Algorithmically Efficient Syntactic Characterization of Possibility Domains

We call domain any arbitrary subset of a Cartesian power of the set {0,1} when we think of it as reflecting abstract rationality restrictions on vectors of two-valued judgments on a number of issues. In Computational Social Choice Theory, and in particular in the theory of judgment aggregation, a domain is called a possibility domain if it admits a non-dictatorial aggregator, i.e. if for some k there exists a unanimous (idempotent) function F:D^k → D which is not a projection function. We prove that a domain is a possibility domain if and only if there is a propositional formula of a certain syntactic form whose set of satisfying truth assignments, or models, comprise the domain. A formula whose set of satisfying truth assignments is equal to a given arbitrary domain D is sometimes referred to as an integrity constraint for D. So we call possibility integrity constraints the formulas of the specific syntactic type we define. Given a possibility domain D, we show how to construct a possibility integrity constraint for D efficiently, i.e, in polynomial time in the size of the domain. We also show how to distinguish formulas that are possibility integrity constraints in linear time in the size of the input formula. Our result falls in the realm of classical results that give syntactic characterizations of logical relations that have certain closure properties, like e.g. the result that logical relations component-wise closed under logical AND are precisely the models of Horn formulas. However, our techniques draw from results in judgment aggregation theory as well from results about propositional formulas and logical relations.

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1 Introduction

We call domain any arbitrary subset of a Cartesian power () when we think of it as the set of yes/no ballots, or accept/reject judgment vectors on issues that are “rational” in the sense manifested by being a member of the subset. A domain has a non-dictatorial aggregator if for some there is a unanimous (idempotent) function that is not a projection function. The theory of judgment aggregation was put in this abstract framework by Wilson [20], and then elaborated by several others (see e.g. the work by Dietrich [3] and Dokow and Holzman [5, 4]). It can be trivially shown that non-dictatorial aggregators always exist unless we demand that is defined on an issue by issue fashion (see next section for formal definitions). Such aggregators are called Independent of Irrelevant Alternatives (IIA). In this work aggregators are assumed to be IIA.

It is a well known fact from elementary Propositional Logic that for every subset of , , i.e. for every domain, there is a Boolean formula in Conjunctive Normal Form (CNF) whose set of satisfying truth assignments, or models, denoted by , is equal to (see e.g. Enderton [7, Theorem 15B]). Zanuttini and Hébrard [22] give an algorithm that finds such a formula and runs in polynomial-time with respect to the size of the representation of as input. Following Grandi and Endriss [8], we call such a an integrity constraint and think of it as expressing the “rationality” of (the term comes from databases, see e.g. [6]).

We prove that a domain is a possibility domain, if and only if it admits an integrity constraint of a certain syntactic form to be precisely defined, which we call a possibility integrity constraint. Very roughly, possibility integrity constraints are formulas that belong to one of three types, the first two of which correspond to “easy” cases of possibility domains: (i) formulas whose variables can be partitioned into two subsets so that no clause contains variables from both sets and (ii) formulas whose clauses are exclusive OR’s of their literals. The most interesting third type is comprised of formulas such that if we change the logical sign of some of their variables, we get formulas that have a Horn part and whose remaining clauses contain only negative occurrences of the variables in the Horn part. We call such formulas renamable partially Horn, whereas we call partially Horn111A weaker notion of Horn formulas has appeared before in the work of Yamasaki and Doshita [21]; however our notion is incomparable with theirs, in the sense that the class of partially Horn formulas in neither a subset nor a superset (nor equal) to the class they define. the formulas that belong to the third type without having to rename any variables. Furthermore, we show that any integrity constraint for a possibility domain that is prime (i.e. we cannot further simplify its clauses; see Definition 2) is a possibility integrity constraint. Actually, in addition to the syntactical characterization of possibility domains, we give two algorithms: the first on input a formula decides whether it is a possibility integrity constraint in time linear in the length of the formula (notice that the definition of possibility integrity constraint entails searching over all subsets of variables of the formula); the second on input a domain halts in time polynomial in the size of and either decides that is not a possibility domain or otherwise returns a possibility integrity constraint that describes . It should be noted that the satisfiability problem remains NP-complete even when restricted to formulas that are partially Horn. However in Computational Social Choice, domains are considered to be non-empty (see paragraph preceding Example 2).

