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Indistinguishability Quotient
In combinatorial game theory, and particularly in the theory of impartial games in misère play, an indistinguishability quotient is a commutative monoid that generalizes and localizes the Sprague–Grundy theorem for a specific game's rule set. In the specific case of misere-play impartial games, such commutative monoids have become known as misere quotients. Example: Misere Nim variant Suppose the game of Nim is played as usual with heaps of objects, but that at the start of play, every heap is restricted to have either one or two objects in it. In the normal-play convention, players take turns to remove any number of objects from a heap, and the last player to take an object from a heap is declared the winner of the game; in Misere play, that player is the loser of the game. Regardless of whether the normal or misere play convention is in effect, the outcome of such a position is necessarily of one of two types: ; N : The Next player to move has a forced win in best play; ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Combinatorial Game Theory
Combinatorial game theory is a branch of mathematics and theoretical computer science that typically studies sequential games with perfect information. Study has been largely confined to two-player games that have a ''position'' that the players take turns changing in defined ways or ''moves'' to achieve a defined winning condition. Combinatorial game theory has not traditionally studied games of chance or those that use imperfect or incomplete information, favoring games that offer perfect information in which the state of the game and the set of available moves is always known by both players. However, as mathematical techniques advance, the types of game that can be mathematically analyzed expands, thus the boundaries of the field are ever changing. Scholars will generally define what they mean by a "game" at the beginning of a paper, and these definitions often vary as they are specific to the game being analyzed and are not meant to represent the entire scope of the field. C ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Impartial Game
In combinatorial game theory, an impartial game is a game in which the allowable moves depend only on the position and not on which of the two players is currently moving, and where the payoffs are symmetric. In other words, the only difference between player 1 and player 2 is that player 1 goes first. The game is played until a terminal position is reached. A terminal position is one from which no moves are possible. Then one of the players is declared the winner and the other the loser. Furthermore, impartial games are played with perfect information and no chance moves, meaning all information about the game and operations for both players are visible to both players. Impartial games include Nim, Sprouts, Kayles, Quarto, Cram, Chomp, Subtract a square, Notakto, and poset games. Go and chess are not impartial, as each player can only place or move pieces of their own color. Games such as poker, dice or dominos are not impartial games as they rely on chance. Impartial games c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Misère
Misère ( French for "destitution"), misere, bettel, betl, or (German for "beggar"; equivalent terms in other languages include , , ) is a bid in various card games, and the player who bids misère undertakes to win no tricks or as few as possible, usually at no trump, in the round to be played. This does not allow sufficient variety to constitute a game in its own right, but it is the basis of such trick-avoidance games as Hearts, and provides an optional contract for most games involving an auction. The term or category may also be used for some card game of its own with the same aim, like Black Peter. A misère bid usually indicates an extremely poor hand, hence the name. An open or lay down misère, or misère ouvert is a 500 bid where the player is so sure of losing every trick that they undertake to do so with their cards placed face-up on the table. Consequently, 'lay down misère' is Australian gambling slang for a predicted easy victory. In Skat, the bidding can ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Commutative Monoid
In abstract algebra, a branch of mathematics, a monoid is a set equipped with an associative binary operation and an identity element. For example, the nonnegative integers with addition form a monoid, the identity element being 0. Monoids are semigroups with identity. Such algebraic structures occur in several branches of mathematics. The functions from a set into itself form a monoid with respect to function composition. More generally, in category theory, the morphisms of an object to itself form a monoid, and, conversely, a monoid may be viewed as a category with a single object. In computer science and computer programming, the set of strings built from a given set of characters is a free monoid. Transition monoids and syntactic monoids are used in describing finite-state machines. Trace monoids and history monoids provide a foundation for process calculi and concurrent computing. In theoretical computer science, the study of monoids is fundamental for automata theor ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sprague–Grundy Theorem
In combinatorial game theory, the Sprague–Grundy theorem states that every impartial game under the normal play convention is equivalent to a one-heap game of nim, or to an infinite generalization of nim. It can therefore be represented as a natural number, the size of the heap in its equivalent game of nim, as an ordinal number in the infinite generalization, or alternatively as a nimber, the value of that one-heap game in an algebraic system whose addition operation combines multiple heaps to form a single equivalent heap in nim. The Grundy value or nim-value of any impartial game is the unique nimber that the game is equivalent to. In the case of a game whose positions are indexed by the natural numbers (like nim itself, which is indexed by its heap sizes), the sequence of nimbers for successive positions of the game is called the nim-sequence of the game. The Sprague–Grundy theorem and its proof encapsulate the main results of a theory discovered independently by R ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Normal Play Convention
