|
Linear Speedup Theorem
In computational complexity theory, the linear speedup theorem for Turing machines states that given any real ''c'' > 0 and any ''k''-tape Turing machine solving a problem in time ''f''(''n''), there is another ''k''-tape machine that solves the same problem in time at most , where ''k'' > 1. If the original machine is non-deterministic, then the new machine is also non-deterministic. The constants 2 and 3 in 2''n'' + 3 can be lowered, for example, to ''n'' + 2. The theorem also holds for Turing machines with 1-way, read-only input tape and k\ge 1 work tapes. For single-tape Turing machines, linear speedup holds for machines with execution time at least n^2. It provably does not hold for machines with time t(n)\in \Omega(n\log n)\cap o(n^2). Proof The construction is based on packing several tape symbols of the original machine ''M'' into one tape symbol of the new machine ''N''. It has a similar effect as using longer words and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Computational Complexity Theory
In theoretical computer science and mathematics, computational complexity theory focuses on classifying computational problems according to their resource usage, and explores the relationships between these classifications. A computational problem is a task solved by a computer. A computation problem is solvable by mechanical application of mathematical steps, such as an algorithm. A problem is regarded as inherently difficult if its solution requires significant resources, whatever the algorithm used. The theory formalizes this intuition, by introducing mathematical models of computation to study these problems and quantifying their computational complexity, i.e., the amount of resources needed to solve them, such as time and storage. Other measures of complexity are also used, such as the amount of communication (used in communication complexity), the number of logic gate, gates in a circuit (used in circuit complexity) and the number of processors (used in parallel computing). O ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Turing Machine
A Turing machine is a mathematical model of computation describing an abstract machine that manipulates symbols on a strip of tape according to a table of rules. Despite the model's simplicity, it is capable of implementing any computer algorithm. The machine operates on an infinite memory tape divided into discrete mathematics, discrete cells, each of which can hold a single symbol drawn from a finite set of symbols called the Alphabet (formal languages), alphabet of the machine. It has a "head" that, at any point in the machine's operation, is positioned over one of these cells, and a "state" selected from a finite set of states. At each step of its operation, the head reads the symbol in its cell. Then, based on the symbol and the machine's own present state, the machine writes a symbol into the same cell, and moves the head one step to the left or the right, or halts the computation. The choice of which replacement symbol to write, which direction to move the head, and whet ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Real Number
In mathematics, a real number is a number that can be used to measure a continuous one- dimensional quantity such as a duration or temperature. Here, ''continuous'' means that pairs of values can have arbitrarily small differences. Every real number can be almost uniquely represented by an infinite decimal expansion. The real numbers are fundamental in calculus (and in many other branches of mathematics), in particular by their role in the classical definitions of limits, continuity and derivatives. The set of real numbers, sometimes called "the reals", is traditionally denoted by a bold , often using blackboard bold, . The adjective ''real'', used in the 17th century by René Descartes, distinguishes real numbers from imaginary numbers such as the square roots of . The real numbers include the rational numbers, such as the integer and the fraction . The rest of the real numbers are called irrational numbers. Some irrational numbers (as well as all the rationals) a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Non-deterministic Turing Machine
In theoretical computer science, a nondeterministic Turing machine (NTM) is a theoretical model of computation whose governing rules specify more than one possible action when in some given situations. That is, an NTM's next state is ''not'' completely determined by its action and the current symbol it sees, unlike a deterministic Turing machine. NTMs are sometimes used in thought experiments to examine the abilities and limits of computers. One of the most important open problems in theoretical computer science is the P versus NP problem, which (among other equivalent formulations) concerns the question of how difficult it is to simulate nondeterministic computation with a deterministic computer. Background In essence, a Turing machine is imagined to be a simple computer that reads and writes symbols one at a time on an endless tape by strictly following a set of rules. It determines what action it should perform next according to its internal ''state'' and ''what symbol it cu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Read-only Turing Machine
A read-only Turing machine or two-way deterministic finite-state automaton (2DFA) is class of models of computability that behave like a standard Turing machine and can move in both directions across input, except cannot write to its input tape. The machine in its bare form is equivalent to a deterministic finite automaton in computational power, and therefore can only parse a regular language. Theory We define a standard Turing machine by the 9-tuple M = (Q, \Sigma, \Gamma, \vdash, \_, \delta, s, t, r) where * Q is a finite set of ''states''; * \Sigma is the finite set of the ''input alphabet''; * \Gamma is the finite ''tape alphabet''; * \vdash \in \Gamma - \Sigma is the ''left endmarker''; * \_ \in \Gamma - \Sigma is the ''blank symbol''; * \delta : Q \times \Gamma \rightarrow Q \times \Gamma \times \ is the ''transition function''; * s \in Q is the ''start state''; * t \in Q is the ''accept state''; * r \in Q, ~ r \ne t is the ''reject state''. So given initial state q read ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kenneth W
Kenneth is a given name of Gaelic origin. The name is an Anglicised form of two entirely different Gaelic personal names: ''Cainnech'' and '' Cináed''. The modern Gaelic form of ''Cainnech'' is ''Coinneach''; the name was derived from a byname meaning "handsome", "comely". Etymology The second part of the name ''Cinaed'' is derived either from the Celtic ''*aidhu'', meaning "fire", or else Brittonic ''jʉ:ð'' meaning "lord". People Fictional characters * Kenneth Widmerpool, character in Anthony Powell's novel sequence ''A Dance to the Music of Time'' * Kenneth Parcell from 30 Rock Places In the United States: * Kenneth, Minnesota * Kenneth City, Florida In Scotland: * Inch Kenneth Inch Kenneth () is a small grassy island off the west coast of the Isle of Mull, in Scotland. It is at the entrance of Loch na Keal, to the south of Ulva. It is part of the Loch na Keal National Scenic Area, one of 40 in Scotland. It is within ..., an island off the west coast of the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pointer Machine
In theoretical computer science, a pointer machine is an atomistic abstract computational machine whose storage structure is a graph. A pointer algorithm could also be an algorithm restricted to the pointer machine model. Some particular types of pointer machines are called a linking automaton, a KU-machine, an SMM, an atomistic LISP machine, a tree-pointer machine, etc. Amir Ben-Amram (1995). What is a "Pointer machine"?, SIGACT News (ACM Special Interest Group on Automata and Computability Theory), volume 26, 1995. Pointer machines do not have arithmetic instructions. Computation proceeds only by reading input symbols, modifying and doing various tests on its storage structure—the pattern of nodes and pointers, and outputting symbols based on the tests. In this sense, the model is similar to the Turing machine. Types of "pointer machines" Both Gurevich and Ben-Amram list a number of very similar "atomistic" models of "abstract machines"; Yuri Gurevich (2000), ''Sequential ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Theorems In Computational Complexity Theory
In mathematics and formal logic, a theorem is a statement that has been proven, or can be proven. The ''proof'' of a theorem is a logical argument that uses the inference rules of a deductive system to establish that the theorem is a logical consequence of the axioms and previously proved theorems. In mainstream mathematics, the axioms and the inference rules are commonly left implicit, and, in this case, they are almost always those of Zermelo–Fraenkel set theory with the axiom of choice (ZFC), or of a less powerful theory, such as Peano arithmetic. Generally, an assertion that is explicitly called a theorem is a proved result that is not an immediate consequence of other known theorems. Moreover, many authors qualify as ''theorems'' only the most important results, and use the terms ''lemma'', ''proposition'' and ''corollary'' for less important theorems. In mathematical logic, the concepts of theorems and proofs have been formalized in order to allow mathematical reason ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
