Monadic Second-order Sentences Research Articles

Order-theoretic Trees: Monadic Second-order Descriptions and Regularity

An order-theoretic forest is a countable partial order such that the set of elements larger than any element is linearly ordered. It is an order-theoretic tree if any two elements have an upper-bound. The order type of a branch (a maximal linearly ordered subset) can be any countable linear order. Such generalized infinite trees yield convenient definitions of the rank-width and the modular decomposition of countable graphs. We define an algebra based on only four operations that generate up to isomorphism and via infinite terms these order-theoretic trees and forests.We prove that the associated regular objects, i.e., those defined by regular terms, are exactly the ones that are the unique models of monadic second-order sentences. We adapt some tools that we have used in a previous article for proving a similar result for join-trees, i.e., for order-theoretic trees such that any two nodes have a least upper-bound.

Betweenness of partial orders

We construct a monadic second-order sentence that characterizes the ternary relations that are the betweenness relations of finite or infinite partial orders. We prove that no first-order sentence can do that. We characterize the partial orders that can be reconstructed from their betweenness relations. We propose a polynomial time algorithm that tests if a finite relation is the betweenness of a partial order.

Logical laws for short existential monadic second-order sentences about graphs

In 2001, Le Bars proved that there exists an existential monadic second-order (EMSO) sentence such that the probability that it is true on [Formula: see text] does not converge and conjectured that, for EMSO sentences with two first-order variables, the zero–one law holds. In this paper, we prove that the conjecture fails for [Formula: see text], and give new examples of sentences with fewer variables without convergence (even for [Formula: see text]).

A Disproof the Le Bars Conjecture about the Zero–One Law for Existential Monadic Second-Order Sentences

The Le Bars conjecture (2001) states that the binomial random graph G(n, $$\frac{1}{2}$$ ) obeys the zero–one law for existential monadic sentences with two first-order variables. This conjecture is disproved. Moreover, it is proved that there exists an existential monadic sentence with a single monadic variable and two first-order variables whose truth probability does not converge.

On quasi-planar graphs: Clique-width and logical description

Motivated by the construction of FPT graph algorithms parameterized by clique-width or tree-width, we study graph classes for which tree-width and clique-width are linearly related. This is the case for all graph classes of bounded expansion, but in view of concrete applications, we want to have ”small” constants in the comparisons between these width parameters.We focus our attention on graphs that can be drawn in the plane with limited edge crossings, for an example, at most p crossings for each edge. These graphs are called p-planar. We consider a more general situation where the graph of edge crossings must belong to a fixed class of graphs D. For p-planar graphs, D is the class of graphs of degree at most p. We prove that tree-width and clique-width are linearly related for graphs drawable with a graph of crossings of bounded average degree.We prove that the class of 1-planar graphs, although conceptually close to that of planar graphs, is not characterized by a monadic second-order sentence. We identify two subclasses that are.

Logical laws for existential monadic second-order sentences with infinite first-order parts

We consider existential monadic second-order sentences ∃X φ(X) about undirected graphs, where ∃X is a finite sequence of monadic quantifiers and φ(X) ∈ +∞ωω is an infinite first-order formula. We prove that there exists a sentence (in the considered logic) with two monadic variables and two first-order variables such that the probability that it is true on G(n, p) does not converge. Moreover, such an example is also obtained for one monadic variable and three first-order variables.

Automata for the verification of monadic second-order graph properties

The model-checking problem for monadic second-order logic on graphs is fixed-parameter tractable with respect to tree-width and clique-width. The proof constructs finite automata from monadic second-order sentences. These automata recognize the terms over fixed finite signatures that define graphs satisfying the given sentences. However, this construction produces automata of hyper-exponential sizes, and is thus impossible to use in practice in many cases. To overcome this difficulty, we propose to specify the transitions of automata by programs instead of tables. Such automata are called fly-automata. By using them, we can check certain monadic second-order graph properties with limited quantifier alternation depth, that are nevertheless interesting for Graph Theory. We give explicit constructions of automata relative to graphs of bounded clique-width, and we report on experiments.

