On the Monadic Second-Order Transduction Hierarchy

We compare classes of finite relational structures via monadic second-order transductions. More precisely, we study the preorder where we set C \subseteq K if, and only if, there exists a transduction {\tau} such that C\subseteq{\tau}(K). If we only consider classes of incidence structures we can completely describe the resulting hierarchy. It is linear of order type {\omega}+3. Each level can be characterised in terms of a suitable variant of tree-width. Canonical representatives of the various levels are: the class of all trees of height n, for each n \in N, of all paths, of all trees, and of all grids

Decidability of Querying First-Order Theories via Countermodels of Finite Width

We propose a generic framework for establishing the decidability of a wide range of logical entailment problems (briefly called querying), based on the existence of countermodels that are structurally simple, gauged by certain types of width measures (with treewidth and cliquewidth as popular examples). As an important special case of our framework, we identify logics exhibiting width-finite finitely universal model sets, warranting decidable entailment for a wide range of homomorphism-closed queries, subsuming a diverse set of practically relevant query languages. As a particularly powerful width measure, we propose Blumensath's partitionwidth, which subsumes various other commonly considered width measures and exhibits highly favorable computational and structural properties. Focusing on the formalism of existential rules as a popular showcase, we explain how finite partitionwidth sets of rules subsume other known abstract decidable classes but -- leveraging existing notions of stratification -- also cover a wide range of new rulesets. We expose natural limitations for fitting the class of finite unification sets into our picture and provide several options for remedy

Logic and Automata

Mathematical logic and automata theory are two scientific disciplines with a fundamentally close relationship. The authors of Logic and Automata take the occasion of the sixtieth birthday of Wolfgang Thomas to present a tour d'horizon of automata theory and logic. The twenty papers in this volume cover many different facets of logic and automata theory, emphasizing the connections to other disciplines such as games, algorithms, and semigroup theory, as well as discussing current challenges in the field

Structures of bounded partition width

In the present thesis we study the question of which monadic theories are simple. In particular, we are looking for theories that are decidable or at least simple enough that we are able to derive structure theorems. We propose to draw the line between simple and complicated theories by defining that a structure has a simple monadic theory if and only if it can be interpreted in some (possibly infinite) coloured tree. The class of structures obtained this way generalises the class of graphs of bounded clique width which was originally defined by Courcelle, Engelfriet, and Rozenberg via graph grammars. Analogously to the results of Courcelle et.al., we will introduce terms denoting arbitrary relational structures and we show that a structure can be denoted by a term if and only if it is interpretable in some (possibly infinite) tree. Furthermore, we obtain an equivalent characterisation via hierarchical decompositions of the structure which can be used to define a complexity measure, called partition width, which provides our generalisation of the notion of clique width. The intuitive idea that structures interpretable in a tree have a simple monadic theory is supported by several model theoretic results we obtain for this class. Finiteness of partition width is preserved by elementary embeddings and we will prove a compactness theorem for structures of finite partition width. Furthermore, no such structure has the independence property or, equivalently, infinite VC-dimension, that is, in no structure of finite partition width it is possible to encode, in a first-order way, all subsets of some infinite set by single elements. After having obtained a class of simple structures the obvious next question is whether this characterisation is precise. That is, we would like to prove that all other structures have a complicated monadic theory. We conjecture that every structure of infinite partition width contains arbitrarily large finite MSO-definable grids. This would imply that the full second-order theory of the class of finite sets can be interpreted in the monadic second-order theory of every structure of infinite partition width. In particular, every such structure would have an undecidable MSO-theory. Therefore, a proof of this conjecture would settle the conjecture of Seese which states that every graph with decidable MSO-theory has finite clique width. We try to obtain an answer to this question by developing a theory of connectedness based on cuts and separations that is symmetric with regard to edges and non-edges. After sufficient preparations of this kind we are able to translate the core of the original proof of Robertson and Seymour's Excluded Grid Theorem from tree width to partition width. Despite these encouraging results, both, a full analogue of the Excluded Grid Theorem and the conjecture itself remain open. In the second part of the thesis we turn to the investigation of subclasses consisting of structures with decidable monadic theory that, furthermore, admit a finite representation. Mainly, we will consider the class of structures that can be interpreted in the complete binary tree without additional unary predicates. We will study algebraic properties of these structures including a characterisation of all linear orders contained in this class. We will also show that every such structure can be finitely axiomatised in guarded second-order logic with cardinality quantifiers