A normal play convention in a game is the method of determining the winner that is generally regarded as standard. For example: *Preventing the other player from being able to move *Being the first player to achieve a target *Holding the highest value hand *Taking the most card tricks In combinatorial game theory, the normal play convention of an impartial game is that the last player able to move is the winner. By contrast "misère game Misère ( French for "destitution"), misere, bettel, betl, or (German for "beggar"; equivalent terms in other languages include , , ) is a bid in various card games, and the player who bids misère undertakes to win no tricks or as few as possi ...s" involve upsetting the convention and declaring a winner the individual who would normally be considered the loser. Gaming {{Game-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nimber
In mathematics, the nimbers, also called ''Grundy numbers'', are introduced in combinatorial game theory, where they are defined as the values of heaps in the game Nim. The nimbers are the ordinal numbers endowed with ''nimber addition'' and ''nimber multiplication'', which are distinct from ordinal addition and ordinal multiplication. Because of the Sprague–Grundy theorem In combinatorial game theory, the Sprague–Grundy theorem states that every impartial game under the normal play convention is equivalent to a one-heap game of nim, or to an infinite generalization of nim. It can therefore be represented as ... which states that every impartial game is equivalent to a Nim heap of a certain size, nimbers arise in a much larger class of impartial games. They may also occur in partisan games like Domineering. Nimbers have the characteristic that their Left and Right options are identical, following a certain schema, and that they are their own negatives, such that a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Klein Four-group
In mathematics, the Klein four-group is a Group (mathematics), group with four elements, in which each element is Involution (mathematics), self-inverse (composing it with itself produces the identity) and in which composing any two of the three non-identity elements produces the third one. It can be described as the symmetry group of a non-square rectangle (with the three non-identity elements being horizontal and vertical reflection and 180-degree rotation), as the group of bitwise operation, bitwise exclusive or operations on two-bit binary values, or more abstract algebra, abstractly as , the Direct product of groups, direct product of two copies of the cyclic group of Order (group theory), order 2. It was named ''Vierergruppe'' (meaning four-group) by Felix Klein in 1884. It is also called the Klein group, and is often symbolized by the letter V or as K4. The Klein four-group, with four elements, is the smallest group that is not a cyclic group. There is only one other group ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Group Presentation
In mathematics, a presentation is one method of specifying a group. A presentation of a group ''G'' comprises a set ''S'' of generators—so that every element of the group can be written as a product of powers of some of these generators—and a set ''R'' of relations among those generators. We then say ''G'' has presentation :\langle S \mid R\rangle. Informally, ''G'' has the above presentation if it is the "freest group" generated by ''S'' subject only to the relations ''R''. Formally, the group ''G'' is said to have the above presentation if it is isomorphic to the quotient of a free group on ''S'' by the normal subgroup generated by the relations ''R''. As a simple example, the cyclic group of order ''n'' has the presentation :\langle a \mid a^n = 1\rangle, where 1 is the group identity. This may be written equivalently as :\langle a \mid a^n\rangle, thanks to the convention that terms that do not include an equals sign are taken to be equal to the group identity. S ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Presentation Of A Monoid
In algebra, a presentation of a monoid (or a presentation of a semigroup) is a description of a monoid (or a semigroup) in terms of a set of generators and a set of relations on the free monoid (or the free semigroup ) generated by . The monoid is then presented as the quotient of the free monoid (or the free semigroup) by these relations. This is an analogue of a group presentation in group theory. As a mathematical structure, a monoid presentation is identical to a string rewriting system (also known as a semi-Thue system). Every monoid may be presented by a semi-Thue system (possibly over an infinite alphabet).Book and Otto, Theorem 7.1.7, p. 149 A ''presentation'' should not be confused with a '' representation''. Construction The relations are given as a (finite) binary relation on . To form the quotient monoid, these relations are extended to monoid congruences as follows: First, one takes the symmetric closure of . This is then extended to a symmetric relation by ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Octal Games
The octal games are a class of two-player games that involve removing tokens (game pieces or stones) from heaps of tokens. They have been studied in combinatorial game theory as a generalization of Nim, Kayles, and similar games. Revised and reprinted as Octal games are impartial meaning that every move available to one player is also available to the other player. They differ from each other in the numbers of tokens that may be removed in a single move, and (depending on this number) whether it is allowed to remove an entire heap, reduce the size of a heap, or split a heap into two heaps. These rule variations may be described compactly by a coding system using octal numerals. Game specification An octal game is played with tokens divided into heaps. Two players take turns moving until no moves are possible. Every move consists of selecting just one of the heaps, and either * removing all of the tokens in the heap, leaving no heap, * removing some but not all of the tokens, l ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Genus Theory
{{multiple issues, {{confusing, date=March 2009 {{more citations needed, date=January 2019 In the mathematical theory of games, genus theory in impartial games is a theory by which some games played under the misère play convention can be analysed, to predict the outcome class of games. Genus theory was first published in the book ''On Numbers and Games'', and later in '' Winning Ways for your Mathematical Plays'' Volume 2. Unlike the Sprague–Grundy theory for normal play impartial games, genus theory is not a complete theory for misère play impartial games. Genus of a game The genus of a game is defined using the mex (minimum excludant) of the options of a game. g+ is the grundy value or nimber of a game under the normal play convention. g- or ''λ''0 is the outcome class of a game under the misère play convention. More specifically, to find g+, *0 is defined to have g+ = 0, and all other games have g+ equal to the mex of its options. To find g−, *0 has g− = 1, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
