Monadic Second-Order Logic for Graphs: Algorithmic and Language Theoretical Applications

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

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

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

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 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.

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 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.

The main idea of graph grammars and graph transformation is the rule-based graph rewriting, where the application of every rule graph rule leads to a graph transformation step. Graph grammars can be used to generate graph languages which is similar to Chomsky grammars in formal language theory. Moreover, graphs can be used to model the states of all kinds of systems, which allows one to use graph transformation to model state changes in these systems. A rule-based transformation system may show two kinds of nondeterminism: (1) for each rule can exist several matches, and (2) several rules can be applicable. This paper concentrates on the second non-determinism, studying a method to detect the rules which are not applicable in graph transformation systems, proposing a backward graph transformation conflict detecting algorithm. At first the computation of conflict and dependency of two rules is available by use of AGG . Secondly the set of dependencies has to be tested by Warshall algorithm and the set of conflicts has to be computed by backward algorithm. At last, the termination of graph transformation systems and validity of rules can be detected as well.

Graph transformation theory uses rules to perform a graph transformation. However, there is no a way to choose between such different transformations in the case where several of them are applicable. A way to get the choice is suggested here based on the comparing of the values of implications which correspond to different transformation variants. The relationship between the topos of bundles, and the set of graphs with the same vertices, is introduced to include logic into graph transformation theory. Thus, one can use the special type of implication and the truth-values set of such a topos to estimate different variants of graph transformations. In this approach, the maximal part of the initial graph towards the terminal one is conserved in the chosen variant. Analysis of self-adaptive systems uses some graph grammars. Self-adaptive systems autonomously perform an adaptation to changes both in user needs and in their operational environments, while still maintaining some desired properties. The suggested way to choose such graph transformation variants may be used to make a choice between different graph grammars in such systems modeling. This approach is illustrated in a model of some business processes, that result in the automated choice of the business process adaptation under the assumption that the process changes are minimal towards the terminal state.

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

On the expressiveness of second-order spider diagrams

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.

Grammar inference deals with determining (preferably simple) models/grammars consistent with a set of observations. There is a large body of research on grammar inference within the theory of formal languages. However, there is surprisingly little known on grammar inference for graph grammars. In this article, we take a further step in this direction and work within the framework of node label controlled (NLC) graph grammars. Specifically, given a graph G and a set 𝒮 of disjoint and isomorphic subgraphs of G, we characterize whether or not there is a graph grammar consisting of one production such that G may be derived from G0, the graph obtained from G by ‘contraction’ of each subgraph in 𝒮 to a node labelled by N. This generalizes a previous result that assumes boundary NLC graph grammars, and leads one to consider the more involved ‘non-confluent’ graph grammar rules.

Groups, graphs, languages, automata, games and second-order monadic logic

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.