On the Satisfiability Problem for a 4-level Quantified Syllogistic and Some Applications to Modal Logic

Abstract

We introduce a multi-sorted stratified syllogistic, called 4LQS R , admitting variables of four sorts and a restricted form of quantification over variables of the first three sorts, and prove that it has a solvable satisfiability problem by showing that it enjoys a small model property. Then, we consider the fragments (4LQS R ) h of 4LQS R , consisting of 4LQS R -formulae whose quantifier prefixes have length bounded by h ≥ 2 and satisfying certain additional syntactical constraints, and prove that each of them has an NP-complete satisfiability problem. Finally we show that the modal logic K45 can be expressed in (4LQS R ) 3 . 1 The term 'multi-level syllogistic' was originally introduced by Jack Schwartz to denote decidable fragments of set theory, as he considered them generalizations of Aristotelian syllogistics. 428 D. Cantone and M. Nicolosi Asmundo / On the Satisfiability Problem for a 4-level Quantified Syllogistic...universe of sets (see [8,11] for a thorough account of the state-of-the-art until 2001). Only a few stratified syllogistics, where variables of different sorts are allowed, have been investigated, despite the fact that in many fields of computer science and mathematics one often has to deal with multi-sorted languages. For instance, in modal logics, one has to consider entities of different types, such as worlds, formulae, and accessibility relations.In [14] an efficient decision procedure was presented for the satisfiability of the two-level syllogistic 2LS . 2LS has variables of two sorts and admits propositional connectives, the basic set-theoretic operators ∪, ∩, \, and the predicate symbols =, ∈, and ⊆. Then,in [4], it was shown that the extension of 2LS with the singleton operator and the Cartesian product operator is decidable. Tarski's and Presburger's arithmetics extended with sets have been analyzed in [6]. Subsequently, in [5], the three-sorted language 3LSSPU (involving singleton, powerset, and the general union operators) has been proved decidable. Recently,in [9], it was shown that the language 3LQS R (Three-Level Quantified Syllogistic with Restricted quantifiers) has a decidable satisfiability problem. 3LQS R admits variables of three sorts and a restricted form of quantification. Its vocabulary contains only the predicate symbols = and ∈. But in spite of that, 3LQS R allows one to express several constructs of set theory. Among them, the most comprehensive one is the set-former operator, which in turn enables one to express other operators like the powerset operator, the singleton operator, and so on. In [9] it is also shown that the modal logic S5 can be expressed in a fragment of 3LQS R , whose satisfiability problem is NP-complete. D. Cantone and M. Nicolosi Asmundo / On the Satisfiability Problem for a 4-level Quantified Syllogistic... 429 The unrestricted language 4LQSSyntax of 4LQS . The four-level quantified language 4LQS involves four collections of variables, V 0 , V 1 , V 2 , and V 3 . Each V i contains variables of sort i, denoted by X i , Y i , Z i , etc. When we ...

The Satisfiability Problem for Boolean Set Theory with a Choice Correspondence

“…We also intend to find decidable extensions with choice correspondence terms of the three-sorted fragment of set theory 3LQST R 0 (see [7]) and of the four-sorted fragments (4LQS R ) h , with h ∈ N (see [6]), which admit a restricted form of quantification over individual and set variables (in the case of 3LQST R 0 ), and also over collection variables (in the case of (4LQS R ) h . The resulting decision procedures would allow to reason automatically on very expressive properties in choice theory.…”

A Set-Theoretic Approach to ABox Reasoning Services

“…Quantifier-free atomic formulae and purely universal formulae are evaluated in a standard way according to the usual meaning of the predicates '∈' and '='. The interpretation of quantifier-free atomic formulae and of purely universal formulae is given in [7]. Finally, compound formulae are interpreted according to the standard rules of propositional logic.…”

“…Decidability of the consistency problem for DL 4,× D -knowledge bases was proved in [6] via a reduction to the satisfiability problem for formulae of a four level quantified syllogistic called 4LQS R . The latter problem was proved decidable in [7]. Some considerations on the expressive power of DL 4,× D are in order.…”

“…In particular, the decidable four-level stratified fragment of set theory 4LQS R , involving variables of four sorts, pair terms, and a restricted form of quantification over variables of the first three sorts (cf. [7]), has been used in [6] to represent the description logic DL 4LQS R (D) (more simply referred to as DL 4 D ). The logic DL 4 D admits concept constructs such as full negation, union and intersection of concepts, concept domain and range, existential quantification and min cardinality on the left-hand side of inclusion axioms.…”

An Optimized KE-Tableau-Based System for Reasoning in the Description Logic $${\mathcal {DL}}_{{\mathbf {D}}}^{4,\!\times }$$

“…In addition, it permits to express the Semantic Web Rule Language (SWRL), an extension of the Ontology Web Language (OWL). Decidability of the Conjunctive Query Answering (CQA) problem for DL 4,× D has been proved in [2] via a reduction to the CQA problem for 4LQS R , whose decidability easily follows from that of 4LQS R (see [1]). In [2], the authors provided a terminating KE-tableau based procedure that, given a DL 4,× D -query Q and a DL 4,× D -knowledge base KB represented in set-theoretic terms, determines the answer set of Q with respect to KB.…”