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Cyclic Tag System
A tag system is a deterministic computational model published by Emil Leon Post in 1943 as a simple form of a Post canonical system. A tag system may also be viewed as an abstract machine, called a Post tag machine (not to be confused with Post–Turing machines)—briefly, a finite-state machine whose only tape is a FIFO queue of unbounded length, such that in each transition the machine reads the symbol at the head of the queue, deletes a constant number of symbols from the head, and appends to the tail a symbol-string that depends solely on the first symbol read in this transition. Because all of the indicated operations are performed in a single transition, a tag machine strictly has only one state. Definitions A ''tag system'' is a triplet (''m'', ''A'', ''P''), where * ''m'' is a positive integer, called the ''deletion number''. * ''A'' is a finite alphabet of symbols, one of which is a special ''halting symbol''. All finite (possibly empty) strings on ''A'' are called ' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Turing-complete
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 Turi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Computational Model
A computational model uses computer programs to simulate and study complex systems using an algorithmic or mechanistic approach and is widely used in a diverse range of fields spanning from physics, chemistry and biology to economics, psychology, cognitive science and computer science. The system under study is often a complex nonlinear system for which simple, intuitive analytical solutions are not readily available. Rather than deriving a mathematical analytical solution to the problem, experimentation with the model is done by adjusting the parameters of the system in the computer, and studying the differences in the outcome of the experiments. Operation theories of the model can be derived/deduced from these computational experiments. Examples of common computational models are weather forecasting models, earth simulator models, flight simulator models, molecular protein folding models, and neural network models. See also * Computational cognition *Reversible computing ... [...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] |
Emil Post
Emil Leon Post (; February 11, 1897 – April 21, 1954) was an American mathematician and logician. He is best known for his work in the field that eventually became known as computability theory. Life Post was born in Augustów, Suwałki Governorate, Congress Poland, Russian Empire (now Poland) into a Polish-Jewish family that immigrated to New York City in May 1904. His parents were Arnold and Pearl Post. Post had been interested in astronomy, but at the age of twelve lost his left arm in a car accident. This loss was a significant obstacle to being a professional astronomer, leading to his decision to pursue mathematics rather than astronomy. Post attended the Townsend Harris High School and continued on to graduate from City College of New York in 1917 with a B.S. in Mathematics. After completing his Ph.D. in mathematics in 1920 at Columbia University, supervised by Cassius Jackson Keyser, he did a post-doctorate at Princeton University in the 1920–1921 academic year. Pos ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Marvin Minsky
Marvin Lee Minsky (August 9, 1927 – January 24, 2016) was an American cognitive and computer scientist concerned largely with research of artificial intelligence (AI), co-founder of the Massachusetts Institute of Technology's AI laboratory, and author of several texts concerning AI and philosophy. Minsky received many accolades and honors, including the 1969 Turing Award. Biography Marvin Lee Minsky was born in New York City, to an eye surgeon father, Henry, and to a mother, Fannie (Reiser), who was a Zionist activist. His family was Jewish. He attended the Ethical Culture Fieldston School and the Bronx High School of Science. He later attended Phillips Academy in Andover, Massachusetts. He then served in the US Navy from 1944 to 1945. He received a B.A. in mathematics from Harvard University in 1950 and a Ph.D. in mathematics from Princeton University in 1954. His doctoral dissertation was titled "Theory of neural-analog reinforcement systems and its application to the brain- ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Queue Automaton
A queue machine, queue automaton, or pullup automaton (PUA) is a finite state machine with the ability to store and retrieve data from an infinite-memory queue. It is a model of computation equivalent to a Turing machine, and therefore it can process the same class of formal languages. Theory A queue machine can be defined as a six-tuple :M = (Q, \Sigma, \Gamma, \$, s, \delta) where * \,Q is a finite set of ''states''; * \,\Sigma \subset \Gamma is the finite set of the ''input alphabet''; * \,\Gamma is the finite ''queue alphabet''; * \,\$ \in \Gamma \setminus \Sigma is the ''initial queue symbol''; * \,s \in Q is the ''start state''; * \,\delta : Q \times \Gamma \rightarrow Q \times \Gamma^* is the ''transition function''. A machine ''configuration'' is an ordered pair of its state and queue contents \,(q,\gamma)\in Q\times\Gamma^*, where \,\Gamma^* denotes the Kleene closure of \,\Gamma. The starting configuration on an input string \,x is defined as \,(s,x\$), and the transi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rule 110 Cellular Automaton
The Rule 110 cellular automaton (often called simply Rule 110) is an elementary cellular automaton with interesting behavior on the boundary between stability and chaos. In this respect, it is similar to Conway's Game of Life. Like Life, Rule 110 with a particular repeating background pattern is known to be Turing complete. This implies that, in principle, any calculation or computer program can be simulated using this automaton. Definition In an elementary cellular automaton, a one-dimensional pattern of 0s and 1s evolves according to a simple set of rules. Whether a point in the pattern will be 0 or 1 in the new generation depends on its current value, as well as on those of its two neighbors. The Rule 110 automaton has the following set of rules: The name "Rule 110" derives from the fact that this rule can be summarized in the binary sequence 01101110; interpreted as a binary number, this corresponds to the decimal value 110. History In 2004, Matthew Cook published a pr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Matthew Cook
Matthew Cook (born February 7, 1970) is a mathematician and computer scientist who is best known for having proved Stephen Wolfram's conjecture that the Rule 110 cellular automaton is Turing-complete. Biography Cook was born in Morgantown, West Virginia and grew up in Evanston, Illinois. He is an alumnus of the Hampshire College Summer Studies in Mathematics program and completed his undergraduate studies at the University of Illinois and the Budapest Semesters in Mathematics program. In 1987, Cook qualified as a member of the six-person US team to the International Mathematical Olympiad and won a bronze medal. In 1990, Cook went to work for Wolfram Research, makers of the computer algebra system Mathematica. He did his doctoral work in Computation and Neural Systems at Caltech from 1999 to 2005. He is now at the Institute of Neuroinformatics at Zurich in Switzerland. Work with Stephen Wolfram In the 1990s Cook worked as a research assistant to Stephen Wolfram, assisting with wor ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tag (game)
Tag (also called touch and go AG'', tig, it, tiggy, tips, tick, tip) is a playground game involving two or more players chasing other players in an attempt to "tag" and mark them out of play, usually by touching with a hand. There are many variations; most forms have no teams, scores, or equipment. Usually, when a person is tagged, the tagger says, "Tag, you're 'it'!" The last one tagged during tag is "it" for the next round. The game is known by other names in various parts of the world, including "running and catching" in India and "catch and cook" in the Middle East. Basic rules Players (two or more) decide who is going to be "it", often using a counting-out game such as eeny, meeny, miny, moe. The player selected to be "it" then chases the others, attempting to "tag" one of them (by touching them with a hand) as the others try to avoid being tagged. A tag makes the tagged player "it". In some variations, the previous "it" is no longer "it" and the game can continue indefini ... [...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 mat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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] |
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] |
