Related papers: A Homological Separation of $\mathbf{P}$ from $\ma…
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.…
In computational complexity, a complexity class is given by a set of problems or functions, and a basic challenge is to show separations of complexity classes $A \not= B$ especially when $A$ is known to be a subset of $B$. In this paper we…
We show the problem of counting homomorphisms from the fundamental group of a homology $3$-sphere $M$ to a finite, non-abelian simple group $G$ is #P-complete, in the case that $G$ is fixed and $M$ is the computational input. Similarly,…
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…
In this paper we explore fundamental concepts in computational complexity theory and the boundaries of algorithmic decidability. We examine the relationship between complexity classes \textbf{P} and \textbf{NP}, where $L \in \textbf{P}$…
Representing graphs by their homomorphism counts has led to the beautiful theory of homomorphism indistinguishability in recent years. Moreover, homomorphism counts have promising applications in database theory and machine learning, where…
Treating a conjecture, P^#P != NP, on the separation of complexity classes as an axiom, an implication is found in three manifold topology with little obvious connection to complexity theory. This is reminiscent of Harvey Friedman's work on…
Computation of homology or cohomology is intrinsically a problem of high combinatorial complexity. Recently we proposed a new efficient algorithm for computing cohomologies of Lie algebras and superalgebras. This algorithm is based on…
The subject logic in computer science should entail proof theoretic applications. So the question arises whether open problems in computational complexity can be solved by advanced proof theoretic techniques. In particular, consider the…
This paper argues that the requirement of applicableness of quantum linearity to any physical level from molecules and atoms to the level of macroscopic extensional world, which leads to a main foundational problem in quantum theory…
We introduce a notion of complexity of diagrams (and in particular of objects and morphisms) in an arbitrary category, as well as a notion of complexity of functors between categories equipped with complexity functions. We discuss several…
The central problem in computational algebraic topology is the computation of the homotopy groups of a given space, represented as a simplicial set. Algorithms have been found which achieve this, but the running times depend on the size of…
This paper lays the foundations of an approach to applying Gromov's ideas on quantitative topology to topological data analysis. We introduce the "contiguity complex", a simplicial complex of maps between simplicial complexes defined in…
In recent work by Johnson et al. (2022), a framework was described for the study of graph problems over classes specified by omitting each of a finite set of graphs as subgraphs. If a problem falls into the framework then its computational…
This chapter delves into the realm of computational complexity, exploring the world of challenging combinatorial problems and their ties with statistical physics. Our exploration starts by delving deep into the foundations of combinatorial…
Topological complexity $\TC{B}$ of a space $B$ is introduced by M. Farber to measure how much complex the space is, which is first considered on a configuration space of a motion planning of a robot arm. We also consider a stronger version…
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…
Category theory provides a means through which many far-ranging fields of mathematics can be related by their similar structure. In a paper by Robinson [2], this interconnectivity afforded by categorical perspectives allowed for the…
Topological complexity is a numerical homotopy invariant that measures the instability of motion planning in a space. To study the topological complexity of non-simply connected spaces, Costa and Farber introduced a cohomology class whose…
Homomorphisms between relational structures are not only fundamental mathematical objects, but are also of great importance in an applied computational context. Indeed, constraint satisfaction problems (CSPs), a wide class of algorithmic…