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Unambiguous Turing Machine
In theoretical computer science, a Turing machine is a theoretical machine that is used in thought experiments to examine the abilities and limitations of computers. An unambiguous Turing machine is a special kind of non-deterministic Turing machine, which, in some sense, is similar to a deterministic Turing machine. Formal definition A ''non-deterministic Turing machine'' is represented formally by a 6-tuple, M=(Q, \Sigma, \iota, \sqcup, A, \delta), as explained in the page ''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'' comp ...''. An ''unambiguous Turing machine'' is a non-deterministic Turing machine such that, for every input w=a_1,a_2,...,a_n, there exists at most one sequence of configurations c_0,c_1,...,c_m with the following conditions: # c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Theoretical Computer Science
Theoretical computer science (TCS) is a subset of general computer science and mathematics that focuses on mathematical aspects of computer science such as the theory of computation, lambda calculus, and type theory. It is difficult to circumscribe the theoretical areas precisely. The Association for Computing Machinery, ACM's ACM SIGACT, Special Interest Group on Algorithms and Computation Theory (SIGACT) provides the following description: History While logical inference and mathematical proof had existed previously, in 1931 Kurt Gödel proved with his incompleteness theorem that there are fundamental limitations on what statements could be proved or disproved. Information theory was added to the field with a 1948 mathematical theory of communication by Claude Shannon. In the same decade, Donald Hebb introduced a mathematical model of Hebbian learning, learning in the brain. With mounting biological data supporting this hypothesis with some modification, the fields of n ... [...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 cells, each of which can hold a single symbol drawn from a finite set of symbols called the 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 and which direction to move is based on a finite table that specifies what to do for each co ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Thought Experiment
A thought experiment is a hypothetical situation in which a hypothesis, theory, or principle is laid out for the purpose of thinking through its consequences. History The ancient Greek ''deiknymi'' (), or thought experiment, "was the most ancient pattern of mathematical proof", and existed before Euclidean mathematics, where the emphasis was on the conceptual, rather than on the experimental part of a thought-experiment. Johann Witt-Hansen established that Hans Christian Ørsted was the first to use the German term ' (lit. thought experiment) circa 1812. Ørsted was also the first to use the equivalent term ' in 1820. By 1883 Ernst Mach used the term ' in a different way, to denote exclusively the conduct of a experiment that would be subsequently performed as a by his students. Physical and mental experimentation could then be contrasted: Mach asked his students to provide him with explanations whenever the results from their subsequent, real, physical experiment differed ... [...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 cur ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
N-tuple
In mathematics, a tuple is a finite ordered list (sequence) of elements. An -tuple is a sequence (or ordered list) of elements, where is a non-negative integer. There is only one 0-tuple, referred to as ''the empty tuple''. An -tuple is defined inductively using the construction of an ordered pair. Mathematicians usually write tuples by listing the elements within parentheses "" and separated by a comma and a space; for example, denotes a 5-tuple. Sometimes other symbols are used to surround the elements, such as square brackets " nbsp; or angle brackets "⟨ ⟩". Braces "" are used to specify arrays in some programming languages but not in mathematical expressions, as they are the standard notation for sets. The term ''tuple'' can often occur when discussing other mathematical objects, such as vectors. In computer science, tuples come in many forms. Most typed functional programming languages implement tuples directly as product types, tightly associated with algeb ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
UP (complexity)
In complexity theory, UP (unambiguous non-deterministic polynomial-time) is the complexity class of decision problems solvable in polynomial time on an unambiguous Turing machine with at most one accepting path for each input. UP contains P and is contained in NP. A common reformulation of NP states that a language is in NP if and only if a given answer can be verified by a deterministic machine in polynomial time. Similarly, a language is in UP if a given answer can be verified in polynomial time, and the verifier machine only accepts at most ''one'' answer for each problem instance. More formally, a language ''L'' belongs to UP if there exists a two-input polynomial-time algorithm ''A'' and a constant ''c'' such that :if x in ''L'' , then there exists a unique certificate ''y'' with , y, = O(, x, ^c) such that :if x is not in ''L'', there is no certificate ''y'' with , y, = O(, x, ^c) such that :algorithm ''A'' verifies ''L'' in polynomial time. UP (and its complement co ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
NP (complexity)
In computational complexity theory, NP (nondeterministic polynomial time) is a complexity class used to classify decision problems. NP is the set of decision problems for which the problem instances, where the answer is "yes", have proofs verifiable in polynomial time by a deterministic Turing machine, or alternatively the set of problems that can be solved in polynomial time by a nondeterministic Turing machine.''Polynomial time'' refers to how quickly the number of operations needed by an algorithm, relative to the size of the problem, grows. It is therefore a measure of efficiency of an algorithm. An equivalent definition of NP is the set of decision problems ''solvable'' in polynomial time by a nondeterministic Turing machine. This definition is the basis for the abbreviation NP; "nondeterministic, polynomial time". These two definitions are equivalent because the algorithm based on the Turing machine consists of two phases, the first of which consists of a guess ab ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
