Chordality in Matroids: In Search of the Converse to Hliněný's Theorem

<p>Bodlaender et al. [7] proved a converse to Courcelle's Theorem for graphs [15] for the class of chordal graphs of bounded treewidth. Hliněný [25] generalised Courcelle's Theorem for graphs to classes of matroids represented over finite fields and of bounded branchwidth. This thesis has investigated the possibility of obtaining a generalisation of chordality to matroids that would enable us to prove a converse of Hliněný's Theorem [25]. There is a variety of equivalent characterisations for chordality in graphs. We have investigated the relationship between their generalisations to matroids. We prove that they are equivalent for binary matroids but typically inequivalent for more general classes of matroids. Supersolvability is a well studied property of matroids and, indeed, a graphic matroid is supersolvable if and only if its underlying graph is chordal. This is among the stronger ways of generalising chordality to matroids. However, to obtain the structural results that we need we require a stronger property that we call supersolvably saturated. Chordal graphs are well known to induce canonical tree decompositions. We show that supersolvably saturated matroids have the same property. These tree decompositions of supersolvably saturated matroids can be processed by a finite state automaton. However, they can not be completely described in monadic second-order logic. In order to express the matroids and their tree decompositions in monadic second-order logic we need to extend the logic over an extension field for each matroid represented over a finite field. We then use the fact that each maximal round modular flat of the tree decomposition for every matroid represented over a finite field, and in the specified class, spans a point in the vector space over the extension field. This enables us to derive a partial converse to Hliněný's Theorem.</p>

Chordality in Matroids: In Search of the Converse to Hliněný's Theorem

<p>Bodlaender et al. [7] proved a converse to Courcelle's Theorem for graphs [15] for the class of chordal graphs of bounded treewidth. Hliněný [25] generalised Courcelle's Theorem for graphs to classes of matroids represented over finite fields and of bounded branchwidth. This thesis has investigated the possibility of obtaining a generalisation of chordality to matroids that would enable us to prove a converse of Hliněný's Theorem [25]. There is a variety of equivalent characterisations for chordality in graphs. We have investigated the relationship between their generalisations to matroids. We prove that they are equivalent for binary matroids but typically inequivalent for more general classes of matroids. Supersolvability is a well studied property of matroids and, indeed, a graphic matroid is supersolvable if and only if its underlying graph is chordal. This is among the stronger ways of generalising chordality to matroids. However, to obtain the structural results that we need we require a stronger property that we call supersolvably saturated. Chordal graphs are well known to induce canonical tree decompositions. We show that supersolvably saturated matroids have the same property. These tree decompositions of supersolvably saturated matroids can be processed by a finite state automaton. However, they can not be completely described in monadic second-order logic. In order to express the matroids and their tree decompositions in monadic second-order logic we need to extend the logic over an extension field for each matroid represented over a finite field. We then use the fact that each maximal round modular flat of the tree decomposition for every matroid represented over a finite field, and in the specified class, spans a point in the vector space over the extension field. This enables us to derive a partial converse to Hliněný's Theorem.</p>

Plurals and Second-Order Logic

While plural logic can be interpreted in monadic second-order logic, the full system of second-order logic cannot be interpreted in plural logic. This means it is formally possible to eliminate plural logic in favor of monadic second-order logic. However, a number of philosophical considerations militate against such an elimination. The conclusion of this chapter echoes that of the preceding ones: although the two systems can occasionally be used for similar purposes, the notions they represent are different and must be kept apart.

Higher-order Recursion Schemes and Collapsible Pushdown Automata: Logical Properties

This article studies the logical properties of a very general class of infinite ranked trees, namely, those generated by higher-order recursion schemes. We consider, for both monadic second-order logic and modal -calculus, three main problems: model-checking, logical reflection (a.k.a. global model-checking, that asks for a finite description of the set of elements for which a formula holds), and selection (that asks, if exists, for some finite description of a set of elements for which an MSO formula with a second-order free variable holds). For each of these problems, we provide an effective solution. This is obtained, thanks to a known connection between higher-order recursion schemes and collapsible pushdown automata and on previous work regarding parity games played on transition graphs of collapsible pushdown automata.

The Complexity of Model-Checking Tail-Recursive Higher-Order Fixpoint Logic

Higher-Order Fixpoint Logic (HFL) is a modal specification language whose expressive power reaches far beyond that of Monadic Second-Order Logic, achieved through an incorporation of a typed λ-calculus into the modal μ-calculus. Its model checking problem on finite transition systems is decidable, albeit of high complexity, namely k-EXPTIME-complete for formulas that use functions of type order at most k < 0. In this paper we present a fragment with a presumably easier model checking problem. We show that so-called tail-recursive formulas of type order k can be model checked in (k − 1)-EXPSPACE, and also give matching lower bounds. This yields generic results for the complexity of bisimulation-invariant non-regular properties, as these can typically be defined in HFL.

All-Instances Restricted Chase Termination for Linear TGDs

The chase procedure is a fundamental algorithmic tool in database theory with a variety of applications. A key problem concerning the chase procedure is all-instances chase termination: for a given set of tuple-generating dependencies (TGDs), is it the case that the chase terminates for every input database? In view of the fact that this problem is, in general, undecidable, it is natural to ask whether well-behaved classes of TGDs, introduced in different contexts, ensure decidability. It has been recently shown that the problem is decidable for the restricted (a.k.a. standard) version of the chase, and linear TGDs, a prominent class of TGDs that has been introduced in the context of ontological query answering, under the assumption that only one atom appears in TGD-heads. We provide an alternative proof for this result based on Monadic Second-Order Logic, which we believe is simpler that the ones obtained from the literature.