Reachability and LTL model-checking problems for flat counter systems are known to be decidable but whereas the reachability problem can be shown in NP, the best known complexity upper bound for the latter problem is made of a tower of several exponentials. Herein, we show that the problem is only NP-complete even if LTL admits past-time operators and arithmetical constraints on counters. For instance, adding past-time operators to LTL immediately leads to complications; an NP upper bound cannot be deduced by translating formulae into Büchi automata. Actually, the NP upper bound is shown by adequately combining a new stuttering theorem for Past LTL and the property of small integer solutions for quantifier-free Presburger formulae. Other complexity results are proved, for instance for restricted classes of flat counter systems such as path schemas. Our NP upper bound extends known and recent results on model-checking weak Kripke structures with LTL formulae as well as reachability problems for flat counter systems.Keywords: linear-time temporal logic, stuttering, model-checking, counter system, flatness, complexity, system of equations, small solution, Presburger arithmetic.Email addresses: demri@lsv.ens-cachan.fr (Stéphane Demri), dhar@liafa.univ-paris-diderot.fr (Amit Kumar Dhar), sangnier@liafa.univ-paris-diderot.fr (Arnaud Sangnier)Supported by ANR project REACHARD ANR-11-BS02-001. This is the completed version of [8].

Ontologies provide features like a common vocabulary, reusability, machine-readable content, and also allows for semantic search, facilitate agent interaction and ordering & structuring of knowledge for the Semantic Web (Web 3.0) application. However, the challenge in ontology engineering is automatic learning, i.e., the there is still a lack of fully automatic approach from a text corpus or dataset of various topics to form ontology using machine learning techniques. In this paper, two topic modeling algorithms are explored, namely LSI & SVD and Mr.LDA for learning topic ontology. The objective is to determine the statistical relationship between document and terms to build a topic ontology and ontology graph with minimum human intervention. Experimental analysis on building a topic ontology and semantic retrieving corresponding topic ontology for the user"s query demonstrating the effectiveness of the proposed approach.

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Abstract. Among the approximation methods for the verification of counter systems, one of them consists in model-checking their flat unfoldings. Unfortunately, the complexity characterization of model-checking problems for such operational models is not always well studied except for reachability queries or for Past LTL. In this paper, we characterize the complexity of model-checking problems on flat counter systems for the specification languages including first-order logic, linear mu-calculus, infinite automata, and related formalisms. Our results span different complexity classes (mainly from PTime to PSpace) and they apply to languages in which arithmetical constraints on counter values are systematically allowed. As far as the proof techniques are concerned, we provide a uniform approach that focuses on the main issues.