1 Introduction
Qualitative conditional preferences on combinatorial domains are an important area of study. The main framework is CPnets [BBD04b]. Given a finite set of features, each with a finite domain of values, the user specifies her preference order over the domain of each feature using rules such as which specify that if the attribute has value and attribute has value then is to be preferred to for attribute . Outcomes (that is, assignment of values to all the features) are ordered according to the socalled ceteris paribus interpretation: an outcome is preferred to another outcome if they differ on the value of just one feature, and the value in is preferred to the one in (given the rest of ). Thus in CPnets, preferences are always of the form “I prefer fish to meat, all else being equal”. The key computational tasks are checking for consistency (preference ordering is acyclic), as well as optimizing and comparing outcomes.
CPnets represent one end of the expressiveness/tractability spectrum. While useful in practice, CPnets are limited in expressiveness: a rule may specify a preference for exactly one value over another (in the same feature domain). General CPnets [GLTW08] define CPnets that can be incomplete or locally inconsistent. CPtheories [Wil04] are a specialized formalism, with its own ad hoc syntax (rules are of the form ) and semantics, in which preferences may be conditioned on indifference to certain features (those in the set above). For example, one may say “I prefer fish to meat, no matter what the dessert is”. Comparative preference theories [Wil09] permit preferences to be defined on a set of features simultaneously. Along another direction, [BBD04a] and [DPR06] introduce the idea of adding (hard and soft) constraints to CPnets. Roughly speaking, the added constraints need to be respected by valid outcomes.
This paper develops the fundamental idea that conditional preferences can be directly expressed in standard first order logic, as constrained Datalog theories [CGT89, KKR95, Tom98] involving a binary preference relation d(_,_) on pairs of outcomes. For instance, in the context of preferences over entrees and desserts, the rule “I prefer fish to meat, all else being equal” may be written as:^{1}^{1}1The examples in this paper are written in Prolog and run in the XSB Prolog system, using tabling. Note that the clause is strictly speaking not a Datalog clause because it uses a function symbol o/n. However this function symbol is used just for convenience, and can be eliminated at the cost of increasing the arity of predicates such as d and dom.
d(o(fish,X),o(meat,X)).
and the rule “I prefer fish to meat, no matter what the dessert is” is written as:
d(o(fish,X),o(meat,Y)).
More generally, a Logical Conditional Preference (LCP) rule is of the form:
d(o(X1,...Xn),o(Y1,...,Yn)) : c, g1,...,gn.
where c is a constraint (possibly involving equalities), and g1, …, gn are possibly recursively defined predicates involving d/2. A predicate dom/2 is defined to be the transitive closure of d/2 and represents the dominance relation on outcomes.
dom(X,Z): d(X,Z), outcome(X), outcome(Z). dom(X,Z): d(X,Y), dom(Y,Z).
The theory can be checked for consistency by simply ensuring that given the clause
inconsistent : dom(X,X).
the goal inconsistent cannot be established. Hard constraints C are specified by adding them to the body of the clause defining legal outcomes:
outcome(o(X1,...,Xn)): C, d1(X1), ..., dn(Xn).
The fundamental advantage of introducing conditional preference theories as Datalog programs is that Datalog’s rich semantic, algorithmic and implementation framework is now available in service of conditional preferences. The semantics of LCP theories is that of (constrained) firstorder logic theories. The framework is rich enough to express CPnets and each of its extensions discussed above, including algorithms for consistency, dominance and optimality (Section 3). Using outcomes rather than assertions of preference over individual features permits the formalization of the semantics (e.g. ceteris paribus or indifference) internally, as just a certain pattern of quantification over variables.
Constraints fit in naturally and do not have to be introduced after the fact in an ad hoc fashion as in [DPR06, BBD04a, PRVW04]. For example, [PRVW04] provides a dominance algorithm using the notion of a consistent flipping worsening sequence: they allow worsening flips only between consistent outcomes. In our formulation the constraints are already built into basic definitions (constraints are additional goals in the body of preference clauses, and in the body of the clause defining outcomes) and no changes are necessary.
