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Universal Turing Machine
In computer science, a universal Turing machine (UTM) is a Turing machine that can simulate an arbitrary Turing machine on arbitrary input. The universal machine essentially achieves this by reading both the description of the machine to be simulated as well as the input to that machine from its own tape. Alan Turing introduced the idea of such a machine in 1936–1937. This principle is considered to be the origin of the idea of a stored-program computer used by John von Neumann in 1946 for the "Electronic Computing Instrument" that now bears von Neumann's name: the von Neumann architecture. Martin Davis, ''The universal computer : the road from Leibniz to Turing'' (2017) In terms of computational complexity, a multi-tape universal Turing machine need only be slower by logarithmic factor compared to the machines it simulates. Introduction Every Turing machine computes a certain fixed partial computable function from the input strings over its alphabet. In that sense it be ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Computer Science
Computer science is the study of computation, automation, and information. Computer science spans theoretical disciplines (such as algorithms, theory of computation, information theory, and automation) to Applied science, practical disciplines (including the design and implementation of Computer architecture, hardware and Computer programming, software). Computer science is generally considered an area of research, academic research and distinct from computer programming. Algorithms and data structures are central to computer science. The theory of computation concerns abstract models of computation and general classes of computational problem, problems that can be solved using them. The fields of cryptography and computer security involve studying the means for secure communication and for preventing Vulnerability (computing), security vulnerabilities. Computer graphics (computer science), Computer graphics and computational geometry address the generation of images. Progr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Microcode
In processor design, microcode (μcode) is a technique that interposes a layer of computer organization between the central processing unit (CPU) hardware and the programmer-visible instruction set architecture of a computer. Microcode is a layer of hardware-level instructions that implement higher-level machine code instructions or internal finite-state machine sequencing in many digital processing elements. Microcode is used in general-purpose central processing units, although in current desktop CPUs, it is only a fallback path for cases that the faster hardwired control unit cannot handle. Microcode typically resides in special high-speed memory and translates machine instructions, state machine data, or other input into sequences of detailed circuit-level operations. It separates the machine instructions from the underlying electronics so that instructions can be designed and altered more freely. It also facilitates the building of complex multi-step instructions, while red ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Recursively Enumerable Language
In mathematics, logic and computer science, a formal language is called recursively enumerable (also recognizable, partially decidable, semidecidable, Turing-acceptable or Turing-recognizable) if it is a recursively enumerable subset in the set of all possible words over the alphabet of the language, i.e., if there exists a Turing machine which will enumerate all valid strings of the language. Recursively enumerable languages are known as type-0 languages in the Chomsky hierarchy of formal languages. All regular, context-free, context-sensitive and recursive languages are recursively enumerable. The class of all recursively enumerable languages is called RE. Definitions There are three equivalent definitions of a recursively enumerable language: # A recursively enumerable language is a recursively enumerable subset in the set of all possible words over the alphabet of the language. # A recursively enumerable language is a formal language for which there exists a Turing mach ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Recursive Language
In mathematics, logic and computer science, a formal language (a set of finite sequences of symbols taken from a fixed alphabet) is called recursive if it is a recursive subset of the set of all possible finite sequences over the alphabet of the language. Equivalently, a formal language is recursive if there exists a total Turing machine (a Turing machine that halts for every given input) that, when given a finite sequence of symbols as input, accepts it if it belongs to the language and rejects it otherwise. Recursive languages are also called decidable. The concept of decidability may be extended to other models of computation. For example, one may speak of languages decidable on a non-deterministic Turing machine. Therefore, whenever an ambiguity is possible, the synonym used for "recursive language" is Turing-decidable language, rather than simply ''decidable''. The class of all recursive languages is often called R, although this name is also used for the class RP. This ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rice's Theorem
In computability theory, Rice's theorem states that all non-trivial semantic properties of programs are undecidable. A semantic property is one about the program's behavior (for instance, does the program terminate for all inputs), unlike a syntactic property (for instance, does the program contain an if-then-else statement). A property is ''non-trivial'' if it is neither true for every partial computable function, nor false for every partial computable function. Rice's theorem can also be put in terms of functions: for any non-trivial property of partial functions, no general and effective method can decide whether an algorithm computes a partial function with that property. Here, a property of partial functions is called ''trivial'' if it holds for all partial computable functions or for none, and an effective decision method is called ''general'' if it decides correctly for every algorithm. The theorem is named after Henry Gordon Rice, who proved it in his doctoral dissertation ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Halting Problem
