Improved Algorithms for Allen's Interval Algebra: a Dynamic Programming Approach

The constraint satisfaction problem (CSP) is an important framework in artificial intelligence used to model e.g. qualitative reasoning problems such as Allen's interval algebra A. There is strong practical incitement to solve CSPs as efficiently as possible, and the classical complexity of temporal CSPs, including A, is well understood. However, the situation is more dire with respect to running time bounds of the form O(f(n)) (where n is the number of variables) where existing results gives a best theoretical upper bound 2^O(n * log n) which leaves a significant gap to the best (conditional) lower bound 2^o(n). In this paper we narrow this gap by presenting two novel algorithms for temporal CSPs based on dynamic programming. The first algorithm solves temporal CSPs limited to constraints of arity three in O(3^n) time, and we use this algorithm to solve A in O((1.5922n)^n) time. The second algorithm tackles A directly and solves it in O((1.0615n)^n), implying a remarkable improvement over existing methods since no previously published algorithm belongs to O((cn)^n) for any c. We also extend the latter algorithm to higher dimensions box algebras where we obtain the first explicit upper bound.

Certain and Uncertain Temporal Data Representation and Reasoning in OWL 2

Temporal data given by Alzheimer's patients are mostly uncertain. Many approaches have been proposed to handle certain temporal data and lack uncertain ones. This paper proposes an approach to represent and reason about quantitative time intervals and points and qualitative relations between them. It is suitable to handle certain and uncertain temporal data. It includes three parts. (1) The authors extend the 4D-fluents approach with certain components to represent certain and uncertain temporal data. (2) They extend the Allen's interval algebra to reason about certain and uncertain time intervals. They adapt these relations to relate a time interval and a time point, and two time points. All relations can be used for temporal reasoning by means of transitivity tables. (3) They propose a certain ontology based on the extensions. A prototype is implemented and integrated into an ontology-based memory prosthesis for Alzheimer's patients to handle uncertain data inputs. The evaluation proves the usefulness of the approach as all the inferences are well established and the precision results are promising.

Interval Algebra and Region Connection Calculus for Ontological Spatiotemporal Assembly Product Motion Knowledge Representation

Over the last decades, noticeable efforts have been made to construct design knowledge during the detailed geometric definition phase systematically. However, physical products exhibit functional behaviors, which explain that they evolve over space and time. Hence, there is a need to extend assembly product knowledge towards the spatiotemporal dimension to provide more realistic knowledge models in assembly design. Systematic semantic knowledge representation via ontology enables designers to understand the anticipated product’s behavior in advance. In this article, Interval Algebra (IA) and Region Connection Calculus (RCC) are investigated to formalize and construct ontological spatiotemporal assembly product motion knowledge. IA is commonly used to represent the temporality between two entities, while RCC is more appropriate to represent the ‘part-to-part’ relationships of two topological spaces. This paper discusses the roles of IA and RCC and presents a case study of a nutcracker assembly model’s behavior. The assembly product motion ontology with the aid of IA and RCC is evaluated using a task-based approach. The evaluation shows the added value of the developed ontology compared to others published in the literature.

Dealing with Precise and Imprecise Temporal Data in Crisp Ontology

This article proposes a crisp-based approach for representing and reasoning about concepts evolving in time and of their properties in terms of qualitative relations (e.g., “before”) in addition to quantitative ones, time intervals and points. It is not only suitable to handle precise time intervals and points, but also imprecise ones. It extends the 4D-fluents approach with crisp components to represent handed data. It also extends the Allen's interval algebra. This extension allows reasoning about imprecise time intervals. Compared to related work, it is based on crisp set theory. These relations preserve many properties of the original algebra. Their definitions are adapted to allow relating a time interval and a time point, and two time points. All relations can be used for temporal reasoning by means of transitivity tables. Finally, it proposes a crisp ontology that based on the extended Allen's algebra instantiates the 4D-fluents-based representation.

Formal Representation of Temporal Expressions

In this paper we address the semantics of temporal expressions in natural language (such as vchera,’yesterday’, shestnadcatogo maja, ’on the 16th of May’, tri dnja ’three days’) and the way they interact with some other manifestations of temporality (such as functioning of prepositions and aspectual verb forms). A formal and constituent description of heterogeneous temporal expressions is proposed. We consider the interval algebra presented by James Allen to be the right basis for such a description. The new formal system is compared with the known TimeML project. The latter has weak spots - the meaning of some temporal expressions simply can not be represented in terms of TimeML. We discuss such cases and show how to analyze them in our formal system.

Advanced Temporal Constraints for Business Processes Modelling and Execution

In several applications like healthcare, time in workflow execution is critical. Several control and data dependencies arise that must be specified, validated as conflict free, and maintained during workflow execution. The author models these kinds of dependencies as constraints that impose temporal restrictions on the relative order of execution of the activities. Hence, a finer granularity of activity execution with respect to time is introduced. The author incorporates a subset of interval algebra in the workflow specification model and the author proposes the T-WfMc specification model. The author examines the consistency issues that arise, and the author proposes different correctness criteria.