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Leaf Language
Leaf language is a method in computational complexity theory for characterizing a complexity class by formalizing what it means for a machine to "accept" an input. Complexity classes are typically defined in terms of a polynomial-time nondeterministic Turing machine (NTM). These machines possess multiple computational paths, and the outcomes of these paths determine whether an input is accepted or rejected. Traditionally, an NTM accepts an input if at least one path accepts it, and rejects it only if all paths reject it. In contrast, a co-Nondeterministic Turing Machine (co-NTM) accepts an input only if all paths accept it, and rejects it if any path rejects it. Moreover, more complex notions of acceptance can also be defined. To formalize the characterization of a complexity class, one can examine the formal language In logic, mathematics, computer science, and linguistics, a formal language is a set of strings whose symbols are taken from a set called "alphabet". The al ... [...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] |
Complexity Class
In computational complexity theory, a complexity class is a set (mathematics), set of computational problems "of related resource-based computational complexity, complexity". The two most commonly analyzed resources are time complexity, time and space complexity, memory. In general, a complexity class is defined in terms of a type of computational problem, a model of computation, and a bounded resource like time complexity, time or space complexity, memory. In particular, most complexity classes consist of decision problems that are solvable with a Turing machine, and are differentiated by their time or space (memory) requirements. For instance, the class P (complexity), P is the set of decision problems solvable by a deterministic Turing machine in polynomial time. There are, however, many complexity classes defined in terms of other types of problems (e.g. Counting problem (complexity), counting problems and function problems) and using other models of computation (e.g. probabil ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Polynomial-time
In theoretical computer science, the time complexity is the computational complexity that describes the amount of computer time it takes to run an algorithm. Time complexity is commonly estimated by counting the number of elementary operations performed by the algorithm, supposing that each elementary operation takes a fixed amount of time to perform. Thus, the amount of time taken and the number of elementary operations performed by the algorithm are taken to be related by a constant factor. Since an algorithm's running time may vary among different inputs of the same size, one commonly considers the worst-case time complexity, which is the maximum amount of time required for inputs of a given size. Less common, and usually specified explicitly, is the average-case complexity, which is the average of the time taken on inputs of a given size (this makes sense because there are only a finite number of possible inputs of a given size). In both cases, the time complexity is gener ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nondeterministic 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] |
Formal Language
In logic, mathematics, computer science, and linguistics, a formal language is a set of strings whose symbols are taken from a set called "alphabet". The alphabet of a formal language consists of symbols that concatenate into strings (also called "words"). Words that belong to a particular formal language are sometimes called ''well-formed words''. A formal language is often defined by means of a formal grammar such as a regular grammar or context-free grammar. In computer science, formal languages are used, among others, as the basis for defining the grammar of programming languages and formalized versions of subsets of natural languages, in which the words of the language represent concepts that are associated with meanings or semantics. In computational complexity theory, decision problems are typically defined as formal languages, and complexity classes are defined as the sets of the formal languages that can be parsed by machines with limited computational power. In ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ordered Tree
In graph theory, a tree is an undirected graph in which any two vertices are connected by path, or equivalently a connected acyclic undirected graph. A forest is an undirected graph in which any two vertices are connected by path, or equivalently an acyclic undirected graph, or equivalently a disjoint union of trees. A directed tree, oriented tree,See .See . polytree,See . or singly connected networkSee . is a directed acyclic graph (DAG) whose underlying undirected graph is a tree. A polyforest (or directed forest or oriented forest) is a directed acyclic graph whose underlying undirected graph is a forest. The various kinds of data structures referred to as trees in computer science have underlying graphs that are trees in graph theory, although such data structures are generally rooted trees. A rooted tree may be directed, called a directed rooted tree, either making all its edges point away from the root—in which case it is called an arborescence or out-tree—or m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Computation Tree
A computation tree is a representation for the computation steps of a non-deterministic Turing machine on a specified input.. A computation tree is a rooted tree of nodes and edges. Each node in the tree represents a single computational state, while each edge represents a transition to the next possible computation. The number of nodes of the tree is the size of the tree and the length of the path from the root to a given node is the depth of the node. The largest depth of an output node is the depth of the tree. The leaves of the tree are called output nodes. In a computation tree for a decision problem, each output node is labeled Yes or No. If a tree, T, with an input space X, if x \in X and the path for x ends in node labeled yes, then the input x is accepted. Else it is rejected. The depth of the computation tree for a given input is the computation time for the Turing machine on that input. Computation trees have also been used to study the computational complexity of pro ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
