Formulas Of Monadic Second-order Logic Research Articles

On Model-Checking Higher-Order Effectful Programs

Model-checking is one of the most powerful techniques for verifying systems and programs, which since the pioneering results by Knapik et al., Ong, and Kobayashi, is known to be applicable to functional programs with higher-order types against properties expressed by formulas of monadic second-order logic. What happens when the program in question, in addition to higher-order functions, also exhibits algebraic effects such as probabilistic choice or global store? The results in the literature range from those, mostly positive, about nondeterministic effects, to those about probabilistic effects, in the presence of which even mere reachability becomes undecidable. This work takes a fresh and general look at the problem, first of all showing that there is an elegant and natural way of viewing higher-order programs producing algebraic effects as ordinary higher-order recursion schemes. We then move on to consider effect handlers, showing that in their presence the model checking problem is bound to be undecidable in the general case, while it stays decidable when handlers have a simple syntactic form, still sufficient to capture so-called generic effects. Along the way, we hint at how a general specification language could look like, this way justifying some of the results in the literature, and deriving new ones.

Computability by monadic second-order logic

A binary relation on graphs is recursively enumerable if and only if it can be computed by a formula of monadic second-order logic. The latter means that the formula defines a set of graphs, in the usual way, such that each “computation graph” in that set determines a pair consisting of an input graph and an output graph.

Two-way Automata and Regular Languages of Overlapping Tiles

We consider classes of languages of overlapping tiles, i.e., subsets of the McAlister monoid: the class REG of languages definable by Kleene’s regular expressions, the class MSO of languages definable by formulas of monadic second-order logic, and the class REC of languages definable by morphisms into finite monoids. By extending the semantics of finite-state two-way au- tomata (possibly with pebbles) from languages of words to languages of tiles, we obtain a complete characterization of the classes REG and MSO. In particular, we show that adding pebbles strictly increases the expressive power of two-way automata recognizing languages of tiles, but the hierarchy induced by the number of allowed pebbles collapses to level one.

Polynomial-time recognition of clique-width ≤3 graphs

Clique-width is a relatively new parameterization of graphs, philosophically similar to treewidth. Clique-width is more encompassing in the sense that a graph of bounded treewidth is also of bounded clique-width (but not the converse). For graphs of bounded clique-width, given the clique-width decomposition, every optimization, enumeration or evaluation problem that can be defined by a monadic second-order logic formula using quantifiers on vertices, but not on edges, can be solved in polynomial time.This is reminiscent of the situation for graphs of bounded treewidth, where the same statement holds even if quantifiers are also allowed on edges. Thus, graphs of bounded clique-width are a larger class than graphs of bounded treewidth, on which we can resolve fewer, but still many, optimization problems efficiently.One of the major open questions regarding clique-width is whether graphs of clique-width at most k, for fixed k, can be recognized in polynomial time. In this paper, we present the first polynomial-time algorithm (O(n2m)) to recognize graphs of clique-width at most 3.

A Practical Approach to Courcelle's Theorem

In 1990, Courcelle showed that every problem definable in Monadic Second-Order Logic (MSO) can be solved in linear time on graphs with bounded treewidth. This powerful and important theorem is amongst others the foundation for several fixed parameter tractability results. The standard proof of Courcelle's Theorem is to construct a finite bottom-up tree automaton that recognizes a tree decomposition of the graph. However, the size of the automaton, which is usually hidden as a constant in the Landau-notation, can become extremely large and cannot be bounded by any elemental function unless P=NP (Frick and Grohe, 2004). This makes the problem hard to tackle in practice, because it is just impossible to construct the tree automata. Aiming for a practical implementation, we give a proof of Courcelle's Theorem restricted to Extended MSO formulas of the form optU@?V@f(U), where @f is a first-order formula with vocabulary (adj, U) and opt@?{min,max}. Note that many optimization problems such as Minimum Vertex Cover, Minimum Dominating Set, and Maximum Independent Set can be expressed by such formulas. The proof uses a new technique based on using Hintikka game properties in dynamic programming. To demonstrate the usability of this approach, we present an implementation that solves such formulas on graphs with small pathwidth. It turns out that the large constants can be circumvented on graphs that are not too complex.

The modular decomposition of countable graphs. Definition and construction in monadic second-order logic

We consider the notion of modular decomposition for countable graphs. The modular decomposition of a graph given with an enumeration of its set of vertices can be defined by formulas of monadic second-order logic. Another result is the definition of a representation of modular decompositions by low degree relational structures. Such relational structures can also be defined in the considered graph by monadic second-order formulas.