As examples of similar classical results in the theory of Boolean relations, we mention that domains component-wise closed under or have been identified with the class of domains that are models of Horn or dual-Horn formulas respectively (see Dechter and Pearl [1]). Also it is known that a domain is component-wise closed under the ternary sum if and only if it is the set of models of a formula that is a conjunction of subformulas each of which is an exclusive OR (the term “ternary” refers to the number of bits to be summed). Finally, a domain is closed under the ternary majority operator if and only it is the set of models of a CNF formula where each clause has at most two literals. The latter two results are due to Schaefer [19]. The ternary majority operator is the ternary Boolean function that returns 1 on input three bits if and only if at least two of them are 1. It is also known that the respective formulas for each case can be found in polynomial time with respect to the size of (see Zanuttini and Hébrard [22]).

Our result can be interpreted as verifying that non-dictatorial voting schemes can always be generated by integrity constraints that have a specific, easily recognizable syntactic form. The proofs draw from results in judgment aggregation theory as well from results about propositional formulas and logical relations. Specifically, as stepping stones for our algorithmic syntactic characterization we use three results. First, a theorem implicit in Dokow and Holzman [4] stating that a domain is a possibility domain if and only if it either admits a binary (of arity 2) non-dictatorial aggregator or it is component-wise closed under the ternary direct sum. Second, a theorem by Kirousis et al. [10] which states that the problem of whether a domain is a possibility domain is in the complexity class . Third the “unified framework for structure identification” by Zanuttini and Hébrard [22] (see next section for definitions).

The historically initial approach to judgment theory was to define domains not in an abstract fashion as arbitrary logical relations, but via a sequence of propositional formulas , known as the agenda (see List and Petit [13]). Specifically, if given a formula , we adopt the notation

 ϕx:={ϕ if x=1,¬ϕ if x=0,

then the domain that corresponds to rational judgments on the sequence of propositional formulas is by definition:

 D:={¯x=(x1,…,xn)∈{0,1}n∣∣∣n⋀j=1ϕxjj is satisfiable }.

This approach is sometimes referred to as logic-based, in contrast to the abstract one described above. Dokow and Holzman [4] proved that every domain can be expressed as the set of rational judgments on some comprised of CNF propositional formulas. In the framework of logic-based aggregation, several authors have studied syntactic conditions on the elements of the agenda that are either sufficient or both necessary and sufficient to give an impossibility domain (with varying, non-equivalent, senses of impossibility). In these results, the agenda is assumed to be truth-functional, meaning that all propositional variables should be part of the agenda. This requirement, it must be noted, severely restricts the domains that are considered. As examples of such work, we mention Dokow and Holzman [4], whose work is presented in a framework that is closer to ours, and who give necessary and sufficient conditions for a truth-functional agenda to yield an impossibility domain. Nehring and Puppe [15] give, again for truth functional agendas, necessary and sufficient conditions for another notion of impossibility. Finally Pauly and van Hees [17] and Mogin [14] give only sufficient conditions for notions of impossibility they introduce. However, none of these syntactic conditions is examined from the algorithmic point of view. Actually they invariably entail searching over subsets of the agenda, so they are not algorithmically efficient.

2 Preliminaries

We first give the notation and basic definitions from Propositional Logic and judgment aggregation theory that we will use.

