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

In Tarskiʹs 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 Tarskiʹs formalisation. Then we propose two particular RAs: a binary RA, CYCb , whose universe is a set of (binary) relations on 2D orientations; and a ternary RA, CYCt , whose universe is a set of (ternary) relations on 2D orientations. The RA CYCt , more expressive than CYCb , constitutes a new approach to cyclic ordering of 2D orientations. An atom of CYCt 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. CYCt has 24 atoms and the elements of its universe consist of all possible 224 subsets of the set of all atoms. Amongst other results, 1. we provide for CYCt 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 CYCt 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.