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Hydra Game
In mathematics, specifically in graph theory and number theory, a hydra game is a single-player iterative mathematical game played on a tree (graph theory), mathematical tree called a ''hydra'' where, usually, the goal is to cut off the hydra's "heads" while the hydra simultaneously expands itself. Hydra games can be used to generate large numbers or infinite ordinals or prove the strength of certain mathematical theories. Unlike their combinatorial counterparts like Kruskal's tree theorem, TREE and Friedman's SSCG function, SCG, no search is required to compute these fast-growing function values – one must simply keep applying the transformation rule to the tree until the game says to stop. Introduction A simple hydra game can be defined as follows: * A hydra is a finite rooted tree, which is a network of points and lines with no loops and a single starting point known as the root. The root is referred to as R. * The player selects a leaf node x and a natural number n at eac ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Graph Theory
In mathematics, graph theory is the study of ''graphs'', which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of '' vertices'' (also called ''nodes'' or ''points'') which are connected by '' edges'' (also called ''links'' or ''lines''). A distinction is made between undirected graphs, where edges link two vertices symmetrically, and directed graphs, where edges link two vertices asymmetrically. Graphs are one of the principal objects of study in discrete mathematics. Definitions Definitions in graph theory vary. The following are some of the more basic ways of defining graphs and related mathematical structures. Graph In one restricted but very common sense of the term, a graph is an ordered pair G=(V,E) comprising: * V, a set of vertices (also called nodes or points); * E \subseteq \, a set of edges (also called links or lines), which are unordered pairs of vertices (that is, an edge is associated with t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Number Theory
Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic function, integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Mathematics is the queen of the sciences—and number theory is the queen of mathematics."German original: "Die Mathematik ist die Königin der Wissenschaften, und die Arithmetik ist die Königin der Mathematik." Number theorists study prime numbers as well as the properties of mathematical objects made out of integers (for example, rational numbers) or defined as generalizations of the integers (for example, algebraic integers). Integers can be considered either in themselves or as solutions to equations (Diophantine geometry). Questions in number theory are often best understood through the study of Complex analysis, analytical objects (for example, the Riemann zeta function) that encode properties of the integers, primes ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical Game
A mathematical game is a game whose rules, strategies, and outcomes are defined by clear mathematical parameters. Often, such games have simple rules and match procedures, such as Tic-tac-toe and Dots and Boxes. Generally, mathematical games need not be conceptually intricate to involve deeper computational underpinnings. For example, even though the rules of Mancala are relatively basic, the game can be rigorously analyzed through the lens of combinatorial game theory. Mathematical games differ sharply from mathematical puzzles in that mathematical puzzles require specific mathematical expertise to complete, whereas mathematical games do not require a deep knowledge of mathematics to play. Often, the arithmetic core of mathematical games is not readily apparent to players untrained to note the statistical or mathematical aspects. Some mathematical games are of deep interest in the field of recreational mathematics. When studying a game's core mathematics, arithmetic theory i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tree (graph Theory)
In graph theory In mathematics, graph theory is the study of ''graphs'', which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of '' vertices'' (also called ''nodes'' or ''points'') which are conne ..., a tree is an undirected graph in which any two Vertex (graph theory), vertices are connected by ''exactly one'' Path (graph theory), path, or equivalently a Connected graph, connected Cycle (graph theory), acyclic undirected graph. A forest is an undirected graph in which any two vertices are connected by ''at most one'' path, or equivalently an acyclic undirected graph, or equivalently a Disjoint union of graphs, disjoint union of trees. A polytreeSee . (or directed tree or oriented treeSee .See . or singly connected networkSee .) is a directed acyclic graph (DAG) whose underlying undirected graph is a tree. A polyforest (or directed forest or oriented forest) is a directed acyclic graph whose underlying undirecte ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Infinite Ordinal
In mathematics, transfinite numbers are numbers that are "infinite" in the sense that they are larger than all finite numbers, yet not necessarily absolutely infinite. These include the transfinite cardinals, which are cardinal numbers used to quantify the size of infinite sets, and the transfinite ordinals, which are ordinal numbers used to provide an ordering of infinite sets. The term ''transfinite'' was coined by Georg Cantor in 1895, who wished to avoid some of the implications of the word ''infinite'' in connection with these objects, which were, nevertheless, not ''finite''. Few contemporary writers share these qualms; it is now accepted usage to refer to transfinite cardinals and ordinals as infinite numbers. Nevertheless, the term "transfinite" also remains in use. Definition Any finite natural number can be used in at least two ways: as an ordinal and as a cardinal. Cardinal numbers specify the size of sets (e.g., a bag of five marbles), whereas ordinal numbers specify th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kruskal's Tree Theorem