Let be a set of Boolean variables. A literal is either a variable (positive literal) or a negation of it (negative literal). A clause is a disjunction of literals from different variables. A propositional formula (or just a “formula”, without the specification “propositional”, if clear from the context) in Conjunctive Normal Form (CNF) is a conjunction of clauses. A formula is called -CNF if every clause of it contains exactly literals. A (truth) assignment to the variables is an assignment of either 0 or 1 to each of the variables. We denote by the value of under the assignment . Truth assignments will be identified with elements of , or -sequences of bits. The truth value of a formula for an assignment is computed by the usual rules that apply to logical connectives. The set of satisfying (returning the value 1) truth assignments, or models, of a formula, is denoted by . In what follows, we will assume, except if specifically noted, that denotes the number of variables of a formula and the number of its clauses.

We say that a variable appears positively (resp. negatively) in a clause , if (resp. ) is a literal of . A variable is positively (resp. negatively) pure if it has only positive (resp. negative) appearances in .

A Horn clause is a clause with at most one positive literal. A dual Horn is a clause with at most one negative literal. A formula that contains only Horn (dual Horn) clauses is called Horn (dual Horn, respectively). Generalizing the notion of a clause, we will also call clauses sets of literals connected with exclusive OR (or direct sum), the logical connective that corresponds to summation in . Formulas obtained by considering a conjunction of such clauses are called affine. Finally, bijunctive are called the formulas whose clauses, in inclusive disjunctive form, have at most two literals. A domain is called Horn, dual Horn, affine or bijunctive respectively, if there is a Horn, dual Horn, affine or bijunctive formula of variables such that . In the previous section, we mentioned efficient solutions to classical syntactic characterization problems for classes of relations with given closure properties on one hand, and formulas of the syntactic forms mentioned above on the other.

We have presented the above notions and results without many details, as they are all classical results. For the notions that follow we give more detailed definitions and examples. The first one, as far as we can tell, dates back to 1978 (see Lewis [11]).

A formula whose variables are among the elements of the set is called renamable Horn, if there is a subset so that if we replace every appearance of every negated literal from with the corresponding positive one and vice versa, is transformed to a Horn formula. The process of replacing the literals of some variables with their logical opposite ones, is called a renaming of the variables of . Consider the formulas and , defined over .

The formula is renamable Horn. To see this, let . By renaming these variables, we get the Horn formula . On the other hand, it is easy to check that cannot be transformed into a Horn formula for any subset of , since for the first clause to become Horn, at least two variables from have to be renamed, which will make the second clause not Horn.

It turns out that whether a formula is renamable Horn can be checked in linear time. There are several algorithms that do that in the literature, with the one of del Val [2] being a relatively recent such example. The original non-linear one was given by Lewis [11].

We now proceed with introducing several syntactic types of formulas: A formula is called separable if its variables can be partitioned into two non-empty disjoint subsets so that no clause of it contains literals from both subsets. The formula is separable. Indeed, for the partition , of , we have that no clause of contains variables from both subsets of the partition. On the other hand, there is no such partition of for neither nor of the previous example.

The fact that separable formulas can be recognized in linear time is relatively straightforward (see Proposition 3.1 in Subsection 3.1).

We now introduce the following notions:

A formula is called partially Horn if there is a nonempty subset such that (i) the clauses containing only variables from are Horn and (ii) the variables of appear only negatively (if at all) in a clause containing also variables not in . If a formula is partially Horn, then any non-empty subset that satisfies the requirements of Definition 2 will be called an admissible set of variables. Also the Horn clauses that contain variables only from will be called admissible clauses (the set of admissible clauses might be empty). It is possible to be necessary not to include a Horn clause among the set of admissible Horn clauses (see following example). Such a Horn clause will be called inadmissible.

Notice that a Horn formula is, trivially, partially Horn too, as is a formula that contains at least one negative pure literal. It immedaitely follows that the satisfiability probelm remains NP-complete even when restricted to partially Horn formulas (just add a dummy negative pure literal). However, in Computational Social Choice, domains are considered to be non-empty as a non-degeneracy condition. Actually, it is usually assumed that the projection of a domain to any one of the issues is the set .

We first examine the formulas of the previous examples. is partially Horn, since it contains the negative pure literal . The Horn formula is also trivially partially Horn. On the other hand, and are not, since for every possible , we either get non-Horn clauses containing variables only from , or variables of that appear positively in inadmissible clauses.