Additionally, recursive LCPrules (rules with goals gi in the body) offer a powerful new form of dependent conditional preference statements (Section 3.1), particularly useful in multiagent contexts [RVW04, MMP13]. They support rules such as “If Alice prefers to drive to Oxford today, Bob will prefer to fly to Manchester tomorrow”.
The rich complexity theory developed for Datalog [Var82, FV99, GP03, DEGV01] applies inter alia to conditional preference theories – in particular we discuss the notion of datacomplexity in Section 2.3. General results about (linear) Datalog programs lead to complexity bounds for consistency, dominance and optimization extending current known bounds, and in some cases, providing new, simpler proofs for existing complexity bounds (Section 4). Further, tabled Prolog systems such as XSB Prolog[SW10, SW12a] implement constrained Datalog with sophisticated features such as partial order answer subsumption that are directly usable in an implementation of LCP.
One of the reasons that CPnets are popular in practice is that useful special cases have been identified (acyclic nets, treestructured nets) which can be implemented efficiently. In Section 4 we show how some of these special cases can be extended to the richer language we consider. Further, we provide a compiler for LCP theories that can recognize these special cases and generate custom code for consistency, dominance and optimization (Section 5). We present some scalability numbers.
In summary, we believe our formalization of extensions of CPnets permits an integrated treatment of preferences in constraint (logic) programming, leading to more powerful reasoning systems which can deal with both preferences and hard constraints.
Rest of the paper.
Section 2 introduces the models of GCPnets, CPnets, CPtheories and comparative preference languages, and Datalog and tabled logic programming, and provides basic computational results. Section 3 specifies the LCP formulation, and establishes that LCPtheories conservatively extend GCPnets, CPnets, CPtheories and comparative preference languages. Section 4 studies the computational complexity of outcome optimization, consistency checking and dominance queries. We also provide specific algorithms for the special cases (e.g. acyclic structure and tree structure dependency graphs). Section 5 describes the implementation of the LCP compiler. We conclude in Section 6.
2 Background
Below we assume given a set of features (or variables), . We assume for simplicity that the values in each feature are either (handling multiple values is easy).
2.1 GCPnets and CPnets
GCPnets ([GLTW08] and [DRVW03]) allow a general form of conditional and qualitative preferences to be modeled compactly.
Definition 1
A Generalized CPnet (GCPnet) over Var is a set of conditional preference rules. A conditional preference rule is an expression , where is a literal of some atom and is a propositional formula over Var that does not involve variable . A GCPnet corresponds to a directed graph (dependency graph) where each node is associated with a feature and the edges are pairs where appears in in some rule or . Each node is associated with a CPtable which expresses the user preference over the values of . Each row of the CPtable corresponds to a conditional preference rule.
The CPtables of a GCPnet can be incomplete (i.e. for some values of some variables’ parents, the preferred value of may not be specified) and/or locally inconsistent (i.e. for some values of some variables’ parents, the table may both contain the information and ). CPnets [BBD04b] are a special case of GCPnet in which the preferences are locally consistent and locally complete.
An outcome in a CPnet is a complete assignment to all features. For example, given and binary domains , all the possible outcomes are , , and .
A worsening flip is a change in the value of a feature to a value which is less preferred according to the cpstatement for that feature. This concept defines an order over the set of outcomes such that one outcome is preferred to another outcome () if and only if there is a chain of worsening flips from to . The notion of worsening flip induces a preorder over the set of outcomes. This preorder allows maximal elements that correspond to the socalled optimal outcomes, which are outcomes that have no other outcome better than them.
Given any GCPnet and CPnet the problems of consistency checking and finding optimal outcomes are PSPACEcomplete [GLTW08]. Moreover, there could be several different maximal elements. When the dependency graph has no cycle the CPnet is called acyclic. The optimal outcomes for such nets are unique and can be found in polynomial time in . The procedure used to this purpose is usually called a sweep forward and takes steps [BBD04b].
