The Satisfiability Problem for Boolean Set Theory with a Choice Correspondence

09/07/2017 ∙ by Domenico Cantone, et al. ∙ York University University of Catania 0

Given a set U of alternatives, a choice (correspondence) on U is a contractive map c defined on a family Omega of nonempty subsets of U. Semantically, a choice c associates to each menu A in Omega a nonempty subset c(A) of A comprising all elements of A that are deemed selectable by an agent. A choice on U is total if its domain is the powerset of U minus the empty set, and partial otherwise. According to the theory of revealed preferences, a choice is rationalizable if it can be retrieved from a binary relation on U by taking all maximal elements of each menu. It is well-known that rationalizable choices are characterized by the satisfaction of suitable axioms of consistency, which codify logical rules of selection within menus. For instance, WARP (Weak Axiom of Revealed Preference) characterizes choices rationalizable by a transitive relation. Here we study the satisfiability problem for unquantified formulae of an elementary fragment of set theory involving a choice function symbol c, the Boolean set operators and the singleton, the equality and inclusion predicates, and the propositional connectives. In particular, we consider the cases in which the interpretation of c satisfies any combination of two specific axioms of consistency, whose conjunction is equivalent to WARP. In two cases we prove that the related satisfiability problem is NP-complete, whereas in the remaining cases we obtain NP-completeness under the additional assumption that the number of choice terms is constant.

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1 Introduction

In this paper we examine the decidability of the satisfiability problem connected to rational choice theory, which is a framework to model social and economic behavior. A choice on a set of alternatives is a correspondence associating to “feasible menus” nonempty “choice sets” . This choice can be either total (or full) – i.e, defined for all nonempty subsets of the ground set of alternatives – or partial – i.e., defined only for suitable subsets of .

According to the Theory of Revealed Preferences pioneered by the economist Paul Samuelson [18], preferences of consumers can be derived from their purchasing habits: in a nutshell, an agent’s choice behavior is observed, and her underlying preference structure is inferred. The preference revealed by a primitive choice is typically modeled by a binary relation on . The asymmetric part of this relation is informative of a “strict revealed preference” of an item over another one, whereas its symmetric part codifies a “revealed similarity” of items. Then a choice is said to be rationalizable when the observed behavior can be univocally retrieved by maximizing the relation of revealed preference.

Since the seminal paper of Samuelson, a lot of attention has been devoted to notions of rationality within the framework of choice theory: see, among the many contributions to the topic, the classical papers [16, 3, 17, 15, 20]. (See also the book [2] for the analysis of the links among the theories of choice, preference, and utility. For a very recent contribution witnessing the fervent research on the topic, see [11].) Classically, the rationality of an observed choice behavior is connected to the satisfaction of suitable axioms of choice consistency: these are rules of selections of items within menus, codified by means of sentences of second-order monadic logic, universally quantified over menus. Among the several axioms introduced in the specialized literature, let us recall the following:

standard contraction consistency , introduced by Chernoff [12];

standard expansion consistency , and binary expansion consistency , both due to Sen [20];

the weak axiom of revealed preference (WARP), due to Samuelson [18].
It is well-known that, under suitable assumptions on the domain, a choice is rationalizable if and only if the two standard axioms of consistency and hold. Further, the rationalizing preference satisfies the property of transitivity if and only if axioms and hold if and only if WARP holds: in this case, we speak of a transitively rationalizable choice. Section 2 provides the background to choice theory.

Although the mathematical economics literature on the topic is quite large, there are no contributions which deal with related decision procedures in choice theory. In this paper we start filling this gap. Specifically, we study the satisfiability problem for unquantified formulae of an elementary fragment of set theory (denoted ) involving a choice function symbol , the Boolean set operators , , and the singleton , the predicates equality and inclusion , and the propositional connectives , , , , etc. Here we consider the cases in which the interpretation of is subject to any combination of the axioms of consistency and , whose conjunction is equivalent to WARP. In two cases we prove that the related satisfiability problem is NP-complete, whereas in the remaining cases we obtain NP-completeness only under the additional assumption that the number of choice terms is constant.

