en>Paxinum: /* Definition */ added ref to ehrhart poly

2014-11-12T14:05:34Z

Definition: added ref to ehrhart poly

en>David Eppstein: link Peter McMullen

2013-11-03T20:31:45Z

link Peter McMullen

← Older revision		Revision as of 22:31, 3 November 2013
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	~~Wilber Berryhill~~ is ~~what his wife loves to contact him~~ and ~~he completely enjoys this name~~. ~~She functions~~ as a ~~travel agent but soon she'll~~ be ~~on her personal~~. As a ~~lady what she~~ [~~http~~://~~medialab~~.~~zendesk~~.~~com~~/entries/~~54181460~~-~~Will-You-Often-End-Up-Bored-Try-One-Of-These-Hobby-Ideas- certified psychics~~] ~~really likes~~ is ~~fashion~~ and ~~she's been performing it~~ for ~~fairly~~ a ~~while~~. ~~Mississippi~~ is ~~where her house is but her spouse wants~~ [~~http~~://~~c045~~.~~danah~~.co.kr/~~home~~/~~index~~.~~php?document_srl~~=~~1356970&mid~~=~~qna love psychic~~] ~~them~~ to ~~transfer~~.<br><br>~~Here~~ is ~~my page~~ :~~: tarot card readings~~ - [~~http~~:/~~/jplusfn~~.~~gaplus~~.kr/xe/~~qna~~/~~78647 http~~://~~jplusfn~~.~~gaplus~~.kr/xe/~~qna/78647~~] -		In [[mathematics]] and [[computer science]], the '''probabilistic automaton (PA)''' is a generalization of the [[non-deterministic finite automaton]]; it includes the probability of a given transition into the [[finite state machine\|transition function]], turning it into a [[transition matrix]] or [[stochastic matrix]]. Thus, the probabilistic automaton generalizes the concept of a [[Markov chain]] or [[subshift of finite type]]. The [[formal language\|languages]] recognized by probabilistic automata are called '''stochastic languages'''; these include the [[regular language]]s as a subset. The number of stochastic languages is [[uncountable]].

			The concept was introduced by [[Michael O. Rabin]] in 1963;<ref>M. O Rabin,"Probabilistic Automata", ''Information and Control'' '''6''' (1963) pp. 230–245</ref> a certain special case is sometimes known as the '''Rabin automaton'''. In recent years, a variant has been formulated in terms of quantum probabilities, the [[quantum finite automaton]].

			==Definition==
			The probabilistic automaton may be defined as an extension of a [[non-deterministic finite automaton]] <math>(Q,\Sigma,\delta,q_0,F)</math>, together with two probabilities: the probability <math>P</math> of a particular state transition taking place, and with the initial state <math>q_0</math> replaced by a [[stochastic vector]] giving the probability of the automaton being in a given initial state.

			For the ordinary non-deterministic finite automaton, one has
			* a finite [[Set (mathematics)\|set]] of states <math>Q</math>
			* a finite set of [[input symbol]]s <math>\Sigma</math>
			* a transition function <math>\delta:Q\times\Sigma \to P(Q)</math>
			* a set of states <math>F</math> distinguished as ''accepting'' (or ''final'') ''states'' <math>F\subset Q</math>.

			Here, <math>P(Q)</math> denotes the [[power set]] of <math>Q</math>.

			By use of [[currying]], the transition function <math>\delta:Q\times\Sigma \to P(Q)</math> of a non-deterministic finite automaton can be written as a [[membership function]]

			:<math>\delta:Q\times\Sigma \times Q\to \{0,1\}</math>

			so that <math>\delta(q,a,q^\prime)=1</math> if <math>q^\prime\in \delta(q,a)</math> and <math>\delta(q,a,q^\prime)=0</math> if <math>q^\prime\notin \delta(q,a)</math>. The curried transition function can be understood to be a matrix with matrix entries

			:<math>\left[\theta_a\right]_{qq^\prime}=\delta(q,a,q^\prime)</math>

			The matrix <math>\theta_a</math> is then a square matrix, whose entries are zero or one, indicating whether a transition <math>q\stackrel{a}{\rightarrow} q^\prime</math> is allowed by the NFA. Such a transition matrix is always defined for a non-deterministic finite automaton.

