Decision procedures for elementary sublanguages of set theory

2021

The satisfiability problem for multilevel syllogistic extended with the Cartesian product operator (MLS×) is a long-standing open problem in computable set theory. For long, it was not excluded that such a problem were undecidable, due to its remarkable resemblance with the well-celebrated Hilbert's tenth problem, as it was deemed reasonable that union of disjoint sets and Cartesian product might somehow play the roles of integer addition and multiplication. To dispense with nonessential technical difficulties, we report here about a positive solution to the satisfiability problem for a slight simplified variant of MLS×, yet fully representative of the combinatorial complications due to the presence of the Cartesian product, in which membership is not present and the Cartesian product operator is replaced with its unordered variant. We are very confident that such decidability result can be generalized to full MLS×, though at the cost of considerable technicalities. * We gratefully acknowledge partial support from the projects STORAGE and MEGABIT-Università degli Studi di Catania, PIAno di inCEntivi per la RIcerca di Ateneo 2020/2022 (PIACERI), Linea di intervento 2. initial goal was the mechanical formalisation of mathematics with a proof verifier based on the set-theoretic formalism [OS02, COSU03, OCPS06, SCO11], but soon a foundational interest aimed at the identification of the boundary in set theory between the decidable and the undecidable became more and more compelling. The precursor fragment of set theory investigated for decidability was MLS, which stands for Multi-Level Syllogistic. MLS consists of the quantifierfree formulae of set theory involving only the Boolean set operators ∪, ∩, \ and the relators = and ∈, besides set variables (assumed to be existentially quantified). The satisfiability problem (s.p., briefly) for MLS has been solved in the seminal paper [FOS80], and its NP-completeness has later been proved in [COP90]. Following that, several extensions of MLS with various combinations of the set operators {•} (singleton), pow (power set), (unary union), (unary intersection), rk (rank), etc., and of the set predicates rank comparison, cardinality comparison, finiteness, etc., have been also proved decidable over the years. 1 However, the s.p. for the extension MLS× of MLS with the Cartesian product ×, 2 proposed by the first author since the middle 80s, soon appeared to be very challenging and resisted several efforts to find a solution, either positive or negative. As a matter of fact, for long it was not excluded that the s.p. for MLS× were undecidable (in particular, when restricted to finite models), due to its remarkable resemblance with the well-celebrated Hilbert's Tenth problem (HTP, for short), posed by David Hilbert at the beginning of last century [Hil02]. 3 Indeed, it was deemed reasonable that the union of disjoint sets and the Cartesian product might somehow play the roles of integer addition and multiplication in HTP, respectively, in consideration of the fact that |s ∪ t| = |s| + |t|, for any disjoint sets s and t, and |s × t| = |s| • |t|, for any sets s and t. Attempts to solve the s.p. for MLS× helped shaping the development of computable set theory and led to the introduction of the powerful technique of formative processes, 4 which has been at the base of the highly technical

Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, 1992

The still unsettled decision problem for the restricted purely universal formulae ((V),-formulae) of the first order set-theontic language based over =, E is discussed in relation with the adoption or rejection of the axiom of foundation. Assuming the axiom of foundation, the related finite set-satisfiability problem for the very significant subclass of the (V),-formulae consisting of the formulae involving only nested variables of level 1 is proved to be semidecidable on the ground of a reflection property over the hereditarily finite sets, and various extensions of this result are obtained. When variables are restricted to range only over sets, in universes with infinitely many urelements the set-satisfiability problem is shown to be solvable provided the axiom of foundation is assumed; if it is not, then the decidability of a related derivability problem still holds. That, in turn, suggests the alternative adoption of an antifoundation axiom under which the set-satisfiability problem is also solvable (of course with different answers). Turning to set theory without urelements, assuming a form of Boffa's antifoundation axiom, the complement of the setsatisfiability problem for the full class of do-formulae is shown to be sernidecidable; a result that is known not to hold, for the set-satisfiability problem itself, even for a very restricted subclass of the do-formulae. MSC: 03B25, 03E30.

ACM Transactions on Computational Logic, 2006

Formal Set Theory is traditionally concerned with pure sets; consequently, the satisfiability problem for fragments of set theory was most often addressed (and in many cases positively solved) in the pure framework. In practical applications, however, it is common to assume the existence of a number of primitive objects (sometimes called atoms) that can be members of sets but behave differently from them. If these entities are assumed to be devoid of members, the standard extensionality axiom must be revised; then decidability results can sometimes be achieved via reduction to the pure case and sometimes can be based on direct goal-driven algorithms. An alternative approach to modeling atoms, that allows one to retain the original formulation of extensionality, was proposed by Quine: atoms are self-singletons. In this paper we adopt this approach in coping with the satisfiability problem: we show the decidability of this problem relativized to ∃ * ∀-sentences, and develop a goal-driven unification algorithm. Research funded by MURST/MIUR project Aggregate-and number-reasoning for computing: from decision algorithms to constraint programming with multisets, sets, and maps. This research benefited from collaborations fostered by the European action COST n. 274 (TARSKI, see www.tarski.org).

