Given a subset of $X\subseteq \mathbb{R}^{n}$ we can associate with every point $x\in \mathbb{R}^{n}$ a vector space $V$ of maximal dimension with the property that for some ball centered at $x$, the subset $X$ coincides inside the ball with a union of lines parallel with $V$. A point is singular if $V$ has dimension $0$. In an earlier paper we proved that a $(\mathbb{R}, +,< ,\mathbb{Z})$-definable relation $X$ is actually definable in $(\mathbb{R}, +,< ,1)$ if and only if the number of singular points is finite and every rational section of $X$ is $(\mathbb{R}, +,< ,1)$-definable, where a rational section is a set obtained from $X$ by fixing some component to a rational value. Here we show that we can dispense with the hypothesis of $X$ being $(\mathbb{R}, +,< ,\mathbb{Z})$-definable by assuming that the components of the singular points are rational numbers. This provides a topological characterization of first-order definability in the structure $(\mathbb{R}, +,< ,1)$. It also allows us to deliver a self-definable criterion (in Muchnik's terminology) of $(\mathbb{R}, +,< ,1)$- and $(\mathbb{R}, +,< ,\mathbb{Z})$-definability for a wide class of relations, which turns into an effective criterion provided that the corresponding theory is decidable. In particular these results apply to the class of $k-$recognizable relations on reals, and allow us to prove that it is decidable whether a $k-$recognizable relation (of any arity) is $l-$recognizable for every base $l \geq 2$.

We specify the frontier of decidability for fragments of the first-order theory of ordinal multiplication. We give a NEXPTIME lower bound for the complexity of the existential fragment of [Formula: see text] for every ordinal λ. Moreover, we prove (by reduction from Hilbert Tenth Problem) that the ∃*∀ 6 -fragment of [Formula: see text] is undecidable for every ordinal λ.

Abstract Rationals and countable ordinals are important examples of structures with decidable monadic second-order theories. A chain is an expansion of a linear order by monadic predicates. We show that if the monadic second-order theory of a countable chain C is decidable then C has a non-trivial expansion with decidable monadic second-order theory.

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.

The tableau construct plays a very important role in relational-database theory. The paper shows how this construct can be extended for tuple-timestamped relations. The expressive power of temporal tableau queries is compared with that of a temporal algebra. A temporal extension of the homomorphism theorem is given.

Abstract Let θ, θ′ be two multiplicatively independent Pisot numbers, and letU,U′ be two linear numeration systems whose characteristic polynomial is the minimal polynomial of θ and θ′, respectively. For everyn≥ 1, ifA⊆ ℕnisU-andU′ -recognizable thenAis definable in 〈ℕ: + 〉.

Abstract Let be the restriction of usual order relation to integers which are primes or squares of primes, and let ⊥ denote the coprimeness predicate. The elementary theory of is undecidable. Now denote by <π the restriction of order to primary numbers. All arithmetical relations restricted to primary numbers are definable in the structure (ℕ; ⊥, <π). Furthermore, the structures (ℕ; ∣, <π) (ℕ; =, ×, <π) and (ℕ; =, +, ×) are interdefinable.

Abstract Letkandlbe two multiplicatively independent integers, and letL⊆ ℕnbe al-recognizable set which is not definable in 〈ℕ; +〉. We prove that the elementary theory of 〈ℕ; +,Vk, L〉, whereVk(x)denotes the greatest power ofkdividingx, is undecidable. This result leads to a new proof of the Cobham-Semënov theorem.