(i) The introduction of the concept of transitive closure
according to a given property.
The archetypal example of a relation that cannot be expressed in first-order logic is the transitive closure
of the edges in a graph.
For example, transitive closure
of a relation R may be defined as follows :
Therefore, the transitive closure
of [[PI].sup.m.sub.[DELTA]], denoted by [M.sup.c.sub.[DELTA]], is a strict partial order over [L.sub.[DELTA]] and [??] is a mixed precedence relation over defeasible rules declared in [DELTA].The general idea of this process is depicted in Figure 4, considering an arbitrary situation involving eight labels.
In the sequel, we prove that R is the transitive closure
of R, which also implies that R is an equivalent relation on U.
Since S is a directed tree labeled by a partition of V, its transitive closure
is a preposet on V.
(*) the transitive closure
[[pi].sub.i] of [[rho].sub.i] is contained in [[rho].sub.i+1] and the relation [[rho].sub.i+1]/[[pi].sub.i] induced by [[rho].sub.i+1] on the set of equivalence classes Q/[[pi].sub.i] is a partial ordering.
Notice that the use of a transitive closure
step is not limited to the multi-pass approach [MMT+, 03].
Fuzzy transitive-closure coherence axiom (FTCCA): for any S [member of] B and x [member of] X, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], where [bar.R] denotes the transitive closure
Thus, the covered knowledge of a class c is the transitive closure
of the relations subclass of class c and image of class c via a certain mapping.
Efficient Algorithms for the Instantiated Transitive Closure
Queries, IEEE Transactions on Software Engineering, 17 (3).
The transitive closure
of a directed graph G = (V, E) is the directed graph G* = (V, [E.sup.*]), which has an edge corresponding to every directed path in G.