Related papers: Physical portrayal of computational complexity
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.…
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…
We devise a hierarchy of computational algorithms to enumerate the microstates of a system comprising N independent, distinguishable particles. An important challenge is to cope with integers that increase exponentially with system size,…
We develop a complexity theory for approximate real computations. We first produce a theory for exact computations but with condition numbers. The input size depends on a condition number, which is not assumed known by the machine. The…
Computational complexity is a particularly important objective. The idea of Landauer principle was extended through mapping three classic problems (sorting,ordered searching and max of N unordered numbers) into Maxwell demon thought…
In spite of the fact that the Newtonian dynsmics of the underlying molecules do not seem to favor one direction of time over its opposite, the non-decreasing entropy of macroscopic physical systems provides a unique direction to 'the arrow…
The purpose of this article is to examine and limit the conditions in which the P complexity class could be equivalent to the NP complexity class. Proof is provided by demonstrating that as the number of clauses in a NP-complete problem…
In this manuscript, we derive the principle of conservation of computational complexity. We measure computational complexity as the number of binary computations (decisions) required to solve a problem. Every problem then defines a unique…
The intuition that a long history is required for the emergence of complexity in natural systems is formalized using the notion of depth. The depth of a system is defined in terms of the number of parallel computational steps needed to…
The resolution of the P vs. NP problem, a cornerstone in computational theory, remains elusive despite extensive exploration through mathematical logic and algorithmic theory. This paper takes a novel approach by integrating information…
Computational complexity theory contains a corpus of theorems and conjectures regarding the time a Turing machine will need to solve certain types of problems as a function of the input size. Nature {\em need not} be a Turing machine and,…
Complexity theory as practiced by physicists and computational complexity theory as practiced by computer scientists both characterize how difficult it is to solve complex problems. Here it is shown that the parameters of a specific model…
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…
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…
We propose a measure of learning efficiency for non-finite state spaces. We characterize the complexity of a learning problem by the metric entropy of its state space. We then describe how learning efficiency is determined by this measure…
Computation is commonly defined as the execution of abstract algorithms over symbolic representations, with physical systems treated as substrates that realise predefined operations. While effective for engineered machines, this separation…
A measure called Physical Complexity is established and calculated for a population of sequences, based on statistical physics, automata theory, and information theory. It is a measure of the quantity of information in an organism's genome.…
Motivated by the fact that information is encoded and processed by physical systems, the P versus NP problem is examined in terms of physical processes. In particular, we consider P as a class of deterministic, and NP as nondeterministic,…
Computing is a high-level process of a physical system. Recent interest in non-standard computing systems, including quantum and biological computers, has brought this physical basis of computing to the forefront. There has been, however,…
Quantum superposition, a cornerstone of quantum mechanics, enables systems to exist in multiple states simultaneously, giving rise to probabilistic outcomes. In quantum information science, conditional entropy has become a key metric for…