Efficient Genomic Interval Queries Using Augmented Range Trees

Efficient large-scale annotation of genomic intervals is essential for personal genome interpretation in the realm of precision medicine. There are 13 possible relations between two intervals according to Allen's interval algebra. Conventional interval trees are routinely used to identify the genomic intervals satisfying a coarse relation with a query interval, but cannot support efficient query for more refined relations such as all Allen's relations. We design and implement a novel approach to address this unmet need. Through rewriting Allen's interval relations, we transform an interval query to a range query, then adapt and utilize the range trees for querying. We implement two types of range trees: a basic 2-dimensional range tree (2D-RT) and an augmented range tree with fractional cascading (RTFC) and compare them with the conventional interval tree (IT). Theoretical analysis shows that RTFC can achieve the best time complexity for interval queries regarding all Allen's relations among the three trees. We also perform comparative experiments on the efficiency of RTFC, 2D-RT and IT in querying noncoding element annotations in a large collection of personal genomes. Our experimental results show that 2D-RT is more efficient than IT for interval queries regarding most of Allen's relations, RTFC is even more efficient than 2D-RT. The results demonstrate that RTFC is an efficient data structure for querying large-scale datasets regarding Allen's relations between genomic intervals, such as those required by interpreting genome-wide variation in large populations.Comment: 4 figures, 4 table

A decidable weakening of Compass Logic based on cone-shaped cardinal directions

We introduce a modal logic, called Cone Logic, whose formulas describe properties of points in the plane and spatial relationships between them. Points are labelled by proposition letters and spatial relations are induced by the four cone-shaped cardinal directions. Cone Logic can be seen as a weakening of Venema's Compass Logic. We prove that, unlike Compass Logic and other projection-based spatial logics, its satisfiability problem is decidable (precisely, PSPACE-complete). We also show that it is expressive enough to capture meaningful interval temporal logics - in particular, the interval temporal logic of Allen's relations "Begins", "During", and "Later", and their transposes

A Global Constraint for a Tractable Class of Temporal Optimization Problems

International audienceThis paper is originally motivated by an application where the objective is to generate a video summary, built using intervals extracted from a video source. In this application, the constraints used to select the relevant pieces of intervals are based on Allen's algebra. The best state-of-the-art results are obtained with a small set of ad hoc solution techniques, each specific to one combination of the 13 Allen's relations. Such techniques require some expertise in Constraint Programming. This is a critical issue for video specialists. In this paper, we design a generic constraint, dedicated to a class of temporal problems that covers this case study, among others. ExistAllen takes as arguments a vector of tasks, a set of disjoint intervals and any of the 2 13 combinations of Allen's relations. ExistAllen holds if and only if the tasks are ordered according to their indexes and for any task at least one relation is satisfied , between the task and at least one interval. We design a propagator that achieves bound-consistency in O(n + m), where n is the number of tasks and m the number of intervals. This propagator is suited to any combination of Allen's relations, without any specific tuning. Therefore, using our framework does not require a strong expertise in Constraint Programming. The experiments, performed on real data, confirm the relevance of our approach

Decidability of the interval temporal logic ABBar over the natural numbers

In this paper, we focus our attention on the interval temporal logic of the Allen's relations "meets", "begins", and "begun by" (ABBar for short), interpreted over natural numbers. We first introduce the logic and we show that it is expressive enough to model distinctive interval properties,such as accomplishment conditions, to capture basic modalities of point-based temporal logic, such as the until operator, and to encode relevant metric constraints. Then, we prove that the satisfiability problem for ABBar over natural numbers is decidable by providing a small model theorem based on an original contraction method. Finally, we prove the EXPSPACE-completeness of the proble

Crossing the Undecidability Border with Extensions of Propositional Neighborhood Logic over Natural Numbers

Propositional Neighborhood Logic (PNL) is an interval temporal logic featuring two modalities corresponding to the relations of right and left neighborhood between two intervals on a linear order (in terms of Allen's relations, meets and met by). Recently, it has been shown that PNL interpreted over several classes of linear orders, including natural numbers, is decidable (NEXPTIME-complete) and that some of its natural extensions preserve decidability. Most notably, this is the case with PNL over natural numbers extended with a limited form of metric constraints and with the future fragment of PNL extended with modal operators corresponding to Allen's relations begins, begun by, and before. This paper aims at demonstrating that PNL and its metric version MPNL, interpreted over natural numbers, are indeed very close to the border with undecidability, and even relatively weak extensions of them become undecidable. In particular, we show that (i) the addition of binders on integer variables ranging over interval lengths makes the resulting hybrid extension of MPNL undecidable, and (ii) a very weak first-order extension of the future fragment of PNL, obtained by replacing proposition letters by a restricted subclass of first-order formulae where only one variable is allowed, is undecidable (in contrast with the decidability of similar first-order extensions of point-based temporal logics)

