜 (cid:15) 𝜑 𝑜 ′ , thenwe have the answer to the initial query; if the answer is that 𝑜 ′ (cid:15) 𝜑 𝑜 , since we know that 𝑜 and 𝑜 ′ are not incomparable,it must be the case that 𝑜 (cid:23) 𝜑 𝑜 ′ .The remaining properties are elementary. (cid:3) Proof of Proposition 12.
That ≻ -comparison is NP hard forthe language of fully acyclic CP-nets follows from the fact that (cid:23) -comparison is hard for this language [4, Theorems 15, 16] andfrom property 3 in lemma 3. Property 2 shows that ⊲⊳ -comparison too is NP hard for this language because ordering is easy for thislanguage [4, Theorems 5] and (cid:23) -comparison is hard. These twoqueries are tractable for polytree CP-nets because (cid:23) -comparison is (property 1 in lemma 3). ∼ -comparison is easy for (cid:8) lex because lexico-compatible formulas are linearisable. (cid:3) Proposition 13. equivalence is coNP -hard for ⋫ ∧6 (cid:8) , andfor ∧ , both restricted to binary attributes. Given a propositional language P we define P ∨ to be the set offinite disjunctions of formulas in P , and: ⋫ P is ⋫ restricted to those statements such that thecondition is in P P is restricted to those LP-trees such that the condi-tion of every rule is in P .The proof of the proposition is based on the following lemma,which a formalizes the intuition suggested by Example 9. Lemma 4.
Given a propositional language P closed for conjunc-tion, equivalence for P ∨ (in the sense of propositional logic), reducesto equivalence for ⋫ P restricted to fully acyclic formulas, andto equivalence for P . Proof.
Consider two formulas 𝛼 = Ô 𝑖 ∈ 𝐼 𝛼 𝑖 and 𝛼 ′ = Ô 𝑖 ∈ 𝐼 ′ 𝛼 ′ 𝑖 over a set X of binary attributes, with all 𝛼 𝑖 ’s and 𝛼 ′ 𝑖 ’s in P ; takesome binary attribute 𝑋 ∉ X , with values 𝑥 and ¯ 𝑥 , and let 𝜑 = { 𝛼 𝑖 : 𝑥 ≥ ¯ 𝑥 | 𝑖 ∈ 𝐼 } and 𝜑 ′ = { 𝛼 ′ 𝑖 : 𝑥 ≥ ¯ 𝑥 | 𝑖 ∈ 𝐼 ′ } . Note that 𝜑, 𝜑 ′ ∈ ⋫ P , that they are acyclic, and that they can be computed intime polynomial in | 𝛼 | + | 𝛼 ′ | . Then 𝜑 ∗ = {( 𝑜𝑥, 𝑜 ¯ 𝑥 ) | 𝑜 ∈ X , 𝑜 | = 𝛼 } and for every 𝑜 , 𝑜 ∈ X , for every 𝑥 , 𝑥 ∈ 𝑋 , 𝑜 𝑥 (cid:23) 𝜑 𝑜 𝑥 if andonly if 𝑜 = 𝑜 , 𝑥 = 𝑥 , 𝑥 = ¯ 𝑥 and 𝑜 | = 𝛼 ; similarly, 𝑜 𝑥 (cid:23) 𝜑 ′ 𝑜 𝑥 if and only if 𝑜 = 𝑜 , 𝑥 = 𝑥 , 𝑥 = ¯ 𝑥 and 𝑜 | = 𝛼 ′ . Thus 𝛼 ≡ 𝛼 ′ ifand only if for every 𝑜 ∈ X , 𝑜 | = 𝛼 ⇔ 𝑜 | = 𝛼 ′ , iff for every 𝑜 ∈ X , 𝑜𝑥 (cid:23) 𝜑 𝑜 ¯ 𝑥 ⇔ 𝑜𝑥 (cid:23) 𝜑 ′ 𝑜 ¯ 𝑥 , if and only if 𝜑 ≡ 𝜑 ′ .Similarly, we can define two linear 1-LP-trees 𝜓 and 𝜓 ′ as fol-lows: the top | X | nodes are labelled with attributes from X , inany order and with no rule; then there is one node labelled with 𝑋 ,and the same preference rules as above. (cid:3) Proposition 15. top- 𝑝 can be answered in time which is polyno-mial in the size of 𝜑 and the size of 𝑆 for k-lexico-compatible formulas(for fixed 𝑘 ); and for LPT . roof. It suffices to prove the result for the