2.2 CPtheories and comparative preference languages
CPtheories are introduced in [Wil04] as a logic of conditional preference which generalizes CPnets.
Definition 2
Given a set of variables with domains , , the language is defined by all the statements of the form: where is an assignment of a set of variables , and is a set of variables such that .
Definition 3
Given a language as defined above, a conditional preference theory (CPtheory) on Var is a subset of . generates a set of preferences that corresponds to the set where given , is defined as .
A CPnet is a particular case of a CPtheory where for all .
Two graphs are associated to a CPtheory: and .
The semantics of CPtheories depends on the notion of a worsening swap, which is a change in the assignment of a set of variables to an assignment which is less preferred by a rule . We say that one outcome is better than another outcome () if and only if there is a chain of worsening swaps (a worsening swapping sequence) from to .
Definition 4
A CPtheory is locally consistent if and only if for all and in the graph , is irreflexive.
Local consistency can be determined in time proportional to . Given a CPtheory , if the graph is acyclic, is consistent if and only if is locally consistent, thus global consistency has the same complexity as local consistency given an acyclic graph .
Comparative preference theories [Wil09] are an extension of CPtheories.
Definition 5
The comparative preference language is defined by all statements of the form: where , and are subsets of Var and and are assignments respectively of the variables in and in .
Definition 6
Given a language as defined above, a comparative preference theory on Var is a subset of . generates a set of preferences that corresponds to the set where if , is defined as a pair of outcomes such that extends and extends and and agree on : .
2.3 Datalog and tabled logic programming
A Datalog program consists of a collection of definite clauses in a language with no function symbols, hence a finite Herbrand domain. Datalog programs can be implemented using tabled Logic Programming (TLP). Tabling maintains a memo table of subgoals produced in a query evaluation and their answers. If a subgoal is reached again then the information in the table can be reused, without recomputing the subgoal. This method ensures termination and improves the computational complexity for a large class of problems [SW12a] (at the expense of additional space consumption). Answer subsumption extends the functionality of tabling.
Answer variance
adds a new answer to a table only if the new answer is not a variant of any other answer already in the table. Partial order answer subsumption adds a new answer to a table only if the new answer is maximal with respect to the answers in the table, given a partial order po/2:: table predicate(_,_,partialOrder(po/2)).
Traditionally, predicates are divided into extensional and intensional predicates. The extensional predicates define a database, and intensional predicates define (possibly recursively defined) queries over the database. In our context, d/2 and outcome/1 will be considered extensional predicates (in Flat LCP) and other predicates such as dom/2, inconsistent and userdefined predicates are considered intensional. Given an intensional program , database and query , datacomplexity [Var82] is the complexity of answering as a function of the size of and (thus the program is considered fixed). Combined complexity is the complexity of answering as a function of the size of , and (thus nothing is taken to be fixed). The basic results are that for datacomplexity, general Datalog programs are PTIME
complete, and linear programs are
NLOGSPACEcomplete [GP03]. For combined complexity, general Datalog programs are EXPTIMEcomplete, and linear programs are PSPACEcomplete.3 Logical Conditional Preference Theories
We assume given a set of features, and a logical vocabulary with unary predicates d1, …, dN (corresponding to the domains of the features), constants for every value in the domains di, a single function symbol o/N, and a single binary predicate d/2.
The user specifies preferences between two outcomes S,T (expressed as o/N terms) by supplying clauses for the atom d(S,T):
d(S,T) : C, g1, ..., gk.
where C is a constraint (possibly involving equality), and g1, …, gk are possibly recursively defined predicates involving d/2. The clause is said to be flat if k=0, else it is recursive. The user specifies hard constraints on features by providing clauses for the outcome/1 predicate, typically of the form
outcome(o(X1,...,Xn)) : C, d1(X1), ..., dn(Xn).
where C is a constraint and the di are domain predicates.
