Range and k-nearest neighbor searching are core problems in pattern recognition. Given a database S of objects in a metric space M and a query object q in M, in a range searching problem the goal is to find the objects of S within some threshold distance to q, whereas in a k-nearest neighbor searching problem, the k elements of S closest to q must be produced. These problems can obviously be solved with a linear number of distance calculations, by comparing the query object against every object in the database. However, the goal is to solve such problems much faster. We combine and extend ideas from the M-Tree, the Multivantage Point structure, and the FQ-Tree to create a new structure in the "bisector tree" class, called the Antipole Tree. Bisection is based on the proximity to an "Antipole" pair of elements generated by a suitable linear randomized tournament. The final winners a; b of such a tournament are far enough apart to approximate the diameter of the splitting set. If distða; bÞ is larger than the chosen cluster diameter threshold, then the cluster is split. The proposed data structure is an indexing scheme suitable for (exact and approximate) best match searching on generic metric spaces. The Antipole Tree outperforms by a factor of approximately two existing structures such as List of Clusters, M-Trees, and others and, in many cases, it achieves better clustering properties.Index Terms-Indexing methods, similarity measures, information search and retrieval.

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 ...

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In this paper we consider the most common ABox reasoning services for the description logic DL 4LQS R,× (D) (DL 4,× D , for short) and prove their decidability via a reduction to the satisfiability problem for the set-theoretic fragment 4LQS R . The description logic DL 4,× D is very expressive, as it admits various concept and role constructs, and data types, that allow one to represent rule-based languages such as SWRL. Decidability results are achieved by defining a generalization of the conjunctive query answering problem, called HOCQA (Higher Order Conjunctive Query Answering), that can be instantiated to the most widespread ABox reasoning tasks. We also present a KE-tableau based procedure for calculating the answer set from DL 4,× D knowledge bases and higher order DL 4,× D conjunctive queries, thus providing means for reasoning on several well-known ABox reasoning tasks. Our calculus extends a previously introduced KE-tableau based decision procedure for the CQA problem.

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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. Specifically, we introduce the description logic DL 4LQS R (D)-admitting features such as min/max cardinality constructs on the left-hand/righthand side of inclusion axioms, role chain axioms, and datatypes-which turns out to be quite expressive if compared with SROIQ(D), the description logic underpinning the Web Ontology Language OWL. Then we show that the consistency problem for DL 4LQS R (D)-knowledge bases is decidable by reducing it, through a suitable translation process, to the satisfiability problem of the stratified fragment 4LQS R of set theory, involving variables of four sorts and a restricted form of quantification. We prove also that, under suitable not very restrictive constraints, the consistency problem for DL 4LQS R (D)-knowledge bases is NP-complete. Finally, we provide a 4LQS R -translation of rules belonging to the Semantic Web Rule Language (SWRL).

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