Let 
be a semantic network retrieval model. The set 
consists of
objects each of which is a node in a directed graph G with two types of arcs: isa
and partof. An isa-arc denotes the subclass-superclass relationship
between the nodes it connects; a partof-arcs denotes the part-whole
relationship between the nodes (Fitzgerald 1995). Let 
be the 
matrix such
that 
if there is an isa-arc from 
and 
, and 
,
if there is no such arc. Let 
be a similar matrix for the partof-relationship.
An object 
abstracts an object 
iff G has a path of isa-arcs
from to . When abstracts , is
an abstraction of . An object specializes an
object iff G has a path of isa-arcs from to .
Thus, abstracts iff specializes .
Any object both abstracts and specializes itself.
Associated with each node is a single set of labels. A label 
is a sequence
of elements such that for all i, 
. Thus, labels may contain not only
tokens but also object ids. If 
, then 
is the set of labels
associated with . If , and 
, 
. An expectation
is a 3-tuple 
such that 
. For example, if 
, then 
, 
, 
,
and 
are expectations. Intuitively, an expectation reflects how completed a
label is with respect to an object. If 
is an expectation, 
, 
, 
,
and key(z) = head(ecseq(z)).
, where is the set of labels associated with , 
,
and 
. Note that 
and 
can be empty. For example,
if 
, 
. Let 
and . The object
similarity between and 
is 
, where 
, 
, 
completes 
, 
and 
. The object activates
the object iff there exists a label and a label such that completes .
If there is no such that completes 
, then 
.
Let [x] be a label associated with a node . The token closure of [x],
, is the set of all token sequences that complete [x]. For example, let 
and
be two nodes such that 
and 
, where all 
.
Then 
. An algorithm for computing token closures is given in Listing 1 of the
Appendix. Intuitively, the token closure of a label is the set of all token sequences
through which the label can be completed, and, consequently, the label's node activated.
For example, the first two elements of 
complete 
associated in 
,
thus activating . When is activated, 
is advanced one element to
the right, which allows the rest of the label, i.e., 
to be completed by the
last two elements of , thus activating . The token
closure of a node , 
, is the union of the
token closures of its labels. Thus, the token closure of a node is the set of all token
sequences that can possibly activate the node. The union of the token closures of nodes in
is called -closure. Formally, -closure 
.