The LCP runtime supplies the following definition for the dom/2 predicate, expressing (tabled) transitive closure over d/2, and for consistency and optimal outcomes:
: table(dom(_,_)). dom(X,Y): d(X,Y), outcome(X), outcome(Y). dom(X,Y): d(X,Z), dom(Z,Y). consistent : \+ dom(X,X). : table(optimal(po(dom/2))). optimal(X): outcome(X).
Note that the clauses above are linear. Below, given an LCP theory (Datalog program) , by we will mean together with the LCP runtime clauses specified above.
Given these definitions, the problem solver may use consistent to determine whether the supplied preference clauses are consistent, dom(S,T) to determine whether outcome S is preferred to T, and optimal(S) (where S may be a partially instantiated o/N structure) to determine an optimal completion of S.
Example 1 (Dinner, modified from [Bbd04b])
Two components of a meal are the soup (fish or veg) and wine (white or red). I prefer fish to veg. If I am having fish, I prefer white wine to red. I simply do not want to consider veg with red. This may be formulated as the LCP theory:
soup(fish). soup(veg). wine(white). wine(red). outcome(o(X,Y)): soup(X), wine(Y), (X\== veg; Y\==red). d(o(fish, X), o(veg, X)). d(o(fish, white),o(fish, red)).
On this theory, the query ?consistent. returns yes. The dom/2 predicates order outcomes as:
o(fish,white) > o(fish,red) o(fish,white) > o(veg,white)
Note because of hard constraints the outcomes are not totally ordered. The query optimal(X) returns the single answer X=o(fish,white); the query optimal(o(fish,X)) returns X=white; optimal(o(X,red)) returns X=fish, etc. The behavior of the optimal/1 queries will be explained later; for now observe that an optimal(O) query returns that (in this case, unique) instantiation of O which is highest in the dom/2 order.
Example 2 (Holiday Planning, [Wil04])
There are three features: time, with values l and n for later and now; place, with values m and o for Manchester and Oxford, and mode with values f and d for fly and drive. The preference “All else being equal, I would prefer to go to Manchester” is formulated as clause 1 below. The rule “I would prefer to fly rather than drive, unless I go later in the year to Manchester, where the weather will be warmer, and a car would be useful for touring around” translates to clauses 23. The CPtheory rule “I would prefer to go next week, regardless of other choices.” corresponds translated to clause 4, and the comparative preference rule “All other things being equal, I would prefer to fly now, rather than to drive later.” to clause 5:
time(n). time(l). place(o). place(m). mode(f). mode(d). outcome(o(T,P,M)): time(T), place(P), mode(M). /*1*/ d(o(X,m,Y),o(X,o,Y)). /*2*/ d(o(T,P,f),o(T,P,d)): T=n;P=o. /*3*/ d(o(l,m,d),o(l,m,f)). /*4*/ d(o(n,_,_),o(l,_,_)). /*5*/ d(o(n,X,f),o(l,X,d)).
Proposition 1 (Normal form for Flat LCP rules)
Let be a flat LCPrule d(o(), o()) : c. where c is an equality constraint. For appropriate choices of disjoint index sets , , and s.t. , and given and disjoint subsets of and given constants , , and , is logically equivalent to the clause d(o(), o()). where are defined by: , , , , , .
The set corresponds to the parent variables, the set to the ceteris paribus variables, the set to the variables that change the value from to and to the variables that are less important than the variables in . We call the lowerbound set, and the upperbound set.
Example 3
Given three variables main, drink and dessert with domains {meat, fish,veg}, {water,wine} and {cake,fruit} respectively. The rule “If I eat cake as dessert I prefer to drink water and I prefer to not eat meat”, gives Z,U,K=, J={dessert}, M={main, drink}, and L={A} (clause 1). The rule “Given the same dessert, I always prefer fish as main course, regardless of the drink.”, gives J,L= Z={drink}, K={dessert}, and M=U={main} (clause 2):
/*1*/ d(o(X1, water, cake), o(meat, wine, cake)). /*2*/ d(o(fish, X2, X3),o(Y1, Y2, X3)).