By depriving the -language of the choice function symbol , we obtain the fragment 2LSS (here denoted ) whose decidability was known since the birth of Computable Set Theory in the late 70’s. In Section 4.3 we rediscover such result as a by-product of the solution to the satisfiability problem of under the WARP-semantics: the latter is based on a novel term-oriented non-clausal approach. The reader can find extensive information on Computable Set Theory in the monographs [4, 9, 19, 10].

For our purposes, it will be relevant to solve the following lifting problem: Given a partial choice satisfying some axioms of consistency, can we suitably characterize whether it is extendable to a total choice satisfying the same axioms? The lifting problem for the various combinations of axioms and is addressed in depth in Section 3. In particular, in the case of finite choice correspondences, our characterizations turn out to be effective and, with only one exception, expressible in the same -language. This facilitates the design of effective procedures for the solution of the satisfiability problems of our concern. The syntax and semantics of the -language, as well as the solutions of the satisfiability problem for -formulae under the various combinations of axioms and are presented in Section 4. Finally, in Section 5, we draw our conclusions and hint at future developments.

2 Preliminaries on choice theory

Hereafter, we fix a nonempty set (the “universe”). Let be the family of all subsets of , and the subfamily . The next definition collects some basic notions in choice theory.

Definition 1.

Let be nonempty. A map is contractive if for each . A choice correspondence on is a contractive map that is never empty-valued, i.e.,

In this paper, we denote a choice correspondence on by , and simply refer to it as a choice. The family is the choice domain of , sets in are (feasible) menus, and elements of a menu are items. Further, we say that is total (or full) if , and partial otherwise. The rejection map associated to is the contractive function defined by for all .         

Given a choice , the choice set of a menu collects the elements of that are deemed selectable by an economic agent. Thus, in case contains more than one element, the selection of a single element of is deferred to a later time, usually with a different procedure (according to additional information or “subjective randomization”, e.g., flipping a coin). Notice that the rejection map associated to a choice may fail to be a choice, since the rejection set of some menu can be empty.

The next definition recalls the classical notion of a rationalizable choice.

Definition 2.

A choice is rationalizable (or binary) if there exists a binary relation on such that the equality111Recall that , where means and . holds for all menus .         

The revealed preference theory approach postulates that preferences can be derived from choices. The preference revealed by a primitive choice is modeled by a suitable binary relation on the set of alternatives. Then a choice is rationalizable whenever the observed behavior can be fully explained (i.e., retrieved) by constructing a binary relation of revealed preference.

The rationalizability of choice is traditionally connected to the satisfaction of suitable axioms of choice consistency. These axioms codify rules of coherent behavior of an economic agent. Among the several axioms that are considered in the literature, the following are relevant to our analysis (a universal quantification on all the involved menus is implicit):

axiom [standard contraction consistency]:
axiom [standard expansion consistency]:
axiom [symmetric expansion consistency]:
axiom [standard replacement consistency]:
WARP [weak axiom of revealed preference]: .

Axiom was studied by Chernoff [12], whereas axioms and are due to Sen [20]. WARP was introduced by Samuelson in [18]. Axiom has been recently introduced in [5], in connection to the transitive structure of the relation of revealed preference.

Upon reformulating these properties in terms of items, their semantics becomes clear. Chernoff’s axiom states that any item selected from a menu is still selected from any submenu containing it. Sen’s axiom says that any item selected from two menus and is also selected from the menu (if feasible). The expansion axiom can be equivalently written as follows: if , and , then . In this form, says that if two items are selected from a menu , then they are simultaneously either selected or rejected in any larger menu . Axiom can be equivalently written as follows: if , then . In this form, says that if an item is selected from a menu but not from the larger menu , then the new item is selected from . WARP summarizes features of contraction and expansion consistency in a single – and rather strong, despite its name – axiom, in fact it is equivalent to the conjunction of and  [20].