The formula is partially Horn. Its first three clauses are Horn, though the third has to be put in every inadmissible set, since appears positively in the forth clause which is not Horn. The first two clauses though constitute an admissible set of Horn clauses. Finally, let . Indeed, since all its variables appear positively in some clause, we need at least one clause to be admissible. The first two clauses of are Horn, but we will show that they both have to be included in an inadmissible set. Indeed, the second has to belong to every inadmissible set since appears positively in the third, not Horn, clause. Furthermore, appears positively in the second clause, which we just showed to belong to every inadmissible set. Thus, the first clause also has to be included in every inadmissible set, and therefore is not partially Horn.

Accordingly to the case of renamable Horn formulas, we define: A formula is called renamable partially Horn if some of its variables can be renamed (in the sense of Definition 2) so that it becomes partially Horn. Observe that any Horn, renamable horn or partially Horn formula is trivially renamable partially Horn. Also, a formula with at least one pure positive literal is renamable partially Horn, since by renaming the corresponding variable, we get a formula with a pure negative literal. All formulas of the previous examples are renamable partially Horn: , and correspond to the trivial cases we discussed above, whereas , and all contain the pure positive literal .

Lastly, we examine two more formulas: is easily not partially Horn, but by renaming , we obtain the partially Horn formula , where is the set of admissible variables. One the other hand, the formula is not renamable partially Horn. Indeed, whichever variables we rename, we end up with one Horn and one non-Horn clause, with at least one variable of the Horn clause appearing positively in the non-Horn clause. We prove, by Theorem 3.1 in Subsection 3.1 that checking whether a formula is renamable partially Horn can be done in linear time in the length of the formula. Let be a renamable partially Horn formula, and let be a partially Horn formula obtained by renaming some of the variables of , with being the admissible set of variables. Let also be an admissible set of Horn clauses in . We can assume that only variables of have been renamed, since the other variables are not involved in the definition of being partially Horn. Also, we can assume that a Horn clause of whose variables appear only in clauses in belongs to . Indeed, if not, we can add it to . A formula is called a possibility integrity constraint if it is either separable, or renamable partially Horn or affine. From the above and the fact that checking whether a formula is affine is easy we get Theorem 3.1 in Subsection 3.1, which states that checking whether a formula is a possibility integrity constraint can be done in polynomial time in the size of the formula.

Finally, give a clause of a formula , we say that a sub-clause of is any non-empty clause created by deleting at least one literal of . In Quine [18] and Zanuttini and Hébrard [22], we find the following definitions: A clause of a formula is a prime implicate of if no sub-clause of is logically implied by . Furthermore, is prime if all its clauses are prime implicates of it. In sub-section 3.2, we show that any prime formula logically equivalent to a possibility integrity constraint, is itself a possibility integrity constraint.

We come now to some notions from Social Choice Theory (for an introduction, see e.g. List [12]). In the sequel, we will deal with sequences of -bit-vectors, each of which belongs to a fixed domain . It is convenient to present such sequences with an matrix with bits as entries. The rows of this matrix are denoted by and the columns by . Each row represents a row-vector of 0/1 decisions on issues by one of individuals. Each column represents the column-vector of the positions of all individuals on a particular issue.

In Social Choice Theory, is said to have a -ary (of arity ) unanimous aggregator if there are exists a sequence of -ary Boolean functions , such that

• all are unanimous, i.e if are equal bits, then

 fj(b1,…,bk)=b1=⋯=bk, and
• if for a matrix that represents the opinions of individuals on issues we have that the row-vectors for all , then

 (f1(x1),…,fn(xn))∈D.

Notice that in the second bullet above, the ’s are applied to column-vectors, which have dimension . The ’s are called the components of the aggregator . Intuitively, an aggregator is a sequence of functions that when applied onto some rational opinion vectors of individuals on issues, in a issue-by-issue fashion, they return a row-vector that is still rational. From now on, we will refer to unanimous aggregators, simply as aggregators.