A GCPnet rule is simply a Flat LCPrule such that , , . A CPtheory rule is a Flat LCPrule with , . A comparative preference rule is a Flat LCPrule with .
The following theorems establish that LCPtheories conservatively extend these sublanguages. Proofs are straightforward, we have essentially just used standard logical notions to formalize the sublanguages:
Theorem 3.1 (Logical characterization of ceteris paribus and general ceteris paribus)
Given a CPnet , consider the set of flat LCPrules which represent all rows of its CPtables. Then, for any two outcomes and , in iff .
Theorem 3.2 (Logical characterization of CPtheories and comparative preference languages)
Given a CPtheory , consider the set of flat LCP rules modeling all the rules of the CPtheory. Given two outcomes and , iff .
3.1 Recursive LCPtheories
Recursive or dependent rules are particularly useful in multiagent contexts, where different agents may influence each other by stating their preferences depending on the preferences of some other agent [MMP13]. Here we illustrate with an extension to the Holiday planning example:
Example 4 (Holiday planning)
John and Mary work for the same office and need to travel separately. “If John prefers Oxford to Manchester (all other things being equal), then Mary prefers Manchester to Oxford (all other things being equal).”
jPlace(X,Y) : dom(o(J1, X, J3, M1, M2, M3), o(J1, Y, J3, M1, M2, M3)). d(o(J1, J2, J3, M1, m, M3), o(J1, J2, J3, M1, o, M3)).
The predicate jPlace(X,Y) may be read as “John prefers place X to place Y, all other things being equal.” The second clause may be read as saying “Mary prefers m to o, all other things being equal, provided that jPlace(o,m) holds.
Example 5
Let us consider two agents ranking features “appetizer” (rolls or bread), “main dish” (pasta or fish) and dessert (tiramisu or breadpudding). We can formulate “If Alice doesn’t prefer pasta, I would like to take pasta” as:
d(o(AA, AM, AD, MA, pasta, MD), o(AA, AM, AD, MA, fish, MD)) : dom(o(_, fish,_,_,_,_),o(_, pasta,_,_,_,_)).
Note we do not assume acyclicity in variable ordering.
4 Algorithmic properties
The main algorithmic tasks regarding a preference theory are dominance queries, consistency checking, and outcome optimization (also of interest are ordering queries, but we rule them out of scope because of limitations of space). Below we fix a set of features Var with cardinality and an LCPtheory over Var.
Checking the dominance over a pair of outcomes corresponds to finding a swapping sequence in CPtheories or a flipping sequence in CPnets. For LCPtheories this is determined by firstorder derivability: the dominance query (s,t) succeeds iff .
The problem of consistency checking is checking whether there is any outcome s such that .
The problem of outcome optimization corresponds to find the most preferred outcome given an assignment to a (possibly empty) subset of features.
Definition 7 (Optimal outcome)
An optimal outcome is an outcome s such that there is no other outcome t such that (i.e. s is an undominated outcome).
Definition 8 (Optimal completion)
Given a (possibly nonground) term s (representing a partial outcome) an optimal completion of s is a ground term t instantiating s s.t. for no other ground term t1 instantiating s is it the case that .
Note that an acyclic CPnet and CPtheory has a unique optimal outcome, but an LCPtheory may have several optimal outcomes.
Our algorithmic approach is based on analyzing whether the input LCPtheory corresponds to a special case (e.g. acyclic CPnet, treestructured dependency graph, etc.). If so, optimal algorithms for the special case are used. Otherwise general Datalog procedures are used.