3 Liftings

In this section we examine the “lifting problem”: this corresponds to finding necessary and sufficient conditions such that a partial choice satisfying some axioms of consistency can be extended to a total choice satisfying the same axioms. We shall exploit such conditions in the decision results to be presented in Section 4.

The next definition makes the notion of lifting formal.

Definition 3.

Let be a choice. Given a nonempty set of sentences of second-order monadic logic, we say that has the -lifting property if there is a total choice extending (i.e., ) and satisfying all formulae in . In this case, is called an -lifting of . (Of course, we are interested in cases such that is a family of axioms of choice consistency.) Whenever is a single formula, we simplify notation and write, e.g., -lifting, WARP-lifting, etc. Similarly, we say that has the rational lifting property if there is a total choice that is rational and extends .

Notice that whenever is a nonempty set of axioms of choice consistency (which are formulae in prenex normal form where all quantifiers are universal), if a choice has the -lifting property, then it automatically satisfies all axioms in . The same reasoning applies for the rational lifting property, since it is based on the existence of a binary relation of revealed preference that is fully informative of the choice.

On the other hand, it may happen that a partial choice satisfies some axioms in but there is no lifting to a total choice satisfying the same axioms. The next examples exhibit two instances of this kind. (To simplify notation, we underline all items that are selected within a menu: for instance and stand for, respectively, and . Obviously, we always have for any , so we can safely omit defining for singletons.)

Example 1.

Let and . Define a partial choice by , , and . This choice is rationalizable by the (cyclic) preference defined by . However, does not admit any rational lifting to a total choice , since we would have .

Example 2.

Let and . Define a partial choice by One can easily check that satisfies axiom (but it fails to be rationalizable). On the other hand, admits no -lifting, since violates axiom for any choice extending to the full menu .

3.1 Lifting of axiom

In this section we characterize the choices that are -liftable. To that end, it is convenient to reformulate axiom in terms of the monotonicity of the rejection map. We need the following preliminary result, whose simple proof is omitted, and a technical definition.

Lemma 1.

Let . For any pair of sets , we have

Definition 4.

Let be a choice. Given a menu , the relativized choice domain w.r.t.  is the collection of all submenus of , that is, such that ; in symbols,

(1)

A set of menus is -closed w.r.t.  if , for every such that .         

In view of Lemma 1, axiom can be equivalently rewritten as follows:

(2)

In this form, axiom just asserts that enlarging the set of alternatives may only cause the set of neglected members to grow. As announced, we have:

Theorem 1.

A partial choice has the -lifting property if and only if the following two conditions hold:

  1. [label=(), start=1]

  2. ,  for all ;

  3. , for every such that is -closed w.r.t. .

Proof.

For necessity, assume that can be extended to a total choice on satisfying axiom , and let be the associated rejection map of . Since , condition 1 follows immediately from axiom for . To prove that satisfies condition 2 as well, let be a nonempty -closed subset of . By the equivalent formulation (2) of axiom , we obtain for every , where is the associated rejection map of . Hence

holds. It follows that

thus showing that 2 holds. This completes the proof of necessity.

For sufficiency, assume that 1 and 2 hold for the choice . For each , define

where we recall that is the relativized choice domain w.r.t. to (cf. Definition 4). In what follows we prove that the map is a well-defined choice, which extends and satisfies axiom .

Since the map is obviously contractive by definition, to prove that it is a well-defined choice it suffices to show that it is never empty-valued. Toward a contradiction, assume that for some . The definition of readily yields which implies and , since is -closed w.r.t. . However, this contradicts 2. Thus, is a well-defined (total) choice.

Next, we show that extends . Let . Plainly, we have and for each . Property 1 yields , and so by Lemma 1. Thus, we obtain

which proves the claim.

