Compositional and inductive semantic definitions in fixpoint, equational, constraint, closure-condition, rule-based and game-theoretic form

We present a language and semantics-independent, compositional and inductive method for specifying formal semantics or semantic properties of programs in equivalent fixpoint, equational, constraint, closure-condition, rule-based and game-theoretic form. The definitional method is obtained by extending set-theoretic definitions in the context of partial orders. It is parameterised by the language syntax, by the semantic domains and by the semantic transformers corresponding to atomic and compound program components. The definitional method is shown to be preserved by abstract interpretation in either fixpoint, equational, constraint, closure-condition, rule-based or game-theoretic form. The features common to all possible instantiations are factored out thus allowing for results of general scope such as well-definedness, semantic equivalence, soundness and relative completeness of abstract interpretations, etc. to be proved compositionally in a general language and semantics-independent framework.