An aggregator is called dictatorial if there is a such that , where is the -ary projection function on the ’th coordinate.

A -ary aggregator is called a projection aggregator if each of its components is a projection function , for some .

Notice that it is conceivable to have non-dictatorial aggregators that are projection aggregators.

A binary (of arity 2) Boolean function is called symmetric if for all pairs of bits , we have that . A binary aggregator is called symmetric if all its components are symmetric. Let us mention here the easily to check fact that the only unanimous binary functions are the , and the two projection functions . Of those four, only the first two are symmetric. It is convenient to define: The description of a binary aggregator is a quaternary vector of dimension that for each determines wether is .

A domain is called a possibility domain if it has a (unanimous) non-dictatorial aggregator of some arity. Notice that the search space for such an aggregator is large, as the arity is not restricted. However, from [9, Theorem 3.7] (a result that follows from Dokow and Holzman [4], but without being explicitly mentioned there), we can easily get that: [Dokow and Holzman [4]] A domain is a possibility domain if and only if one of the following is true:

1. admits a non-dictatorial binary projection aggregator.

2. admits a non-projection binary aggregator (i.e. with at least one component that is either or ).

3. admits a ternary aggregator all components of which are the binary addition .

Furthermore, by [10, Theorem 1] we have that:

[Kirousis, Kolaitis and Livieratos [10]] Let . There is a polynomial-time algorithm that determines if admits a binary non-dictatorial aggregator and, if it does, produces one.

The above results will be used in the proofs of our results.

3 Syntactic characterization of possibility domains by possibility integrity constraints

3.1 Identifying possibility integrity constraints

In this subsection, we show that identifying possibility integrity constraints can be done in time linear in the length of the input formula. By Definition 2, it suffices to show that for separable and renamable partially Horn formulas, since the corresponding problem for affine formulas is trivial.

In all that follows, we assume that we have a set of variables and a formula defined on that is a conjunction of clauses , where , , and is a positive or negative literal of , . We denote the set of variables corresponding to the literals of a clause by . Also, we say that a formula is a sub-formula of , if any clause of is a clause of .

We begin with the result for separable formulas: There is an algorithm that, on input a formula , halts in time linear in the length of and either returns that the formula is not separable, or alternatively produces a partition of in two non-empty and disjoint subsets , such that no clause of contains variables from both and .

Proof.

Suppose the variables of each clause are ordered by the indices of their corresponding literals in the clause. Thus, we say that are consecutive in , if , .

Given a formula , construct an undirected graph , where :

• is the set of variables of , and

• two vertices are connected if they appear consecutively in a common clause of .

Thus each clause , where induces the path in . For the sake of completeness, we also provide the pseudocode of this algorithm (see Algorithm 1). For the proof of linearity, notice that the set of edges can be constructed in linear time with respect to the length of , since we simply need to read once each clause of and connect its consecutive vertices. Also, there are standard techniques to check connectivity in linear time in the number of edges (e.g. by a depth-first search algorithm).

The correctness of the algorithm is derived by noticing that two connected vertices of cannot be separated in . Indeed, consider a path in (this need not be a path induced by a clause). Then, each couple of vertices in belongs in a common clause of , . Thus, is separable if and only if is not connected. ∎

To deal with renamable partially Horn formulas, we will start with Lewis’ idea [11] of creating, for a formula , a 2Sat formula whose satisfiability is equivalent to being renamable Horn. Then, we can simply check if is satisfiable by one of the known algorithms for 2Sat. However, here we need to (i) look for a renaming that might transform only some clauses into Horn and (ii) deal with inadmissible Horn clauses, since such clauses can cause other Horn clauses to become inadmissible too. For every formula , there is a 2Sat formula such that is renamable partially Horn if and only if is satisfiable. Before delving into the proof, we introduce some notation. Assume that after a renaming of some of the variables in , we get the partially Horn formula , with being the admissible set of variables. Let be an admissible set of clauses for . We assume below that only a subset has been renamed and that all Horn clauses of with variables exclusively from belong to (see Remark 2). Also, let . The clauses of , which are in a one to one correspondence with those of , are denoted by , where corresponds to , .

