The paradigm of type-based termination is explored for functional programming with recursive data types. The article introduces $\boldsymbol{\Lambda_\mu^+}$, a lambda-calculus with recursion, inductive types, subtyping and bounded quantification. Decorated type variables representing approximations of inductive types are used to track the size of function arguments and return values. The system is shown to be type safe and strongly normalizing. The main novelty is a bidirectional type checking algorithm whose soundness is established formally.

The question of how to combine monads arises naturally in many areas with much recent interest focusing on the coproduct of two monads. In general, the coproduct of arbitrary monads does not always exist. Although a rather general construction was given by Kelly  [Bull.  Austral. Math. Soc.22 (1980) 1–83], its generality is reflected in its complexity which limits the applicability of this construction. Following our own research [C. Lüth and N. Ghani, Lect. Notes Artif. Intell.2309 (2002) 18–32], and that of Hyland, Plotkin and Power [IFIP Conf. Proc.223 (2002) 474–484], we are looking for specific situations when simpler constructions are available. This paper uses fixed points to give a simple construction of the coproduct of two ideal monads.

We study the succinctness of monadic second-order logic and a variety of monadic fixed point logics on trees. All these languages are known to have the same expressive power on trees, but some can express the same queries much more succinctly than others. For example, we show that, under some complexity theoretic assumption, monadic second-order logic is non-elementarily more succinct than monadic least fixed point logic, which in turn is non-elementarily more succinct than monadic datalog.
Succinctness of the languages is closely related to the combined and parameterised complexity of query evaluation for these languages.

This paper proposes an operational semantics for value recursion in the context of monadic metalanguages. Our technique for combining value recursion with computational effects works uniformly for all monads. The operational nature of our approach is related to the implementation of recursion in Scheme and its monadic version proposed by Friedman and Sabry, but it defines a different semantics and does not rely on assignments. When contrasted to the axiomatic approach proposed by Erkök and Launchbury, our semantics for the continuation monad invalidates one of the axioms, adding to the evidence that this axiom is problematic in the presence of continuations.