In the following sections we analyze the algorithms for dominance, consistency and optimality from a general point of view, and then considering the special cases in which the algorithms are faster. We take the viewpoint of combined complexity, i.e. assuming , the clauses of the LPC theory, and the given query are supplied at runtime.
The results are summarized in Table 1.
General structure  Acyclic  Tree  
Dominance  
CPnets:  PSPACEcomp  PSPACEcomp  Polynomial 
[GLTW08]  [GLTW08]  [BBD04b, BFMZ13]  
CPtheories:  PSPACEcomp  PSPACEcomp  Polynomial 
[Wil04]  [Wil04]  
Flat LCPtheories:  PSPACEcomp  PSPACEcomp  ? 
Recursive LCPtheories:  EXPTIMEcomp  PSPACEcomp  ? 
Consistency/Optimality  
CPnets:  PSPACEcomplete  Polynomial  Polynomial 
[GLTW08]  [BBD04b]  [BBD04b]  
CPtheories:  PSPACEcomplete  Polynomial  Polynomial 
[Wil04]  [Wil04]  [Wil04]  
Flat LCPtheories:  PSPACEcomplete/?  Polynomial*  Polynomial * 
Recursive LCPtheories:  EXPTIMEcomplete  PSPACEcomplete  ? 
Bold results are provided in this paper, *with some constraint on the form of the rule, ? means the corresponding problem is open.
It is important to notice that for Recursive LCPtheories optimality and consistency procedures never have a lower computational complexity then the dominance procedure, because a recursive LCP rule also contains a dominance query.
We note in passing that for Flat LCP theories, datacomplexity is also of interest. Recall that datacomplexity for a Datalog programs is the complexity of determining, for a fixed program , and input database and query , whether (as a function of the size of and ).
What is the distinction between and for LCP theories? For Flat LCP theories, is simply the clauses for dom/2, and consistent/0. Once , the number of features is fixed, this program is fixed. Thus data complexity for consistency of LCP theories corresponds to the complexity of determining for fixed , whether , as a function of the number of rules in the program. For Flat LCP (=linear Datalog) the datacomplexity is NLogSpacecomplete (see e.g. [GP03]).
4.1 Dominance
Theorem 4.1
Given a flat LCP theory over features, deciding dom(s,t) is PSPACEcomplete in .
Proof
Since flat LCP theories can encode the GCPnets of [GLTW08], the dominance problem is at least PSPACEhard. That the problem is in PSPACE can be established in a form similar to the proof of Theorem 4.4 in [GP03]. Since has but a single rule, and the rule is linear, we can build up the proof nondeterministically using a polynomial amount of space. In fact, we need to keep space only for two ground facts of the form dom(s, t) where the s,t are constants. We start by using the base clause for dom/2 to nondeterministically establish a dom/2 fact, using some fact for d/2, and scratch space linear in . Then we use the recursive clause for dom/2 to nondeterministically generate a new dom/2 fact. From this new fact, we can generate another, and delete the old fact. We stop when dom(s,t) is established.
Theorem 4.2
Given a recursive LCP theory over features, deciding dom(s,t) is EXPTIMEcomplete in .
Proof
The proof is as above, except that the d/2 clauses may no longer be linear, hence the combined complexity for full Datalog comes into play.
We note in passing – and document in the supplementary material of this paper – that the connection with Datalog allows for a simple and direct proof of the PSPACEhardness of dominance for CPnets. We show that the wellknown PSPACEhard
problem of determining whether a deterministic Turing Machine can accept the empty string without ever moving out of the first
tape cells can be reduced to checking dominance queries for CPnets by modifying slightly the proof for Datalog in [GP03, Theorem 4.5].In practice, the solutions of the dominance problem can be found using tabling on the dom/2 predicate.
Note that in tree structured CPnets a dominance query can be computed in time linear in [BBD04b, BFMZ13]. We observe that a procedure similar to [BFMZ13] can be used for CPtheories. We use the generalized dependency graph described in [Wil04] (see Section 2.2, this includes “importance” edges and dependency edges), and when is a tree, we apply the dominance procedure of [BFMZ13].