Proof.

For each variable , we introduce a new variable . Intuitively, setting means that is renamed (and therefore ), whereas setting means that is in , but is not renamed. Finally we set both and equal to in case is not in . Obviously, we should not not allow the assignment (a variable in cannot be renamed and not renamed). Let .

Consider the 2Sat formula below, with variable set . For each clause of and for each : if appears positively in , introduce the literals and and if it appears negatively, the literals and . is the conjunction of the following clauses: for each clause of and for each two variables , contains the disjunctions of the positive with the negative literals introduced above. Thus:

• if contains the literals , then contains the clauses and ,

• if contains the literals , then contains the clauses and (accordingly if contains ) and

• if contains the literals , then contains the clauses and .

Finally, we add the following clauses to :

• , and

• .

The clauses of items (i)–(iv) correspond to the intuition we explained in the beginning. For example, consider the case where a clause of has the literals . If we add to without renaming it, we should not rename , since we would have two positive literals in a clause of . Also, we should not add the latter to , since we would have a variable of appearing positively in a clause containing a variable of . Thus, we have that , which is expressed by the equivalent clause of item (ii). The clauses of item (iv) exclude the assignment for any . Finally, since we want to be non-empty, we need at least one variable of to be set to .

To complete the proof of Proposition 3.1, we now proceed as follows.

( First, suppose is renamable partially Horn. Let , , and as above. Suppose also that .

Set to be the following assignment of values to the variables of :

 a(x)={1, if x∈V∗,0, else, and a(x′)={0, % if x∈V∗∪V1,1, else,

for all . To obtain a contradiction, suppose does not satisfy .

Obviously, the clauses of items (iv) and (v) above are satisfied, by the definition of and the fact that is not empty.

Now, consider the remaining clauses of items (i)–(iii) above and suppose for example that some is not satisfied. By the definition of , there exists a clause which, before the renaming takes place, contains the literals (see item (iii)). Since the clause is not satisfied, and , which in turn means that and . If , contains, after the renaming, a variable in and a positive appearance of a variable in . If , contains two positive literals of variables in . Contradiction. The remaining cases can be proven analogously and are left to the reader.

() Suppose now that is an assignment of values to the variables of that satisfies and let:

 V∗={x∣a(x)=1}, V0={x∣a(x)=1 or a(x′)=1} and V1={x∣a(x)=a(x′)=0}.

Let be the formula obtained by , after renaming the variables of .

Obviously, is not empty, since satisfies the clause of item (v).

Suppose that a clause , containing only variables from , is not Horn. Then, contains two positive literals . If , then neither variable was renamed and thus also contains the literals . This means that, by item (i) above, contains the clauses and . Now, since , it holds that and . Then, does not satisfy these two clauses. Contradiction. In the same way, we obtain contradictions in cases that at least one of and is in .

Finally, suppose that there is a variable that appears positively in a clause . Let be a variable in (there is at least one such variable, lest ). Suppose also that appears positively in .

Assume . Then, contains the literals . Thus, by item (ii), contains the clause . Furthermore, since , and since , . Thus the above clause is not satisfied. Contradiction. In the same way, we obtain contradictions in all the remaining cases. ∎

Although checking the satisfiability of a 2Sat formula can be done in linear time, to compute from , one would need quadratic time in the length of . Thus, we introduce the following linear algorithm that decides if a formula is renamable partially Horn. Note that is never computed and is only used to make the proof easier to follow.

There is an algorithm that, on input a formula , halts in time linear in the length of and either returns that is not renamable partially Horn or alternatively produces a subset such that the formula obtained from by renaming the literals of variables in is partially Horn. To prove Theorem 3.1, we define a directed bipartite graph , i.e. a directed graph whose set of vertices is partitioned in two sets such that no vertices belonging in the same part are adjacent. Then, by computing its strongly connected components (scc), i.e. its maximal sets of vertices such that every two of them are connected by a directed path, we show that at least one of them is not bad (does not contain a pair of vertices we will specify below) if and only if is renamable partially Horn.

