Related papers: Some structural complexity results for $\exists\ma…
We study the complexity of computational problems arising from existence theorems in extremal combinatorics. For some of these problems, a solution is guaranteed to exist based on an iterated application of the Pigeonhole Principle. This…
The constraint satisfaction problem (CSP) of a first-order theory T is the computational problem of deciding whether a given conjunction of atomic formulas is satisfiable in some model of T. We study the computational complexity of CSP$(T_1…
We present the MEoP problem that decides the existence of solutions to certain modular equations over prime numbers and show how this separates the complexity class NP from its subclass P
We provide the first evidence for the inherent difficulty of finding complex sets with optimal proof systems. For this, we construct oracles $O_1$ and $O_2$ with the following properties, where $\mathrm{RE}$ denotes the class of recursively…
The class $\mathcal{UP}$ of `ultimate polynomial time' problems over $\mathbb C$ is introduced; it contains the class $\mathcal P$ of polynomial time problems over $\mathbb C$. The $\tau$-Conjecture for polynomials implies that…
Classification theory of elementary classes deals with first order (elementary) classes of structures (i.e. fixing a set T of first order sentences, we investigate the class of models of T with the elementary submodel notion). It tries to…
In this paper we survey the history of, and recent developments on, two major conjectures originating in Zilber's model-theoretic work on complex exponentiation -- Existential Closedness and Zilber-Pink. The main focus is on the modular…
This article presents a general solution to the problem of computational complexity. First, it gives a historical introduction to the problem since the revival of the foundational problems of mathematics at the end of the 19th century.…
We study the decidability and complexity of equational theories of the existential calculus of relations with transitive closure (ECoR*) and its fragments, where ECoR* is the positive calculus of relations with transitive closure extended…
The purpose of this paper is to clarify the relationship between various conditions implying essential undecidability: our main result is that there exists a theory $T$ in which all partially recursive functions are representable, yet $T$…
The paper deals with two issues: the existence of universal models of a theory T and related properties when cardinal arithmetic does not give this existence offhand. In the first section we prove that simple theories (e.g., theories…
By Fagin's Theorem, NP contains precisely those problems that can be described by formulas starting with an existential second-order quantifier, followed by only first-order quantifiers (ESO formulas). Subsequent research refined this…
Let $\RR_S$ denote the expansion of the real ordered field by a family of real-valued functions $S$, where each function in $S$ is defined on a compact box and is a member of some quasianalytic class which is closed under the operations of…
Continuing the study of complexity theory of Koepke's Ordinal Turing Machines (OTMs) that was started by Rin, L\"owe and the author, we prove the following results: (1) An analogue of Ladner's theorem for OTMs holds: That is, there are…
In this paper we investigate formal verification problems for Neural Network computations. Various reachability problems will be in the focus, such as: Given symbolic specifications of allowed inputs and outputs in form of Linear…
Theoretical complexity is a vital subfield of computer science that enables us to mathematically investigate computation and answer many interesting queries about the nature of computational problems. It provides theoretical tools to assess…
Our main results are in the following three sections: 1. We prove new relations between proof complexity conjectures that are discussed in \cite{pu18}. 2. We investigate the existence of p-optimal proof systems for $\mathsf{TAUT}$, assuming…
We introduce and study a natural class of fields in which certain first-order definable sets are existentially definable, and characterise this class by a number of equivalent conditions. We show that global fields belong to this class, and…
We show that determining the rank of a tensor over a field has the same complexity as deciding the existential theory of that field. This implies earlier NP-hardness results by H{\aa}stad~\cite{H90}. The hardness proof also implies an…
Although $S_\infty$ (the group of all permutations of $\mathbb{N}$) is size continuum, both it and its closed subgroups can be presented as the set of paths through a countable tree. The subgroups of $S_\infty$ that can be presented this…