case where | 𝑆 | = LPT in this case is a simpleconsequence of the fact that ≻ -comparison is tractable for LPT .For 𝑘 -lexico-compatible formulas, given some 𝜑 ∈ CP known tobe 𝑘 -lexico-compatible, and given a pair of alternatives { 𝑜, 𝑜 ′ } , wecan run the algorithm proposed in Section 3.6 that builds a com-plete LP tree 𝜓 that extends 𝜑 , but build one branch only, the onethat corresponds to the pair { 𝑜, 𝑜 ′ } , as long as the chosen attributeshave equal values for 𝑜 and 𝑜 ′ : when reaching a node where thechosen set of attributes 𝑇 si such that 𝑜 [ 𝑇 ] ≠ 𝑜 ′ [ 𝑇 ] , the nodedecides the pair, and 𝑜 and 𝑜 ′ can be ordered accordingly, as inthe case of LP trees: if 𝑜 [ 𝑇 ] > 𝑜 ′ [ 𝑇 ] , then 𝑜 ≻ 𝜓 𝑜 ′ , thus 𝑜 ′ (cid:15) 𝜓 𝑜 ,hence 𝑜 ′ (cid:15) 𝜑 𝑜 ; similarly, if 𝑜 ′ [ 𝑇 ] > 𝑜 [ 𝑇 ] , then it cannot be thecase that 𝑜 ≻ 𝜑 𝑜 ′ . The algorithm will not return FAILURE because 𝜑 is known to be 𝑘 -lexico-compatible. The branch has no morethat | X | nodes, and there is, for fixed 𝑘 , a polynomial number ofpossible labels to try at each node. (cid:3) Proposition 16. undominated check is in P for CP . Proof.
Alternative 𝑜 is not undominated if and only if thereis some 𝑜 ′ such that 𝑜 ′ (cid:23) 𝜑 𝑜 , which can happen if and only if 𝜑 contains a statement 𝛼 | 𝑉 : 𝑤 > 𝑤 ′ with 𝑜 | = 𝛼 and 𝑜 [ Var ( 𝑤 ′ )] = 𝑤 ′ : this can be checked in polynomial time. (cid:3) Proposition 17. (cid:23) -cut extraction is tractable for CP . ≻ -cutcounting and ≻ -cut extraction are PSPACE -hard for ⋫ ∧ .For CP (cid:8) lex 𝑘 , ≻ -cut extraction is equivalent to (cid:23) -cut extraction and is tractable. ≻ -cut counting is tractable for LP-trees. Proof. (cid:23) -cut extraction is easy for CP : given 𝑜 and 𝜑 , inorder to return an element of CUT (cid:23) ,𝑜 ( 𝜑 ) , it is sufficient to findone statement in 𝜑 which sanctions an improving swap for 𝑜 .Note that alternative 𝑜 is weakly undominated iff CUT ≻ ,𝑜 ( 𝜑 ) = ∅ , iff | CUT ≻ ,𝑜 ( 𝜑 ) | =
0; therefore, ≻ -cut counting and ≻ -cutextraction are at least as hard as weakly undominated check ,they are therefore PSPACE -hard for ⋫ ∧ . For acyclic CP-nets,and more generally for cuc-acyclic CP theories, ≻ is the irreflexivepart of (cid:23) , thus ≻ -cut extraction is equivalent to (cid:23) -cut extrac-tion and is easy. Finally, ≻ -cut counting is tractable for LP-trees:given some 𝑜 , when going down some LP-tree 𝜓 in the branch thatcorresponds to 𝑜 , it is possible, at each node 𝑁 encountered, la-belled with 𝑇 , to count the number of 𝑡 ′ in 𝑇 such that 𝑡 > 𝑜 [ 𝑡 ′ ] (according to the preference rule 𝛽 : ≥ 𝛽 at 𝑁 such that 𝑜 | = 𝛽 ),and to multiply this number by the sizes of the domains of the at-tributes that have not been encountered yet; adding these sums ofproducts along the branch will give the number of alternatives 𝑜 such that 𝑜 ′ ≻ 𝜓 𝑜 ..