For a directed graph , we will denote a directed edge from a vertex to a vertex by . A (directed) path from to , containing the vertices , will be denoted by and its existence by . If both and exist, we will sometimes write .

Recall that given a directed graph , there are known algorithms that can compute the scc of in time , where denotes the number of vertices of and that of its edges. By identifying the vertices of each scc, we obtain a directed acyclic graph (DAG). An ordering of the vertices of a graph is called topological if there are no edges such that , for all .

Proof.

Given defined on , whose set of clauses is and let again . We define the graph , with vertex set and edge set such that, if and , then:

• if appears negatively in , contains and ,

• if appears positively in , contains and and

• contains no other edges.

Intuitively, if , then a path corresponds to the clause which is logically equivalent to . The intuition behind and is exactly the same as in Proposition 3.1. We will thus show that the bipartite graph defined above, contains all the necessary information to decide if is satisfiable, with the difference that can obviously be constructed in time linear in the length of the input formula.

There is a slight technicality arising here since, by the construction above, always contains either the path or , for any clause and , whereas neither nor are ever clauses of . Thus, from now on, we will assume that no path can contain the vertices , and or , and consecutively, for any clause and .

Observe that by construction, (i) or is an edge of if and only if , and (ii) (resp. ) is an edge of if and only if (resp.) is one too.

We now prove several claims concerning the structure of . To make notation less cumbersome, assume that for an , . Consider the formula of Proposition 3.1. Let . For and , it holds that is a path of if and only if , , and are all clauses of . Proof of Claim. Can be easily proved inductively to the length of the path, by recalling that a path corresponds to the clause , for all and . Let . If , then . Proof of Claim. Since , there exist and , such that is a path of . By Claim 3.1, , , and are all clauses of . By Proposition 3.1, so do , , and and the result is obtained by using Claim 3.1 again.

We can obtain the scc’s  of using a variation of a depth-first search (DFS) algorithm, that, whenever it goes from a vertex (resp. ) to a vertex , it cannot then go to (resp. ) at the next step. Since the algorithm runs in time linear in the number of the vertices and the edges of , it is also linear in the length of the input formula .

Let be a scc of . We say that is bad, if, for some , contains both and . We can decide if each of the scc’s is bad or not again in time linear in the length of the input formula. Let be a bad scc of and be a vertex of . Then, is in . Proof of Claim. Since is bad, there exist two vertices of in . If we have nothing to prove, so we assume that . Then, we have that , which, by Claim 3.1 implies that . Since , we get that . That can be proven analogously. .

Let the scc’s  of , in reverse topological order, be . We describe a process of assigning values to the variables of :

1. Set every variable that appears in a bad scc of to .

2. For each assign value to every variable of that has not already received one (if is bad no such variable exists). If some of takes value , then assign value to .

3. Let be the resulting assignment to the variables of .

Now, the last claim we prove is the following: There is at least one variable that does not appear in a bad scc of if and only if is satisfiable. Proof of Claim. ( We prove that every clause of type (i)–(v) is satisfied. First, by the construction of , every clause , , of type (iv) is obviously satisfied. Also, since by the hypothesis, is not in a bad scc, it holds, by step 2 above, that either or are set to . Thus, the clause of type (v) is also satisfied.

Now, suppose some clause (type (i)) of is not satisfied. Then and . Furthermore, there is a vertex such that and are edges of . By the construction of , and are also edges of .

Since , it must hold either that is in a bad scc of , or that . In the former case, we have that , which, together with and gives us that . Contradiction, since then should be . In the latter case, we have that there are two scc’s , of such that , and in their topological order. But then, there is some such that in . Now, if , we obtain a contradiction due to the existence of , else, due to .

The proof for the rest of the clauses of types (i)–(iii) are left to the reader.

