Amar Isli

A new approach to cyclic ordering of 2D orientations using ternary relation algebras

Abstract In Tarskis formalisation, the universe of a relation algebra (RA) consists of a set of binary relations. A first contribution of this work is the introduction of RAs whose universe is a set of ternary relations: these support rotation as an operation in addition to those present in Tarskis formalisation. Then we propose two particular RAs: a binary RA, CYC b , whose universe is a set of (binary) relations on 2D orientations; and a ternary RA, CYC t , whose universe is a set of (ternary) relations on 2D orientations. The RA CYC t , more expressive than CYC b , constitutes a new approach to cyclic ordering of 2D orientations. An atom of CYC t expresses for triples of orientations whether each of the three orientations is equal to, to the left of, opposite to, or to the right of each of the other two orientations. CYC t has 24 atoms and the elements of its universe consist of all possible 2 24 subsets of the set of all atoms. Amongst other results, 1. we provide for CYC t a constraint propagation procedure computing the closure of a problem under the different operations, and show that the procedure is polynomial, and complete for a subset including all atoms; 2. we prove that another subset, expressing only information on parallel orientations, is NP-complete; 3. we show that provided that a subset S of CYC t includes two specific elements, deciding consistency for a problem expressed in the closure of S can be polynomially reduced to deciding consistency for a problem expressed in S ; and 4. we derive from the previous result that for both RAs we “jump” from tractability to intractability if we add the universal relation to the set of all atoms. A comparison to the most closely related work in the literature indicates that the approach is promising.

A System Handling RCC-8 Queries on 2D Regions Representable in the Closure algebra of Half-Planes

The paper describes an algebraic framework for representing and reasoning about 2D spatial regions. The formalism is based on a Closure Algebra (CA) of half-plaries -i.e., a Boolean Algebra augmented with a closure operator. The CA provides a flexible representation for polygonal regions and for expressing topological constraints among such regions. The paper relates these constraints to relations defined in the 1st-order Region Connection Calculus (RCC). This theory allows the definition of a set of eight topological relations (RCC-8) which forms a partition of all possible relations between two regions. We describe an implemented algorithm for determining which of the RCC-8 relations holds between any two regions representable in the CA. One application of such a system is in Geographical Information Systems (GIS), where often the data is represented quantitatively, but it would be desirable for queries to be expressed qualitatively in a high level language such as that of the RCC theory.

A Topological Calculus for Cartographic Entities

Qualitative spatial reasoning (QSR) has many and varied applications among which reasoning about cartographic entities. We focus on reasoning about topological relations for which two approaches can be found in the literature: region-based approaches, for which the basic spatial entity is the spatial region; and point-set approaches, for which spatial regions are viewed as sets of points. We will follow the latter approach and provide a calculus for reasoning about point-like, linear and areal entities in geographic maps. The calculus consists of a constraint-based approach to the calculus-based method (CBM) in (Clementini et al., 1993). It is presented as an algebra alike to Allens (1983) temporal interval algebra. One advantage of presenting the CBM calculus in this way is that Allens incremental constraint propagation algorithm can then be used to reason about knowledge expressed in the calculus. The algorithm is guided by composition tables and a converse table provided in this contribution.

Temporal Constraint Satisfaction Techniques in Job ShopScheduling Problem Solving

We describe a restriction of Dechter, Meiri and Pearls TCSPs (Temporal Constraint Satisfaction Problems) sufficiently expressive to represent any job shop scheduling problem. A solver based on the restriction is then described, which is similar to Ladkin and Reinefelds qualitative interval network solver; except, however, that the filtering method used during the search is not path consistency but either ULT (Upper-Lower Tightening) or LPC (Loose Path- Consistency), which are both less effective but have the advantage of getting rid of the so-called “fragmentation problem”.

Study of symmetry in qualitative temporal interval networks

Symmetry has been studied in both propositional calculus and discrete constraint satisfaction problems. This has been shown to reduce considerably the search space. In this paper, we extend the study to qualitative interval networks. We provide experimental tests on the performances of a variant of Ladkin and Reinefelds search algorithm in the following two cases: (1) the algorithm as provided by its authors, with no advantage of symmetry, and (2) the algorithm to which is added symmetry detection during the search. The experiments show that symmetries are profitable for hard problems.