We study definability problems and algorithmic issues for infinite structures that are finitely presented. After a brief overview over different classes of finitely presentable structures, we focus on structures presented by automata or by model-theoretic interpretations.

This is a survey on the relationship between logical definability and computational complexity on finite structures. Particular emphasis is given to game-based evaluation algorithms for various logical formalisms and to logics capturing complexity classes. In addition to the

We present several characterisations of the class of prefix-recognisable

This paper addresses the question of whether a given algebraic structure has an automatic presentation and, in the case that it has, how similar its presentations are. That is, what is the complexity of finding out whether two presentations stand for the same algebraic structure. So the first topic is that of characterising the class of automatic structures. The automatic Boolean algebras are characterised, and it is proven that the free Abelian group of infinite rank and many Frasse limits do not have automatic presentations. In particular, the countably infinite random graph and the universal partial order do not have automatic presentations. Furthermore, no infinite integral domain is automatic. The second topic of the paper is the isomorphism problem. We prove that the complexity of the isomorphism problem for the class of all automatic ... -complete.

We study analogs of classical relational calculus in the context of strings. We start by studying string logics. Taking a classical model-theoretic approach, we fix a set of string operations and look at the resulting collection of definable relations. These form an algebra - a class of n-ary relations for every n, closed under projection and Boolean operations. We show that by choosing the string vocabulary carefully, we get string logics that have desirable properties: computable evaluation and normal forms. We identify five distinct models and study the differences in their model-theory and complexity of evaluation. We identify a subset of these models which have additional attractive properties, such as finite VC dimension and quantifier elimination. Once you have a logic,

Besides the very successful concept of tree-width (see [Bodlaender, H. and Koster, A. (2007) Combinatorial optimisation on graphs of bounded treewidth. These are special issues on Parameterized Complexity]), many concepts and parameters measuring the similarity or dissimilarity of structures compared to trees have been born and studied over the past years. These concepts and parameters have proved to be useful tools in many applications, especially in the design of efficient algorithms. Our presented novel look at the contemporary developments of these ‘width ’ parameters in combinatorial structures delivers—besides traditional tree-width and derived dynamic programming schemes—also a number of other useful parameters like branch-width, rank-width (clique-width) or hypertree-width. In this contribution, we demonstrate how ‘width ’ parameters of graphs and generalized structures (such as matroids or hypergraphs), can be used to improve the design of parameterized algorithms and the structural analysis in other applications on an abstract level.

We study algebras of de nable string relations - classes of regular n-ary relations that arise as the definable sets within a model whose carrier is the set of all strings. We show that the largest such algebra - the collection of regular relations - has some quite undesirable computational and model-theoretic properties. In contrast, we exhibit several definable relation algebras that have much tamer behavior: for example, they admit quantifier elimination, and have finite VC dimension. We show that the properties of a definable relation algebra are not at all determined by the one-dimensional definable sets. We give models whose definable sets are all star-free, but whose binary relations are quite complex, as well as models whose definable sets include all regular sets, but which are much more restricted and tractable than the full algebra of regular relations.

We study relational calculi with support for string operations. Most prior proposals were based on adding the operation of concatenation to rst-order logic. Such an extension is problematic as the relational calculus becomes computationally complete, which in turn implies strong limits on the ability to perform optimization and static analysis of properties such as query safety. In contrast, we look at extensions of relational calculus that have nice expressiveness, decidability, and safety properties, while corresponding to sets of string operations used in SQL. We start with an extension based on the string ordering and LIKE predicates. We then extend this basic model to include string length comparison. While both of these share some of the attractive properties of relational calculus (low data complexity for generic queries, eective syntax for safe queries, correspondence with an algebra), there is a large gap between these calculi in expressive power and complexity. The smaller...

We introduce the class of tree-interpretable structures which generalises the notion of a prefix-recognisable graph to arbitrary relational structures. We prove that every tree-interpretable structure is finitely axiomatisable in guarded second-order logic with cardinality quantifiers.

The rst-order theory of an automatic structure is known to be decidable but there are examples of automatic structures with nonelementary rst-order theories. We prove that the rst-order theory of an automatic structure of bounded degree (meaning that the corresponding Gaifman-graph has bounded degree) is elementary decidable. More precisely, we prove an upper bound of triply exponential alternating time with a linear number of alternations. We also present an automatic structure of bounded degree such that the corresponding rst-order theory has a lower bound of doubly exponential time with a linear number of alternations. We prove similar results also for tree automatic structures.