() Suppose that all appear in bad scc’s of . Then, , for all and thus the clause of type (v) is not satisfied.

By Proposition 3.1, we have seen that is renamable partially Horn if and only if is satisfiable. Also, in case is satisfiable, a variable is renamed if and only if .

Thus, by the above and Claim 3.1, is renamable partially Horn if and only if there is some variable that does not appear in a bad scc of . Furthermore, the process described in order to obtain assignment is linear in the length of the input formula, and provides the information about which variables to rename. ∎

Because checking whether a formula is affine can be trivially done in linear time, we get: There is an algorithm that, on input a formula , halts in linear time in the length of and either returns that is not a possibility integrity constraint, or alternatively, (i) in case is separable, it produces two non-empty and disjoint subsets such that no clause of contains variables from both and and (ii) in case is renamable partially Horn, it produces a subset such that the formula obtained from by renaming the literals of variables in is partially Horn.

3.2 Syntactic Characterization of possibility domains

In this subsection, we provide a syntactic characterization for possibility domains, by proving they are the models of possibility integrity constraints. Furthermore, we show that given a possibility domain , we can produce a possibility integrity constraint, whose set of models is , in time polynomial in the size of .

To obtain the characterization, we proceed as follows. We separately show that each type of a possibility integrity constraint of Definition 2 corresponds to one of the conditions of Theorem 2:

• Domains admitting non-dictatorial binary projection aggragators are the sets of models of separable formulas.

• Domains admitting non-projection binary aggregators are the sets of models of renamable partially Horn formulas.

• Affine domains are the sets of models of affine formulas.

We will need some additional notation. For a set of indices , let be the projection of to the indices of and . Also, for two (partial) vectors , and , we define their concatenation to be the vector ). Finally, given two subsets , we say that if we can obtain by permuting the coordinates of , i.e. if , where .

We begin with characterizing the domains closed under a non-dictatorial projection aggregator as the models of separable formulas. Recall that a partition of a set is a collection of non-empty and pairwise disjoint subsets of such that . We will need the following fairly straightforward Lemma.

is closed under a binary non-dictatorial projection aggregator if and only if there exists a partition of such that .

Proof.

() Let be a binary non-dictatorial projection aggregator for . Assume, without loss of generality, that , and , . Let also and . Since , is a partition of . To prove that , it suffices to prove that (the reverse inclusion is always true).

Let and . It holds that there exists an and a such that both . Thus:

 (f1,…,fn)(ab′,a′b)=ab∈D,

since is an aggregator for , , and , .

() Suppose that , where is a partition of . Assume, without loss of generality, that , and (thus ). Let also , where and .

Obviously, if is an -tuple of projections, such that , and , , then , since and . Thus is a non-dictatorial projection aggregator for . ∎

We are now ready to prove the syntactic characterization for domains closed under binary non-dictatorial projection aggregators.

admits a binary non-dictatorial projection aggregator if and only if there exists a separable formula whose set of models equals . Furthermore, there is an algorithm that, on input and given the description of , halts in time and computes a separable formula containing clauses, whose set of models is .

Proof.

() Since admits a binary non-dictatorial projection aggregator , by Lemma 3.2, , where is a partition of such that and .

Using Zanuttini and Hébrard’s result [22], we can efficiently compute, in time , two formulas , where is defined on and on , such that and and each formula contains clauses.

Let . It is straightforward to observe that, since and contain no common variables:

 Mod(ϕ)≈Mod(ϕ1)×Mod(ϕ2)=DI×DJ≈D.

() Assume that is separable and that . Since is separable, we can find a partition of , a formula defined on and a defined on , such that . Easily, it holds that:

 Mod(ϕ)≈Mod(ϕ1)×Mod(ϕ2)=DI×DJ≈D.

The required now follows by Lemma 3.2. ∎

We now turn our attention to domains closed under binary non projection aggregators. Again, we first need a Lemma. Suppose admits a binary aggregator such that, for some , , for all . For each , let be such that:

 d∗j={1−dj if j∈J,dj else,

for