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

We prove that one can express the vertex-minor relation on finite undirected graphs by formulas of monadic second-order logic (with no edge set quantification) extended with a predicate expressing that a set has even cardinality. We obtain a slight weakening of a conjecture by Seese stating that sets of graphs having a decidable satisfiability problem for monadic second-order logic have bounded clique-width. We also obtain a polynomial-time algorithm to check that the rank-width of a graph is at most k for any fixed k. The proofs use isotropic systems.

Dot-depth, monadic quantifier alternation, and first-order closure over grids and pictures

This paper presents results from two different areas. The first area is monadic second-order logic (MSO) over finite structures, in particular over the so-called grids. These are structures whose elements can be arranged as a matrix and which have two binary relations corresponding to vertical and horizontal successors. For this logic, we study the expressive power of the alternation of existential and universal monadic second-order quantifiers, i.e., set quantifiers. In Matz et al. (Information and Computation, LICS’ 97, 1999, to appear) it had been shown that these alternations cannot be limited to a fixed number without loss of expressiveness. In this paper, we strengthen this result in several ways. Firstly, we show that there are MSO formulas that have a very restricted form of k+1 set quantifiers but are not equivalent to a formula with k quantifiers. Secondly, we show that if we fix the number of such alternations, the expressive power of formulas that start with a block of universal quantifiers differs from the power of those that start with an existential one—this was previously known only for coloured grids. Thirdly, we investigate how an additional prefix of first-order (i.e., element) quantifiers influences the expressive power of MSO formulas. The second area that this paper is concerned with is two-dimensional formal language theory. We study how the alternation of (first- and monadic second-order) quantifications, on the one hand, relates to the dot-depth measure of two-dimensional (i.e., picture) languages, on the other hand. That measure is the two-dimensional version of the classical notion of dot-depth for (one-dimensional) starfree word languages. We show that the hierarchy induced by this dot-depth cuts through the hierarchy given by monadic second-order quantifications. In particular, beyond each level of the monadic second-order alternation hierarchy, there is a starfree picture language.

The monadic second-order logic of graphs XII: planar graphs and planar maps

We prove that we can specify by formulas of monadic second-order logic the unique planar embedding of a 3-connected planar graph. If the planar graph is not 3-connected but given with a linear order of its set of edges, we can also define a planar embedding by monadic second-order formulas. We cannot do so in general without the ordering, even for 2-connected planar graphs. The planar embedding of a graph can be specified by a relational structure called a map, which is a graph enriched with a circular ordering of the edges incident with each vertex. This circular ordering, called a rotation system, represents a planar embedding of the neighbourhood of each vertex. For each connected map one can define a linear order on its vertices by formulas of monadic second-order logic. Hence, we have for planar graphs, some kind of equivalence between linear orderings and planar embeddings.

The monadic second-order logic of graphs XI: Hierarchical decompositions of connected graphs

We prove that the unique decomposition of connected graphs defined by Tutte is definable by formulas of monadic second-order logic. This decomposition has two levels: every connected graph is a tree of “2-connected components” called blocks; every 2-connected graph is a tree of so-called 3- blocks. Our proof uses 2- dags which are certain acyclic orientations of the considered graphs. We also obtain a unique decomposition theorem for 2-dags and a definability of this decomposition in monadic second-order logic.

Monadic second-order definable graph transductions: a survey

Formulas of monadic second-order logic can be used to specify graph transductions, i.e., multi-valued functions from graphs to graphs. We obtain in this way classes of graph transductions, called monadic second-order definable graph transductions (or, more simply, definable transductions) that are closed under composition and preserve the two known classes of context-free sets of graphs, namely the class of hyperedge replacement (HR) and the class of vertex replacement (VR) sets. These two classes can be characterized in terms of definable transductions and recognizable sets of finite trees, independently of the rewriting mechanisms used to define the HR and VR grammars. When restricted to words, the definable transductions are strictly more powerful than the rational transductions such that the image of every finite word is finite; they do not preserve context-free languages. We also describe the sets of discrete (edgeless) labelled graphs that are the images of HR and VR sets under definable transductions: this gives a version of Parikh's theorem (i.e., the characterization of the commutative images of context-free languages) which extends the classical one and applies to HR and VR sets of graphs

The monadic second-order logic of graphs IV: definability properties of equational graphs

We establish that every equational graph can be characterized, up to isomorphism, by a formula of monadic second-order logic. It follows that the isomorphism of two equational graphs is decidable. We also establish that a graph specified in an equational graph by monadic second-order formulas is equational.