An interpretation of consists of a domain D which is the set of all feature graphs built from L, C and V, and a solution mapping . which will be defined below. An -assignment is a mapping from the set of variables to D. I write ASS for the set of all -assignments. Variables and constants will denote feature graphs, relative to some assignment.
The denotation of a variable X w.r.t. assignment is simply (X). The denotation of a constant c is the feature graph (c,) (for any assignment). The denotation of a descriptor sp is defined as Fp where F is the denotation of s; i.e. the denotation of sp is the subgraph at p of the graph denoted by s. Note that the denotation of some descriptors is undefined. Summarizing:
As an example, consider the feature graph F2 in figure 2.2.
An interpretation satisfies an atomic constraint = d1 d2, relative to an assignment , written if the denotation of descriptors d1, d2 are both defined and the same, i.e.:
Hence, satisfies the equation Xl1l3 Xl2l3 with respect to the assignment that maps X to the feature graph F2 defined above. As another example, also satisfies the equation Xl1l3l1 c2, with the same .
The solutions of a constraint are all assignments that give satisfaction. Constraints are thus seen as restrictions on the values the variables in the constraints can take. The solutions of an atomic constraint are defined as follows:
The set of solutions of a constraint ,..., is defined as the intersection of the solutions of its atomic constraints:
A constraint is satisfiable iff it has at least one solution. Two constraints are equivalent iff they have the same solutions. A constraint is valid iff its solutions are all possible assignments ASS.