In every undirected graph or, more generally, in every undirected hypergraph of bounded rank, one can specify an orientation of the edges or hyperedges by monadic second-order formulas using quantifications on sets of edges or hyperedges. The proof uses an extension to hypergraphs of the classical notion of a depth-first spanning tree. Applications are given to the characterization of the classes of graphs and hypergraphs having decidable monadic theories.

The classical Feferman–Vaught Theorem for First Order Logic explains how to compute the truth value of a first order sentence in a generalized product of first order structures by reducing this computation to the computation of truth values of other first order sentences in the factors and evaluation of a monadic second order sentence in the index structure. This technique was later extended by Läuchli, Shelah and Gurevich to monadic second order logic. The technique has wide applications in decidability and (...) definability theory. Here we give a unified presentation, including some new results, of how to use the Feferman–Vaught Theorem, and some new variations thereof, algorithmically in the case of Monadic Second Order Logic MSOL . We then extend the technique to graph polynomials where the range of the summation of the monomials is definable in MSOL . Here the Feferman–Vaught Theorem for these polynomials generalizes well known splitting theorems for graph polynomials. Again, these can be used algorithmically. Finally, we discuss extensions of MSOL for which the Feferman–Vaught Theorem holds as well. (shrink)

Aiming for applications in monadic second-order model theory, we study first-order theories without definable pairing functions. Our main results concern forking-properties of sequences of indiscernibles. These turn out to be very well-behaved for the theories under consideration.

We prove that a finitely generated group is context-free whenever its Cayley-graph has a decidable monadic second-order theory. Hence, by the seminal work of Muller and Schupp, our result gives a logical characterization of context-free groups and also proves a conjecture of Schupp. To derive this result, we investigate general graphs and show that a graph of bounded degree with a high degree of symmetry is context-free whenever its monadic second-order theory is decidable. Further, it is shown that the word (...) problem of a finitely generated group is decidable if and only if the first-order theory of its Cayley-graph is decidable. (shrink)

We generalise the concept of clique width to structures of arbitrary signature and cardinality. We present characterisations of clique width in terms of decompositions of a structure and via interpretations in trees. Several model-theoretic properties of clique width are investigated including VC-dimension and preservation of finite clique width under elementary extensions and compactness.