Order-Invariant MSO is Stronger than Counting MSO in the Finite

We compare the expressiveness of two extensions of monadic second-order logic (MSO) over the class of finite structures. The first, counting monadic second-order logic (CMSO), extends MSO with first-order modulo-counting quantifiers, allowing the expression of queries like ``the number of elements in the structure is even''. The second extension allows the use of an additional binary predicate, not contained in the signature of the queried structure, that must be interpreted as an arbitrary linear order on its universe, obtaining order-invariant MSO. While it is straightforward that every CMSO formula can be translated into an equivalent order-invariant MSO formula, the converse had not yet been settled. Courcelle showed that for restricted classes of structures both order-invariant MSO and CMSO are equally expressive, but conjectured that, in general, order-invariant MSO is stronger than CMSO. We affirm this conjecture by presenting a class of structures that is order-invariantly definable in MSO but not definable in CMSO.Comment: Revised version contributed to STACS 200

Definability and model checking : the role of orders and compositionality

Finite model theory and descriptive complexity theory are concerned with assessing the expressive power of logics over finite models and with relating the descriptive resources needed for defining a class of structures to the computational complexity of the corresponding decision problem. In recent years, also the model theory and computational handling (e.g. of the model checking problem) of finitely representable infinite structures have gained much interest. In the first part of this thesis we study how the expressive power of logics over finite structures is affected by the presence of orders. We will first review the concept of order-invariant definability where we allow formulae to use an additional order-relation that is not present in a given structure in such a way that the truth of the formula in the structure does not depend on the actual interpretation of the order relation, and show that, in the context of monadic second-order logic, order-invariance yields more expressive power than adding modulo-counting quantifiers. Second, we investigate structures with weaker forms of orderings, namely locally ordered graphs, in which only the neighbourhoods of the vertices are linearly ordered. Using recent results on reachability algorithms by Reingold, we are able to show that the transitive closure of the edge relation in such graphs is definable in deterministic transitive closure logic (DTC) in a two-sorted setting, and this observation is the key to linking the descriptive power of DTC with counting to the computational power of logspace-bounded Turing machines. The second part of the thesis is concerned with techniques for model checking weak monadic second-order logic (WMSO) on the class of so-called inductive structures that allow for a finite representation via systems of equations, and which includes structures relevant for verification purposes such as the infinite binary tree, infinite lists, etc. Our new approach presents an algorithmic alternative to automata-theoretic methods, which exhibit certain drawbacks, and is based on a purely logical decomposition technique using the defining equations. Further, the deployed techniques can be extended to obtain a model checking algorithm for the extension of WMSO by an unbounding quantifier, and thus establish the decidability of its model checking problem on the class of inductive structures

Heuristic Methods for Hypertree Decompositions

In this paper we propose new algorithms for generating generalized hypertree decompositions. The well known heuristics for generating tree decompositions based on vertex ordering have been extended to produce hypertree decompositions. We investigate the generation of hypertree decompositions based on the tree decompositions of the primal and the dual graph of the hypergraph. Further, we propose a method for generating hypertree decompositions using hypergraph partitioning. We use different algorithms for partitioning hypergraphs. The proposed algorithms are experimentally evaluated in benchmark problems from the literature and industry. Using the proposed algorithms we improve the best existing upper bounds for hypertree width for many problems