|
Search Problem
In computational complexity theory and computability theory, a search problem is a type of computational problem represented by a binary relation. If ''R'' is a binary relation such that field(''R'') ⊆ Γ+ and ''T'' is a Turing machine, then ''T'' calculates ''R'' if: * If ''x'' is such that there is some ''y'' such that ''R''(''x'', ''y'') then ''T'' accepts ''x'' with output ''z'' such that ''R''(''x'', ''z'') (there may be multiple ''y'', and ''T'' need only find one of them) * If ''x'' is such that there is no ''y'' such that ''R''(''x'', ''y'') then ''T'' rejects ''x'' Intuitively, the problem consists in finding structure "y" in object "x". An algorithm is said to solve the problem if at least one corresponding structure exists, and then one occurrence of this structure is made output; otherwise, the algorithm stops with an appropriate output ("Item not found" or any message of the like). Such problems occur very frequently in graph theory, for example, where searching g ... [...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 relating these classes to each other. 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 gates in a circuit (used in circuit complexity) and the number of processors (used in parallel computing). One of the roles of compu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
State (computer Science)
In information technology and computer science, a system is described as stateful if it is designed to remember preceding events or user interactions; the remembered information is called the state of the system. The set of states a system can occupy is known as its state space. In a discrete system, the state space is countable and often finite. The system's internal behaviour or interaction with its environment consists of separately occurring individual actions or events, such as accepting input or producing output, that may or may not cause the system to change its state. Examples of such systems are digital logic circuits and components, automata and formal language, computer programs, and computers. The output of a digital circuit or deterministic computer program at any time is completely determined by its current inputs and its state. Digital logic circuit state Digital logic circuits can be divided into two types: combinational logic, whose output signals are ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Function Problem
In computational complexity theory, a function problem is a computational problem where a single output (of a total function) is expected for every input, but the output is more complex than that of a decision problem. For function problems, the output is not simply 'yes' or 'no'. Formal definition A functional problem P is defined as a relation R over strings of an arbitrary alphabet \Sigma: : R \subseteq \Sigma^* \times \Sigma^*. An algorithm solves P if for every input x such that there exists a y satisfying (x, y) \in R, the algorithm produces one such y. Examples A well-known function problem is given by the Functional Boolean Satisfiability Problem, FSAT for short. The problem, which is closely related to the SAT decision problem, can be formulated as follows: :Given a boolean formula \varphi with variables x_1, \ldots, x_n, find an assignment x_i \rightarrow \ such that \varphi evaluates to \text or decide that no such assignment exists. In this case the relati ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Counting Problem (complexity)
In computational complexity theory and computability theory, a counting problem is a type of computational problem. If ''R'' is a search problem then :c_R(x)=\vert\\vert \, is the corresponding counting function and :\#R=\ denotes the corresponding decision problem. Note that ''cR'' is a search problem while #''R'' is a decision problem, however ''cR'' can be ''C'' Cook-reduced to #''R'' (for appropriate ''C'') using a binary search (the reason #''R'' is defined the way it is, rather than being the graph of ''cR'', is to make this binary search possible). Counting complexity class If ''NX'' is a complexity class associated with non-deterministic machines then ''#X'' = is the set of counting problems associated with each search problem in ''NX''. In particular, #P is the class of counting problems associated with NP search problems. Just as NP has NP-complete problems via many-one reductions, #P has complete problems via parsimonious reductions, problem transformations ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Optimization Problem
In mathematics, computer science and economics, an optimization problem is the problem of finding the ''best'' solution from all feasible solutions. Optimization problems can be divided into two categories, depending on whether the variables are continuous or discrete: * An optimization problem with discrete variables is known as a '' discrete optimization'', in which an object such as an integer, permutation or graph must be found from a countable set. * A problem with continuous variables is known as a ''continuous optimization'', in which an optimal value from a continuous function must be found. They can include constrained problems and multimodal problems. Continuous optimization problem The '' standard form'' of a continuous optimization problem is \begin &\underset& & f(x) \\ &\operatorname & &g_i(x) \leq 0, \quad i = 1,\dots,m \\ &&&h_j(x) = 0, \quad j = 1, \dots,p \end where * is the objective function to be minimized over the -variable vector , * are called ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Decision Problem
In computability theory and computational complexity theory, a decision problem is a computational problem that can be posed as a yes–no question of the input values. An example of a decision problem is deciding by means of an algorithm whether a given natural number is prime. Another is the problem "given two numbers ''x'' and ''y'', does ''x'' evenly divide ''y''?". The answer is either 'yes' or 'no' depending upon the values of ''x'' and ''y''. A method for solving a decision problem, given in the form of an algorithm, is called a decision procedure for that problem. A decision procedure for the decision problem "given two numbers ''x'' and ''y'', does ''x'' evenly divide ''y''?" would give the steps for determining whether ''x'' evenly divides ''y''. One such algorithm is long division. If the remainder is zero the answer is 'yes', otherwise it is 'no'. A decision problem which can be solved by an algorithm is called ''decidable''. Decision problems typically appear in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Unbounded Search Operator
