A decidable quantified fragment of set theory with ordered pairs and some undecidable extensions^{†}^{†}thanks: Work partially supported by the INdAM/GNCS 2012 project “Specifiche insiemistiche eseguibili e loro verifica formale” and by Network Consulting Engineering Srl.
Abstract
In this paper we address the decision problem for a fragment of set theory with restricted quantification which extends the language studied in [5] with pair related quantifiers and constructs, in view of possible applications in the field of knowledge representation. We will also show that the decision problem for our language has a nondeterministic exponential time complexity. However, for the restricted case of formulae whose quantifier prefixes have length bounded by a constant, the decision problem becomes NPcomplete. We also observe that in spite of such restriction, several useful settheoretic constructs, mostly related to maps, are expressible. Finally, we present some undecidable extensions of our language, involving any of the operators domain, range, image, and map composition.
M. Faella, A. Murano (Eds.): Games, Automata, Logics and Formal Verification (GandALF 2012) EPTCS 96, 2012, pp. A decidable quantified fragment of set theory with ordered pairs and some undecidable extensions^{†}^{†}thanks: Work partially supported by the INdAM/GNCS 2012 project “Specifiche insiemistiche eseguibili e loro verifica formale” and by Network Consulting Engineering Srl.–LABEL:LastPage, doi:10.4204/EPTCS.96.17 © D. Cantone & C. Longo This work is licensed under the Creative Commons Attribution License.
A decidable quantified fragment of set theory with ordered pairs and some undecidable extensions^{†}^{†}thanks: Work partially supported by the INdAM/GNCS 2012 project “Specifiche insiemistiche eseguibili e loro verifica formale” and by Network Consulting Engineering Srl.
Domenico Cantone \IfArrayPackageLoaded  





cantone@dmi.unict.it and Cristiano Longo \IfArrayPackageLoaded  




cristiano.longo@nce.eu 
1 Introduction
The intuitive formalism of set theory has helped providing solid and unifying foundations to such diverse areas of mathematics as geometry, arithmetic, analysis, and so on. Hence, positive solutions to the decision problem for fragments of set theory can have considerable applications to the automation of mathematical reasoning and therefore in any area which can take advantage of automated deduction capabilities.
The decision problem in set theory has been intensively studied in the context of Computable Set Theory (see [6, 10, 19]), and decision procedures or undecidability results have been provided for several sublanguages of set theory. MultiLevel Syllogistic (in short , cf. [13]) was the first unquantified sublanguage of set theory that has been shown to have a solvable satisfiability problem. We recall that is the Boolean combinations of atomic formulae involving the set predicates , , , and the Boolean set operators , , . Numerous extensions of with various combinations of operators (such as singleton, powerset, unionset, etc.) and predicates (on finiteness, transitivity, etc.) have been proved to be decidable. Sublanguages of set theory admitting explicit quantification (see for example [5, 17, 18, 7]) are of particular interest, since, as reported in [5], they allow one to express several settheoretical constructs using only the basic predicates of membership and equality among sets.
Applications of Computable Set Theory to knowledge representation have been recently investigated in [9, 7], where some interrelationships between (decidable) fragments of set theory and description logics have been exploited.^{1}^{1}1We recall that description logics are a wellestablished framework for knowledge representation; see [2] for an introduction. As knowledge representation mainly focuses on representing relationships among items of a particular domain, any settheoretical language of interest to knowledge representation should include a suitable collection of operators on multivalued maps. ^{2}^{2}2According to [20], we use the term ‘maps’ to denote sets of ordered pairs.
Nondeterministic exponential time decision procedures for two unquantified fragments of set theory involving map related constructs have been provided in [14, 11]. As in both cases the map domain operator is allowed together with all the constructs of , it turns out that both fragments have an ExpTimehard decision problem (cf. [8]). On the other hand, the somewhat less expressive fragment has been shown to have an NPcomplete decision problem in [8], where is a twosorted language with set and map variables, which involves various map constructs like Cartesian product, map restrictions, map inverse, and Boolean operators among maps, and predicates for singlevaluedness, injectivity, and bijectivity of maps.
In [5], an extension of the quantified fragment (studied in the same paper—here the subscript ‘’ denotes that quantification is restricted) with singlevalued maps, the map domain operator, and terms of the form , with a functionfree term, was considered. We recall that formulae are propositional combinations of restricted quantified prenex formulae , where is a Boolean combination of atoms of the types , , and quantified variables nesting is not allowed, in the sense that any quantified variable can not occur at the righthand side of a membership symbol in the same quantifier prefix (roughly speaking, no can be a ). More recently, a decision procedure for a new fragment of set theory, called , has been presented in [7]. The superscript “” denotes the presence of operators related to ordered pairs. Formulae of the fragment , to be reviewed in Section 4, involve the operator , which intuitively represents the collection of the nonpair members of its argument, and terms of the form , for ordered pairs. The predicates and allowed in it can occur only within atoms of the forms , , and ; quantifiers in formulae are restricted to the forms and , and, much as in the case of the fragment , quantified variables nesting is not allowed.
In this paper we solve the decision problem for the extension of the fragment with ordered pairs and prove that, under particular conditions, our decision procedure runs in nondeterministic polynomial time. is a twosorted (as indicated by the second subscript “”) quantified fragment of set theory which allows restricted quantifiers of the forms , , , , and literals of the forms , , , , where , are set variables and , are map variables. Considerably many settheoretic constructs are expressible in it, as shown in Table 1. In fact, the language is also an extension of . However, as will be shown in Section 5, it is not strong enough to express inclusions like , , , and , but only those in which the operators domain, range, (multi)image, and map composition are allowed to appear on the lefthand side of the inclusion operator .
The paper is organized as follows. Section 2 provides some preliminary notions and definitions. In Section 3 we give the precise syntax and semantics of the language . Decidability and complexity of reasoning in the language are addressed in Section 4. Some undecidable extensions of are then presented in Section 5. Finally, in Section 6 we draw our conclusions and provide some hints for future works.
























































