Related papers: A fixed point for the jump operator on structures
In this paper we consider a fragment of the first-order theory of the real numbers that includes systems of equations of continuous functions in bounded domains, and for which all functions are computable in the sense that it is possible to…
Consider the set of unitary operators on a complex separable Hilbert space $\hilh$, denoted as $\mathcal{U}(\hilh)$. Consider $1<p<\infty$. We establish that a function $f$ defined on the unit circle $\cir$ is $n$ times continuously…
We prove several results about the relationship between the word complexity function of a subshift and the set of Turing degrees of points of the subshift, which we call the Turing spectrum. Among other results, we show that a Turing…
The Turing machine is one of the simple abstract computational devices that can be used to investigate the limits of computability. In this paper, they are considered from several points of view that emphasize the importance and the…
We examine various categorical structures that can and cannot be constructed. We show that total computable functions can be mimicked by constructible functors. More generally, whatever can be done by a Turing machine can be constructed by…
In this paper, we take an in-depth look at the complexity of a hitherto unexplored Multiobjective Spanner (MSp) problem. The MSp is a multiobjective generalization of the well-studied Minimum t-Spanner problem. This multiobjective approach…
For a very general class of unbounded self-adjoint operator function we prove upper bounds for eigenvalues which lie within arbitrary gaps of the essential spectrum. These upper bounds are given by triple variations. Furthermore, we find…
This paper builds a cumulative tower of Grothendieck universes that provides a precise size discipline for higher type theory. Starting from an increasing sequence of inaccessible cardinals, we give an inductive-recursive definition of…
We seek progress in the study of subtoposes of the effective topos. First we treat Van Oosten's result that local operators on the effective topos are internally NNO-indexed joins of what we shall call 'basic' local operators. Our main…
The relationship between the complexity classes P and NP is a question that has not yet been answered by the Theory of Computation. The existence of a language in NP, proven not to belong to P, is sufficient evidence to establish the…
Let \pi : X -> S be a morphism of algebraic stacks that is locally of finite presentation with affine stabilizers. We prove that there is an algebraic S-stack, the Hilbert stack, parameterizing proper algebraic stacks mapping quasi-finitely…
We present an algebraic characterization of the complexity classes Logspace and NLogspace, using an algebra with a composition law based on unification. This new bridge between unification and complexity classes is inspired from proof…
It is well known that several classical geometry problems (e.g., angle trisection) are unsolvable by compass and straightedge constructions. But what kind of object is proven to be non-existing by usual arguments? These arguments refer to…
Abstract axiomatic formulation of mathematical structures are extensively used to describe our physical world. We take here the reverse way. By making basic assumptions as starting point, we reconstruct some features of both geometry and…
We construct a topology on a given algebraically closed field with a distinguished subfield which is also algebraically closed. This topology is finer than Zariski topology and it captures the sets definable in the pair of algebraically…
The universal-algebraic approach has proved a powerful tool in the study of the complexity of CSPs. This approach has previously been applied to the study of CSPs with finite or (infinite) omega-categorical templates, and relies on two…
We consider analogues of the Bernstein-Gelfand-Gelfand resolution in a highest weight category $\mathscr{P}$. We prove the resulting category of complexes is a chain-level lift of the heart of the constructible $t$-structure on its bounded…
Claims about recursive self-improvement in AI often slide from repeated internal revision to the possibility of qualitatively stronger capability without clearly distinguishing the underlying computational regimes. This paper gives a formal…
We investigate the complexity of explicit construction problems, where the goal is to produce a particular object of size $n$ possessing some pseudorandom property in time polynomial in $n$. We give overwhelming evidence that $\bf{APEPP}$,…
For each type of number, structures that differ by arbitrary scaling factors and are isomorphic to one another are described. The scaling of number values in one structure, relative to the values in another structure, must be compensated…