Related papers: Uniform first-order definitions in finitely genera…

Let $F_q$ be a field with $q$ elements, where $q$ is a power of a prime number $p\geq 5$. For any integer $m\geq 2$ and $a\in F_q^*$ such that the polynomial $x^m-a$ is irreducible in $F_q[x]$, we combine two different methods to construct…

We provide a denotational semantics for first-order logic that captures the two-level view of the computation process typical for constraint programming. At one level we have the usual program execution. At the other level an automatic…

We describe a deterministic algorithm for finding a generating element of the multiplicative group of the finite field $\mathbb{F}_{p^n}$ where $p$ is a prime. In time polynomial in $p$ and $n$, the algorithm either outputs an element that…

We give first-order definitions of Campana and Darmon points in algebraic function fields in one variable over number fields. These sets are geometric generalizations of $n$-full integers (integers whose nonzero valuations are at least $n$)…

A field $K$ in a ring language $\mathcal{L}$ is finitely undecidable if $\mbox{Cons}(\Sigma)$ is undecidable for every nonempty finite $\Sigma \subseteq \mbox{Th}(K; \mathcal{L})$. We extend a construction of Ziegler and (among other…

We obtain several results concerning the concept of isotypic structures. Namely we prove that any field of finite transcendence degree over a prime subfield is defined by types; then we construct isotypic but not isomorphic structures with…

We prove some general estimates for exponential sums over subsets of finite fields which are definable in the language of rings. This generalizes both the classical exponential sum estimates over varieties over finite fields due to Weil,…

We consider sets/relations/computations defined by *Elementary Inference Systems* I, which are obtained from Smullyan's *elementary formal systems* using Gentzen's notation for inference rules, and proof trees for atoms P(t_1,...,t_n),…

We prove that a complete local or graded one-dimensional domain of prime characteristic has finite F-representation type if its residue field is algebraically closed or finite, and present examples of a complete local or graded…

First-order logic is the basis for many knowledge representation formalisms and methods. Providing technological support for learning to write first-order formulas for natural language specifications requires methods to test formulas for…

Building on work of J. Robinson and A. Shlapentokh, we develop a general framework to obtain definability and decidability results of large classes of infinite algebraic extensions of $\mathbb{F}_p(t)$. As an application, we show that for…

In this paper, we give two proofs of the wellfoundedness of recursive notation systems for $\Pi_N$-reflecting ordinals. One is based on $\Pi_{N-1}^0$-inductive definitions, and the other is based on distinguished classes.

This paper gives a thorough overview of what is known about first-order logic with counting quantifiers and with arithmetic predicates. As a main theorem we show that Presburger arithmetic is closed under unary counting quantifiers.…

We show pro-definability of spaces of definable types in various classical complete first order theories, including complete o-minimal theories, Presburger arithmetic, $p$-adically closed fields, real closed and algebraically closed valued…

We apply the {\it first-order formalism} method to obtaining BPS equations for Alice string. This is done by generalizing the well-known first-order formalism to the case of non-Abelian strings. We do not assume any specific gauge group nor…

Over the last century, the principle of "induction on the continuum" has been studied by different authors in different formats. All of these different readings are equivalent to one of the three versions that we isolate in this paper. We…

The first-order theory of a string automatic structure is known to be decidable, but there are examples of string automatic structures with nonelementary first-order theories. We prove that the first-order theory of a string automatic…

A policy describes the conditions under which an action is permitted or forbidden. We show that a fragment of (multi-sorted) first-order logic can be used to represent and reason about policies. Because we use first-order logic, policies…

A new description of free massless superfields of arbitrary superspin $Y$ ($Y>1/2$) is proposed. Following the first-order philosophy, we relax some of the properties (reality, gauge redundancy) of the unconstrained higher spin…

In the classification of complete first-order theories, many dividing lines have been defined in order to understand the complexity and the behavior of some classes of theories. In this paper, using the concept of patterns of consistency…