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Second-order Cellular Automaton
A second-order cellular automaton is a type of reversible cellular automaton (CA) invented by Edward Fredkin. Reprinted in . where the state of a cell at time depends not only on its neighborhood at time , but also on its state at time .. General technique In general, the evolution rule for a second-order automaton may be described as a function that maps the neighborhood of a cell to a permutation on the states of the automaton. In each time step , for each cell of the automaton, this function is applied to the neighborhood of to give a permutation . Then, this permutation is applied to the state of cell at time , and the result is the state of the cell at time . In this way, the configuration of the automaton at each time step is computed from two previous time steps: the immediately previous step determines the permutations that are applied to the cells, and the time step before that one gives the states on which these permutations operate. The reversed time dynamics ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Reversible Cellular Automaton
A reversible cellular automaton is a cellular automaton in which every configuration has a unique predecessor. That is, it is a regular grid of cells, each containing a state drawn from a finite set of states, with a rule for updating all cells simultaneously based on the states of their neighbors, such that the previous state of any cell before an update can be determined uniquely from the updated states of all the cells. The time-reversed dynamics of a reversible cellular automaton can always be described by another cellular automaton rule, possibly on a much larger neighborhood. Several methods are known for defining cellular automata rules that are reversible; these include the block cellular automaton method, in which each update partitions the cells into blocks and applies an invertible function separately to each block, and the second-order cellular automaton method, in which the update rule combines states from two previous steps of the automaton. When an automaton is not ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Edward Fredkin
Edward Fredkin (born October 2, 1934) is a distinguished career professor at Carnegie Mellon University (CMU), and an early pioneer of digital physics. Fredkin's primary contributions include work on reversible computing and cellular automata. While Konrad Zuse's book, ''Calculating Space'' (1969), mentioned the importance of reversible computation, the Fredkin gate represented the essential breakthrough. In recent work, he uses the term ''digital philosophy'' (DP). During his career, Fredkin has been a professor of computer science at the Massachusetts Institute of Technology, a Fairchild Distinguished Scholar at Caltech, and Research Professor of Physics at Boston University. Early life and education At age 19, Fredkin left California Institute of Technology (Caltech) after a year to join the United States Air Force (USAF) to become a fighter pilot. Fredkin’s computer career started in 1956 when the air force assigned him to MIT Lincoln Laboratory where we worked on the SA ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
A New Kind Of Science
''A New Kind of Science'' is a book by Stephen Wolfram, published by his company Wolfram Research under the imprint Wolfram Media in 2002. It contains an empirical and systematic study of computational systems such as cellular automata. Wolfram calls these systems ''simple programs'' and argues that the scientific philosophy and methods appropriate for the study of simple programs are relevant to other fields of science. Contents Computation and its implications The thesis of ''A New Kind of Science'' (''NKS'') is twofold: that the nature of computation must be explored experimentally, and that the results of these experiments have great relevance to understanding the physical world. Simple programs The basic subject of Wolfram's "new kind of science" is the study of simple abstract rules—essentially, elementary computer programs. In almost any class of a computational system, one very quickly finds instances of great complexity among its simplest cases (after a time serie ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Permutation
In mathematics, a permutation of a set is, loosely speaking, an arrangement of its members into a sequence or linear order, or if the set is already ordered, a rearrangement of its elements. The word "permutation" also refers to the act or process of changing the linear order of an ordered set. Permutations differ from combinations, which are selections of some members of a set regardless of order. For example, written as tuples, there are six permutations of the set , namely (1, 2, 3), (1, 3, 2), (2, 1, 3), (2, 3, 1), (3, 1, 2), and (3, 2, 1). These are all the possible orderings of this three-element set. Anagrams of words whose letters are different are also permutations: the letters are already ordered in the original word, and the anagram is a reordering of the letters. The study of permutations of finite sets is an important topic in the fields of combinatorics and group theory. Permutations are used in almost every branch of mathematics, and in many other fields of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Identity Permutation