Finally, we show that satisfies . Let . Since and , we have:

This proves that axiom holds for , and the proof is complete. ∎

3.2 Lifting of axiom

Here we prove that any partial choice satisfying axiom can be always lifted to a total choice still satisfying axiom . To that end, we need the notion of the intersection graph associated to a family of sets : this is the undirected graph whose nodes are the sets belonging to , and whose edges are the pairs of distinct intersecting sets (i.e., such that ).

Theorem 2.

A partial choice has the -lifting property if and only it satisfies axiom .

Proof.

Clearly, axiom holds for any choice that admits an extension to a total choice satisfying . Thus, it suffices to prove that any choice satisfying has the -lifting property. For every , pick an element , subject only to the condition that whenever . If (where ), then let be the connected component of the intersection graph associated to the family such that . Then, for , set

By definition, is a total contractive map on that is never empty-valued. In addition, if , then . It follows that is a well-defined total choice that extends .

To complete the proof, we only need to show that satisfies . Let be such that and . If , then plainly . On the other hand, if , then , hence and . Thus, we obtain again , as claimed. ∎

3.3 Lifting of Warp

Finally, we characterize choices that have the WARP-lifting property. This characterization will be obtained in terms of the existence of a suitable Noetherian total preorder on the collection of the Euler’s regions of the union of the choice domain with its image under the given choice. (Recall that a preorder is a binary relation that is reflexive and transitive. Further, a relation on is Noetherian if the converse relation is well-founded, i.e., if every nonempty subset of has an -maximal element.)

Thus, let be a partial choice. Denote by the Euler’s diagram of the family namely the partition

of formed by all the nonempty sets of the form , for . Further, for each , denote by the envelope of in , namely, the collection of regions in intersecting ; formally, Observe that, for each , we have . It turns out that the choice can be lifted to a total choice satisfying WARP if and only if there exists a suitable Noetherian total preorder on such that

More precisely, we have:

Theorem 3.

A partial choice has the WARP-lifting property if and only if there exists a total Noetherian preorder on the collection of Euler’s regions of such that, for all and , the following conditions hold:

  1. [label=()]

  2. if and , then ;

  3. if is -maximal in , then .

The (constructive) proof of Theorem 3 is technical and rather long. In order to maintain focus on the main goal of the paper, we omit including it here. The interested reader can find it in the extended version of the paper available at the following URL:   https://arxiv.org/abs/1708.06121.

4 The satisfiability problem in presence of a choice correspondence

We are now ready to define the syntax and semantics of the Boolean set-theoretic language extended with a choice correspondence, denoted by , of which we shall study the satisfiability problem.

4.1 Syntax of

The language involves

  • denumerable collections and of individual and set variables, respectively;

  • the constant (empty set);

  • operation symbols: , , , , (choice map);

  • predicate symbols: , , .

Set terms of are recursively defined as follows:

  • set variables and the constant are set terms;

  • if are set terms and is an individidual variable, then are set terms.

The atomic formulae (or atoms) of have one of the following two forms where are set terms. Atoms and their negations are called literals.
Finally, -formulae are propositional combinations of -atoms by means of the usual logical connectives , , , , .

We regard as a shorthand for the set term . Likewise, and are regarded as shorthands for and , respectively.

Choice terms are -terms of type , whereas choice-free terms are -terms which do not involve the choice map (at any level of nesting). We refer to -formulae containing only choice-free terms as -formulae.222Up to minor syntactic differences, -formulae are essentially 2LSS-formulae, whose decision problem has been solved (see, for instance, [9, Exercise 10.5]).

4.2 Semantics of

We first describe the unrestricted semantics of , when the choice operator is not required to satisfy any particular consistency axiom.

A set assignment is a pair , where is any nonempty collection of objects, called the domain or universe of , and is an assignment over the variables of such that

  • , for each individual variable ;

  • ;

  • , for each set variable ;

  • is a total choice correspondence over .

