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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] |
Characterization (mathematics)
In mathematics, a characterization of an object is a set of conditions that, while different from the definition of the object, is logically equivalent to it. To say that "Property ''P'' characterizes object ''X''" is to say that not only does ''X'' have property ''P'', but that ''X'' is the ''only'' thing that has property ''P'' (i.e., ''P'' is a defining property of ''X''). Similarly, a set of properties ''P'' is said to characterize ''X'', when these properties distinguish ''X'' from all other objects. Even though a characterization identifies an object in a unique way, several characterizations can exist for a single object. Common mathematical expressions for a characterization of ''X'' in terms of ''P'' include "''P'' is necessary and sufficient for ''X''", and "''X'' holds if and only if ''P''". It is also common to find statements such as "Property ''Q'' characterizes ''Y'' up to isomorphism". The first type of statement says in different words that the extension of ''P'' is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cellular Automaton
A cellular automaton (pl. cellular automata, abbrev. CA) is a discrete model of computation studied in automata theory. Cellular automata are also called cellular spaces, tessellation automata, homogeneous structures, cellular structures, tessellation structures, and iterative arrays. Cellular automata have found application in various areas, including physics, theoretical biology and microstructure modeling. A cellular automaton consists of a regular grid of ''cells'', each in one of a finite number of '' states'', such as ''on'' and ''off'' (in contrast to a coupled map lattice). The grid can be in any finite number of dimensions. For each cell, a set of cells called its ''neighborhood'' is defined relative to the specified cell. An initial state (time ''t'' = 0) is selected by assigning a state for each cell. A new ''generation'' is created (advancing ''t'' by 1), according to some fixed ''rule'' (generally, a mathematical function) that determines the new state of e ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Theorems In Discrete Mathematics
In mathematics, a theorem is a statement that has been proved, or can be proved. The ''proof'' of a theorem is a logical argument that uses the inference rules of a deductive system to establish that the theorem is a logical consequence of the axioms and previously proved theorems. In the mainstream of mathematics, the axioms and the inference rules are commonly left implicit, and, in this case, they are almost always those of Zermelo–Fraenkel set theory with the axiom of choice, or of a less powerful theory, such as Peano arithmetic. A notable exception is Wiles's proof of Fermat's Last Theorem, which involves the Grothendieck universes whose existence requires the addition of a new axiom to the set theory. Generally, an assertion that is explicitly called a theorem is a proved result that is not an immediate consequence of other known theorems. Moreover, many authors qualify as ''theorems'' only the most important results, and use the terms ''lemma'', ''proposition'' and '' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Surjunctive Group
In mathematics, a surjunctive group is a group such that every injective cellular automaton with the group elements as its cells is also surjective. Surjunctive groups were introduced by . It is unknown whether every group is surjunctive. Definition A cellular automaton consists of a regular system of cells, each containing a symbol from a finite alphabet, together with a uniform rule called a ''transition function'' for updating all cells simultaneously based on the values of neighboring cells. Most commonly the cells are arranged in the form of a line or a higher-dimensional integer grid, but other arrangements of cells are also possible. What is required of the cells is that they form a structure in which every cell "looks the same as" every other cell: there is a symmetry of both the arrangement of cells and the rule set that takes any cell to any other cell. Mathematically, this can be formalized by the notion of a group, a set of elements together with an associative and inv ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Developments In Language Theory
DLT, the International Conference on Developments in Language Theory is an academic conference in the field of computer science held annually under the auspices of the European Association for Theoretical Computer Science. Like most theoretical computer science conferences its contributions are strongly peer-reviewed; the articles appear in proceedings published in Springer Lecture Notes in Computer Science. Extended versions of selected papers of each year's conference appear in international journals, such as Theoretical Computer Science and International Journal of Foundations of Computer Science. Topics of the conference Typical topics include: * grammars, acceptors and transducers for words, trees and graphs * algebraic theories of automata * algorithmic, combinatorial and algebraic properties of words and languages * variable length codes * symbolic dynamics * cellular automata * polyominoes and multidimensional patterns * decidability questions * image manipulation and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Group Action (mathematics)
In mathematics, a group action on a space is a group homomorphism of a given group into the group of transformations of the space. Similarly, a group action on a mathematical structure is a group homomorphism of a group into the automorphism group of the structure. It is said that the group ''acts'' on the space or structure. If a group acts on a structure, it will usually also act on objects built from that structure. For example, the group of Euclidean isometries acts on Euclidean space and also on the figures drawn in it. For example, it acts on the set of all triangles. Similarly, the group of symmetries of a polyhedron acts on the vertices, the edges, and the faces of the polyhedron. A group action on a vector space is called a representation of the group. In the case of a finite-dimensional vector space, it allows one to identify many groups with subgroups of , the group of the invertible matrices of dimension over a field . The symmetric group acts on any set wit ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tychonoff's Theorem
