On factorisation forests

The theorem of factorisation forests shows the existence of nested factorisations -- a la Ramsey -- for finite words. This theorem has important applications in semigroup theory, and beyond. The purpose of this paper is to illustrate the importance of this approach in the context of automata over infinite words and trees. We extend the theorem of factorisation forest in two directions: we show that it is still valid for any word indexed by a linear ordering; and we show that it admits a deterministic variant for words indexed by well-orderings. A byproduct of this work is also an improvement on the known bounds for the original result. We apply the first variant for giving a simplified proof of the closure under complementation of rational sets of words indexed by countable scattered linear orderings. We apply the second variant in the analysis of monadic second-order logic over trees, yielding new results on monadic interpretations over trees. Consequences of it are new caracterisations of prefix-recognizable structures and of the Caucal hierarchy.Comment: 27 page

Monadic second-order definable graph orderings

We study the question of whether, for a given class of finite graphs, one can define, for each graph of the class, a linear ordering in monadic second-order logic, possibly with the help of monadic parameters. We consider two variants of monadic second-order logic: one where we can only quantify over sets of vertices and one where we can also quantify over sets of edges. For several special cases, we present combinatorial characterisations of when such a linear ordering is definable. In some cases, for instance for graph classes that omit a fixed graph as a minor, the presented conditions are necessary and sufficient; in other cases, they are only necessary. Other graph classes we consider include complete bipartite graphs, split graphs, chordal graphs, and cographs. We prove that orderability is decidable for the so called HR-equational classes of graphs, which are described by equation systems and generalize the context-free languages

Order Invariance on Decomposable Structures

Order-invariant formulas access an ordering on a structure's universe, but the model relation is independent of the used ordering. Order invariance is frequently used for logic-based approaches in computer science. Order-invariant formulas capture unordered problems of complexity classes and they model the independence of the answer to a database query from low-level aspects of databases. We study the expressive power of order-invariant monadic second-order (MSO) and first-order (FO) logic on restricted classes of structures that admit certain forms of tree decompositions (not necessarily of bounded width). While order-invariant MSO is more expressive than MSO and, even, CMSO (MSO with modulo-counting predicates), we show that order-invariant MSO and CMSO are equally expressive on graphs of bounded tree width and on planar graphs. This extends an earlier result for trees due to Courcelle. Moreover, we show that all properties definable in order-invariant FO are also definable in MSO on these classes. These results are applications of a theorem that shows how to lift up definability results for order-invariant logics from the bags of a graph's tree decomposition to the graph itself.Comment: Accepted for LICS 201

Expansions of MSO by cardinality relations

We study expansions of the Weak Monadic Second Order theory of (N,<) by cardinality relations, which are predicates R(X1,...,Xn) whose truth value depends only on the cardinality of the sets X1, ...,Xn. We first provide a (definable) criterion for definability of a cardinality relation in (N,<), and use it to prove that for every cardinality relation R which is not definable in (N,<), there exists a unary cardinality relation which is definable in (N,<,R) and not in (N,<). These results resemble Muchnik and Michaux-Villemaire theorems for Presburger Arithmetic. We prove then that + and x are definable in (N,<,R) for every cardinality relation R which is not definable in (N,<). This implies undecidability of the WMSO theory of (N,<,R). We also consider the related satisfiability problem for the class of finite orderings, namely the question whether an MSO sentence in the language {<,R} admits a finite model M where < is interpreted as a linear ordering, and R as the restriction of some (fixed) cardinality relation to the domain of M. We prove that this problem is undecidable for every cardinality relation R which is not definable in (N,<).Comment: to appear in LMC

Covering of ordinals

The paper focuses on the structure of fundamental sequences of ordinals smaller than $\epsilon_0$. A first result is the construction of a monadic second-order formula identifying a given structure, whereas such a formula cannot exist for ordinals themselves. The structures are precisely classified in the pushdown hierarchy. Ordinals are also located in the hierarchy, and a direct presentation is given.Comment: Accepted at FSTTCS'0

Some natural zero one laws for ordinals below ε0

We are going to prove that every ordinal α with ε_0 > α ≥ ω^ω satisfies a natural zero one law in the following sense. For α < ε_0 let Nα be the number of occurences of ω in the Cantor normal form of α. (Nα is then the number of edges in the unordered tree which can canonically be associated with α.) We prove that for any α with ω ω  ≤ α < ε_0 and any sentence ϕ in the language of linear orders the asymptotic density of ϕ along α is an element of  {0,1}. We further show that for any such sentence ϕ the asymptotic density along ε_0 exists although this limit is in general in between 0 and 1. We also investigate corresponding asymptotic densities for ordinals below ω^ω