Lecture Notes in Computer Science, 1998

In solving a query, the SLD proof procedure for definite programs sometimes searches an infinite space for a non existing solution. For example, querying a planner for an unreachable goal state. Such programs motivate the development of methods to prove the absence of a solution. Considering the definite program and the query ← Q as clauses of a first order theory, one can apply model generators which search for a finite interpretation in which the program clauses as well as the clause false ← Q are true. This paper develops a new approach which exploits the fact that all clauses are definite. It is based on a goal directed abductive search in the space of finite pre-interpretations for a pre-interpretation such that Q is false in the least model of the program based on it. Several methods for efficiently searching the space of pre-interpretations are presented. Experimental results confirm that our approach find solutions with less search than with the use of a first order model generator.

Motivated by model-theoretic properties of the Bernays-Schönfinkel-Ramsey (BSR) class, we present a family of semantic classes of FO formulae with finite or co-finite spectra over a relational vocabulary Σ. A class in this family is denoted EBS Σ (σ), where σ ⊆ Σ. Formulae in EBS Σ (σ) are preserved under substructures modulo a bounded core and modulo re-interpretation of predicates in Σ \ σ. We study several properties of the family EBS Σ = {EBS Σ (σ) | σ ⊆ Σ}. For example, classes in EBS Σ are spectrally indistinguishable, the smallest class, EBS Σ (Σ), is semantically equivalent to BSR over Σ, and the largest class, EBS Σ (∅), is the set of all FO formulae over Σ with finite or co-finite spectra. Furthermore, (EBS Σ , ⊆) forms a lattice that is isomorphic to the powerset lattice (℘(Σ), ⊆). We also show that if Σ contains at least one predicate of arity ≥ 2, there exist semantic gaps between EBS Σ (σ 1 ) and EBS Σ (σ 2 ) if σ 1 = σ 2 . This gives a natural semantic generalization of BSR as ascending chains in the lattice (EBS Σ , ⊆).

2019

We recently undertook an investigation aimed at identifying small fragments of set theory (which in most cases are sublanguages of Multi-Level Syllogistic) endowed with polynomial-time satisfiability decision tests, potentially useful for automated proof verification. Leaving out of consideration the membership relator ∈ for the time being, in this note we provide a complete taxonomy of the polynomial and the NP-complete fragments involving, besides variables intended to range over the von Neumann set-universe, the Boolean operators ∪,∩, \, the Boolean relators ⊆, 6⊆,=, 6=, and the predicates ‘· = ∅’ and ‘Disj(·, ·)’, meaning ‘the argument set is empty’ and ‘the arguments are disjoint sets’, along with their opposites ‘· 6= ∅’ and ‘¬Disj(·, ·)’.

Information and Computation/information and Control, 2002

In this paper we solve the satisfiability problem for a quantifier-free fragment of set theory involving the powerset and the singleton operators and a finiteness predicate, besides the basic Boolean set operators of union, intersection, and difference. The more restricted fragment obtained by dropping the finiteness predicate has been shown to have a solvable satisfiability problem in a previous paper, by establishing for it a small model property. To deal with the finiteness predicate we have formulated and proved a small model witness property for our fragment of set theory, namely a property which asserts that any satisfiable formula of our fragment has a model admitting a "small" representation.

Fundamenta Informaticae, 2017

In the last decades, several fragments of set theory have been studied in the context of the so-called Computable Set Theory. In general, the semantics of set-theoretical languages di↵ers from the canonical first-order semantics in that the interpretation domain of set-theoretical terms is fixed to a given universe of sets, as for instance the von Neumann standard cumulative hierarchy of sets, i.e., the class consisting of all the pure sets. Because of this, theoretical results and various machinery developed in the context of first-order logic cannot be easily adapted to work in the set-theoretical realm. Recently, quantified fragments of set-theory which allow one to explicitly handle ordered pairs have been studied for decidability purposes, in view of applications in the field of knowledge representation. Among other results, a NexpTime decision procedure for satisfiability of formulae in one of these fragments, 8 ⇡ 0 , has been provided. In this paper we exploit the main features of such a decision procedure to reduce the satisfiability problem for the fragment 8 ⇡ 0 to the problem of Herbrand satisfiability for a first-order language extending it. In addition, it turns out that such reduction maps formulae of the Disjunctive Datalog subset of 8 ⇡ 0 into Disjunctive Datalog programs.

Lecture Notes in Computer Science, 1996

We consider reasoning and rewriting with set-relations: inclusion, nonempty intersection and singleton identity, each of which satis es only two among the three properties of the equivalence relations. The paper presents a complete inference system which is a generalization of ordered paramodulation and superposition calculi. Notions of rewriting proof and con uent rule system are de ned for such nonequivalence relations. Together with the notions of forcing and redundancy they are applied in the completeness proof. Ground completeness cannot be lifted to the nonground case because substitution for variables is restricted to deterministic terms. To overcome the problems of restricted substitutivity and hidden (in relations) existential quanti cation, uni cation is de ned as a three step process: substitution of determistic terms, introduction of bindings and \on-line" skolemisation. The inference rules based on this uni cation derive non-ground clauses even from the ground ones, thus making an application of a standard lifting lemma impossible. The completness theorem is proved directly without use of such a lemma.

Annals of Pure and Applied Logic, 1989

The complexity of subclasses of Magical theories (mainly Presburger and Skolem arithmetic) is studied. The subclasses are defined by the structure of the quantifier prefix.