4.2 Consistency and Outcome optimization
Consistency is determined by invoking theq query ?consistent. This takes advantage of the tabling of the dom/2 predicate.
The following theorem affirms that consistency remains in PSPACE even when the language for preferences is extended beyond CPnets to flat LCP rules, and it is a generalization of Theorem 3 in [GLTW08].
Theorem 4.3
Given a flat LCP theory over features, deciding consistency is PSPACEcomplete in .
The proof follows directly from the proof of Theorem 4.1 since consistency is reduced to checking entailment.
Theorem 4.4
Given a recursive LCP theory over features, deciding consistency is EXPTIMEcomplete in .
As above, noting that the combined complexity for full Datalog is EXPTIME.
For optimality, the user invokes the query ? optimal(s). Note that optimal/1 uses partial order answer subsumption (see Section 2.3). In theory this may result in an exponential number of calls to dom/2 atoms, with each check taking exponential time. This leads to:
Theorem 4.5
Given a recursive LCP theory over features, deciding optimality is in EXPTIME over .
For now we leave as open the corresponding problem for flat recursive theories (note that this problem is PSPACEcomplete for CPnets).
4.2.1 Optimization and Consistency for acyclic dependency graphs
In acyclic CPnets the sweepforward procedure [BBD04b] finds the unique optimal outcome (or completion) in polynomial time. A similar result holds for [Wil04].
Under the assumptions of consistency and acyclicity, an optimal outcome (and completion) can also be found for Flat LCP theories in polynomial time, using the following algorithm AcyclicLCPOpt generalizing sweepforward (see Section 5 for implementation details):

Given a set of LCPrules we compute the dependency graph , and we check if is acyclic.

We compute a total order over Var as a linearization of the topological order defined by .

For each variable , chosen following , we consider a set such that it contains all the variables that change the value jointly with in at least one rule ( could contain only ). Using the rules in the LCPtheory that involve the variables in , and given the assignments for the variables , we generate an ordering over the partial outcomes defined on the variables in . We assign to the value of the top element of the ordering that satisfies outcome/1 for at least one completion (the completion has the given assignment to and an arbitrary assignment to ).

We repeat the previous step for all the features in Var (following ), never changing the value of an assigned variable.
Proposition 2
The outcome obtained with the AcyclicLCPOpt procedure is an optimal outcome for acyclic LCPtheories.
Proof
We suppose by contradiction that exists an outcome such that . This implies that exists a chain of outcomes such that exists a rule that implies . Considering a linearization of the graph associated to our set of rules , we take the first variable in this order such that . Thus there exist a rule and an index such that and . This implies that is not the maximal element in the set of rules that involve , that is a contradiction.
In LCPtheories it is possible to have many different optimal outcomes: we can obtain the whole set of optimal outcomes using the AcyclicLCPOpt procedure. If there is more then one optimal outcome this means that there exists some variable that in the third step of the algorithm has more then one top element. Running in parallel all these possible assignments we obtain the whole set of optimal outcomes.
The complexity of the procedure is where : the third step of the procedure could involve all the partial outcomes defined on . If the program bounds , the procedure become linear in . Note that if the LCPtheory corresponds to a CPnet or a CPtheory then and the algorithm coincides with the sweepforward procedure for CPnets, and the procedure introduced in [Wil04] for CPtheories.
We can use this procedure also to compute the optimal completion of a given partial outcome, simply considering the partial outcome as preassigned values for a subset of features.
Recursive LCPtheories may require dominance queries, where is the number of dominance goals in the body of the input preference rules. Since we use tabling, the time cost is amortized over all calls from the problemsolver (in lieu of space). With this change, the procedure described above can be used. Because of the dominance queries, the optimality procedure is PSpacecomplete.