2 Preliminaries
We briefly review basic notions from set theory and introduce also some definitions which will be used throughout the paper.
Let and be two infinite disjoint collections of set and map variables, respectively. As we will see, map variables will be interpreted as maps (i.e., sets of ordered pairs). We put . For a formula , we write for the collection of variables occurring free (i.e., not bound by any quantifier) in , and put and .
Semantics of most of the languages studied in the context of Computable Set Theory are based on the von Neumann standard cumulative hierarchy of sets , which is the class containing all the pure sets (i.e., all sets whose members are recursively based on the empty set ). The von Neumann hierarchy is defined as follows:
where is the powerset operator and denotes the class of all ordinals. The rank of a set is defined as the least ordinal such that . We will refer to mappings from to as assignments.
Next we introduce some notions related to pairing functions and ordered pairs. Let be a binary operation over the universe . The Cartesian product of two sets , relative to , is defined as . When it is clear from the context, for the sake of conciseness we will omit to specify the binary operation and simply write ‘’ in place of ‘’. A binary operation over sets in is said to be a pairing function if

, and

the Cartesian product (relative to ) is a set of , for all .
In view of the replacement axiom, condition (ii) is obvioulsy met when is expressible by a settheoretic term. This, for instance, is the case for Kuratowski’s ordered pairs, defined by , for all . Given a pairing function and a set , we denote with the collection of the pairs in (with respect to ), namely .
A pairaware interpretation consists of a pairing function and an assignment such that holds for every map variable (i.e., map variables can only be assigned sets of ordered pairs, or the empty set). For conciseness, in the rest of the paper we will refer to pairaware interpretations just as interpretations. An interpretation associates sets to variables and pair terms, respectively, as follows:
(1) 
for all . Let be a finite collection of variables, and let be two assignments. We say that is a variant of if for all . For two interpretations and , we say that is a variant of if is a variant of and .
In the next section we introduce the precise syntax and semantics of the language .
3 The language
The language consists of the denumerable infinity of variables , the binary pairing operator , the predicate symbols , the Boolean connectives of propositional logic , , , , , parentheses, and restricted quantifiers of the forms , , , and . Atomic formulae are expressions of the following four types
(2) 
with and . Quantifierfree formulae are propositional combinations of atomic formulae. Prenex formulae are expressions of the following two forms
(3)  
(4) 
where , , and is a quantifierfree formula. We will refer to the variables as the domain variables of the formulae (3) and (4). Notice that quantifierfree formulae can also be regarded as prenex formulae with an empty quantifier prefix. A prenex formula is said to be simple if nesting among quantified variables is not allowed, i.e., if no quantified variable can occur also as a domain variable. Finally, formulae are Boolean combinations of simpleprenex formulae.
Semantics of formulae is given in terms of interpretations. An interpretation evaluates a formula into a truth value in the following recursive manner. First of all, interpretation of quantifierfree formulae is carried out following the rules of propositional logic, where atomic formulae (2) are interpreted according to the standard meaning of the predicates and in set theory and the pair operator is interpreted as in (1). Thus, for instance, , provided that either or . Then, evaluation of simpleprenex formulae is defined recursively as follows:

, provided that , for every variant of such that ;

, provided that , for every variant of such that ;

, provided that ; and

, provided that .
Finally, evaluation of formulae is carried out following the rules of propositional logic.
If an interpretation evaluates a formula to we say that is a model for (and write ). A formula is said to be satisfiable if and only if it admits a model. Two formulae are said to be equivalent if they have exactly the same models. Two formulae and are said to be equisatisfiable provided that is satisfiable if and only if so is . The satisfiability problem (s.p., for short) for the theory is the problem of establishing algorithmically whether any given formula is satisfiable or not.
By way of a simple normalization procedure based on disjunctive normal form, the s.p. for formulae can be reduced to that for conjunctions of simpleprenex formulae of the types (3) and (4). Moreover, since any such conjunction of the form
is equisatisfiable with , where is obtained from the quantifierfree formula
by a suitable renaming of the (quantified) variables , it turns out that the s.p. for formulae can be reduced to the s.p. for conjunctions of simpleprenex formulae of the type (3) only, which we call normalized conjunctions.
Satisfiability of normalized conjunctions does not depend strictly on the pairing function of the interpretation, provided that suitable conditions hold, as proved in the following technical lemma.
Lemma 1.
Let be a normalized conjunction, and let and be two interpretations such that

, for all ,

, for all and .
Then .
Proof.
It is enough to prove that
(5) 
holds, for every (universal) simpleprenex conjunct occurring in . We shall proceed by induction on the length of the quantifier prefix of . We begin with observing that, by 1, and evaluate to the same truth values all atomic formulae of the types and , for all . Likewise,
follow directly from 1 and 2. Thus (5) follows easily when is quantifierfree, i.e., when the length of its quantifier prefix is .
Next, let , for some , where is a universally quantified simpleprenex formula with one less quantifier than and containing no quantified occurrence of . We prove that is a model for if and only if so is , for every , where and denote, respectively, the variants of and such that . But, for each , and satisfy conditions 1 and 2 of the lemma, so that, by inductive hypothesis, we have . Hence .
The case in which , with , , and a universally quantified simpleprenex formula containing no quantified occurrence of and , can be dealt with much in the same manner, thus concluding the proof of the lemma. ∎
In the following section we show that the s.p. for normalized conjunctions is solvable.
4 A decision procedure for
We solve the s.p. for formulae by reducing the s.p. for normalized conjunctions to the s.p. for the fragment of set theory , studied in [7]. Following [7], formulae are finite conjunctions of simpleprenex formulae, namely expressions of the form
where , for , no domain variable can occur quantified, and is a quantifierfree Boolean combination of atomic formulae of the types , , , with .^{3}^{3}3Thus, normalization is already builtin into formulae, and we could have called them normalized conjunctions. Intuitively, a term of the form represents the set of the nonpair members of . Notice that formulae involve only set variables.
Semantics for formulae is given by extending interpretations also to terms of the form as indicated below:
where . Evaluation of formulae is carried out much in the same way as for formulae. In particular, we also put , provided that , for every variant of such that .
We recall that satisfiability of formulae can be tested in nondeterministic exponential time. Additionally, the s.p. for formulae with quantifier prefixes of length at most , for any fixed constant , is NPcomplete (cf. [7]).
The s.p. for normalized conjunctions can be reduced to the s.p. for formulae. To begin with, we define a syntactic transformation on normalized conjunctions. More specifically, is obtained from a given normalized conjunction by replacing

