Definability in Rationals with Real Order in the Background

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.

On the expressiveness of second-order spider diagrams

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.

We study probabilistic complexity classes and questions of derandomisation from a logical point of view. For each logic L we introduce a new logic BPL, bounded error probabilistic L, which is defined from L in a similar way as the complexity class BPP, bounded error probabilistic polynomial time, is defined from PTIME. Our main focus lies on questions of derandomisation, and we prove that there is a query which is definable in BPFO, the probabilistic version of first-order logic, but not in Cinf, finite variable infinitary logic with counting. This implies that many of the standard logics of finite model theory, like transitive closure logic and fixed-point logic, both with and without counting, cannot be derandomised. Similarly, we present a query on ordered structures which is definable in BPFO but not in monadic second-order logic, and a query on additive structures which is definable in BPFO but not in FO. The latter of these queries shows that certain uniform variants of AC0 (bounded-depth polynomial sized circuits) cannot be derandomised. These results are in contrast to the general belief that most standard complexity classes can be derandomised. Finally, we note that BPIFP+C, the probabilistic version of fixed-point logic with counting, captures the complexity class BPP, even on unordered structures.

In this chapter we present a method for obtaining lower bounds on the computational complexity of logical theories, and give several illustrations of its use. This method is an extension of widely used procedures for proving the recursive undecidability of logical theories. (See Rabin [1965] and Eršov et al. [1965].) One important aspect of this method is that it is based on a family of inseparability results for certain logical problems, closely related to the well-known inseparability result of Trakhtenbrot (as refined by Vaught), that no recursive set separates the logically valid sentences from those which are false in some finite model, as long as the underlying language has at least one non-unary relation symbol. By using these inseparability results as a foundation, we are able to obtain hereditary lower bounds, i.e., bounds which apply uniformly to all subtheories of the theory. The second important aspect of this method is that we use interpretations to transfer lower bounds from one theory to another. By doing this we eliminate the need to code machine computations into the models of the theory being studied. (The coding of computations is done once and for all in proving the inseparability results.) By using interpretations, attention is centred on much simpler definability considerations, viz., what kinds of binary relations on large finite sets can be defined using short formulas in models of the theory. This is conceptually much simpler than other approaches that have been proposed for obtaining lower bounds, such as the method of bounded concatenations of Fleischmann et al. [1977]. We will deal primarily with theories in first-order logic and monadic second-order logic.

The Specker–Blatter theorem does not hold for quaternary relations

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.

We investigate local and global paradigms of reasoning about distributed behaviours, modelled as Mazurkiewicz traces, in the context of first-order and monadic second-order logics. We describe new normal forms for properties expressible in these logics. The first normal form, surprisingly, yields a decomposition of a global property as a boolean combination of local properties. The second normal form strengthens McNaughton's theorem and states that global properties of infinite behaviours may also be described as boolean combinations of recurring properties of finite local histories of the behaviours. We briefly touch upon some of the interesting applications of these normal forms.

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.

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

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

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

In this chapter we make a case for the monadic second-order logic (that is to say, for the extension of first-order logic allowing quantification over monadic predicates) as a good source of theories that are both expressive and manageable. We illustrate two powerful decidability techniques here. One makes use of automata and games. The other is an offshot of a composition theory where one composes models as well as their theories. Monadic second-order logic appears to be the most natural match for the composition theory. Undecidability proofs must be thought out anew in this area; for, whereas true first-order arithmetic is reducible to the monadic theory of the real line R, it is nevertheless not interpretable in the monadic theory of R. A quite unusual undecidability method is another subject of this chapter. In the last section we briefly review the history of the methods thus far developed and mention numerous results obtained by the methods.

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.