Related papers: Undecidability in number theory
The show that the upper-left-corner problem and upper-right-corner problem for matrix groups with rational entries are undecidable. To reach this aim, we answer a question of Dixon from 1985 by proving the undecidability of the stabilizer…
We provide a sufficient condition for a polynomial ring, not necessarily commutative, to have a first-order definition for the rational integers.
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…
To prove that Hilbert's tenth problem over a ring R has a negative answer, usually the integers or another ring for which Hilbert's tenth problem has a negative solution is modelled inside the ring of interest. In this paper, we formalize…
We consider the logic MSO+U, which is monadic second-order logic extended with the unbounding quantifier. The unbounding quantifier is used to say that a property of finite sets holds for sets of arbitrarily large size. We prove that the…
We prove the undecidability of determining whether a Turing machine yields an eventually periodic trajectory. From this, we deduce the undecidability of orbit finiteness in the polynomial dynamical system on infinite tuples of integers.
This article critically reappraises arguments in support of Cantor's theory of transfinite numbers. The following results are reported: i) Cantor's proofs of nondenumerability are refuted by analyzing the logical inconsistencies in…
In this paper we give elementary conditions completely characterising when the theory of modules of a Pr\"ufer domain is decidable. Using these results, we show that the theory of modules of the ring of integer valued polynomials is…
In this work, we show that very natural, apparently simple problems in quantum measurement theory can be undecidable even if their classical analogues are decidable. Undecidability hence appears as a genuine quantum property here. Formally,…
We prove the universality of the regular realizability problems for several classes of filters. The filters are encodings of finite relations on the set of non-negative integers in the format proposed by P. Wolf and H. Fernau. The…
Models of a generalized nondeterminism are defined by limitations on nonde- terministic behavior of a computing device. A regular realizability problem is a problem of verifying existence of a special sort word in a regular language. These…
Taking the view that computation is after all physical, we argue that physics, particularly quantum physics, could help extend the notion of computability. Here, we list the important and unique features of quantum mechanics and then…
For a degree $n$ polynomial $f$ over the rationals, the elements in the fiber $f^{-1}(a)$ are of degree $n$ over $\mathbb Q$ for most rational values $a$ by Hilbert's irreducibility theorem. Determining the set of exceptional $a$'s without…
In this paper, a computably definable predicate is defined and characterized. Then, it is proved that every separable infinite-dimensional Hilbert structure in an effectively presented language is computable. Moreover, every definable…
We relate the decidability problem for BS with unordered cartesian product with Hilbert's Tenth problem and prove that BS with unordered cartesian product is NP-complete.
Recent work has examined how undecidable problems can arise in quantum information science. We augment this by introducing three new undecidable problems stated in terms of tensor networks. These relate to ideas of Penrose about the…
Descent via an isogeny on an elliptic curve is used to construct two subrings of the field of rational numbers, which are complementary in a strong sense, and for which Hilbert's Tenth Problem is undecidable. This method further develops…
We resolve three long-standing open problems, namely the (algorithmic) decidability of network coding, the decidability of conditional information inequalities, and the decidability of conditional independence implication among random…
We consider the termination/non-termination property of a class of loops. Such loops are commonly used abstractions of real program pieces. Second-order logic is a convenient language to express non-termination. Of course, such property is…
The notion of a k-automatic set of integers is well-studied. We develop a new notion - the k-automatic set of rational numbers - and prove basic properties of these sets, including closure properties and decidability.