each restricted universal quantifier in by the quantifier ,

each atomic formula in by the literal , and

each map variable occurring in by a fresh set variable , thus identifying an application from into , which will be referred to as mapvariable renaming for .
Thus, for instance, if
then
where is a set variable distinct from , , and .
The following lemma provides a useful semantic relation between universal simpleprenex formulae and their corresponding formula via .
Lemma 2.
Let be a universal simpleprenex formula and let be an interpretation such that

(i.e., is not a pair, for any free variable of ), and

, for every domain variable of .
Then if and only if .
Proof.
We proceed by induction on the quantifier prefix length of the formula . To begin with, we observe that in force of 1 we have if and only if , for any two free variables and of , so that, given any atomic formula involving only variables in , if and only if . Hence the lemma follows directly from propositional logic if is quantifierfree, i.e., .
Next, let , where is a universal simpleprenex formula with quantifiers, are set variables occurring neither as domain nor as bound variables in . Observe that, by 2, , since is a domain variable of . Thus it will be enough to prove that
(6) 
holds for every variant of such that , with . But can not be a pair (with respect to the pairing function ), as it is a member of and is a domain variable of . Thus (6) follows by applying the inductive hypothesis to and to every interpretation such that .
Finally, the case in which , where is a universal simpleprenex formula, are set variables not occurring as domain variables in , and is a map variable, can be dealt with much in the same way as the previous case, and is left to the reader. ∎
In the following theorem we use the transformation to reduce the s.p. for normalized conjunctions to the s.p. for formulae.
Theorem 1.
The s.p. for normalized conjunctions can be reduced in linear time to the s.p. for formulae, and therefore it is in NExpTime.
Proof.
We prove the theorem by showing that, given any normalized conjunction , we can construct in linear time a corresponding formula which is equisatisfiable with .
So, let be a normalized conjunction and let be the mapvariable renaming for . We define the corresponding formula as follows:
where is a fresh set variable. Plainly, the size of is linear in the size of .
Let us first assume that admits a model . For each we have , as , for . Likewise, for each we have , as , for . Finally, for each , we have , so that . We define as the variant of such that , for . Plainly, so that, by Lemma 2, as well.
For the converse direction, let be a model for . We shall exhibit an interpretation which satisfies . To begin with, we define a new pairing function by putting
for every , where is the Kuratowski’s pairing function and . Then we define as the variant of the assignment such that , for each . From Lemma 1, it follows that the interpretation satisfies . Moreover, we have
(7) 
for each . Indeed, if for some and we had , then
contradicting the regularity axiom of set theory. Next, let and let be the variant of , where , for , and . In view of (7), it is an easy matter to verify that
(8) 
From (7), we have immediately that , so that
(9) 
Likewise, by reasoning much in the same manner as for the proof of (7), one can prove that
(10) 
From (8), (9), and (10), it follows at once that , completing the proof that and are equisatisfiable.
Since the s.p. for formulae is in NExpTime, as was shown in [7, Section 3.1], it readily follows that the s.p. for normalized conjunctions is in NExpTime as well. ∎
Corollary 1.
The s.p. for formulae is in NExpTime.
Proof.
Let be a satisfiable formula. We may assume without loss of generality that all existential simpleprenex formulae of the form (4) have already been rewritten in terms of equivalent universal simpleprenex formulae of the form (3), so that is a propositional combination of universal simpleprenex formulae. In addition, by suitably renaming variables, we may assume that all quantified variables in are pairwise distinct and that they are also distinct from free variables.
Let be the collection of the universal simpleprenex formulae occurring in . By traversing the syntax tree of , one can find in linear time the propositional skeleton of and a substitution from the propositional variables of into , such that , where is the result of substituting each propositional variable in by the universal simpleprenex formula . Then to check the satisfiability of one can perform the following nondeterministic procedure:

guess a Boolean valuation of the propositional variables of such that