Then, recursively, we put

  • , where are set terms and ;

  • , where is an individual variable;

  • , where is a set term.

Satisfiability of any -formula by (written ) is defined by putting

for -atoms (where are set terms and ), and by interpreting logical connectives according to their classical meaning.

For a -formula , if (i.e., satisfies ), then is said to be a -model for . A -formula is said to be satisfiable if it has a -model. Two -formulae and are equivalent if they share exactly the same -models; they are equisatisfiable if one is satisfiable if and only if so is the other (possibly by different -models).

The satisfiability problem (or decision problem) for asks for an effective procedure (or decision procedure) to establish whether any given -formula is satisfiable or not.

We shall also address the satisfiability problem for under other semantics: specifically, the -semantics, the -semantics, and the WARP-semantics (whose satisfiability relations are denoted by , , and , respectively). These differ from the unrestricted semantics in that the interpreted choice map is required to satisfy axiom in the first case, axiom in the second case, and axioms and conjunctively (namely WARP) in the latter case.

4.3 The decision problem for -formulae

The satisfiability problem for - and -formulae under the various semantics are NP-hard, as the satisfiability problem for propositional logic can readily be reduced to any of them (in linear time). In the cases of - and WARP-semantics, we shall prove NP-completeness only under the additional hypothesis that the number of choice terms is constant, otherwise, in both cases, we have to content ourselves with a NEXPTIME complexity. As a by-product, it will follow that the satisfiability problem for is NP-complete. On the other hand, we shall prove that the satisfiability problem for -formulae under the unrestricted and the -semantics can be reduced polynomially to the satisfiability problem for -formulae, thereby proving their NP-completeness.

Let be a -formula, and the collections of individual and set variables occurring in , respectively, and the collection of the set terms occurring in . For convenience, we shall assume that . Let also

(3)

be the distinct choice terms occurring in , with (when , is a -formula).

Without loss of generality, we may assume that is in choice-flat form, namely that all the terms in (3) are choice-free. In fact, if this were not the case, then, for each choice term in occurring inside the scope of a choice symbol and such that is choice-free, we could replace in all occurrences of by a newly introduced variable and add the conjunct to , until no choice term is left which properly contains a choice subterm. It is an easy matter to check that the resulting formula is in choice-flat form, it is equisatisfiable with (under any of our semantics), and its size is linear in the size of .

Without disrupting satisfiability, we may add to the following formulae:

choice conditions:

,   for ;

single-valuedness conditions:

,   for all distinct ,

since they are plainly true in any -assignment. In this case the size of could have up to a quadratic increase. However, the total number of terms remains unchanged.333See Footnote 4. For the sake of simplicity, we shall assume that includes its choice and single-valuedness conditions, and thereby say that it is complete.

Notice that the above considerations hold irrespectively of the semantics adopted.

In the sections which follow, we study the satisfiability problem for complete -formulae under the various semantics described earlier. We start our course with the WARP-semantics.

4.3.1 Warp-semantics

We first derive some necessary conditions for to be satisfiable and later prove their sufficiency. Hence, to begin with, let us assume that is satisfiable under the WARP-semantics and let be a model for it. Let be the Euler’s diagram of . Notice that, for each region and term , either or . Thus, to each , there corresponds a Boolean map over (where we have identified the truth values true and false with and , respectively) such that

(4)

Let . Hence, we have:

  1. [label=()]

  2. , for each ;

  3. , for each map and set term in ;

  4. , for each map and set term in ;

  5. , for each map and set term in .

In addition, we have , for every .

By uniformly replacing the atomic formulae in with propositional variables, in such a way that different occurrences of the same atomic formula are replaced by the same propositional variable and different atomic formulae are replaced by distinct propositional variables, we can associate to its propositional skeleton (up to variables renaming). For instance, the propositional skeleton of

(5)

is the propositional formula

Plainly, a necessary condition for to be satisfiable (by a -model) is that its skeleton is propositionally satisfiable (however, the converse does not hold in general).