In mathematics, Kruskal's tree theorem states that the set of finite trees over a well-quasi-ordered set of labels is itself well-quasi-ordered under homeomorphic embedding. History The theorem was conjectured by Andrew Vázsonyi and proved by ; a short proof was given by . It has since become a prominent example in reverse mathematics as a statement that cannot be proved within ATR0 (a form of arithmetical transfinite recursion), and a finitary application of the theorem gives the existence of the fast-growing TREE function. In 2004, the result was generalized from trees to graphs as the Robertson–Seymour theorem, a result that has also proved important in reverse mathematics and leads to the even-faster-growing SSCG function. Statement The version given here is that proven by Nash-Williams; Kruskal's formulation is somewhat stronger. All trees we consider are finite. Given a tree with a root, and given vertices , , call a successor of if the unique path from the root ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Friedman's SSCG Function
In mathematics, a simple subcubic graph (SSCG) is a finite simple Graph (discrete mathematics), graph in which each vertex (graph theory), vertex has vertex degree, degree at most three. Suppose we have a sequence of simple subcubic graphs ''G''1, ''G''2, ... such that each graph ''G''''i'' has at most ''i'' + ''k'' vertices (for some integer ''k'') and for no ''i'' < ''j'' is ''G''''i'' Homeomorphism (graph theory), homeomorphically embeddable into (i.e. is a graph minor of) ''G''''j''. The Robertson–Seymour theorem proves that subcubic graphs (simple or not) are well-founded by homeomorphic embeddability, implying such a sequence cannot be infinite. So, for each value of ''k'', there is a sequence with maximal length. The function SSCG(''k'') denotes that length for simple subcubic graphs. The function SCG(''k'') denotes that length for (general) subcubic graphs. The ''SCG'' sequence begins SCG(0) = 6, but then explodes to a value equivalent to fε ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical Induction
Mathematical induction is a method for proving that a statement ''P''(''n'') is true for every natural number ''n'', that is, that the infinitely many cases ''P''(0), ''P''(1), ''P''(2), ''P''(3), ... all hold. Informal metaphors help to explain this technique, such as falling dominoes or climbing a ladder: A proof by induction consists of two cases. The first, the base case, proves the statement for ''n'' = 0 without assuming any knowledge of other cases. The second case, the induction step, proves that ''if'' the statement holds for any given case ''n'' = ''k'', ''then'' it must also hold for the next case ''n'' = ''k'' + 1. These two steps establish that the statement holds for every natural number ''n''. The base case does not necessarily begin with ''n'' = 0, but often with ''n'' = 1, and possibly with any fixed natural number ''n'' = ''N'', establishing the truth of the statement for all natu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fast-growing Hierarchy
In computability theory, computational complexity theory and proof theory, a fast-growing hierarchy (also called an extended Grzegorczyk hierarchy) is an ordinal-indexed family of rapidly increasing functions ''f''α: N → N (where N is the set of natural numbers , and α ranges up to some large countable ordinal). A primary example is the Wainer hierarchy, or Löb–Wainer hierarchy, which is an extension to all α < ε0. Such hierarchies provide a natural way to classify s according to rate-of-growth and . ...
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Graham's Number
Graham's number is an immense number that arose as an upper bound on the answer of a problem in the mathematical field of Ramsey theory. It is much larger than many other large numbers such as Skewes's number and Moser's number, both of which are in turn much larger than a googolplex. As with these, it is so large that the observable universe is far too small to contain an ordinary digital representation of Graham's number, assuming that each digit occupies one Planck volume, possibly the smallest measurable space. But even the number of digits in this digital representation of Graham's number would itself be a number so large that its digital representation cannot be represented in the observable universe. Nor even can the number of digits of ''that'' number—and so forth, for a number of times far exceeding the total number of Planck volumes in the observable universe. Thus Graham's number cannot be expressed even by physical universe-scale power towers of the form a ^. How ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Buchholz Hydra
In mathematical logic Mathematical logic is the study of formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory. Research in mathematical logic commonly addresses the mathematical properties of formal ..., specifically in graph theory and number theory, the Buchholz hydra game is a type of hydra game, which is a single-player game based on the idea of chopping pieces off a mathematical tree. The hydra game can be used to generate a rapidly growing function; BH(n), which eventually dominates all recursive functions that are provably total in "\mathsf", and is itself provably total in "\mathsf "transfinite induction with respect to TFB". Rules The game is played on a ''hydra'', a finite, rooted connected mathematical tree A with the following properties: * The root of A has a special label, usually denoted +. * Any other node of A has a label \nu \leq \omega. * All nodes directly above the root of A have ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Goodstein's Theorem
In mathematical logic, Goodstein's theorem is a statement about the natural numbers, proved by Reuben Goodstein in 1944, which states that every ''Goodstein sequence'' eventually terminates at 0. Kirby and Paris showed that it is unprovable in Peano arithmetic (but it can be proven in stronger systems, such as second-order arithmetic). This was the third example of a true statement that is unprovable in Peano arithmetic, after the examples provided by Gödel's incompleteness theorem and Gerhard Gentzen's 1943 direct proof of the unprovability of ε0-induction in Peano arithmetic. The Paris–Harrington theorem gave another example. Laurence Kirby and Jeff Paris introduced a graph-theoretic hydra game with behavior similar to that of Goodstein sequences: the "Hydra" (named for the mythological multi-headed Hydra of Lerna) is a rooted tree, and a move consists of cutting off one of its "heads" (a branch of the tree), to which the hydra responds by growing a finite number of new ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
