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

Adding modular predicates to first-order fragments

We investigate the decidability of the definability problem for fragments of first order logic over finite words enriched with modular predicates. Our approach aims toward the most generic statements that we could achieve, which successfully covers the quantifier alternation hierarchy of first order logic and some of its fragments. We obtain that deciding this problem for each level of the alternation hierarchy of both first order logic and its two-variable fragment when equipped with all regular numerical predicates is not harder than deciding it for the corresponding level equipped with only the linear order and the successor. For two-variable fragments we also treat the case of the signature containing only the order and modular predicates.Relying on some recent results, this proves the decidability for each level of the alternation hierarchy of the two-variable first order fragmentwhile in the case of the first order logic the question remains open for levels greater than two.The main ingredients of the proofs are syntactic transformations of first order formulas as well as the algebraic framework of finite categories

The Church Synthesis Problem with Parameters

For a two-variable formula &psi;(X,Y) of Monadic Logic of Order (MLO) the Church Synthesis Problem concerns the existence and construction of an operator Y=F(X) such that &psi;(X,F(X)) is universally valid over Nat. B\"{u}chi and Landweber proved that the Church synthesis problem is decidable; moreover, they showed that if there is an operator F that solves the Church Synthesis Problem, then it can also be solved by an operator defined by a finite state automaton or equivalently by an MLO formula. We investigate a parameterized version of the Church synthesis problem. In this version &psi; might contain as a parameter a unary predicate P. We show that the Church synthesis problem for P is computable if and only if the monadic theory of is decidable. We prove that the B\"{u}chi-Landweber theorem can be extended only to ultimately periodic parameters. However, the MLO-definability part of the B\"{u}chi-Landweber theorem holds for the parameterized version of the Church synthesis problem

First Order Theories of Some Lattices of Open Sets

We show that the first order theory of the lattice of open sets in some natural topological spaces is $m$-equivalent to second order arithmetic. We also show that for many natural computable metric spaces and computable domains the first order theory of the lattice of effectively open sets is undecidable. Moreover, for several important spaces (e.g., $\mathbb{R}^n$, $n\geq1$, and the domain $P\omega$) this theory is $m$-equivalent to first order arithmetic

The Church Problem for Countable Ordinals

A fundamental theorem of Buchi and Landweber shows that the Church synthesis problem is computable. Buchi and Landweber reduced the Church Problem to problems about &#969;-games and used the determinacy of such games as one of the main tools to show its computability. We consider a natural generalization of the Church problem to countable ordinals and investigate games of arbitrary countable length. We prove that determinacy and decidability parts of the Bu}chi and Landweber theorem hold for all countable ordinals and that its full extension holds for all ordinals < \omega\^\omega

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

What's Decidable About Sequences?

We present a first-order theory of sequences with integer elements, Presburger arithmetic, and regular constraints, which can model significant properties of data structures such as arrays and lists. We give a decision procedure for the quantifier-free fragment, based on an encoding into the first-order theory of concatenation; the procedure has PSPACE complexity. The quantifier-free fragment of the theory of sequences can express properties such as sortedness and injectivity, as well as Boolean combinations of periodic and arithmetic facts relating the elements of the sequence and their positions (e.g., "for all even i's, the element at position i has value i+3 or 2i"). The resulting expressive power is orthogonal to that of the most expressive decidable logics for arrays. Some examples demonstrate that the fragment is also suitable to reason about sequence-manipulating programs within the standard framework of axiomatic semantics.Comment: Fixed a few lapses in the Mergesort exampl