Decidability of Definability Issues in the Theory of Real Addition

Given a subset of X ⊆ ℝn we can associate with every point x ∈ vector space V maximal dimension the property that for some ball centered at x, coincides inside union lines parallel to V. A is singular if has 0. In an earlier paper proved 〈ℝ, +, <, ℤ〉-definable relation 1〉-definable and only number points finite rational section 〈R, 1〉-definable, where set obtained from by fixing component value. Here show dispense hypothesis being requiring components be numbers. This provides topological characterization first-order definability in structure 1〉. It also allows us deliver self-definable criterion (in Muchnik’s terminology) 1〉- <,ℤ〉-definability wide class relations, which turns into effective provided corresponding theory decidable. particular these results apply so-called k–recognizable relations are defined Muller automata via representation reals integer basis k, allow prove it decidable whether (of any arity) l–recognizable base l ≥ 2.