Incompleteness results for behavioral logics are investigated. We show that there is a basic finite behavioral specification for which the behavioral satisfaction problem is not recursively enumerable, which means that there are no automatic methods for proving all true statements; in particular, behavioral logics do not admit complete deduction systems. This holds for all of the behavioral logics of which we are aware. We also prove that the behavioral satisfaction problem is not co-recursively enumerable, which means that there is no automatic way to refute false statements in behavioral logics. In fact we show stronger results, that all behavioral logics are # 0 2 -hard, and that, for some data algebras, the complexity of behavioral satisfaction is not even arithmetic; matching upper bounds are established for some behavioral logics. In addition, we show for the fixed-data case that if operations mayhave more than one hidden argument, then final models need not exist, so that the coalgebraic flavor of behavioral logic is lost.

Coalgebras of polynomial functors constructed from sets of observable elements have been found useful in modelling various kinds of data types and state-transition systems. This paper presents a calculus of terms for operations on such coalgebras, based on a simple type theory, and develops its semantics. The terms admit a single state-valued parameter, but may also have state-valued variables. In a "rigid" term all state-variables are bound. Boolean