Related papers: Complexity Classes as Mathematical Axioms
The second author previously discussed how classical complexity separation conjectures, we call them "axioms", have implications in three manifold topology: polynomial length stings of operations which preserve certain Jones polynomial…
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…
This paper establishes the separation of complexity classes $\mathbf{P}$ and $\mathbf{NP}$ through a novel homological algebraic approach grounded in category theory. We construct the computational category $\mathbf{Comp}$, embedding…
Many of the properties of sectional category, topological complexity and homotopic distance are in fact derived from a small number of basic properties, which, once established, lead to all the others without further recourse to topology.…
We study the complexity of reasoning in abstracts argumentation frameworks close to graph classes that allow for efficient reasoning methods, i.e.\ to one of the classes of acyclic, noeven, biparite and symmetric AFs. In this work we show…
Harvey Friedman, in his remarkable paper Finite functions and the necessary use of large cardinals, Ann. Math. 148:803-893, 1998 and in a technical report, Applications of large cardinals to graph theory, Ohio State University, 1997,…
We introduce a hierarchy of fast-growing complexity classes and show its suitability for completeness statements of many non elementary problems. This hierarchy allows the classification of many decision problems with a non-elementary…
We observe that many of the separation axioms of topology (including $T_0-T_4$) can be expressed concisely and uniformly in terms of category theory as lifting properties (in the sense of Quillen model categories) with respect to (usually…
We develop a theory of complexity for numerical computations that takes into account the condition of the input data and allows for roundoff in the computations. We follow the lines of the theory developed by Blum, Shub, and Smale for…
We survey a collective achievement of a group of researchers: the PCP Theorems. They give new definitions of the class \np, and imply that computing approximate solutions to many \np-hard problems is itself \np-hard. Techniques developed to…
Theories of classification distinguish classes with some good structure theorem from those for which none is possible. Some classes (dense linear orders, for instance) are non-classifiable in general, but are classifiable when we consider…
In this article, we shall describe some of the most interesting topics in the subject of Complexity Science for a general audience. Anyone with a solid foundation in high school mathematics (with some calculus) and an elementary…
We survey results on the formalization and independence of mathematical statements related to major open problems in computational complexity theory. Our primary focus is on recent findings concerning the (un)provability of complexity…
Complexity theory provides a wealth of complexity classes for analyzing the complexity of decision and counting problems. Despite the practical relevance of enumeration problems, the tools provided by complexity theory for this important…
We present an application of the basic mathematical concept of complex functions as topological solitons, a most interesting area of research in physics. Such application of complex theory is virtually unknown outside the community of…
Defined by a single axiom, finite abstract simplicial complexes belong to the simplest constructs of mathematics. We look at a a few theorems.
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…
The central conjecture of parameterized complexity states that FPT is not equal to W[1], and is generally regarded as the parameterized counterpart to P != NP. We revisit the issue of the plausibility of FPT != W[1], focusing on two…
We develop theory concerning non-uniform complexity in a setting in which the notion of single-pass instruction sequence considered in program algebra is the central notion. We define counterparts of the complexity classes P/poly and…
We prove two sets of results concerning computational complexity classes. The first concerns a variation of the random oracle hypothesis posed by Bennett and Gill after they showed that relative to a randomly chosen oracle, P not equal NP…