Bi-inductive structural semantics

We propose a simple order-theoretic generalization, possibly non monotone, of settheoretic inductive definitions. This generalization covers inductive, co-inductive and bi-inductive definitions and is preserved by abstraction. This allows structural operational semantics to describe simultaneously the finite/terminating and infinite/diverging behaviors of programs. This is illustrated on grammars and the structural bifinitary small/big-step trace/relational/operational semantics of the call-by-value λ-calculus (for which co-induction is shown to be inadequate).