Achim Blumensath

Finite Presentations of Infinite Structures:Automata and Interpretations

Abstract 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. These two ways of presenting a structure are related. Indeed, a structure is automatic if, and only if, it is first-order interpretable in an appropriate expansion of Presburger arithmetic or, equivalently, in the infinite binary tree with prefix order and equal length predicate. Similar results hold for ω-automatic structures and appropriate expansions of the real ordered group. We also discuss the relationship to automatic groups. The model checking problem for FO(∃ω), first-order logic extended by the quantifier “there are infinitely many”, is proved to be decidable for automatic and ω-automatic structures. Further, the complexity for various fragments of first-order logic is determined. On the other hand, several important properties not expressible in FO, such as isomorphism or connectedness, turn out to be undecidable for automatic structures. Finally, we investigate methods for proving that a structure does not admit an automatic presentation, and we establish that the class of automatic structures is closed under Feferman–Vaught-like products.

On the structure of graphs in the Caucal hierarchy

We investigate the structure of graphs in the Caucal hierarchy. We provide criteria concerning the degree of vertices or the length of paths which can be used to show that a given graph does not belong to a certain level of this hierarchy. Each graph in the Caucal hierarchy corresponds to the configuration graph of some higher-order pushdown automaton. The main part of the paper consists of a study of such configuration graphs. We provide tools to decompose and reassemble their runs, and we prove a pumping lemma for higher-order pushdown automata.

The monadic theory of tree-like structures

Initiated by the work of Buchi, Lauchli, Rabin, and Shelah in the late 60s, the investigation of monadic second-order logic (MSO) has received continuous attention. The attractiveness of MSO is due to the fact that, on the one hand, it is quite expressive subsuming - besides first-order logic - most modal logics, in particular the μ-calculus. On the other hand, MSO is simple enough such that model-checking is still decidable for many structures. Hence, one can obtain decidability results for several logics by just considering MSO.

Axiomatising Tree-Interpretable Structures

Abstract 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.

ON THE MONADIC SECOND-ORDER TRANSDUCTION HIERARCHY

We compare classes ofnite relational structures via monad ic second-order transductions. More precisely, we study the preorder where we set C v K if, and only if, there exists a transductionsuch that C � �(K). If we only consider classes of incidence structures we can completely describe the resulting hierarchy. It is linear of order type !+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 2 N, of all paths, of all trees, and of all grids.

DECIDABILITY RESULTS FOR THE BOUNDEDNESS PROBLEM

We prove decidability of the boundedness problem for monadic least fixed-point recursion based on positive monadic second-order (MSO) formulae over trees. Given an MSO-formula phi(X,x) that is positive in X, it is decidable whether the fixed-point recursion based on phi is spurious over the class of all trees in the sense that there is some uniform finite bound for the number of iterations phi takes to reach its least fixed point, uniformly across all trees. We also identify the exact complexity of this problem. The proof uses automata-theoretic techniques. This key result extends, by means of model-theoretic interpretations, to show decidability of the boundedness problem for MSO and guarded second-order logic (GSO) over the classes of structures of fixed finite tree-width. Further model-theoretic transfer arguments allow us to derive major known decidability results for boundedness for fragments of first-order logic as well as new ones.

A model-theoretic characterisation of clique width

Abstract 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.

Boundedness of Monadic Second-Order Formulae over Finite Words

We prove that the boundedness problem for monadic second-order logic over the class of all finite words is decidable.

Two-way cost automata and cost logics over infinite trees

Regular cost functions provide a quantitative extension of regular languages that retains most of their important properties, such as expressive power and decidability, at least over finite and infinite words and over finite trees. Much less is known over infinite trees. We consider cost functions over infinite trees defined by an extension of weak monadic second-order logic with a new fixed-point-like operator. We show this logic to be decidable, improving previously known decidability results for cost logics over infinite trees. The proof relies on an equivalence with a form of automata with counters called quasi-weak cost automata, as well as results about converting two-way alternating cost automata to one-way alternating cost automata.

Recognisability for algebras of infinite trees

We develop an algebraic language theory for languages of infinite trees. We define a class of algebras called @w-hyperclones and we show that a language of infinite trees is regular if, and only if, it is recognised by a finitary path-continuous @w-hyperclone.