We survey the properties of sets of integers recognizable by automata when they are written in p-ary expansions. We focus on Cobham’s theorem which characterizes the sets recognizable in different bases p and on its generalization to N m due to Semenov. We detail the remarkable proof recently given by Muchnik for the theorem of Cobham-Semenov, the original proof being published in Russian. 1

Expansions of the natural number ordering by unary predicates are studied, using logics which in expressive power are located between first-order and monadic second-order logic. Building on the modeltheoretic composition method of Shelah, we give two characterizations of the decidable theories of this form, in terms of effectiveness conditions on two types of “homogeneous sets”. We discuss the significance of these characterizations, show that the first-order theory of successor with extra predicates is not covered by this approach, and indicate how analogous results are obtained in the semigroup theoretic and the automata theoretic framework.

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Abstract. The following problem is known as the Church Synthesis problem: Input: an MLO formula ψ(X, Y). Task: Check whether there is an operator Y = F (X) such that

Abstract. We generalize a question of Büchi: Let R be an integral domain and k ≥ 2 an integer. Is there an algorithm to solve in R any given system of polynomial equations, each of which is linear in the k−th powers of the unknowns? We examine variances of this problem for k = 2, 3 and for R a field of rational functions of characteristic zero. We obtain negative answers, provided that the analogous problem over Z has a negative answer. In particular we prove that the generalization of Büchi’s question for fields of rational functions over a real-closed field F, for k = 2, has a negative answer if the analogous question over Z has a negative answer. 1

We survey de nability and decidability issues related to rst-order fragments of arithmetic, with a special emphasis on Presburger and Skolem arithmetic and their (un)decidable extensions.

The paper studies different variants of almost periodicity notion. We introduce the class of eventually strongly almost periodic sequences where some suffix is strongly almost periodic (=uniformly recurrent). The class of almost periodic sequences includes the class of eventually strongly almost periodic sequences, and we prove this inclusion to be strict. We prove that the class of eventually strongly almost periodic sequences is closed under finite automata mappings and finite transducers. Moreover, an effective form of this result is presented. Finally we consider some algorithmic questions concerning almost periodicity. 1

Dedicated to Yuri Gurevich on the occasion of his seventieth birthday Abstract. Let M =(A, <, P)where(A, <) is a linear ordering and P denotes a finite sequence of monadic predicates on A. We show that if A contains an interval of order type ω or −ω, and the monadic second-order theory of M is decidable, then there exists a non-trivial expansion M ′ of M by a monadic predicate such that the monadic second-order theory of M ′ is still decidable.

Abstract. For a two-variable formula B(X,Y) of Monadic Logic of Order (MLO) the Church Synthesis Problem concerns the existence and construction of a finite-state operator Y=F(X) such that B(X,F(X)) is universally valid over Nat. Büchi and Landweber (1969) proved that the Church synthesis problem is decidable. We investigate a parameterized version of the Church synthesis problem. In this extended version a formula B and a finite-state operator F might contain as a parameter a unary predicate P. A large class of predicates P is exhibited such that the Church problem with the parameter P is decidable. Our proofs use Composition Method and game theoretical techniques. 1

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