A retrieval model 
is embedded in a universe of objects. An object
is a 2-tuple 
, where 
is the object's unique id, and 
is
the object's unique representation. The exact definition of representation depends
on specific retrieval tasks (Kulyukin 1999a; Kulyukin 1999b). The universe in which
M is embedded is the set of all objects, and is denoted by 
. The finite
set of objects retrievable by M is denoted by 
. The set 
contains the
ids of objects in 
. Since there is a bijection between and 
,
when the context permits, objects are referred to by their ids. M's primitives are
called tokens. The precise definition of a token depends on the context (Kulyukin
1998a; Kulyukin 1998b). Tokens can be keywords, keyword collocations, nodes in a semantic
network, etc. The set of all possible tokens is denoted by 
.
If 
is M's set of representations, M's representation
function is 
. If 
, 
. The token weight
function 
assigns weights to tokens in objects. The object similarity function
computes the similarity between two objects in . The rank function 
imposes an ordering on 's objects. The rank of 
with respect to
is denoted by 
. If 
, then 
, and 
. Thus, the
ranking of objects is determined by 
and their initial ordering in .
A retrieval sequence returned by M in response to is denoted by 
,
and is a permutation 
of the ids of objects in such that 
.
Let 
and 
. 
and 
are equivalent
under ranked retrieval ( 
) iff 
= 
, 
, 
,
... , 
, 
, 
, 
= , 
, ,
... , , , , and 
.