Decidable Monadic Second-order Theory Research Articles

Tree Automata and Pigeonhole Classes of Matroids: I

Hliněný’s Theorem shows that any sentence in the monadic second-order logic of matroids can be tested in polynomial time, when the input is limited to a class of {mathbb {F}}-representable matroids with bounded branch-width (where {mathbb {F}} is a finite field). If each matroid in a class can be decomposed by a subcubic tree in such a way that only a bounded amount of information flows across displayed separations, then the class has bounded decomposition-width. We introduce the pigeonhole property for classes of matroids: if every subclass with bounded branch-width also has bounded decomposition-width, then the class is pigeonhole. An efficiently pigeonhole class has a stronger property, involving an efficiently-computable equivalence relation on subsets of the ground set. We show that Hliněný’s Theorem extends to any efficiently pigeonhole class. In a sequel paper, we use these ideas to extend Hliněný’s Theorem to the classes of fundamental transversal matroids, lattice path matroids, bicircular matroids, and H-gain-graphic matroids, where H is any finite group. We also give a characterisation of the families of hypergraphs that can be described via tree automata: a family is defined by a tree automaton if and only if it has bounded decomposition-width. Furthermore, we show that if a class of matroids has the pigeonhole property, and can be defined in monadic second-order logic, then any subclass with bounded branch-width has a decidable monadic second-order theory.

On countable chains having decidable monadic theory

AbstractRationals and countable ordinals are important examples of structures with decidable monadic second-order theories. A chain is an expansion of a linear order by monadic predicates. We show that if the monadic second-order theory of a countable chain C is decidable then C has a non-trivial expansion with decidable monadic second-order theory.

LOGICAL ASPECTS OF CAYLEY-GRAPHS: THE MONOID CASE

Cayley-graphs of monoids are investigated under a logical point of view. It is shown that the class of monoids, for which the Cayley-graph has a decidable monadic second-order theory, is closed under free products. This result is derived from a result of Walukiewicz, stating that the decidability of monadic second-order theories is preserved under tree-like unfoldings. Concerning first-order logic, it is shown that the class of monoids, for which the Cayley-graph has a decidable first-order theory, is closed under arbitrary graph products, which generalize both, free and direct products. For the proof of this result, tree-like unfoldings are generalized to so-called factorized unfoldings. It is shown that the decidability of the first-order theory of a structure is preserved by factorized unfoldings. Several additional results concerning factorized unfoldings are shown.

Trees, grids, and MSO decidability: From graphs to matroids

Monadic second order (MSO) logic has proved to be a useful tool in many areas of application, reaching from decidability and complexity to picture processing, correctness of programs and parallel processes. To characterize the structural borderline between decidability and undecidability is a classical research problem here. This problem is related to questions in computational complexity, especially to the model checking problem, for which many tools developed in the area of decidability have proved to be useful. For more than two decades it was conjectured in [D. Seese, The structure of the models of decidable monadic theories of graphs, Ann. Pure Appl. Logic 53 (1991) 169–195] that decidability of monadic theories of countable structures implies that the theory can be reduced via interpretability to a theory of trees. It is one of the main goals of this article to prove a variant of this conjecture for matroids representable over a finite field. (Matroids can be viewed as a wide generalization of graphs, and they seem to capture some second order properties in a more suitable way than graphs themselves, cf. the recent development in matroid structure theory [J.F. Geelen, A.H.M. Gerards, G.P. Whittle, Branch-width and well-quasi-ordering in matroids and graphs, J. Combin. Theory Ser. B 84 (2002) 270–290; J.F. Geelen, A.H.M. Gerards, N. Robertson, G.P. Whittle, Excluding a planar graph from a GF ( q ) -representable matroid, manuscript, 2003].) More exactly we prove, for every finite field F , that any class of F -representable matroids with a decidable MSO theory must have uniformly bounded branch-width. Moreover, we show that bounding the branch-width of all matroids in general is not sufficient to obtain a decidable MSO theory. Our paper gives a (rather detailed) introduction to these different subjects, and shows that a blend of ideas and methods from logic together with structural matroid theory can lead to new tools and algorithms, and can shed light on some old open problems.

The monadic second-order logic of graphs XV: On a conjecture by D. Seese

A conjecture by D. Seese states that if a set of graphs has a decidable monadic second-order theory, then it is the image of a set of trees under a transformation of relational structures defined by monadic second-order formulas, or equivalently, has bounded clique-width. We prove that this conjecture is true if and only if it is true for the particular cases of bipartite undirected graphs, of directed graphs, of partial orders and of comparability graphs. We also prove that it is true for line graphs, for interval graphs and for partial orders of dimension 2. Our treatment of certain countably infinite graphs uses a representation of countable linear orders by binary trees that can be constructed by monadic second-order formulas. By using a counting argument, we show the intrinsic limits of the methods used here to handle this conjecture.

Logical aspects of Cayley-graphs: the group case

We prove that a finitely generated group is context-free whenever its Cayley-graph has a decidable monadic second-order theory. Hence, by the seminal work of Muller and Schupp, our result gives a logical characterization of context-free groups and also proves a conjecture of Schupp. To derive this result, we investigate general graphs and show that a graph of bounded degree with a high degree of symmetry is context-free whenever its monadic second-order theory is decidable. Further, it is shown that the word problem of a finitely generated group is decidable if and only if the first-order theory of its Cayley-graph is decidable.

A Monadic Second-Order Definition of the Structure of Convex Hypergraphs

We consider finite hypergraphs with hyperedges defined as sets of vertices of unbounded cardinality. Each such hypergraph has a unique modular decomposition, which is a tree, the nodes of which correspond to certain subhypergraphs (induced by certain sets of vertices called strong modules) of the considered hypergraph. One can define this decomposition by monadic second-order (MS) logical formulas. Such a hypergraph is convex if the vertices are linearly ordered in such a way that the hyperedges form intervals. Our main result says that the unique linear order witnessing the convexity of a prime hypergraph (i.e., of one, the modular decomposition of which is trivial) can be defined in MS logic. As a consequence, we obtain that if a set of bipartite graphs that correspond (in the usual way) to convex hypergraphs has a decidable monadic second-order theory (which means that one can decide whether a given MS formula is satisfied in some graph of the set) then it has bounded clique-width. This yields a new case of validity of a conjecture which is still open.

Model checking the full modal mu-calculus for infinite sequential processes

It is known that pushdown processes have a decidable monadic second-order theory (Muller and Schupp, Theoret. Comput. Sci. 37 (1985) 51–;75) and that this result covers the model-checking problem for the modal mu-calculus. Unfortunately, the corresponding decidability procedure is not practical due to its nonelementary complexity. Recently, however, a very intricate elementary algorithm for model checking the full modal mu-calculus for pushdown processes based on games was presented by Walukiewicz (CAV'96, Lecture Notes in Computer Science, vol. 1102, Springer, Berlin, 1996, pp. 62–;74). Lifting the classical finite-state model checking technique to second-order, we develop here a more structural and transparent elementary algorithm for model-checking infinite sequential processes, including context-free processes, pushdown processes, and regular graphs, that captures the full modal mu-calculus as well. Whereas the actual model-checking algorithm simply resorts to backtracking in order to capture alternation, the corresponding correctness proof requires to introduce the stronger framework of dynamic environments which are modelled by finite-state automata.