Lagrangian Floer homology in a general case has been constructed by Fukaya,
Oh, Ohta and Ono, where they construct an $\AI$-algebra or an $\AI$-bimodule
from Lagrangian submanifolds, and studied the obstructions and deformation
theories. But for obstructed Lagrangian submanifolds, standard Lagrangian Floer
homology can not be defined.

We explore several well-known cohomology theories on these $\AI$-objects and
explore their properties, which are well-defined and invariant even in the
obstructed cases. These are Hochschild and cyclic homology of an $\AI$-objects
and Chevalley-Eilenberg or cyclic Chevalley-Eilenberg homology of their
underlying $\LI$ objects. We explain how the existence of $m_0$ effects the
usual homological algebra of these homology theories. We also provide some
computations. We show that for an obstructed $\AI$-algebra with a non-trivial
primary obstruction, Chevalley-Eilenberg Floer homology vanishes, whose proof
is inspired by the comparison with cluster homology theory of Lagrangian
submanifolds by Cornea and Lalonde.

In contrast, we also provide an example of an obstructed case whose cyclic
Floer homology is non-vanishing.