The above characterizations express only the case that for parsing and generation the same relation has to be defined by some grammar G. But this does not exclude the possibility that G is compiled into specific parsing and generation grammars, as long as they define the same relation R (providing the compilation step is correct, e.g., [Strzalkowski1989, Block1991, Dymetman et al. 1990]).
In order to distinguish the two cases, where a reversible grammar G is used only during compile-time or is used during run-time for performing parsing and generation the terms weakly reversible grammars and strongly reversible grammars are introduced. We will say, that a reversible grammar G is strongly reversible iff P enumerates the respective sets using G during run-time, otherwise G is weakly reversible. In this thesis, we are only interested in strongly reversible grammars.
If a sentence s has been associated with more than one
interpretation, say 
, the relation R defined by G
will contain pairs 
and analogously for a meaning representation lf we will get
a set of pairs 
, of all possible sentences that have the same interpretation.
Accordingly, the sets are denoted as R(s) or R(lf). The cardinality
card(R(s)) of R(s) is defined as the degree of ambiguity of
s and the cardinality card(R(lf)) of R(lf) as the degree
of paraphrases of lf.
Suppose that for some s there exists exactly one semantic expression lf,
i.e., card(R(s)) = 1. Then, it is not valid to deduce that if
generation is performed starting with lf the resulting set R(lf) is
. However, it is guaranteed that 
(see also
figure 3.3).
Figure 3.3: The relationship between paraphrases and ambiguities.
Of course, this kind of ``reversibility'' is an intrinsic property of
each relation. But, if two separate grammars for parsing and generation
are used in a natural language system it has to be proven that they
describe the same relation
; otherwise it would be possible that a
sentence which is parse-able cannot be generated and vice versa. Grammar
reversibility is very important in practice because it ensures that
ambiguous structures and its paraphrases are interrelated. If this is
not the case then important aspects of performance like
self-monitoring or generation of paraphrases in order to disambiguate
ambiguous sentences cannot be modelled properly (in chapter
5 we discuss this problem in more detail).
Thus viewed, understanding and generation are dual processes, in the sense that each sentence which can be understood should also be producible and vice versa. This kind of duality is naturally captured if reversible grammars are used.