Related papers: Complexity Science for Simpletons
This paper calls attention to the current state of the probability (P) domain which presents weak points at the mathematical level and more significant flaws at the application level. Popper notices how fundamental issues raised in quantum…
I want to show one possibility to proof the Collatz conjecture, also called 3n+1 conjecture, for any natural number N. For this, I limit my analysis on the direct odd follower of every natural odd number and show the connections between the…
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…
The term complexity derives etymologically from the Latin plexus, which means interwoven. Intuitively, this implies that something complex is composed by elements that are difficult to separate. This difficulty arises from the relevant…
This paper illustrates the richness of the concept of regular sets of time bounds and demonstrates its application to problems of computational complexity. There is a universe of bounds whose regular subsets allow to represent several time…
This paper demonstrates the relativity of Computability and Nondeterministic; the nondeterministic is just Turing's undecidable Decision rather than the Nondeterministic Polynomial time. Based on analysis about TM, UM, DTM, NTM, Turing…
Some representation-theoretic multiplicities, such as the Kostka and the Littlewood-Richardson coefficients, admit a combinatorial interpretation that places their computation in the complexity class #P. Whether this holds more generally is…
We prove that it is NP-complete to decide whether a given (3-dimensional) simplicial complex is collapsible. This work extends a result of Malgouyres and Franc\'{e}s showing that it is NP-complete to decide whether a given simplicial…
In the well-known complexity class NP are combinatorial problems, whose optimization counterparts are important for many practical settings. These problems typically consider full knowledge about the input. In practical settings, however,…
We prove a complexity dichotomy theorem for all non-negative weighted counting Constraint Satisfaction Problems (CSP). This caps a long series of important results on counting problems including unweighted and weighted graph homomorphisms…
The quest for quantum computers is motivated by their potential for solving problems that defy existing, classical, computers. The theory of computational complexity, one of the crown jewels of computer science, provides a rigorous…
A computational problem exhibits a "gap property" when there is no tractable boundary between two disjoint sets of instances. We establish a Gap Trichotomy Theorem for a family of constraint problem variants, completely classifying the…
The basic problem in the PAC model of computational learning theory is to determine which hypothesis classes are efficiently learnable. There is presently a dearth of results showing hardness of learning problems. Moreover, the existing…
Complexity theory can be viewed as the study of the relationship between computation and applications, understood the former as complexity classes and the latter as problems. Completeness results are clearly central to that view. Many…
Not only did Turing help found one of the most exciting areas of modern science (computer science), but it may be that his contribution to our understanding of our physical reality is greater than we had hitherto supposed. Here I explore…
I try to clarify several confusions in the popular literature concerning chaos, determinism, the arrow of time, entropy and the role of probability in physics. Classical ideas going back to Laplace and Boltzmann are explained and defended…
In this lecture I will talk about three mathematical puzzles involving mathematics and computation that have preoccupied me over the years. The first puzzle is to understand the amazing success of the simplex algorithm for linear…
These are notes on discrete mathematics for computer scientists. The presentation is somewhat unconventional. Indeed I begin with a discussion of the basic rules of mathematical reasoning and of the notion of proof formalized in a natural…
A dichotomy theorem for counting problems due to Creignou and Hermann states that or any nite set S of logical relations, the counting problem #SAT(S) is either in FP, or #P-complete. In the present paper we show a dichotomy theorem for…
Clarithmetics are number theories based on computability logic (see http://www.csc.villanova.edu/~japaridz/CL/ ). Formulas of these theories represent interactive computational problems, and their "truth" is understood as existence of an…