In mathematics, Tychonoff's theorem states that the product of any collection of compact topological spaces is compact with respect to the product topology. The theorem is named after Andrey Nikolayevich Tikhonov (whose surname sometimes is transcribed ''Tychonoff''), who proved it first in 1930 for powers of the closed unit interval and in 1935 stated the full theorem along with the remark that its proof was the same as for the special case. The earliest known published proof is contained in a 1935 article of Tychonoff, A., "Uber einen Funktionenraum", Mathematical Annals, 111, pp. 762–766 (1935). (This reference is mentioned in "Topology" by Hocking and Young, Dover Publications, Ind.) Tychonoff's theorem is often considered as perhaps the single most important result in general topology (along with Urysohn's lemma). The theorem is also valid for topological spaces based on fuzzy sets.Joseph Goguen, "The Fuzzy Tychonoff Theorem", Journal of Mathematical Analysis and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Compact Space
In mathematics, specifically general topology, compactness is a property that seeks to generalize the notion of a closed and bounded subset of Euclidean space by making precise the idea of a space having no "punctures" or "missing endpoints", i.e. that the space not exclude any ''limiting values'' of points. For example, the open interval (0,1) would not be compact because it excludes the limiting values of 0 and 1, whereas the closed interval ,1would be compact. Similarly, the space of rational numbers \mathbb is not compact, because it has infinitely many "punctures" corresponding to the irrational numbers, and the space of real numbers \mathbb is not compact either, because it excludes the two limiting values +\infty and -\infty. However, the ''extended'' real number line ''would'' be compact, since it contains both infinities. There are many ways to make this heuristic notion precise. These ways usually agree in a metric space, but may not be equivalent in other topologic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Open Cover
In mathematics, and more particularly in set theory, a cover (or covering) of a set X is a collection of subsets of X whose union is all of X. More formally, if C = \lbrace U_\alpha : \alpha \in A \rbrace is an indexed family of subsets U_\alpha\subset X, then C is a cover of X if \bigcup_U_ = X. Thus the collection \lbrace U_\alpha : \alpha \in A \rbrace is a cover of X if each element of X belongs to at least one of the subsets U_. Cover in topology Covers are commonly used in the context of topology. If the set X is a topological space, then a ''cover'' C of X is a collection of subsets \_ of X whose union is the whole space X. In this case we say that C ''covers'' X, or that the sets U_\alpha ''cover'' X. Also, if Y is a (topological) subspace of X, then a ''cover'' of Y is a collection of subsets C=\_ of X whose union contains Y, i.e., C is a cover of Y if :Y \subseteq \bigcup_U_. That is, we may cover Y with either open sets in Y itself, or cover Y by open sets in the p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Equivariant
In mathematics, equivariance is a form of symmetry for functions from one space with symmetry to another (such as symmetric spaces). A function is said to be an equivariant map when its domain and codomain are acted on by the same symmetry group, and when the function commutes with the action of the group. That is, applying a symmetry transformation and then computing the function produces the same result as computing the function and then applying the transformation. Equivariant maps generalize the concept of invariants, functions whose value is unchanged by a symmetry transformation of their argument. The value of an equivariant map is often (imprecisely) called an invariant. In statistical inference, equivariance under statistical transformations of data is an important property of various estimation methods; see invariant estimator for details. In pure mathematics, equivariance is a central object of study in equivariant topology and its subtopics equivariant cohomology and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Shift Map
In mathematics, and in particular functional analysis, the shift operator also known as translation operator is an operator that takes a function to its translation . In time series analysis, the shift operator is called the lag operator. Shift operators are examples of linear operators, important for their simplicity and natural occurrence. The shift operator action on functions of a real variable plays an important role in harmonic analysis, for example, it appears in the definitions of almost periodic functions, positive-definite functions, derivatives, and convolution. Shifts of sequences (functions of an integer variable) appear in diverse areas such as Hardy spaces, the theory of abelian varieties, and the theory of symbolic dynamics, for which the baker's map is an explicit representation. Definition Functions of a real variable The shift operator (where ) takes a function on R to its translation , : T^t f(x) = f_t(x) = f(x+t)~. A practical operational calculus ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Function (mathematics)
In mathematics, a function from a set to a set assigns to each element of exactly one element of .; the words map, mapping, transformation, correspondence, and operator are often used synonymously. The set is called the domain of the function and the set is called the codomain of the function.Codomain ''Encyclopedia of Mathematics'Codomain. ''Encyclopedia of Mathematics''/ref> The earliest known approach to the notion of function can be traced back to works of Persian mathematicians Al-Biruni and Sharaf al-Din al-Tusi. Functions were originally the idealization of how a varying quantity depends on another quantity. For example, the position of a planet is a ''function'' of time. Historically, the concept was elaborated with the infinitesimal calculus at the end of the 17th century, and, until the 19th century, the functions that were considered were differentiable (that is, they had a high degree of regularity). The concept of a function was formalized at the end of the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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