Monadic datalog over finite structures of bounded treewidth

Bounded treewidth and monadic second-order (MSO) logic have proved to be key concepts in establishing fixed-parameter tractability results. Indeed, by Courcelle's Theorem we know that any property of finite structures, which is expressible by an MSO sentence, can be decided in linear time (data complexity) if the structures have bounded treewidth. In principle, Courcelle's Theorem can be applied directly to construct concrete algorithms by transforming the MSO evaluation problem into a tree language recognition problem. The latter can then be solved via a finite tree automaton (FTA). However, this approach has turned out to be problematical, since even relatively simple MSO formulae may lead to a “state explosion” of the FTA. In this work we propose monadic datalog (i.e., datalog where all intentional predicate symbols are unary) as an alternative method to tackle this class of fixed-parameter tractable problems. We show that if some property of finite structures is expressible in MSO then this property can also be expressed by means of a monadic datalog program over the decomposed structure : we mean by this that the original structure is augmented with new elements and new relations that encode one of its tree decompositions. In the first place, we thus compare the expressive power of two query languages. However, we also show that the resulting fragment of datalog can be evaluated in linear time (both with respect to the program size and with respect to the data size). Hence, our transformation of an MSO query into a monadic datalog program yields an alternative proof of Courcelle's Theorem. In order to actually construct efficient algorithms for problems whose tractability is due to Courcelle's Theorem, we propose to use a fragment of full (i.e., not necessarily monadic) datalog which allows for a succinct representation of the corresponding monadic datalog programs and for an efficient execution. This new approach is put to work by devising datalog programs for the 3-Colorability problem of graphs and for the PRIMALITY problem of relational schemas (i.e., testing if some attribute in a relational schema is part of a key). We also report on experimental results with a prototype implementation.

Closure properties of locally finite ω-languages

Locally finite ω-languages, defined via second-order quantifications followed by a first-order locally finite sentence, were introduced by Ressayre (J. Symbolic Logic 53(4) (1988) 1009–1026). They enjoy very nice properties and extend ω-languages accepted by finite automata or defined by monadic second-order sentences. We study here closure properties of the family LOC ω of locally finite omega languages. In particular, we show that the class LOC ω is neither closed under intersection nor under complementation, giving an answer to a question of Ressayre (Question posed during a Working Group on Automata and Logic at University Paris 7, 1989).

Logic over Words on Denumerable Ordinals

The main result of this paper is the extension of the theorem of Schützenberger, McNaughton, and Papert on star-free sets of finite words to languages of words of countable length. We also give another proof of the theorem of Büchi which establishes the equivalence between automata and monadic second-order sentences for defining sets of words of denumerable length.

Locally finite languages

We investigate properties of locally finite languages introduced by Ressayre (J. Symbolic Logic 53 (4) (1988) 1009–1026). These languages are defined by locally finite sentences and generalize languages recognized by automata or defined by monadic second-order sentences. We give many examples, showing that numerous context free languages are locally finite. Then we study closure properties of the family LOC of locally finite languages, and show that most undecidability results that hold for context free languages may be extended to locally finite languages. In a second part, we consider an extension of these languages to infinite and transfinite length words. We prove that each α-language which is recognized by a Büchi automaton (where α is an ordinal and ω⩽α<ω ω ) is defined by a locally finite sentence. This result, combined with a preceding one of (Finkel and Ressayre (J. Symbolic Logic 61 (2) (1996) 563–585), provides a generalization of Büchi's result about decidability of monadic second-order theory of the structure (α,<).

Coloring rules for finite trees, and probabilities of monadic second order sentences

A system of coloring rules is a set of rules for coloring the vertices of any finite rooted tree, starting at its leaves, which has the property that the color assigned to each vertex depends only on how many of its immediate predecessors there are of each color. The asymptotic behavior of the fraction of n vertex trees with a given root color is investigated. As a primary application, if φ is any monadic second-order sentence, then the fractions μn(φ), νn(φ) of labeled and unlabeled rooted trees satisfying φ, converge to limiting probabilities. An extension of the idea shows that for a particular model of Boolean formulas in k variables, k fixed, the fraction of such formulas of size n which compute a given Boolean function approaches a positive limit as n→∞. © 1997 John Wiley & Sons, Inc. Random Struct. Alg., 10, 453–485, 1997

Reachability is harder for directed than for undirected finite graphs

AbstractAlthough it is known that reachability in undirected finite graphs can be expressed by an existential monadic second-order sentence, our main result is that this is not the case for directed finite graphs (even in the presence of certain “built-in” relations, such as the successor relation). The proof makes use of Ehrenfeucht-Fraïssé games, along with probabilistic arguments. However, we show that for directed finite graphs with degree at most k, reachability is expressible by an existential monadic second-order sentence.

On the definability of properties of finite graphs

This paper considers the definability of graph-properties by restricted second-order and first-order sentences. For example, it is shown that the class of Hamiltonian graphs cannot be defined by monadic second-order sentences (i.e., if quantification over the subsets of vertices is allowed); any first-order sentence that defines Hamiltonian graphs on n vertices must contain at least 1 2 n quantifiers. The proofs use Fraïssé-Ehrenfeucht games and ultraproducts.

Complexity classes and theories of finite models

LetL $$ \subseteq $$ Σ* be accepted in timef(n) by a nondeterministic Turing machine. Then there is a monadic existential second-order sentence σ in the language of + such that for everyx∈Σ*,x∈L if and only if a certain structureU of cardinalityf(|x|) satisfies σ. It follows that ifL is accepted in nondeterministic timen d, d a natural number, then there is a sentence whose relational symbols ared-ary or less, whose finite spectrum isL.