Monadic Second-order Formulas Research Articles

Solving Infinite Games in the Baire Space

Infinite games (in the form of Gale-Stewart games) are studied where a play is a sequence of natural numbers chosen by two players in alternation, the winning condition being a subset of the Baire space ωω. We consider such games defined by a natural kind of parity automata over the alphabet ℕ, called ℕ-MSO-automata, where transitions are specified by monadic second-order formulas over the successor structure of the natural numbers. We show that the classical Büchi-Landweber Theorem (for finite-state games in the Cantor space 2ω) holds again for the present games: A game defined by a deterministic parity ℕ-MSO-automaton is determined, the winner can be computed, and an ℕ-MSO-transducer realizing a winning strategy for the winner can be constructed.

Reversible Regular Languages: Logical and Algebraic Characterisations

We present first-order (FO) and monadic second-order (MSO) logics with predicates ‘between’ and ‘neighbour’ that characterise the class of regular languages that are closed under the reverse operation and its subclasses. The ternary between predicate bet(x, y, z) is true if the position y is strictly between the positions x and z. The binary neighbour predicate N(x, y) is true when the the positions x and y are adjacent. It is shown that the class of reversible regular languages is precisely the class definable in the logics MSO(bet) and MSO(N). Moreover the class is definable by their existential fragments EMSO(bet) and EMSO(N), yielding a normal form for MSO formulas. In the first-order case, the logic FO(bet) corresponds precisely to the class of reversible languages definable in FO(<). Every formula in FO(bet) is equivalent to one that uses at most 3 variables. However the logic FO(N) defines only a strict subset of reversible languages definable in FO(+1). A language-theoretic characterisation of the class of languages definable in FO(N), called locally-reversible threshold-testable (LRTT), is given. In the second part of the paper we show that the standard connections that exist between MSO and FO logics with order and successor predicates and varieties of finite semigroups extend to the new setting with the semigroups extended with an involution operation on its elements. The case is different for FO(N) where we show that one needs an additional equation that uses the involution operator to characterise the class. While the general problem of characterising FO(N) is open, an equational characterisation is shown for the case of neutral letter languages.

Model-Checking on Ordered Structures

We study the model-checking problem for first- and monadic second-order logic on finite relational structures. The problem of verifying whether a formula of these logics is true on a given structure is considered intractable in general, but it does become tractable on interesting classes of structures, such as on classes whose Gaifman graphs have bounded treewidth. In this article, we continue this line of research and study model-checking for first- and monadic second-order logic in the presence of an ordering on the input structure. We do so in two settings: the general ordered case, where the input structures are equipped with a fixed order or successor relation, and the order-invariant case, where the formulas may resort to an ordering, but their truth must be independent of the particular choice of order. In the first setting we show very strong intractability results for most interesting classes of structures. In contrast, in the order-invariant case we obtain tractability results for order-invariant monadic second-order formulas on the same classes of graphs as in the unordered case. For first-order logic, we obtain tractability of successor-invariant formulas on classes whose Gaifman graphs have bounded expansion. Furthermore, we show that model-checking for order-invariant first-order formulas is tractable on coloured posets of bounded width.

Graph-based representation of design properties in creating building floorplans

This paper proposes a framework for supporting the design process by defining design requirements over graph-based representations of designs. First, hierarchical layout hypergraphs (HL-graphs) representing designs and hierarchical layout graph grammars generating them are specified. Then, local and global graph requirements over HL-graphs, which correspond to design constraints, are defined. The proposed ontological interpretations transform first-order and monadic second-order logic formulas expressing design criteria into equivalent local and global graph requirements. The satisfiability of graph requirements by representations of designs allows for checking correctness of design solutions. The approach is illustrated on examples of designing floor layouts of buildings.

From tree-decompositions to clique-width terms

Tree-width and clique-width are two important graph complexity measures that serve as parameters in many fixed-parameter tractable algorithms. We give two algorithms that transform tree-decompositions represented by normal trees into clique-width terms (a rooted tree is normal for a graph if its nodes are the vertices of the graph and every two adjacent vertices are on a path of the tree starting at the root). As a consequence, we obtain that, for certain classes of sparse graphs, clique-width is polynomially bounded in terms of tree-width. It is even linearly bounded for planar graphs and incidence graphs. These results are useful in the construction of model-checking algorithms for problems described by monadic second-order formulae, including those allowing edge set quantifications.

