where a and b are the measures being combined. This T-norm is commonly referred to as the probabilistic and (PAND) function [Bonissone Decker1986]. The immediate parent category node directly receives an associated measure by combining the measures of the functional property nodes using the same T-norm.
For example, the functional property provides_sittable_surface is defined
by six primitives. For simplicity, we'll denote the evaluation
measures returned by these six primitives as 
through 
. The functional
property stability is defined by a single primitive, which also returns an
evaluation measure (
). To determine the overall evaluation measure of a
shape for the category conventional chair we compute
conventional chair ::= provides_sittable_surface PAND
stability
where
provides_sittable_surface ::= 
PAND 
PAND 
PAND
PAND 
PAND 
and
stability := 
Since the definition of a (sub)category is a conjunction of required functional properties, the cumulative measure should be dominated by the ``weakest link" in the individual primitive evaluation measures, a property of the PAND function. So, an evaluation measure of 0 for any one primitive physical property will result in a cumulative evaluation measure of 0. An evaluation measure of 1 indicates that the primitive physical property has been ideally satisfied, and the shape may belong to the object category. The final result depends on the evaluation of other primitive physical properties.
It would seem that each category could simply be defined by the knowledge primitives without using the notion of functional properties. The functional property level was introduced into the representation hierarchy for two reasons. First, the subgroupings of functional properties intuitively follow the levels of named categorization typical of human concepts of function. Secondly, most functional property evaluations result in the labeling of the functional elements of the object (i.e., the portions of the structure) that fulfill the functional requirement.
Since the subcategory definition represents an increasingly specialized
definition, evidence for belonging to the subcategory should result in an
increased measure for the object belonging to the subcategory as opposed
to just the parent category. The combination of the functional property
measurement of a subcategory node, a, with its parent node's evaluation
measure, b, is computed using the T-conorm:
This T-conorm is commonly referred to as the probabilistic or (POR) function [Bonissone Decker1986]. While the T-conorm is used to combine measures at a subcategory node, the final subcategory evaluation measure is actually computed as:
where T is a user defined threshold. Thus, the functional property
measurement of a subcategory node, a, must be greater than some
minimum in order for a shape to receive a non-zero evaluation
measure for the subcategory. For the purposes of this work, a value
of T=0 is assumed, indicating that a shape can be assigned to a
subcategory as long as there is some non-zero evidence that it meets
the additional functional requirements associated with the subcategory.
In practice, a final classification decision might require much
stronger evidence, say 
, before a shape is assigned to a subcategory.
For example, to determine the overall evaluation measure of a shape for the
category straightback chair, we first compute the overall evaluation
measure for the category conventional chair, as previously described.
The functional property provides_back_support is defined by 8 primitives.
Denoting the measurements returned by the 8 primitives as 
through
, the overall evaluation measure (assuming the measure for 
) for the category straightback chair is computed as:
straightback chair ::= conventional chair POR
provides_back_support
where
provides_back_support ::= 
PAND 
PAND 
PAND
PAND 
PAND 
PAND 
PAND 
An object that can function as a straightback chair can also by definition function as a conventional chair. The T-conorm will give the object a higher evaluation measure for the subcategory straightback chair since there is some evidence in addition to the ``minimal" amount of evidence required for the shape to belong to the parent category conventional chair. Thus, GRUFF performs recognition of a shape by selecting the (sub)category with the highest overall evaluation measure. This should correspond to the most specific applicable subcategory. One exception occurs when the parent category has an evaluation measure of 1 and there is non-zero evidence supporting the subcategory functional requirements. In this case, the T-conorm assigns an evaluation measure of 1 to both the category and subcategory.
The particular T-norm/T-conorm pair utilized in this paper was chosen from among representative T-norm/T-conorm possibilities (including non-probabilistic formulations) described by Bonissone and Decker (1986) after analyzing their performance in conjunction with GRUFF across a set of example shapes [Stark, Hall, Bowyer 1993a].