相关论文: Complexity Science for Simpletons
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…
The classical PCP theorem is arguably the most important achievement of classical complexity theory in the past quarter century. In recent years, researchers in quantum computational complexity have tried to identify approaches and develop…
Can the computational complexity theory of computer science and mathematics say something new about unresolved problems in quantum physics? Particularly, can the P versus NP question in the computational complexity theory be a factor in the…
For many fundamental problems in computational topology, such as unknot recognition and $3$-sphere recognition, the existence of a polynomial-time solution remains unknown. A major algorithmic tool behind some of the best known algorithms…
Transcript of G.J. Chaitin's 2 March 2000 Carnegie Mellon University School of Computer Science Distinguished Lecture. The notion of randomness is taken from physics and applied to pure mathematics in order to shed light on the…
Satisfiability problems play a central role in computer science and engineering as a general framework for studying the complexity of various problems. Schaefer proved in 1978 that truth satisfaction of propositional formulas given a…
The Boolean satisfiability problem (SAT) is a well-known example of monotonic reasoning, of intense practical interest due to fast solvers, complemented by rigorous fine-grained complexity results. However, for non-monotonic reasoning,…
We use the powerful tools of counting complexity and generic oracles to help understand the limitations of the complexity of quantum computation. We show several results for the probabilistic quantum class BQP. 1. BQP is low for PP, i.e.,…
In this paper we study the complexity of solving a problem when a solution of a similar instance is known. This problem is relevant whenever instances may change from time to time, and known solutions may not remain valid after the change.…
The Riemann hypothesis is one of the most famous unresolved problems in modern mathematics. The discussion here will present an overview of past methods that prove the Riemann hypothesis is a $\Pi_1^0$ sentence. We also end with some…
NP complete problem is one of the most challenging issues. The question of whether all problems in NP are also in P is generally considered one of the most important open questions in mathematics and theoretical computer science as it has…
In this paper the reason why entropy reduction (negentropy) can be used to measure the complexity of any computation was first elaborated both in the aspect of mathematics and informational physics. In the same time the equivalence of…
The 3n+1, or Collatz problem, is one of the hardest math problems, yet still unsolved. The Collatz conjecture is to prove or disprove that the Collatz sequences COL(n) always eventually reach the number of 1, for all n belongs to N+ (all…
Proposed in 1937, the Collatz conjecture has remained in the spotlight for mathematicians and computer scientists alike due to its simple proposal, yet intractable proof. In this paper, we propose several novel theorems, corollaries, and…
Computational complexity is a core theory of computer science, which dictates the degree of difficulty of computation. There are many problems with high complexity that we have to deal, which is especially true for AI. This raises a big…
In this paper we generalize the classical Proth's theorem for integers of the form $N=Kp^n+1$. For these families, we present a primality test whose computational complexity is $\widetilde{O}(\log^2(N))$ and, what is more important, that…
Chaitin's work, in its depth and breadth, encompasses many areas of scientific and philosophical interest. It helped establish the accepted mathematical concept of randomness, which in turn is the basis of tools that I have developed to…
Except for crystalline or random structures, an agreed definition of complexity for intermediate and hence interesting cases does not exist. We fill this gap with a notion of complexity that characterises shapes formed by any finite number…
A major challenge of interdisciplinary description of complex system behaviour is whether real systems of higher complexity levels can be understood with at least the same degree of objective, "scientific" rigour and universality as…
After a brief review of current scenarios for the resolution and/or avoidance of the Big Bang, an alternative hypothesis is put forward implying an infinite increase in complexity towards the initial singularity. This may result in an…