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Garden Of Eden Theorem
In a cellular automaton, a Garden of Eden is a configuration that has no predecessor. It can be the initial configuration of the automaton but cannot arise in any other way. John Tukey named these configurations after the Garden of Eden in Abrahamic religions, which was created out of nowhere. A Garden of Eden is determined by the state of every cell in the automaton (usually a one- or two-dimensional infinite square lattice of cells). However, for any Garden of Eden there is a finite pattern (a subset of cells and their states, called an ''orphan'') with the same property of having no predecessor, no matter how the remaining cells are filled in. A configuration of the whole automaton is a Garden of Eden if and only if it contains an orphan. For one-dimensional cellular automata, orphans and Gardens of Eden can be found by an efficient algorithm, but for higher dimensions this is an undecidable problem. Nevertheless, computer searches have succeeded in finding these patterns in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Garden Of Eden Pattern
In a cellular automaton, a Garden of Eden is a configuration that has no predecessor. It can be the initial configuration of the automaton but cannot arise in any other way. John Tukey named these configurations after the Garden of Eden in Abrahamic religions, which was created out of nowhere. A Garden of Eden is determined by the state of every cell in the automaton (usually a one- or two-dimensional infinite square lattice of cells). However, for any Garden of Eden there is a finite pattern (a subset of cells and their states, called an ''orphan'') with the same property of having no predecessor, no matter how the remaining cells are filled in. A configuration of the whole automaton is a Garden of Eden if and only if it contains an orphan. For one-dimensional cellular automata, orphans and Gardens of Eden can be found by an efficient algorithm, but for higher dimensions this is an undecidable problem. Nevertheless, computer searches have succeeded in finding these patterns in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nondeterministic Finite Automaton
In automata theory, a finite-state machine is called a deterministic finite automaton (DFA), if * each of its transitions is ''uniquely'' determined by its source state and input symbol, and * reading an input symbol is required for each state transition. A nondeterministic finite automaton (NFA), or nondeterministic finite-state machine, does not need to obey these restrictions. In particular, every DFA is also an NFA. Sometimes the term NFA is used in a narrower sense, referring to an NFA that is ''not'' a DFA, but not in this article. Using the subset construction algorithm, each NFA can be translated to an equivalent DFA; i.e., a DFA recognizing the same formal language. Like DFAs, NFAs only recognize regular languages. NFAs were introduced in 1959 by Michael O. Rabin and Dana Scott, who also showed their equivalence to DFAs. NFAs are used in the implementation of regular expressions: Thompson's construction is an algorithm for compiling a regular expression to an NFA that ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Product Topology
In topology and related areas of mathematics, a product space is the Cartesian product of a family of topological spaces equipped with a natural topology called the product topology. This topology differs from another, perhaps more natural-seeming, topology called the box topology, which can also be given to a product space and which agrees with the product topology when the product is over only finitely many spaces. However, the product topology is "correct" in that it makes the product space a categorical product of its factors, whereas the box topology is too fine; in that sense the product topology is the natural topology on the Cartesian product. Definition Throughout, I will be some non-empty index set and for every index i \in I, let X_i be a topological space. Denote the Cartesian product of the sets X_i by X := \prod X_ := \prod_ X_i and for every index i \in I, denote the i-th by \begin p_i :\;&& \prod_ X_j &&\;\to\; & X_i \\ .3ex && \left(x_j\r ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Discrete Topology
In topology, a discrete space is a particularly simple example of a topological space or similar structure, one in which the points form a , meaning they are '' isolated'' from each other in a certain sense. The discrete topology is the finest topology that can be given on a set. Every subset is open in the discrete topology so that in particular, every singleton subset is an open set in the discrete topology. Definitions Given a set X: A metric space (E,d) is said to be '' uniformly discrete'' if there exists a ' r > 0 such that, for any x,y \in E, one has either x = y or d(x,y) > r. The topology underlying a metric space can be discrete, without the metric being uniformly discrete: for example the usual metric on the set \left\. Properties The underlying uniformity on a discrete metric space is the discrete uniformity, and the underlying topology on a discrete uniform space is the discrete topology. Thus, the different notions of discrete space are compatible with one ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Continuous Function
In mathematics, a continuous function is a function such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. This means that there are no abrupt changes in value, known as '' discontinuities''. More precisely, a function is continuous if arbitrarily small changes in its value can be assured by restricting to sufficiently small changes of its argument. A discontinuous function is a function that is . Up until the 19th century, mathematicians largely relied on intuitive notions of continuity, and considered only continuous functions. The epsilon–delta definition of a limit was introduced to formalize the definition of continuity. Continuity is one of the core concepts of calculus and mathematical analysis, where arguments and values of functions are real and complex numbers. The concept has been generalized to functions between metric spaces and between topological spaces. The latter are the mo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Curtis–Hedlund–Lyndon Theorem