Boundedness or bounded may refer to: Economics * Bounded rationality, the idea that human rationality in decision-making is bounded by the available information, the cognitive limitations, and the time available to make the decision * Bounded emotionality, a concept within communication theory that stems from emotional labor and bounded rationality Linguistics * Boundedness (linguistics), whether a situation has a clearly defined beginning or end Mathematics * Boundedness axiom, the axiom schema of replacement * Bounded deformation, a function whose distributional derivatives are not quite well-behaved-enough to qualify as functions of bounded variation, although the symmetric part of the derivative matrix does meet that condition * Bounded growth, occurs when the growth rate of a mathematical function is constantly increasing at a decreasing rate * Bounded operator, a linear transformation ''L'' between normed vector spaces for which the ratio of the norm of ''L''(''v'') to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Successor Function
In mathematics, the successor function or successor operation sends a natural number to the next one. The successor function is denoted by ''S'', so ''S''(''n'') = ''n'' +1. For example, ''S''(1) = 2 and ''S''(2) = 3. The successor function is one of the basic components used to build a primitive recursive function. Successor operations are also known as zeration in the context of a zeroth hyperoperation: H0(''a'', ''b'') = 1 + ''b''. In this context, the extension of zeration is addition, which is defined as repeated succession. Overview The successor function is part of the formal language used to state the Peano axioms, which formalise the structure of the natural numbers. In this formalisation, the successor function is a primitive operation on the natural numbers, in terms of which the standard natural numbers and addition is defined. For example, 1 is defined to be ''S''(0), and addition on natural numbers is defined recursively by: : This can be ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Goal State
A goal is an idea of the future or desired result that a person or a group of people envision, plan and commit to achieve. People endeavour to reach goals within a finite time by setting deadlines. A goal is roughly similar to a purpose or aim, the anticipated result which guides reaction, or an end, which is an object, either a physical object or an abstract object, that has intrinsic value. Goal setting Goal-setting theory was formulated based on empirical research and has been called one of the most important theories in organizational psychology. Edwin A. Locke and Gary P. Latham, the fathers of goal-setting theory, provided a comprehensive review of the core findings of the theory in 2002. In summary, Locke and Latham found that specific, difficult goals lead to higher performance than either easy goals or instructions to "do your best", as long as feedback about progress is provided, the person is committed to the goal, and the person has the ability and knowledge to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Start State
A finite-state machine (FSM) or finite-state automaton (FSA, plural: ''automata''), finite automaton, or simply a state machine, is a mathematical model of computation. It is an abstract machine that can be in exactly one of a finite number of '' states'' at any given time. The FSM can change from one state to another in response to some inputs; the change from one state to another is called a ''transition''. An FSM is defined by a list of its states, its initial state, and the inputs that trigger each transition. Finite-state machines are of two types— deterministic finite-state machines and non-deterministic finite-state machines. A deterministic finite-state machine can be constructed equivalent to any non-deterministic one. The behavior of state machines can be observed in many devices in modern society that perform a predetermined sequence of actions depending on a sequence of events with which they are presented. Simple examples are vending machines, which dispense p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Decision Problem
In computability theory and computational complexity theory, a decision problem is a computational problem that can be posed as a yes–no question of the input values. An example of a decision problem is deciding by means of an algorithm whether a given natural number is prime. Another is the problem "given two numbers ''x'' and ''y'', does ''x'' evenly divide ''y''?". The answer is either 'yes' or 'no' depending upon the values of ''x'' and ''y''. A method for solving a decision problem, given in the form of an algorithm, is called a decision procedure for that problem. A decision procedure for the decision problem "given two numbers ''x'' and ''y'', does ''x'' evenly divide ''y''?" would give the steps for determining whether ''x'' evenly divides ''y''. One such algorithm is long division. If the remainder is zero the answer is 'yes', otherwise it is 'no'. A decision problem which can be solved by an algorithm is called ''decidable''. Decision problems typically appear in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Computability Theory
Computability theory, also known as recursion theory, is a branch of mathematical logic, computer science, and the theory of computation that originated in the 1930s with the study of computable functions and Turing degrees. The field has since expanded to include the study of generalized computability and definability. In these areas, computability theory overlaps with proof theory and effective descriptive set theory. Basic questions addressed by computability theory include: * What does it mean for a function on the natural numbers to be computable? * How can noncomputable functions be classified into a hierarchy based on their level of noncomputability? Although there is considerable overlap in terms of knowledge and methods, mathematical computability theorists study the theory of relative computability, reducibility notions, and degree structures; those in the computer science field focus on the theory of subrecursive hierarchies, formal methods, and formal languages ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