A decidable weakening of Compass Logic based on cone-shaped cardinal directions

We introduce a modal logic, called Cone Logic, whose formulas describeproperties of points in the plane and spatial relationships between them.Points are labelled by proposition letters and spatial relations are induced bythe four cone-shaped cardinal directions. Cone Logic can be seen as a weakeningof Venema's Compass Logic. We prove that, unlike Compass Logic and otherprojection-based spatial logics, its satisfiability problem is decidable(precisely, PSPACE-complete). We also show that it is expressive enough tocapture meaningful interval temporal logics - in particular, the intervaltemporal logic of Allen's relations "Begins", "During", and "Later", and theirtransposes

Temporal and contextual knowledge in model-based expert systems

A basic paradigm that allows representation of physical systems with a focus on context and time is presented. Paragon provides the capability to quickly capture an expert's knowledge in a cognitively resonant manner. From that description, Paragon creates a simulation model in LISP, which when executed, verifies that the domain expert did not make any mistakes. The Achille's heel of rule-based systems has been the lack of a systematic methodology for testing, and Paragon's developers are certain that the model-based approach overcomes that problem. The reason this testing is now possible is that software, which is very difficult to test, has in essence been transformed into hardware

Model Checking the Logic of Allen's Relations Meets and Started-by is P^NP-Complete

In the plethora of fragments of Halpern and Shoham's modal logic of time intervals (HS), the logic AB of Allen's relations Meets and Started-by is at a central position. Statements that may be true at certain intervals, but at no sub-interval of them, such as accomplishments, as well as metric constraints about the length of intervals, that force, for instance, an interval to be at least (resp., at most, exactly) k points long, can be expressed in AB. Moreover, over the linear order of the natural numbers N, it subsumes the (point-based) logic LTL, as it can easily encode the next and until modalities. Finally, it is expressive enough to capture the {\omega}-regular languages, that is, for each {\omega}-regular expression R there exists an AB formula {\phi} such that the language defined by R coincides with the set of models of {\phi} over N. It has been shown that the satisfiability problem for AB over N is EXPSPACE-complete. Here we prove that, under the homogeneity assumption, its model checking problem is {\Delta}^p_2 = P^NP-complete (for the sake of comparison, the model checking problem for full HS is EXPSPACE-hard, and the only known decision procedure is nonelementary). Moreover, we show that the modality for the Allen relation Met-by can be added to AB at no extra cost (AA'B is P^NP-complete as well)

Orientation in French spatial expressions: formal representations and inferences

International audienceIn this paper we propose several formal tools intended to grasp an important aspect of static localisation in language namely orientation. We consider French spatial expressions used in localising an entity in an internal way (Internal Localisation Nouns such as 'haut' (top), 'bas' (bottom), 'devant' (front), 'derrière' (back)) or in an external way (prepositions 'devant' (in front of), 'derrière' (behind), 'au-dessus de' (above), 'au-dessous de' (below)). In order to represent these orientation phenomena, we build a logical framework made up of three levels that we call geometrical, functional and pragmatic. First, we define a geometry based on directions and relative localisation operators. Then, we introduce the functional notions that underly intrinsic orientation processes and we propose several formal definitions which may serve to represent the semantic content of the studied lexemes. These definitions allow us to make a difference between deictic and intrinsic uses of these spatial expressions and to draw interesting deductions and inferences. Finally, we integrate at the pragmatic level various principles governing the interpretation of such orientational expressions. By taking into account the different inferential schemata linked to the use of spatial expressions in discourse, this modular approach constitutes an original contribution to the semantic and cognitive studies of linguistic space

Decidability and complexity of the fragments of the modal logic of Allen's relations over the rationals

Interval temporal logics provide a natural framework for temporal reasoning about interval structures over linearly ordered domains, where intervals are taken as first-class citizens. Their expressive power and computational behaviour mainly depend on two parameters: the set of modalities they feature and the linear orders over which they are interpreted. In this paper, we consider all fragments of Halpern and Shoham's interval temporal logic hs with a decidable satisfiability problem over the rationals, and we provide a complete classification of them in terms of their expressiveness and computational complexity by solving the last few open problems