4.3 Decreasing complexity using evidence specification
We note that if evidence for some variables is given, the resulting simplified LCP theory may have the structure of one of the special cases discussed above, and hence optimality and dominance queries may be answered using specialized, polynomial procedures. To this end, the implementation needs to maintain a dynamic dependency graph that ignores features for which values have been provided.
5 Experimental evaluation
We have developed a compiler for LCP theories, also called LCP. The compiler and associated tooling will be made avaiable on Github as an open source project under the Eclipse Public Licence.
The compiler reads an LCPtheory, builds the dependency graph and checks whether it represents an acyclic CPnet. If so, it performs a linearization of the dependency graph, and produces a predigested representation of the theory. Otherwise it emits the clauses unchanged so that the standard default (tabled) algorithms optimality, consistency and dominance can be used.
In more detail, the compiler captures (a linearization of) the dependency order in a clause
dependency([]). 
where (features are implemented as Prolog integers), and if depends on , then . Suppose depends on . If the input LCP program specifies that if each of the features had values respectively, then the known order of values of is given by (best) (worst), then the compiler emits the fact
preference([], []). 
with as constant, and the remaining as unique variables (occur only once in the clause). Thus we use Prolog unification to select the correct preference clause to use, given the current partial outcome [v_p, …, v_1] specifying values for the first p attributes (in reverse dependency order).
The following code for optimize/1 uses these clauses and implements AcyclicLCPOpt:
optimize(O) : O=o(_,_), dependency(D), reorder(D, O, AList), optimize_a([], AList). reorder([], _, []). reorder([X Xs], O, [V  Vs]) : arg(X, O, V), reorder(Xs, O, Vs). optimize_a(_, []). optimize_a(Upargs, [XsR]) : select(Upargs, Xs), append(Xs, Upargs, Upargs1), optimize_a(Upargs1, R). select(Us, []). select(Us, [X  R]) : nonvar(X), select(Us, R). select(Us, [X  R]) : var(X), preference(Upargs, [X_]), select(Us ,R).
We have also implemented a CPnet generator. Generating CPnets i.i.d. is nontrivial [AGM14] and therefore we use an approximation method that randomly generates acyclic CPnets with features, given a maximum number of dependencies for each feature. We consider a fixed ordering of features. We also take as input the maximum indegree for each feature, . We first generate the acyclic dependency graph. For each feature , we randomly choose its indegree . Next, we randomly choose parents from the features . When the graph is built, we fill in the CPtables choosing randomly one element of the domain (since the domain is binary). The resulting CPnet is written out as an LCP theory, using XSB Version 3.5.0 syntax [SW12b].
We have run two different kinds of experiment: in the first with a fixed upper bound for the number of dependencies for each feature, we varied the number of features from to and measure running time for optimality queries. In the second experiment, fixing the number of features, we varied the upper bound of dependencies from to . In both experiments we asked for the optimal outcome times and then we computed the average elapsed time to output the result.
Figure 1 shows the results for the experiment where the upper bound for the number of dependencies is fixed to . The elapsed time to compute the optimal outcome grows quadratically in the number of features. This is in line with our results as summarized in Table 1. (The runtime is not linear because each step involves checking for the value of parents using unification on terms.)
6 Conclusion and future work
We have presented a new framework for conditional preferences, based on expressing preferences using Datalog. We have shown how dominance, consistency and optimality queries can be formulated directly in Datalog and implemented in modern tabled Prolog systems such as XSB Prolog. We have also analyzed the complexity of the different algorithms, developed efficient procedures for some common use cases, and implemented a translator that exploits these algorithms.
Much work lies ahead. Table 1 contains some open complexity questions. We hope to exploit Datalog theory to develop more efficient special cases. We hope to use the implemented LCP system in reallife applications to determine the adequacy of the LCP system.
Acknowledgements.
We gratefully acknowledge conversations with David S. Warren and Terrance Swift. These conversations led to a small codechange in the XSB compiler to better support LCP theories. We also acknowledge conversations with Francesca Rossi.
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