Jean-François Condotta

Computational Complexity of Propositional Linear Temporal Logics Based on Qualitative Spatial or Temporal Reasoning

We consider the language obtained by mixing the model of the regions and the propositional linear temporal logic. In particular, we propose alternative languages where the model of the regions is replaced by different forms of qualitative spatial or temporal reasoning. In these languages, qualitative formulas describe the movement and the relative positions of spatial or temporal entities in some spatial or temporal universe. This paper addresses the issue of the formal proof that for all forms of qualitative spatial and temporal reasoning such that consistent atomic constraint satisfaction problems are globally consistent, determining of any given qualitative formula whether it is satisfiable or not is PSPACE-complete.

Tractability Results in the Block Algebra

In this paper we define the notion of a block algebra, which is based upon a spatial application of Allen’s interval algebra. In the -dimensional Euclidean space, where , we consider only blocks whose sides are parallel to the axes of some orthogonal basis. The block algebra consists of a set of relations (the block relations) together with the fundamental operations of composition, converse and intersection. The basic relations of this algebra constitute the exhaustive list of the relations possibly holding between two blocks. We are interested in the problem of testing the consistency of a set of spatial constraints between blocks, i.e. a block network. The consistency question for block networks is NP-complete. We first extend the notions of convexity and preconvexity to the block algebra. Similarly to the interval algebra case, convexity leads to a tractable set whereas, contrary to the interval algebra case, preconvexity leads to an intractable set. Nevertheless we characterize a tractable subset of the preconvex relations: the strongly preconvex relations. Moreover we show that strong preconvexity and ORD-Horn representability are the same.

A Generic Toolkit for n-ary Qualitative Temporal and Spatial Calculi

Temporal and spatial reasoning is a central task for numerous applications in many areas of artificial intelligence. For this task, numerous formalisms using the qualitative approach have been proposed. Clearly, these formalisms share a common algebraic structure. In this paper we propose and study a general definition of such formalisms by considering calculi based on basic relations of an arbitrary arity. We also describe the QAT (the qualitative algebra toolkit), a JAVA constraint programming library allowing to handle constraint networks based on those qualitative calculi

A Tractable Subclass of the Block Algebra: Constraint Propagation and Preconvex Relations

We define, in this paper, for every n ≥ 1, n-dimensional block algebra as a set of relations, the block relations, together with the fundamental operations of composition, conversion and intersection. We examine the 13n atomic relations of this algebra which constitute the exhaustive list of the permitted relations that can hold between two blocks whose sides are parallel to the axes of some orthogonal basis in the n-dimensional Euclidean space over the field of real numbers. We organize these atomic relations in ascending order with the intention of defining the concept of convexity as well as the one of preconvexity. We will confine ourselves to the issue of the consistency of block networks which consist of sets of constraints between a finite number of blocks. Showing that the concepts of convexity and preconvexity are preserved by the fundamental operations, we prove the tractability of the problem of the consistency of strongly preconvex block networks, on account of our capacity for deciding it in polynomial time by means of the path-consistency algorithm.

Spatial Reasoning About Points in a Multidimensional Setting

For n ≥ 1, we consider the possible relations between two points of the Euclidean space of dimension n. We define the n-point algebra on the pattern of the point algebra and the cardinal algebra. Generalizing the concept of convexity just as the one of preconvexity, we prove that the consistency problem of convex n-point networks is polynomial for n ≥ 1, whereas the consistency problem of preconvex n-point networks is NP-complete for n ≥ 3. We characterize a subset of the set of all preconvex relations: the set of all strongly preconvex relations, which contains the set of all convex relations. We demonstrate that the consistency problem of strongly preconvex n-point networks can be decided in polynomial time by means of the weak path-consistency method for all n ≥ 1. For n = 3 the set of all strongly preconvex relations is a maximal tractable subclass of the set of all n-point relations. Finally, we prove that the concept of strong preconvexity corresponds to the one of ORD-Horn representability.

Consistency of Triangulated Temporal Qualitative Constraint Networks

In this paper, we introduce for the qualitative constraint networks (QCNs) a new consistency: the partial weak composition consistency. The partial weak composition consistency, similarly to the partial path-consistency, considers triangles of a graph and corresponds to the weak composition consistency restricted to these triangles. We show that for the pre-convex QCNs of the Interval Algebra (IA), the partial weak composition consistency with respect to a triangulation of the graph of constraints is sufficient to decide the consistency problem. From this result, we propose an algorithm allowing to solve QCNs of IA. The experiments that we have conducted show the interest of this algorithm to solve the consistency problem of the QCNs of IA.

Eligible and frozen constraints for solving temporal qualitative constraint networks

In this paper we consider the consistency problem for qualitative constraint networks representing temporal or spatial information. The most efficient method for solving this problem consists in a search algorithm using, on the one hand, the weak composition closure method as a local propagation method, and on the other hand, a decomposition of the constraints into subrelations of a tractable set. We extend this algorithm with the notion of eligibility and the notion of frozen constraints. The first concept allows to characterise constraints which will not be considered during the search. The second one allows to freeze constraints in order to avoid unnecessary updates.

Merging Qualitative Constraints Networks Using Propositional Logic

In this paper we address the problem of merging qualitative constraints networks (QCNs ). We propose a rational merging procedure for QCNs . It is based on translations of QCNs into propositional formulas, and take advantage of propositional merging operators.

On the consistency problem for the INDU calculus

In this paper, we further investigate the consistency problem for the qualitative temporal calculus INDU introduced by A. K. Pujari et al. (1999). We prove the intractability of the consistency problem for the subset of preconvex relations. On the other hand, we show the tractability of strongly preconvex relations. Furthermore, we also define another interesting set of relations for which the consistency problem can be decided by a method similar to the usual path-consistency method.

Tackling Large Qualitative Spatial Networks of Scale-Free-Like Structure

We improve the state-of-the-art method for checking the consistency of large qualitative spatial networks that appear in the Web of Data by exploiting the scale-free-like structure observed in their underlying graphs. We propose an implementation scheme that triangulates the underlying graphs of the input networks and uses a hash table based adjacency list to efficiently represent and reason with them. We generate random scale-free-like qualitative spatial networks using the Barabasi-Albert (BA) model with a preferential attachment mechanism. We test our approach on the already existing random datasets that have been extensively used in the literature for evaluating the performance of qualitative spatial reasoners, our own generated random scale-free-like spatial networks, and real spatial datasets that have been made available as Linked Data. The analysis and experimental evaluation of our method presents significant improvements over the state-of-the-art approach, and establishes our implementation as the only possible solution to date to reason with large scale-free-like qualitative spatial networks efficiently.