Fly-automata for checking [formula omitted] graph properties

A more descriptive but too long title would be: Constructing fly-automata to check properties of graphs of bounded tree-width expressed by monadic second-order formulas written with edge quantifications. Such properties are called MSO2 in short. Fly-automata (FA) run bottom-up on terms denoting graphs and compute “on the fly” the necessary states and transitions instead of looking into huge, actually unimplementable tables. In previous works, we have constructed FA that process terms denoting graphs of bounded clique-width, in order to check their monadic second-order (MSO) properties (expressed by formulas without edge quantifications). Here, we adapt these FA to incidence graphs, so that they can check MSO2 properties of graphs of bounded tree-width. This is possible because: (1) an MSO2 property of a graph is nothing but an MSO property of its incidence graph and (2) the clique-width of the incidence graph of a graph is linearly bounded in terms of its tree-width. Our constructions are actually implementable and usable. We detail concrete constructions of automata in this perspective.

Characterizing weighted MSO for trees by branching transitive closure logics

We introduce the branching transitive closure operator on progressing weighted monadic second-order logic formulas where the branching corresponds in a natural way to the branching inherent in trees. For arbitrary commutative semirings, we prove that weighted monadic second order logics on trees is equivalent to the definability by formulas which start with one of the following operators: (i) a branching transitive closure or (ii) one existential second-order quantifier followed by one universal first-order quantifier; in both cases the operator is applied to step-formulas over (a) Boolean first-order logic enriched by modulo counting or (b) Boolean monadic-second order logic.

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.

Monadic second-order model-checking on decomposable matroids

A notion of branch-width, which generalizes the one known for graphs, can be defined for matroids. We first give a proof of the polynomial time model-checking of monadic second-order formulas on representable matroids of bounded branch-width, by reduction to monadic second-order formulas on trees. This proof is much simpler than the one previously known. We also provide a link between our logical approach and a grammar that allows one to build matroids of bounded branch-width. Finally, we introduce a new class of non-necessarily representable matroids, described by a grammar and on which monadic second-order formulas can be checked in linear time.

On the model-checking of monadic second-order formulas with edge set quantifications

We extend clique-width to graphs with multiple edges. We obtain fixed-parameter tractable model-checking algorithms for certain monadic second-order graph properties that depend on the multiplicities of edges, with respect to this “new” clique-width. We define special tree-width, the variant of tree-width relative to tree-decompositions such that the boxes that contain a vertex are on a path originating from some fixed node. We study its main properties. This definition is motivated by the construction of finite automata associated with monadic second-order formulas using edge set quantifications. These automata yield fixed-parameter linear algorithms with respect to tree-width for the model-checking of these formulas. Their construction is much simpler for special tree-width than for tree-width, for reasons that we explain.

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.

Approximate Satisfiability and Equivalence

Inspired by property testing, for every $\varepsilon>0$ we relax the classical satisfiability $U\models F$ between a finite structure U of a class $\mathbf{K}$ and a formula F, to a notion of $\varepsilon$-satisfiability $U\models_{\varepsilon}F$, and relax the classical equivalence $F_1\equiv F_2$ between two formulas $F_1$ and $F_2$ to $\varepsilon$-equivalence $F_1\equiv_{\varepsilon}F_2$. We consider strings and trees with the norm of the edit distance with moves, and show that, unlike their exact counterparts, these approximate notions can be efficiently decided. We use a statistical embedding of words (resp., trees) into $\ell_1$, which generalizes the original Parikh mapping, obtained by sampling $O(f(\varepsilon))$ finite samples of the words (resp., trees). We give a tester for equality and membership in any regular language, in time independent of the size of the structure. Using our geometrical embedding, we can also test the equivalence between two regular properties over words, defined by regular expressions or monadic second-order formulas. Our equivalence tester has polynomial time complexity in the size of the automaton (or regular expression), for any fixed $\varepsilon$, whereas the exact version of the equivalence problem is PSPACE-complete. We also prove versions of some of these results for trees, but with worse time complexity. Last, we extend the geometric embedding, and hence the testing algorithms, to infinite regular languages and to context-free languages. For context-free languages, the equivalence tester has an exponential time complexity for any fixed $\varepsilon$, whereas the exact version is not even decidable.

