arXiv subjects
Publications and source records attributed to Ali N. Valizadeh.
For any irrational number $ α>\frac{3+\sqrt{5}}{2}\approx2.618$ and given a positive $ n\in\mathbb{N} $, we use elementary number theory to introduce a necessary and sufficient condition for a natural number $ x $ to be in the $n$th iterate of the Beatty sequence of modulus $α$.
By providing a procedure to apply Hrushovski's amalgamation method to the setting of classes of infinite structures, we introduce the notion of \textit{paracollapsed} structures. We show that this approach provides existentially closed generic structures in which the geometric closure is not included in the algebraic closure while the resulting theory is decidable. We show that paracollapsed structures have the strict order property and $\text{TP}_2$.
We introduce a recursive theory that completely axiomatizes the structure $\langle \mathbb{Z},<, +,f,0\rangle$ where $f$ is the function that maps each $x$ to the integer part of $φx $, with $φ$ the golden ratio. We prove that our axiomatization is model-complete in a language expanded with a function which we which we refer as the Fibonacci floor function.
We introduce a model-complete theory which completely axiomatizes the structure $Z_α=(Z, +, 0, 1, f)$ where $f : x \to \lfloorα x \rfloor $ is a unary function with $α$ a fixed transcendental number. When $α$ is computable, our theory is recursively enumerable, and hence decidable as a result of completeness. Therefore, this result fits into the more general theme of adding traces of multiplication to integers without losing decidability.
We show that the Hrushovski-\fraisse limit of certain classes of trees lead to strictly superstable theories of various U-ranks. In fact, for each $ α\inω+1\backslash\{0\} $ we introduce a strictly superstable theory of U-rank $ α. $ Furthermore, we show that these theories are decidable and pseudofinite.
In a relational language consisting of a single relation $ R, $ we investigate pseudofiniteness of certain Hrushovski constructions obtained via predimension functions. It is notable that the arity of the relation $ R $ plays a crucial role in this context. When $ R $ is ternary, by extending the methods developed in [BL12], we interpret $ \langle\mathbb{Q}^{+},<\rangle $ in the $ \langle\mathcal{K}^{+}_{0},\leq^{*}\rangle $-generic and prove that this structure is not pseudofinite. This provides a negative answer to the question posed in [EW09] (Question 2.6). This result, in fact, unfolds another aspect of complexity of this structure, along with undecidability and strict order property proved in [EW09] and [Bl12]. On the other hand, when $ R $ is binary, it can be shown that the $ \langle\mathcal{K}^{+}_{0},\leq^{*}\rangle $-generic is decidable and pseudofinite.
In the context of Hrushovski constructions we take a language $ \mathcal{L} $ with a ternary relation $ R $ and consider the theory of the generic models $ M^{*}_α, $ of the class of finite $ \mathcal{L}$-structures equipped with predimension functions $ δ_α, $ for $ α\in(0,1]\cap\mathbb{Q} $. The theory of generic structures of non-AC smooth classes have been investigated from different points of view, including decidability and their power in interpreting known structures and theories. For a rational $ α\in(0,1], $ first we prove that the theory of $ M^{*}_α $ admits a quantifier elimination down to a meaningful class of formulas, called \textit{closure formulas}; and on the other hand we prove that $ Th(M^{*}_α) $ does not have the finite model property.