A collection of atoms of is said to be promising for if the valuation which maps to true the propositional variables corresponding to the atoms in and to false the remaining ones satisfies the propositional skeleton . For instance, in the case of (5), all collections of its atoms not containing both and are promising for (5).

Let be the collection of the atoms in satisfied by and the collection of the remaining atoms in , namely those that are disproved by . It can be easily checked that is promising. In addition, for every atom in and , we have if and only if is in . Likewise, for every atom in and , we have if and only if is in . Thus, in particular, for every atom in , there exists a map such that . Likewise, for every atom in , there exists a map such that .

Definition 5 (Places).

Any Boolean map on   for which the above properties 14 hold for each set term in is called a place for .

For a given set of atoms occurring in , a set of places for is -ample if

  1. [label=(),start=5]

  2. for each , provided that the atom is not in ;

  3. for each , provided that the atom is not in ;

  4. for some , if the atom belongs to ;

  5. for some , if the atom belongs to .         

Notice that conditions 12 are universal, whereas conditions 34 are existential.

The considerations made just before Definition 5 yield that is an -ample set of places for . However, in order to later establish some tight complexity results, it is convenient to enforce a polynomial bound for the cardinality of the set of places in terms of the size of (where, for instance, could be defined as the number of nodes in the syntax tree of ). We do this as follows: for each atom (resp., ) in , we select a place , with such that (resp., ) holds, and call their collection . Plainly, we have . Notice that is -ample.444A finer construction would yield an -ample set of places such that . Notice that .

Conditions 14 take care of the structure of set terms in but those of the form or , conditions 1 and 2 take care of the atoms in deemed to be positive, whereas conditions 3 and 4 take care of the remaining atoms in , namely those deemed to be negative.

To take care of set terms of the form in , we observe that, for every , there exists a unique Euler’s region such that . Let be the place corresponding to according to (4), namely , and put .

Definition 6 (Places at variables).

Let be an individual variable occurring in . A place (for ) at the variable is any place for such that .         

Next, we take care of choice terms. Thus, let and let be the Euler’s diagram of . Notice that each region in is a disjoint union of regions in ; moreover, the partial choice over the choice domain enjoys the WARP-lifting property. Thus, by Theorem 3, there exists a total Noetherian preorder on such that, for all and , we have:

  1. [label=()]

  2. if and , then ;

  3. if is -maximal in , then .

For each region , let us select a place , such that and , and call their collection . Set . Plainly, , is -ample, and is the sole place in at the variable , for each individual variable in . To ease notation, for , , and , we shall also write (i) for , (ii) for , for every , (iii) for , for some . For each , let , and call their collection. Then, by 1 and 2 above, there exists a total Noetherian preorder on such that, for all and , the following conditions hold:

  1. [label=(’)]

  2. if and , then ;

  3. if is -maximal in , then .

Summing up, we have the following result:

Lemma 2.

Let be a -formula in choice-flat form, the set of individual variables occurring in it, and the choice terms occurring in it. If is satisfiable under the WARP-semantics, then there exist an -ample set of places for such that , for some promising set of atoms in , and a map from into such that is the sole place in at the variable , for . In addition, if is the equivalence relation on such that

and , then there exists a total Noetherian preorder on such that conditions 1 and 2 are satisfied for all and .         

Next we show that the conditions in the preceding lemma are also sufficient for the satisfiability of our -formula under the WARP-semantics. Thus, let , , , , , and be such that the conditions in Lemma 2 are satisfied. Let be any set of cardinality , and any injective map from onto . We define an interpretation over the variables in and the choice terms occurring in , as if they were set variables, by putting:

(6)

Notice that the choice map is not interpreted by . However, it is not hard to check that the set assignment