Linearity and the cnf property in linear fuzzy rule interpolation

It is an important question if rule interpolation is done whether the theoretical shape of the membership function of the calculated conclusion is exactly or approximately linear between two neighboring α-levels in the breakpoint set, or it has a very different shape. In the latter case, interpolation for only a few (as e.g. 0 and 1) levels is not satisfactory, which fact might increase the computational time necessary for generating the conclusion by a large constant factor. It is also examined if the conclusion has a convex and normal membership function, i.e. whether the calculated infima exceed the calculated suprema of the given α-cut or not