Monadic Second-Order Logic, Graphs and Unfoldings of Transition Systems

We study the language-theoretical aspects of parameterized communicating automata (PCAs), in which processes communicate via rendez-vous. A given PCA can be run on any topology of bounded degree such as pipelines, rings, ranked trees, and grids. We show that, under a context bound, which restricts the local behavior of each process, PCAs are effectively complementable. Complementability is considered a key aspect of robust automata models and can, in particular, be exploited for verification. In this paper, we use it to obtain a characterization of context-bounded PCAs in terms of monadic second-order (MSO) logic. As the emptiness problem for context-bounded PCAs is decidable for the classes of pipelines, rings, and trees, their model-checking problem wrt. MSO properties also becomes decidable. While previous work on model checking parameterized systems typically uses temporal logics without next operator, our MSO logic allows one to express several natural next modalities.

We consider monadic second order logic (MSO) and the modal μ-calculus (Lμ) over transition systems (Kripke structures). It is well known that every class of transition systems which is definable by a sentence of Lμ is definable by a sentence of MSO as well. It will be shown that the converse is also true for an important fragment of MSO: every class of transition systems which is MSO-definable and which is closed under bisimulation - i.e., the sentence does not distinguish between bisimilar models - is also Lμ-definable. Hence we obtain the following expressive completeness result: the bisimulation invariant fragment of MSO and Lμ are equivalent. The result was proved by David Janin and Igor Walukiewicz. Our presentation is based on their article [91]. The main step is the development of automata-based characterizations of Lμ over arbitrary transition systems and of MSO over transition trees (see also Chapter 16). It turns out that there is a general notion of automaton subsuming both characterizations, so we obtain a common ground to compare these two logics. Moreover we need the notion of the ω-unravelling for a transition system, on the one hand to obtain a bisimilar transition tree and on the other hand to increase the possibilities of choosing successors.

We establish that every monadic second-order logic (MSO) formula on graphs with bounded treedepth is decidable in a constant number of rounds within the CONGEST model. To our knowledge, this marks the first meta-theorem regarding distributed model-checking. Various optimization problems on graphs are expressible in MSO. Examples include determining whether a graph G has a clique of size k, whether it admits a coloring with k colors, whether it contains a graph H as a subgraph or minor, or whether terminal vertices in G could be connected via vertex-disjoint paths. Our meta-theorem significantly enhances the work of Bousquet et al. [PODC 2022], which was focused on distributed certification of MSO on graphs with bounded treedepth. Moreover, our results can be extended to solving optimization and counting problems expressible in MSO, in graphs of bounded treedepth.

Vertex-minors, monadic second-order logic, and a conjecture by Seese

On the expressiveness of second-order spider diagrams

Graph Operations, Graph Transformations and Monadic Second-Order Logic A survey

Graph Operations, Graph Transformations and Monadic Second-Order Logic:: A survey

The paper deals with logically definable families of sets (or point-sets) of rational numbers. In particular we are interested whether the families definable over the real line with a unary predicate for the rationals are definable over the rational order alone. Let φ(X,Y ) and ψ(Y ) range over formulas in the first-order monadic language of order. Let Q be the set of rationals and F be the family of subsets J of Q such that φ(Q,J) holds over the real line. The question arises whether, for every φ, F can be defined by means of an appropriate ψ(Y ) interpreted over the rational order. We answer the question negatively. The answer remains negative if the firstorder logic is strengthen to weak monadic second-order logic. The answer is positive for the restricted version of monadic second-order logic where set quantifiers range over open sets. The case of full monadic second-order logic remains open.

Every graph can be represented uniquely in a hierarchical way by means of its modular decomposition. We establish that the modular decomposition of a linearly ordered graph is definable in monadic second-order (MS) logic in the graph itself. The modular decomposition does not depend on the linear order of the given graph. A set of graphs is recognizable w.r.t. the operations associated with graph substitution iff it is definable by a formula of an extension of MS logic based on the use of auxilliary linear orderings. This paper is an extended abstract: complete proofs can be found in [7].

We establish zero-one laws and convergence laws for monadic second-order logic (MSO) (and, a fortiori, first-order logic) on a number of interesting graph classes. In particular, we show that MSO obeys a zero-one law on the class of connected planar graphs, the class of connected graphs of tree-width at most $k$ and the class of connected graphs excluding the $k$-clique as a minor. In each of these cases, dropping the connectivity requirement leads to a class where the zero-one law fails but a convergence law for MSO still holds.

Modern biochemistry is producing vast amounts of chemical knowledge. Ontologies, such as the Chemical Entities of Biological Interest (ChEBI) ontology, can help organising this knowledge. With manual classification alone however, ontologies cannot keep up with the growth of their domain. In this work, we propose a novel taxonomy of 67 classes related to peptides, a large branch in ChEBI with nearly 15,000 compounds. The existing natural language definitions in ChEBI have been expanded and specified more precisely. These natural language definitions are accompanied by a logical axiomatisation in monadic second-order logic (MSOL). To use the axiomatisation for automated classification, a methodology has been developed that translates monadic second-order definitions first into partial first-order definitions and finally into an algorithmic classification. This connects three aspects important to ontological definitions: They reflect the opinions of experts, they are unambiguous, and they can be checked automatically. In our evaluation, we compare the results of our classification to the current taxonomy of ChEBI . This reveals potential inconsistencies in ChEBI as well as areas that might benefit from automated extensions. We also evaluate our natural-language definitions in an expert survey.Scientific contribution: This work provides precise natural-language definitions of 14 current ChEBI classes as well as 53 new peptide-related classes. These definitions are formalised in MSOL and come with an efficient implementation that allows for large-scale molecule classification, including a full classification of ChEBI and PubChem.

Our main result is a succinct counterpoint to Courcelle's meta-theorem as follows: every cw-nontrivial monadic second-order (MSO) property is either NP-hard or coNP-hard over graphs given by succinct representations. Succint representations are Boolean circuits computing the adjacency relation. Cw-nontrivial properties are those which have infinitely many models and infinitely many countermodels with bounded cliquewidth. Moreover, we explore what happens when the cw-nontriviality condition is dropped and show that, under a reasonable complexity assumption, the previous dichotomy fails, even for questions expressible in first-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.

The monadic second-order logic of graphs VII: graphs as relational structures

We develop an algebraic model for recognizing languages of words indexed by countable linear orderings. This notion of recognizability is effectively equivalent to definability in monadic second-order (MSO) logic. The proofs also imply the first known collapse result for MSO logic over countable linear orderings.KeywordsConvex SubsetEvaluation TreeLinear OrderingExpressive PowerRegular LanguageThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.