The Curtis–Hedlund–Lyndon theorem is a mathematical characterization of cellular automata in terms of their symbolic dynamics. It is named after Morton L. Curtis, Gustav A. Hedlund, and Roger Lyndon; in his 1969 paper stating the theorem, Hedlund credited Curtis and Lyndon as co-discoverers. It has been called "one of the fundamental results in symbolic dynamics". The theorem states that a function from a shift space to itself represents the transition function of a one-dimensional cellular automaton if and only if it is continuous (with respect to the Cantor topology) and equivariant (with respect to the shift map). More generally, it asserts that the morphisms between any two shift spaces (that is, continuous mappings that commute with the shift) are exactly those mappings which can be defined uniformly by a local rule. The version of the theorem in Hedlund's paper applied only to one-dimensional finite automata, but a generalization to higher dimensional integer lattices wa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bounding Box
In geometry, the minimum or smallest bounding or enclosing box for a point set in dimensions is the box with the smallest measure (area, volume, or hypervolume in higher dimensions) within which all the points lie. When other kinds of measure are used, the minimum box is usually called accordingly, e.g., "minimum-perimeter bounding box". The minimum bounding box of a point set is the same as the minimum bounding box of its convex hull, a fact which may be used heuristically to speed up computation. The terms "box" and "hyperrectangle" come from their usage in the Cartesian coordinate system, where they are indeed visualized as a rectangle (two-dimensional case), rectangular parallelepiped (three-dimensional case), etc. In the two-dimensional case it is called the minimum bounding rectangle. Axis-aligned minimum bounding box The axis-aligned minimum bounding box (or AABB) for a given point set is its minimum bounding box subject to the constraint that the edges of the box are ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Don Woods (programmer)
Donald R. Woods (born April 30, 1954) is an American hacker and computer programmer. He is best known for his role in the development of the ''Colossal Cave Adventure'' game. Biography Early programming career Woods teamed with James M. Lyon while both were attending Princeton in 1972 to produce the unprecedented, excursive INTERCAL programming language. Later, he worked at the Stanford AI lab (SAIL), where among other things he became the SAIL contact for, and a contributor to, the Jargon File. He also co-authored "The Hacker's Dictionary" with Mark Crispin, Raphael Finkel, and Guy L. Steele Jr."The computer contradictionary" by Stan Kelly-Bootle Work on ''Adventure'' Woods discovered the ''Colossal Cave Adventure'' game by accident on a SAIL computer in 1976. After contacting the original author by the (now antiquated) means of sending an e-mail to crowther@''sitename'', where ''sitename'' was every host listed on ARPANET, he heard back from William Crowther shortly afterw ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Honeywell
Honeywell International Inc. is an American publicly traded, multinational conglomerate corporation headquartered in Charlotte, North Carolina. It primarily operates in four areas of business: aerospace, building technologies, performance materials and technologies (PMT), and safety and productivity solutions (SPS). Honeywell is a Fortune 100 company, ranked 94th in 2021. In 2021 the corporation had a global workforce of approximately 99,000 employees, down from 113,000 in 2019. The current chairman and chief executive officer (CEO) is Darius Adamczyk. The corporation's current name, Honeywell International Inc., is a product of the merger of Honeywell Inc. and AlliedSignal in 1999. The corporation headquarters were consolidated with AlliedSignal's headquarters in Morristown, New Jersey; however, the combined company chose the name "Honeywell" because of the considerable brand recognition. Honeywell was a component of the Dow Jones Industrial Average index from 1999 to 200 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Backtracking Search
Backtracking is a class of algorithms for finding solutions to some computational problems, notably constraint satisfaction problems, that incrementally builds candidates to the solutions, and abandons a candidate ("backtracks") as soon as it determines that the candidate cannot possibly be completed to a valid solution. The classic textbook example of the use of backtracking is the eight queens puzzle, that asks for all arrangements of eight chess queens on a standard chessboard so that no queen attacks any other. In the common backtracking approach, the partial candidates are arrangements of ''k'' queens in the first ''k'' rows of the board, all in different rows and columns. Any partial solution that contains two mutually attacking queens can be abandoned. Backtracking can be applied only for problems which admit the concept of a "partial candidate solution" and a relatively quick test of whether it can possibly be completed to a valid solution. It is useless, for example, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Alvy Ray Smith
Alvy Ray Smith III (born September 8, 1943) is an American computer scientist who co-founded Lucasfilm's Computer Division and Pixar, participating in the 1980s and 1990s expansion of computer animation into feature film. Education In 1965, Alvy Smith received his bachelor's degree in electrical engineering from New Mexico State University (NMSU). He created his first computer graphic in 1965 at NMSU. In 1970, he received a Ph.D. in computer science from Stanford University, with a dissertation on cellular automata theory jointly supervised by Michael A. Arbib, Edward J. McCluskey, and Bernard Widrow. Career His first art show was at the Stanford Coffeehouse. From 1969 to 1973 he was an associate professor of Electrical Engineering and Computer Science at New York University, under chairman Herbert Freeman, one of the earliest computer graphics researchers. He taught briefly at the University of California, Berkeley in 1974. While at Xerox PARC in 1974, Smith worked with ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Martin Gardner
Martin Gardner (October 21, 1914May 22, 2010) was an American popular mathematics and popular science writer with interests also encompassing scientific skepticism, micromagic, philosophy, religion, and literatureespecially the writings of Lewis Carroll, L. Frank Baum, and G. K. Chesterton.Martin (2010) He was also a leading authority on Lewis Carroll. ''The Annotated Alice'', which incorporated the text of Carroll's two Alice books, was his most successful work and sold over a million copies. He had a lifelong interest in magic and illusion and in 1999, MAGIC magazine named him as one of the "100 Most Influential Magicians of the Twentieth Century". He was considered the doyen of American puzzlers. He was a prolific and versatile author, publishing more than 100 books. Gardner was best known for creating and sustaining interest in recreational mathematicsand by extension, mathematics in generalthroughout the latter half of the 20th century, principally through his "Mathema ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