In mathematics, a permutation group is a group ''G'' whose elements are permutations of a given set ''M'' and whose group operation is the composition of permutations in ''G'' (which are thought of as bijective functions from the set ''M'' to itself). The group of ''all'' permutations of a set ''M'' is the symmetric group of ''M'', often written as Sym(''M''). The term ''permutation group'' thus means a subgroup of the symmetric group. If then Sym(''M'') is usually denoted by S''n'', and may be called the ''symmetric group on n letters''. By Cayley's theorem, every group is isomorphic to some permutation group. The way in which the elements of a permutation group permute the elements of the set is called its group action. Group actions have applications in the study of symmetries, combinatorics and many other branches of mathematics, physics and chemistry. Basic properties and terminology Being a subgroup of a symmetric group, all that is necessary for a set of permuta ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Boolean Value
In mathematics and mathematical logic, Boolean algebra is a branch of algebra. It differs from elementary algebra in two ways. First, the values of the variables are the truth values ''true'' and ''false'', usually denoted 1 and 0, whereas in elementary algebra the values of the variables are numbers. Second, Boolean algebra uses logical operators such as conjunction (''and'') denoted as ∧, disjunction (''or'') denoted as ∨, and the negation (''not'') denoted as ¬. Elementary algebra, on the other hand, uses arithmetic operators such as addition, multiplication, subtraction and division. So Boolean algebra is a formal way of describing logical operations, in the same way that elementary algebra describes numerical operations. Boolean algebra was introduced by George Boole in his first book ''The Mathematical Analysis of Logic'' (1847), and set forth more fully in his '' An Investigation of the Laws of Thought'' (1854). According to Huntington, the term "Boolean algebra" ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Exclusive Or
Exclusive or or exclusive disjunction is a logical operation that is true if and only if its arguments differ (one is true, the other is false). It is symbolized by the prefix operator J and by the infix operators XOR ( or ), EOR, EXOR, , , , , , and . The negation of XOR is the logical biconditional, which yields true if and only if the two inputs are the same. It gains the name "exclusive or" because the meaning of "or" is ambiguous when both operands are true; the exclusive or operator ''excludes'' that case. This is sometimes thought of as "one or the other but not both". This could be written as "A or B, but not, A and B". Since it is associative, it may be considered to be an ''n''-ary operator which is true if and only if an odd number of arguments are true. That is, ''a'' XOR ''b'' XOR ... may be treated as XOR(''a'',''b'',...). Truth table The truth table of A XOR B shows that it outputs true whenever the inputs differ: Equivalences, elimination, and introd ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stephen Wolfram
Stephen Wolfram (; born 29 August 1959) is a British-American computer scientist, physicist, and businessman. He is known for his work in computer science, mathematics, and theoretical physics. In 2012, he was named a fellow of the American Mathematical Society. He is currently an adjunct professor at the University of Illinois Department of Computer Science. As a businessman, he is the founder and CEO of the software company Wolfram Research where he works as chief designer of Mathematica and the Wolfram Alpha answer engine. Early life Family Stephen Wolfram was born in London in 1959 to Hugo and Sybil Wolfram, both German Jewish refugees to the United Kingdom. His maternal grandmother was British psychoanalyst Kate Friedlander. Wolfram's father, Hugo Wolfram, was a textile manufacturer and served as managing director of the Lurex Company—makers of the fabric Lurex. [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Wolfram Code
Wolfram code is a widely used numbering system for one-dimensional cellular automaton rules, introduced by Stephen Wolfram in a 1983 paper and popularized in his book '' A New Kind of Science''. The code is based on the observation that a table specifying the new state of each cell in the automaton, as a function of the states in its neighborhood, may be interpreted as a ''k''-digit number in the ''S''-ary positional number system, where ''S'' is the number of states that each cell in the automaton may have, ''k'' = S2''n'' + 1 is the number of neighborhood configurations, and ''n'' is the radius of the neighborhood. Thus, the Wolfram code for a particular rule is a number in the range from 0 to ''S''''S'' − 1, converted from ''S''-ary to decimal notation. It may be calculated as follows: # List all the ''S''2''n'' + 1 possible state configurations of the neighbourhood of a given cell. # Interpreting each configuration as a number as d ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Billiard-ball Computer
A billiard-ball computer, a type of conservative logic circuit, is an idealized model of a reversible mechanical computer based on Newtonian dynamics, proposed in 1982 by Edward Fredkin and Tommaso Toffoli. Instead of using electronic signals like a conventional computer, it relies on the motion of spherical billiard balls in a friction-free environment made of buffers against which the balls bounce perfectly. It was devised to investigate the relation between computation and reversible processes in physics. Simulating circuits with billiard balls This model can be used to simulate Boolean circuits in which the wires of the circuit correspond to paths on which one of the balls may travel, the signal on a wire is encoded by the presence or absence of a ball on that path, and the gates of the circuit are simulated by collisions of balls at points where their paths cross. In particular, it is possible to set up the paths of the balls and the buffers around them to form a revers ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