In computability theory, the halting problem is the problem of determining, from a description of an arbitrary computer program and an input, whether the program will finish running, or continue to run forever. Alan Turing proved in 1936 that a general algorithm to solve the halting problem for all possible program–input pairs cannot exist. For any program that might determine whether programs halt, a "pathological" program , called with some input, can pass its own source and its input to ''f'' and then specifically do the opposite of what ''f'' predicts ''g'' will do. No ''f'' can exist that handles this case. A key part of the proof is a mathematical definition of a computer and program, which is known as a Turing machine; the halting problem is '' undecidable'' over Turing machines. It is one of the first cases of decision problems proven to be unsolvable. This proof is significant to practical computing efforts, defining a class of applications which no programming inventi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Undecidable Problem
In computability theory and computational complexity theory, an undecidable problem is a decision problem for which it is proved to be impossible to construct an algorithm that always leads to a correct yes-or-no answer. The halting problem is an example: it can be proven that there is no algorithm that correctly determines whether arbitrary programs eventually halt when run. Background A decision problem is any arbitrary yes-or-no question on an infinite set of inputs. Because of this, it is traditional to define the decision problem equivalently as the set of inputs for which the problem returns ''yes''. These inputs can be natural numbers, but also other values of some other kind, such as strings of a formal language. Using some encoding, such as a Gödel numbering, the strings can be encoded as natural numbers. Thus, a decision problem informally phrased in terms of a formal language is also equivalent to a set of natural numbers. To keep the formal definition simple, it is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Register Machine
In mathematical logic and theoretical computer science a register machine is a generic class of abstract machines used in a manner similar to a Turing machine. All the models are Turing equivalent. Overview The register machine gets its name from its use of one or more " registers". In contrast to the tape and head used by a Turing machine, the model uses multiple, uniquely addressed registers, each of which holds a single positive integer. There are at least four sub-classes found in literature, here listed from most primitive to the most like a computer: * Counter machine – the most primitive and reduced theoretical model of a computer hardware. Lacks indirect addressing. Instructions are in the finite state machine in the manner of the Harvard architecture. *Pointer machine – a blend of counter machine and RAM models. Less common and more abstract than either model. Instructions are in the finite state machine in the manner of the Harvard architecture. *Random-access mach ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Turing Machine Equivalents
A Turing machine is a hypothetical computing device, first conceived by Alan Turing in 1936. Turing machines manipulate symbols on a potentially infinite strip of tape according to a finite table of rules, and they provide the theoretical underpinnings for the notion of a computer algorithm. While none of the following models have been shown to have more power than the single-tape, one-way infinite, multi-symbol Turing-machine model, their authors defined and used them to investigate questions and solve problems more easily than they could have if they had stayed with Turing's ''a''-machine model. Machines equivalent to the Turing machine model Turing equivalence Many machines that might be thought to have more computational capability than a simple universal Turing machine can be shown to have no more power. They might compute faster, perhaps, or use less memory, or their instruction set might be smaller, but they cannot compute more powerfully (i.e. more mathematical funct ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Counter Machine
A counter machine is an abstract machine used in a formal logic and theoretical computer science to model computation. It is the most primitive of the four types of register machines. A counter machine comprises a set of one or more unbounded ''registers'', each of which can hold a single non-negative integer, and a list of (usually sequential) arithmetic and control instructions for the machine to follow. The counter machine is typically used in the process of designing parallel algorithms in relation to the mutual exclusion principle. When used in this manner, the counter machine is used to model the discrete time-steps of a computational system in relation to memory accesses. By modeling computations in relation to the memory accesses for each respective computational step, parallel algorithms may be designed in such a matter to avoid interlocking, the simultaneous writing operation by two (or more) threads to the same memory address. Basic features For a given counter machin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Turing Completeness
In computability theory, a system of data-manipulation rules (such as a computer's instruction set, a programming language, or a cellular automaton) is said to be Turing-complete or computationally universal if it can be used to simulate any Turing machine (devised by English mathematician and computer scientist Alan Turing). This means that this system is able to recognize or decide other data-manipulation rule sets. Turing completeness is used as a way to express the power of such a data-manipulation rule set. Virtually all programming languages today are Turing-complete. A related concept is that of Turing equivalence two computers P and Q are called equivalent if P can simulate Q and Q can simulate P. The Church–Turing thesis conjectures that any function whose values can be computed by an algorithm can be computed by a Turing machine, and therefore that if any real-world computer can simulate a Turing machine, it is Turing equivalent to a Turing machine. A universal Turin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hao Wang (academic)
Hao Wang (; 20 May 1921 – 13 May 1995) was a Chinese-American logician, philosopher, mathematician, and commentator on Kurt Gödel. Biography Born in Jinan, Shandong, in the Republic of China (today in the People's Republic of China), Wang received his early education in China. He obtained a BSc degree in mathematics from the National Southwestern Associated University in 1943 and an M.A. in Philosophy from Tsinghua University in 1945, where his teachers included Feng Youlan and Jin Yuelin, after which he moved to the United States for further graduate studies. He studied logic under W.V. Quine at Harvard University, culminating in a Ph.D. in 1948. He was appointed to an assistant professorship at Harvard the same year. During the early 1950s, Wang studied with Paul Bernays in Zürich. In 1956, he was appointed Reader in the Philosophy of Mathematics at the University of Oxford. In 1959, Wang wrote on an IBM 704 computer a program that in only 9 minutes mechanically pr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