Linear delay enumeration and monadic second-order logic

The results of a query expressed by a monadic second-order formula on a tree, on a graph or on a relational structure of tree-width at most k , can be enumerated with a delay between two outputs proportional to the size of the next output. This is possible by using a preprocessing that takes time O ( n ⋅ log ( n ) ) , where n is the number of vertices or elements. One can also output directly the i -th element with respect to a fixed ordering, however, in more than linear time in its size. These results extend to graphs of bounded clique-width. We also consider the enumeration of finite parts of recognizable sets of terms specified by parameters such as size, height or Strahler number.

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.

Circle graphs and monadic second-order logic

This article is part of a project consisting in expressing, whenever possible, graph properties and graph transformations in monadic second-order logic or in its extensions using modulo p cardinality set predicates or auxiliary linear orders. A circle graph is the intersection graph of a set of chords of a circle. Such a set is called a chord diagram. It can also be described by a word with two occurrences of each letter, called a double occurrence word. If a circle graph is prime for the split ( or join) decomposition defined by Cunnigham, it has a unique representation by a chord diagram, and this diagram can be defined by monadic second-order formulas with the even cardinality set predicate. By using the (canonical) split decomposition of a circle graph, we define in monadic second-order logic with auxiliary linear orders all its chord diagrams. This construction uses the fact that the canonical split decomposition of a graph can be constructed in monadic second-order logic with help of an arbitrary linear order. We prove that the order of first occurrences of the letters in a double occurrence word w that represents a connected circle graph determines this word in a unique way. The word w can be defined by a monadic second-order formula from the word of first occurrences of letters. We also prove that a set of circle graphs has bounded clique-width if and only if all the associated chord diagrams have bounded tree-width.

On decidability of monadic logic of order over the naturals extended by monadic predicates

A fundamental result of Büchi states that the set of monadic second-order formulas true in the structure (Nat, <) is decidable. A natural question is: what monadic predicates (sets) can be added to (Nat, <) while preserving decidability? Elgot and Rabin found many interesting predicates P for which the monadic theory of 〈Nat, <, P〉 is decidable. The Elgot and Rabin automata theoretical method has been generalized and sharpened over the years and their results were extended to a variety of unary predicates. We give a sufficient and necessary model-theoretical condition for the decidability of the monadic theory of (Nat, <, P1, … , Pn).We reformulate this condition in an algebraic framework and show that a sufficient condition proposed previously by O. Carton and W.Thomas is actually necessary. A crucial argument in the proof is that monadic second-order logic has the selection and the uniformization properties over the extensions of (Nat, <) by monadic predicates. We provide a self-contained proof of this result.

The monadic second-order logic of graphs XVI : Canonical graph decompositions

This article establishes that the split decomposition of graphs introduced by Cunnigham, is definable in Monadic Second-Order Logic.This result is actually an instance of a more general result covering canonical graph decompositions like the modular decomposition and the Tutte decomposition of 2-connected graphs into 3-connected components. As an application, we prove that the set of graphs having the same cycle matroid as a given 2-connected graph can be defined from this graph by Monadic Second-Order formulas.

Graph decompositions definable in monadic second-order logic

Abstract We prove that the split decomposition of a graph is definable by Monadic Second-Order formulas. We also prove that the set of graphs having the same cycle matroid as a given graph is definable from this graph by Monadic Second-Order formulas. These results are consequences of general results on the logical definability of graph decompositions of various types.

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.

Learnability and Definability in Trees and Similar Structures

We prove upper bounds for combinatorial parameters of finite relational structures, related to the complexity of learning a definable set. We show that monadic second-order (MSO) formulas with parameters have bounded Vapnik–Chervonenkis dimension over structures of bounded clique-width, and first-order formulas with parameters have bounded Vapnik–Chervonenkis dimension over structures of bounded local clique-width (this includes planar graphs). We also show that MSO formulas of a fixed size have bounded strong consistency dimension over MSO formulas of a fixed larger size, for labeled trees. These bounds imply positive learnability results for the PAC and equivalence query learnability of a definable set over these structures. The proofs are based on bounds for related definability problems for tree automata.