Tomographic reconstruction of 2-convex polyominoes using dual Horn clauses

Among the very many research interests of Maurice Nivat, a special role, according to the produced literature, was played by the study of the algorithmic and combinatorial aspects of connected finite discrete sets of points called polyominoes. In particular, he addressed the problem of a faithful reconstruction of some subclasses of them by imposing convexity constraints. The present study fits in this research line, and relies on a well known algorithm that Maurice Nivat and co-authors defined 1996 for the reconstruction of hv-convex polyominoes by orthogonal projections in polynomial time. Here, we consider a hierarchy on this class of polyominoes, and we continue a longstanding research on the reconstruction of its first levels by specializing the above mentioned result. The algorithm we propose bases on the possibility of characterizing the elements of the second level of the hierarchy, by a logic formula belonging to Dual-Horn and so polynomially solvable. Some related open problems are also presented.