Monadic Second-Order Logic

Descriptive complexity theory is a branch of complexity theory that views the hardness of a problem in terms of the complexity of expressing it in some logical formalism; among the resources considered are the number of object variables, quantifier depth, type, and alternation, sentences length (finite/infinite), etc. In this field we have studied two problems: (i) expressibility in ∃SO and (ii) the descriptive complexity of finite abelian groups. Inspired by Fagin’s result that NP = ∃SO, we have developed a partial framework to investigate expressibility inside ∃SO so as to have a finer look into NP. The framework uses combinatorics derived from second-order Ehrenfeucht-Fraısse games and the notion of game types. Among the results obtained is that for any k, divisibility by k is not expressible by an ∃SO sentence where (1) each second-order variable has arity at most 2, (2) the first-order part has at most 2 first-order variables, and (3) the first-order part has quantifier depth at most 3. In the second project we have investigated the descriptive complexity of finite abelian groups. Using Ehrenfeucht-Fraısse games we find upper and lower bounds on quantifier depth, quantifier alternations, and number of variables of a first-order sentence that distinguishes two finite abelian groups. Our main results are the following. Let G1 and G2 be a pair of non-isomorphic finite abelian groups, and let m be a number that divides one of the two groups’ orders. Then the following hold: (1) there exists a first-order sentence φ that distinguishes G1 and G2 such that φ is existential, has quantifier depth O(log m), and has at most 5 variables and (2) if φ is a sentence that distinguishes G1 and G2 then φ must have quantifier depth Ω(log m). In infinitary model theory we have studied abstract elementary classes. We have defined Galois types over arbitrary subsets of the monster (large enough homogeneous model), have defined a simple notion of splitting, and have proved some properties of this notion such as invariance under isomorphism, monotonicity, reflexivity, existence of non-splitting extensions.

... I'd like also to thank Mohamed Shawky, Mohamed Abdallah, Ahmed Sadek, Karim Sadik, Tarek Ghanem, Yasser Jaradat, Mohamed Farouk, Amr ElSherif and Mohamed Fahmi. ii Page 10. Table of Contents 1 Introduction 1 1.1 Preliminary . . . . . ...

"In this thesis we study the expressive power of variants of monadic second-order logic (MSO) on infinite trees by means of automata. In particular we are interested in weak MSO and well-founded MSO, where the second-order quantifiers range respectively over finite sets and over subsets of well-founded trees. On finitely branching trees, weak and well-founded MSO have the same expressive power and are both strictly weaker than MSO. The associated class of automata (called weak MSO-automata) is a restriction of the class characterizing MSO-expressivity.
We show that, on trees with arbitrary branching degree, weak MSO-automata characterize the expressive power of well-founded MSO, which turns out to be incomparable with weak MSO. Indeed, in this generalized setting, weak MSO gives an account of properties of the ‘horizontal dimension’ of trees, which cannot be described by means of MSO or well-founded MSO formulae.
In analogy with the result of Janin and Walukiewicz for MSO and the modal μ-calculus, this raises the issue of which modal logic captures the bisimulation-invariant fragment of well-founded MSO and weak MSO. We show that the alternation-free fragment of the modal μ-calculus and the bisimulation-invariant fragment of well-founded MSO have the same expressive power on trees of arbitrary branching degree. We motivate the conjecture that weak MSO modulo bisimulation collapses inside MSO and well-founded MSO."

We investigate automatic presentations of infinite words. Starting points of our study are the works of Rigo and Maes, and Carton and Thomas concerning the lexicographic presentation, respectively the decidability of the MSO theory of morphic words. Refining their techniques we observe that the
lexicographic presentation of a (morphic) word is canonical in a certain sense. We then go on to generalize our techniques to a hierarchy of classes of infinite words enjoying the above mentioned properties. We introduce k-lexicographic presentations, and morphisms of level k stacks and show that these are inter-translatable, thus giving rise to the same classes of k-lexicographic or level k morphic words. We prove that these presentations are also canonical, which implies decidability of the MSO theory of every k-lexicographic word as well as closure of these classes under restricted MSO interpretations, e.g. closure under deterministic sequential mappings. The classes of k-lexicographic words are shown to form an infinite hierarchy.

The description of a single state of a modelled system is often complex in practice, but few procedures for synthesis address this problem in depth. We study systems in which a state is described by an arbitrary finite structure, and changes of the state are represented by structure rewriting rules, a generalisation of term and graph rewriting. Both the environment and the controller are allowed to change the structure in this way, and the question we ask is how a strategy for the controller that ensures a given property can be synthesised. We focus on one particular class of structure rewriting rules, namely on separated structure rewriting, a limited syntactic class of rules. To counter this restrictiveness, we allow the property to be ensured by the controller to be specified in a very expressive logic: a combination of monadic second-order logic evaluated on states and the modal mu-calculus for the temporal evolution of the whole system. We show that for the considered class of rules and this logic, it can be decided whether the controller has a strategy ensuring a given property, and in such case a finite-memory strategy can be synthesised. Additionally, we prove that the same holds if the property is given by a monadic second-order formula to be evaluated on the limit of the evolution of the system.

Abstract—Let R be a class of generators of node-labelled infinite trees, and L be a logical language for describing correctness properties of these trees. Given R ∈ R and ϕ ∈ L, we say that Rϕ is a ϕ-reflection of R just if (i) R and Rϕ generate the same underlying tree, and (ii) ...

