Expansions of MSO by cardinality relations

We study expansions of the Weak Monadic Second Order theory of (N,<) by cardinality relations, which are predicates R(X1,...,Xn) whose truth value depends only on the cardinality of the sets X1, ...,Xn. We first provide a (definable) criterion for definability of a cardinality relation in (N,<), and use it to prove that for every cardinality relation R which is not definable in (N,<), there exists a unary cardinality relation which is definable in (N,<,R) and not in (N,<). These results resemble Muchnik and Michaux-Villemaire theorems for Presburger Arithmetic. We prove then that + and x are definable in (N,<,R) for every cardinality relation R which is not definable in (N,<). This implies undecidability of the WMSO theory of (N,<,R). We also consider the related satisfiability problem for the class of finite orderings, namely the question whether an MSO sentence in the language {<,R} admits a finite model M where < is interpreted as a linear ordering, and R as the restriction of some (fixed) cardinality relation to the domain of M. We prove that this problem is undecidable for every cardinality relation R which is not definable in (N,<).Comment: to appear in LMC

Order-Invariant MSO is Stronger than Counting MSO in the Finite

We compare the expressiveness of two extensions of monadic second-order logic (MSO) over the class of finite structures. The first, counting monadic second-order logic (CMSO), extends MSO with first-order modulo-counting quantifiers, allowing the expression of queries like ``the number of elements in the structure is even''. The second extension allows the use of an additional binary predicate, not contained in the signature of the queried structure, that must be interpreted as an arbitrary linear order on its universe, obtaining order-invariant MSO. While it is straightforward that every CMSO formula can be translated into an equivalent order-invariant MSO formula, the converse had not yet been settled. Courcelle showed that for restricted classes of structures both order-invariant MSO and CMSO are equally expressive, but conjectured that, in general, order-invariant MSO is stronger than CMSO. We affirm this conjecture by presenting a class of structures that is order-invariantly definable in MSO but not definable in CMSO.Comment: Revised version contributed to STACS 200

Monadic second order finite satisfiability and unbounded tree-width

The finite satisfiability problem of monadic second order logic is decidable only on classes of structures of bounded tree-width by the classic result of Seese (1991). We prove the following problem is decidable: Input: (i) A monadic second order logic sentence $\alpha$, and (ii) a sentence $\beta$ in the two-variable fragment of first order logic extended with counting quantifiers. The vocabularies of $\alpha$ and $\beta$ may intersect. Output: Is there a finite structure which satisfies $\alpha\land\beta$ such that the restriction of the structure to the vocabulary of $\alpha$ has bounded tree-width? (The tree-width of the desired structure is not bounded.) As a consequence, we prove the decidability of the satisfiability problem by a finite structure of bounded tree-width of a logic extending monadic second order logic with linear cardinality constraints of the form $|X_{1}|+\cdots+|X_{r}|<|Y_{1}|+\cdots+|Y_{s}|$, where the $X_{i}$ and $Y_{j}$ are monadic second order variables. We prove the decidability of a similar extension of WS1S

Decidability Results for the Boundedness Problem

We prove decidability of the boundedness problem for monadic least fixed-point recursion based on positive monadic second-order (MSO) formulae over trees. Given an MSO-formula phi(X,x) that is positive in X, it is decidable whether the fixed-point recursion based on phi is spurious over the class of all trees in the sense that there is some uniform finite bound for the number of iterations phi takes to reach its least fixed point, uniformly across all trees. We also identify the exact complexity of this problem. The proof uses automata-theoretic techniques. This key result extends, by means of model-theoretic interpretations, to show decidability of the boundedness problem for MSO and guarded second-order logic (GSO) over the classes of structures of fixed finite tree-width. Further model-theoretic transfer arguments allow us to derive major known decidability results for boundedness for fragments of first-order logic as well as new ones

Infinite and Bi-infinite Words with Decidable Monadic Theories

We study word structures of the form $(D,<,P)$ where $D$ is either $\mathbb{N}$ or $\mathbb{Z}$, $<$ is the natural linear ordering on $D$ and $P\subseteq D$ is a predicate on $D$. In particular we show: (a) The set of recursive $\omega$-words with decidable monadic second order theories is $\Sigma_3$-complete. (b) Known characterisations of the $\omega$-words with decidable monadic second order theories are transfered to the corresponding question for bi-infinite words. (c) We show that such "tame" predicates $P$ exist in every Turing degree. (d) We determine, for $P\subseteq\mathbb{Z}$, the number of predicates $Q\subseteq\mathbb{Z}$ such that $(\mathbb{Z},\le,P)$ and $(\mathbb{Z},\le,Q)$ are indistinguishable. Through these results we demonstrate similarities and differences between logical properties of infinite and bi-infinite words

An Application of the Feferman-Vaught Theorem to Automata and Logics for<br> Words over an Infinite Alphabet

We show that a special case of the Feferman-Vaught composition theorem gives rise to a natural notion of automata for finite words over an infinite alphabet, with good closure and decidability properties, as well as several logical characterizations. We also consider a slight extension of the Feferman-Vaught formalism which allows to express more relations between component values (such as equality), and prove related decidability results. From this result we get new classes of decidable logics for words over an infinite alphabet.Comment: 24 page