We define a notion of module over a monad and use it to propose a new definition (or semantics) for abstract syntax (with binding constructions). Using our notion of module, we build a category of exponential monads, which can be understood as the category of lambda-calculi, and prove that it has an initial object (the pure untyped lambda-calculus). Our definitions and results are formalized in...

The subject of this paper is monadic second-order logic over two-dimensional grids. We give a game-theoretical proof for the strictness of the monadic second-order quan-tiier alternation hierarchy over grids. Additionally, we can show that monadic second-order logic over coloured grids is expressive enough to deene complete problems for each level of the polynomial time hierarchy.

The monadic fragments of first-order Gödel logics are investigated. It is shown that all finite-valued monadic Gödel logics are decidable; whereas, with the possible exception of one (G↑), all infinitevalued monadic Gödel logics are undecidable. For the missing case G↑ the decidability of an important sub-case, that is well motivated also from an application oriented point of view, is proven. A...

In this paper, we consider existential monadic second-order logic (monadic Σ1) on finite unary graphs, that is finite graphs with functional edge relation. These can be seen as finite encodings of ultimately periodic words. We show that, on these graphs, the bisimulation invariant fragment of monadic Σ1 equals the bisimulation invariant fragment of monadic second order logic itself, and that MS...

We consider an interpretation of monadic second-order logic of order in the continuous time structure of finitely variable signals. We provide a characterization of the expressive power of monadic logic. As a by-product of our characterization we show that many fundamental theorems which hold in the discrete time interpretation of monadic logic are still valid in the continuous time interpretat...

The problem of selecting nodes in unranked trees is the most basic querying problem for XML. We propose stepwise tree automata for querying unranked trees. Stepwise tree automata can express the same monadic queries as monadic Datalog and monadic second-order logic. We prove this result by reduction to the ranked case, via a new systematic correspondence that relates unranked and ranked queries.

Gurevich and Shelah have shown that Peano Arithmetic cannot be interpreted in the monadic second-order theory of short chains (hence, in the monadic secondorder theory of the real line). We will show here that it is consistent that there is no interpretation even in the monadic second-order theory of all chains.

Monadic Second-Order Unification (MSOU) is Second-Order Unification where all function constants occurring in the equations are unary. Here we prove that the problem of deciding whether a set of monadic equations has a unifier is NP-complete. We also prove that Monadic Second-Order Matching is also NP-complete.