  Decidability and synthesis of inductive invariants ranging in a given domain
play an important role in many software and hardware verification systems. We
consider here inductive invariants belonging to an abstract domain $A$ as
defined in abstract interpretation, namely, ensuring the existence of the best
approximation in $A$ of any system property. In this setting, we study the
decidability of the existence of abstract inductive invariants in $A$ of
transition systems and their corresponding algorithmic synthesis. Our model
relies on some general results which relate the existence of abstract inductive
invariants with least fixed points of best correct approximations in $A$ of the
transfer functions of transition systems and their completeness properties.
This approach allows us to derive decidability and synthesis results for
abstract inductive invariants which are applied to the well-known Kildall's
constant propagation and Karr's affine equalities abstract domains. Moreover,
we show that a recent general algorithm for synthesizing inductive invariants
in domains of logical formulae can be systematically derived from our results
and generalized to a range of algorithms for computing abstract inductive
invariants.
