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

An Integrated First-Order Theory of Points and Intervals over Linear Orders (Part I)

There are two natural and well-studied approaches to temporal ontology and reasoning: point-based and interval-based. Usually, interval-based temporal reasoning deals with points as a particular case of duration-less intervals. A recent result by Balbiani, Goranko, and Sciavicco presented an explicit two-sorted point-interval temporal framework in which time instants (points) and time periods (intervals) are considered on a par, allowing the perspective to shift between these within the formal discourse. We consider here two-sorted first-order languages based on the same principle, and therefore including relations, as first studied by Reich, among others, between points, between intervals, and inter-sort. We give complete classifications of its sub-languages in terms of relative expressive power, thus determining how many, and which, are the intrinsically different extensions of two-sorted first-order logic with one or more such relations. This approach roots out the classical problem of whether or not points should be included in a interval-based semantics

An Integrated First-Order Theory of Points and Intervals over Linear Orders (Part II)

There are two natural and well-studied approaches to temporal ontology and reasoning: point-based and interval-based. Usually, interval-based temporal reasoning deals with points as a particular case of duration-less intervals. A recent result by Balbiani, Goranko, and Sciavicco presented an explicit two-sorted point-interval temporal framework in which time instants (points) and time periods (intervals) are considered on a par, allowing the perspective to shift between these within the formal discourse. We consider here two-sorted first-order languages based on the same principle, and therefore including relations, as first studied by Reich, among others, between points, between intervals, and inter-sort. We give complete classifications of its sub-languages in terms of relative expressive power, thus determining how many, and which, are the intrinsically different extensions of two-sorted first-order logic with one or more such relations. This approach roots out the classical problem of whether or not points should be included in a interval-based semantics. In this Part II, we deal with the cases of all dense and the case of all unbounded linearly ordered sets.Comment: This is Part II of the paper `An Integrated First-Order Theory of Points and Intervals over Linear Orders' arXiv:1805.08425v2. Therefore the introduction, preliminaries and conclusions of the two papers are the same. This version implements a few minor corrections and an update to the affiliation of the second autho

Representation of non-convex time intervals and propagation of non-convex relations

For representing natural language expressions with temporal repetition the well known time interval calculus of Allen [Allen 83] is not adequat. The fundamental concept of this calculus is that of convex intervals which have no temporal gaps. However, natural language expressions like "every Summer" or "on each Monday" require the possibility of such temporal gaps. Therefore, we have developed a new calculus based on non-convex intervals and have defined a set of corresponding non-convex relations. The non-convex intervals are sets of convex intervals and contain temporal gaps. The non-convex relations are tripels: a first part for specifying the intended manner of the whole relation, a second part for defining relations between subintervals, and a third part for declaring relations of whole, convexified non-convex intervals. In the non-convex calculus the convex intervals and relations of Allen are also integrated as a special case. Additionally, we have elaborated and fully implemented a constraint propagation algorithm for the non-convex relations. In comparison with the convex case we get a more expressive calculus with same time complexity for propagation and only different by a constant factor

Modal Logics of Topological Relations

Logical formalisms for reasoning about relations between spatial regions play a fundamental role in geographical information systems, spatial and constraint databases, and spatial reasoning in AI. In analogy with Halpern and Shoham's modal logic of time intervals based on the Allen relations, we introduce a family of modal logics equipped with eight modal operators that are interpreted by the Egenhofer-Franzosa (or RCC8) relations between regions in topological spaces such as the real plane. We investigate the expressive power and computational complexity of logics obtained in this way. It turns out that our modal logics have the same expressive power as the two-variable fragment of first-order logic, but are exponentially less succinct. The complexity ranges from (undecidable and) recursively enumerable to highly undecidable, where the recursively enumerable logics are obtained by considering substructures of structures induced by topological spaces. As our undecidability results also capture logics based on the real line, they improve upon undecidability results for interval temporal logics by Halpern and Shoham. We also analyze modal logics based on the five RCC5 relations, with similar results regarding the expressive power, but weaker results regarding the complexity

An integrated first-order theory of points and intervals : expressive power in the class of all linear orders

There are two natural and well-studied approaches to temporal ontology and reasoning, that is, pointbased and interval-based. Usually, interval-based temporal reasoning deals with points as a particular case of duration-less intervals. Recently, a two-sorted point-interval temporal logic in a modal framework in which time instants (points) and time periods (intervals) are considered on a par has been presented. We consider here two-sorted first-order languages, interpreted in the class of all linear orders, based on the same principle, with relations between points, between intervals, and intersort. First, for those languages containing only interval-interval, and only inter-sort relations we give complete classifications of their sub-fragments in terms of relative expressive power, determining how many, and which, are the different two-sorted first-order languages with one or more such relations. Then, we consider the full two-sorted first-order logic with all the above mentioned relations, restricting ourselves to identify all expressively complete fragments and all maximal expressively incomplete fragments, and posing the basis for a forthcoming complete classification

The Corley-Jacobson dispersion relation and trans-Planckian inflation

In this Letter we study the dependence of the spectrum of fluctuations in inflationary cosmology on possible effects of trans-Planckian physics, using the Corley/Jacobson dispersion relations as an example. We compare the methods used in previous work [1] with the WKB approximation, give a new exact analytical result, and study the dependence of the spectrum obtained using the approximate method of Ref. [1] on the choice of the matching time between different time intervals. We also comment on recent work subsequent to Ref. [1] on the trans-Planckian problem for inflationary cosmology.Comment: 6 pages, Revtex

Dispersion Relations for Thermally Excited Waves in Plasma Crystals

Thermally excited waves in a Plasma crystal were numerically simulated using a Box_Tree code. The code is a Barnes_Hut tree code proven effective in modeling systems composed of large numbers of particles. Interaction between individual particles was assumed to conform to a Yukawa potential. Particle charge, mass, density, Debye length and output data intervals are all adjustable parameters in the code. Employing a Fourier transform on the output data, dispersion relations for both longitudinal and transverse wave modes were determined. These were compared with the dispersion relations obtained from experiment as well as a theory based on a harmonic approximation to the potential. They were found to agree over a range of 0.9<k<5, where k is the shielding parameter, defined by the ratio between interparticle distance a and dust Debye length lD. This is an improvement over experimental data as current experiments can only verify the theory up to k = 1.5.Comment: 8 pages, Presented at COSPAR '0