  It is known that a graph $C^*$-algebra $C^*(E)$ is approximately finite
dimensional (AF) if and only if the graph $E$ has no loops. In this paper we
consider the question of when a labeled graph $C^*$-algebra $C^*(E,\CL,\CB)$ is
AF. A notion of loop in a labeled space $(E,\CL,\CB)$ is defined when $\CB$ is
the smallest one among the accommodating sets that are closed under relative
complements and it is proved that if a labeled graph $C^*$-algebra is AF, the
labeled space has no loops. A sufficient condition for a labeled space to be
associated to AF algebra is also given. For graph $C^*$-algebras $C^*(E)$, this
sufficient condition is also a necessary one. Besides, we discuss other
equivalent conditions for a graph $C^*$-algebra to be AF in the setting of
labeled graphs and prove that these conditions are not always equivalent by
invoking various examples.
