Cristiano Longo

Web Ontology Representation and Reasoning via Fragments of Set Theory

In this paper we use results from Computable Set Theory as a means to represent and reason about description logics and rule languages for the semantic web.

A Decidable Quantified Fragment of Set Theory Involving Ordered Pairs with Applications to Description Logics

We present a decision procedure for a quantified fragment of set theory involving ordered pairs and some operators to manipulate them. When our decision procedure is applied to formulae in this fragment whose quantifier prefixes have length bounded by a fixed constant, it runs in nondeterministic polynomial-time. Related to this fragment, we also introduce a description logic which provides an unusually large set of constructs, such as, for instance, Boolean constructs among roles. The set-theoretic nature of the description logics semantics yields a straightforward reduction of the knowledge base consistency problem to the satisfiability problem for formulae of our fragment with quantifier prefixes of length at most 2, from which the NP-completeness of reasoning in this novel description logic follows. Finally, we extend this reduction to cope also with SWRL rules.

A decidable two-sorted quantified fragment of set theory with ordered pairs and some undecidable extensions

In this paper we address the decision problem for a two-sorted fragment of set theory with restricted quantification which extends the language studied in 4 with pair-related quantifiers and constructs. We also show that the decision problem for our language has a nondeterministic exponential-time complexity. However, in the restricted case of formulae whose quantifier prefixes have length bounded by a constant, the decision problem becomes NP-complete. In spite of such restriction, several useful set-theoretic constructs, mostly related to maps, are still expressible. We also argue that our restricted language has applications to knowledge representation, with particular reference to metamodeling issues. Finally, we compare our proposed language with two similar languages in terms of their expressivity and present some undecidable extensions of it, involving any of the domain, range, image, and map composition operators.

A decidable quantified fragment of set theory with ordered pairs and some undecidable extensions

In this paper we address the decision problem for a fragment of set theory with restricted quantification which extends the language studied in [4] 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 non-deterministic exponential time complexity. However, for the restricted case of formulae whose quantifier prefixes have length bounded by a constant, the decision problem becomes NP-complete. We also observe that in spite of such restriction, several useful set-theoretic 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. [4] Michael Breban, Alfredo Ferro, Eugenio G. Omodeo and Jacob T. Schwartz (1981): Decision procedures for elementary sublanguages of set theory. II. Formulas involving restricted quantifiers, together with ordinal, integer, map, and domain notions. Communications on Pure and Applied Mathematics 34, pp. 177-195

Herbrand-satisfiability of a Quantified Set-theoretic Fragment

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 differs 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, ∀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 ∀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 ∀0 into Disjunctive Datalog programs.