We provide a characterization theorem, in the style of van Benthem and Janin-Walukiewicz, for the alternation-free fragment of the modal mu-calculus. For this purpose we introduce a variant of standard monadic second-order logic (MSO), which we call well-founded monadic second-order logic (WFMSO). When interpreted in a tree model, the second-order quantifiers of WFMSO range over subsets of well-founded subtrees. The first main result of the paper states that the expressive power of WFMSO over trees exactly corresponds to that of weak MSO-automata. Using this automata-theoretic characterization, we then show that, over the class of all transition structures, the bisimulation-invariant fragment of WFMSO is the alternation-free fragment of the modal mu-calculus. As a corollary, we find that the logics WFMSO and WMSO (weak monadic second-order logic, where second-order quantification concerns finite subsets), are incomparable in expressive power.

In this paper, we introduce a new fragment of the first-order temporal language, called the monodic fragment, in which all formulas beginning with a temporal operator (Since or Until) have at most one free variable. We show that the satisfiability problem for monodic formulas in various linear time structures can be reduced to the satisfiability problem for a certain fragment

In this paper, we address the decision problem for a sys- tem of monadic second-order logic interpreted over an!- layered temporal structure devoid of both a finest layer and a coarsest one (we call such a structure totally unbounded). We propose an automaton-theoretic method that solves the problem in two steps: first, we reduce the considered prob- lem to the

We investigate automatic presentations of ω-words. Starting points of our study are the works of Rigo and Maes, Caucal, and Carton and Thomas concerning lexicographic presentation, MSOinterpretability in algebraic trees, and the decidability of the MSO theory of morphic words. Refining their techniques we observe that the lexicographic presentation of a (morphic) word is in a certain sense canonical. We then generalize our techniques to a hierarchy of classes of ω-words enjoying the above mentioned definability and decidability properties. We introduce k-lexicographic presentations, and morphisms of level k stacks and show that these are inter-translatable, thus giving rise to the same classes of k-lexicographic or level k morphic words. We prove that these presentations are also canonical, which implies decidability of the MSO theory of every k-lexicographic word as well as closure of these classes under MSO-definable recolorings, e.g. closure under deterministic sequential mapping...

We define tree automata with global equality and disequality constraints (TAGED). TAGEDs can test (dis)equalities between subtrees which may be arbitrarily faraway. In particular, they are equipped with an equality relation and a disequality relation on states, so that whenever two subtrees t and t′ evaluate (in an accepting run) to two states which are in the (dis)equality relation, they must be (dis)equal. We study several properties of TAGEDs, and prove emptiness decidability of for several expressive subclasses of TAGEDs.

A graph polynomial $p(G, \bar{X})$ can code numeric information about the underlying graph G in various ways: as its degree, as one of its specific coefficients or as evaluations at specific points $\bar{X}= \bar{x}_0$ . In this paper we study the question how to prove that a given graph parameter, say ω(G), the size of the maximal clique of G, cannot be a fixed coefficient or the evaluation at any point of the Tutte polynomial, the interlace polynomial, or any graph polynomial of some infinite family of graph polynomials. Our result is very general. We give a sufficient condition in terms of the connection matrix of graph parameter f(G) which implies that it cannot be the evaluation of any graph polynomial which is invariantly definable in CMSOL, the Monadic Second Order Logic augmented with modular counting quantifiers. This criterion covers most of the graph polynomials known from the literature.

We investigate automatic presentations of ω-words. Starting points of our study are the works of Rigo and Maes, Caucal, and Carton and Thomas concerning lexicographic presentation, MSOinterpretability in algebraic trees, and the decidability of the MSO theory of morphic words. Refining their techniques we observe that the lexicographic presentation of a (morphic) word is in a certain sense canonical. We then generalize our techniques to a hierarchy of classes of ω-words enjoying the above mentioned definability and decidability properties. We introduce k-lexicographic presentations, and morphisms of level k stacks and show that these are inter-translatable, thus giving rise to the same classes of k-lexicographic or level k morphic words. We prove that these presentations are also canonical, which implies decidability of the MSO theory of every k-lexicographic word as well as closure of these classes under MSO-definable recolorings, e.g. closure under deterministic sequential mapping...

This paper studies the expressive power that an extra first order quantifier adds to a fragment of monadic second order logic, extending the toolkit of Janin and Marcinkowski [JM01].We introduce an operationexistsn(S) on propertiesSthat says “there arencomponents havingS”. We use this operation to show that under natural strictness conditions, adding a first order quantifier worduto the beginning of a prefix classVincreases the expressive power monotonically inu. As a corollary, if the first order quantifiers are not already absorbed inV, then both the quantifier alternation hierarchy and the existential quantifier hierarchy in the positive first order closure ofVare strict.We generalize and simplify methods from Marcinkowski [Mar99] to uncover limitations of the expressive power of an additional first order quantifier, and show that for a wide class of propertiesS, Scannot belong to the positive first order closure of a monadic prefix classWunless it already belongs toW.We introduc...

We investigate structures that can be represented by omega-automata, so called omega-automatic structures, and prove that relations defined over such structures in first-order logic expanded by the first-order quantifiers `there exist at most aleph_0 many', 'there exist finitely many' and 'there exist k modulo m many' are omega-regular. The proof identifies certain algebraic properties of omega-semigroups. As a consequence an omega-regular equivalence relation of countable index has an omega-regular set of representatives. This implies Blumensath's conjecture that a countable structure with an omega-automatic presentation can be represented using automata on finite words. This also complements a very recent result of Hjörth, Khoussainov, Montalban and Nies showing that there is an omega-automatic structure which has no injective presentation.

We show that the expressive power of the branching time logic CTL coincides with that of the class of bisimula- tion invariant properties expressible in so-called monadi c path logic: monadic second order logic in which set quan- tification is restricted to paths. In order to prove this re- sult, we first prove a new Composition Theorem for trees. This