			The probabilistic automaton replaces this matrix by a [[stochastic matrix]] <math>P</math>, so that the probability of a transition is given by

			:<math>\left[P_a\right]_{qq^\prime}</math>

			A state change from some state to any state must occur with probability one, of course, and so one must have

			:<math>\sum_{q^\prime}\left[P_a\right]_{qq^\prime}=1</math>

			for all input letters <math>a</math> and internal states <math>q</math>. The initial state of a probabilistic automaton is given by a [[row vector]] <math>v</math>, whose components add to unity:

			:<math>\sum_{q}\left[v\right]_{q}=1</math>

			The transition matrix acts on the right, so that the state of the probabilistic automaton, after consuming the input string <math>abc</math>, would be

			:<math>v P_a P_b P_c</math>

			In particular, the state of a probabilistic automaton is always a stochastic vector, since the product of any two stochastic matrices is a stochastic matrix, and the product of a stochastic vector and a stochastic matrix is again a stochastic vector. This vector is sometimes called the '''distribution of states''', emphasizing that it is a [[discrete probability distribution]].

			Formally, the definition of a probabilistic automaton does not require the mechanics of the non-deterministic automaton, which may be dispensed with. Formally, a probabilistic automaton ''PA'' is defined as the tuple <math>(Q,\Sigma,P, v, F)</math>. A '''Rabin automaton''' is one for which the initial distribution <math>v</math> is a [[coordinate vector]]; that is, has zero for all but one entries, and the remaining entry being one.

			==Stochastic languages==
			The set of [[Formal language\|languages]] recognized by probabilistic automata are called '''stochastic languages'''. They include the [[regular language]]s as a subset.

			Let <math>F=Q_\text{accept}\subset Q</math> be the set of "accepting" or "final" states of the automaton. By abuse of notation, <math>Q_\text{accept}</math> can also be understood to be the column vector that is the [[membership function]] for <math>Q_\text{accept}</math>; that is, it has a 1 at the places corresponding to elements in <math>Q_\text{accept}</math>, and a zero otherwise. This vector may be contracted with the internal state probability, to form a [[scalar (mathematics)\|scalar]]. The language recognized by a specific automaton is then defined as

			:<math>L_\eta = \{s\in\Sigma^* \vert vP_s Q_\text{accept} > \eta\}</math>

			where <math>\Sigma^</math> is the set of all [[string (computer science)\|strings]] in the [[alphabet (computer science)\|alphabet]] <math>\Sigma</math> (so that is the [[Kleene star]]). The language depends on the value of the '''cut-point''' <math>\eta</math>, normally taken to be in the range <math>0\le \eta<1</math>.

			A language is called '''η-stochastic''' if and only if there exists some PA that recognizes the language, for fixed <math>\eta</math>. A language is called '''stochastic''' if and only if there is some <math>0\le \eta<1</math> for which <math>L_\eta</math> is η-stochastic.

			A cut-point is said to be an '''isolated cut-point''' if and only if there exists a <math>\delta>0</math> such that

			:<math>\vert vP(s)Q_\text{accept} - \eta \vert \ge \delta</math>

			for all <math>s\in\Sigma^*</math>

			==Properties==
			Every [[regular language]] is stochastic, and more strongly, every regular language is η-stochastic. A weak converse is that every 0-stochastic language is regular; however, the general converse does not hold: there are stochastic languages that are not regular.

			Every η-stochastic language is stochastic, for some <math>0<\eta<1</math>.

			Every stochastic language is representable by a Rabin automaton.

			If <math>\eta</math> is an isolated cut-point, then <math>L_\eta</math> is a regular language.

			==''p''-adic languages==
			The [[p-adic\|''p''-adic]] languages provide an example of a stochastic language that is not regular, and also show that the number of stochastic languages is uncountable. A ''p''-adic language is defined as the set of strings in the letters <math>0,1,2,\ldots,(p-1)</math> such that

			:<math>L_{\eta}(p)=\{0.n_1n_2n_3 \ldots \vert 0\le n_k<p \text{ and }
			0.n_1n_2n_3\ldots > \eta \}</math>

			That is, a ''p''-adic language is merely the set of real numbers, written in base-''p'', such that they are greater than <math>\eta</math>. It is straightforward to show that all ''p''-adic languages are stochastic. However, a ''p''-adic language is regular if and only if <math>\eta</math> is rational. In particular, this implies that the number of stochastic languages is uncountable.

			==Generalizations==
			The probabilistic automaton has a geometric interpretation: the state vector can be understood to be a point that lives on the face of the standard [[simplex]], opposite to the orthogonal corner. The transition matrices form a [[monoid]], acting on the point. This may be generalized by having the point be from some general [[topological space]], while the transition matrices are chosen from a collection of operators acting on the topological space, thus forming a [[semiautomaton]]. When the cut-point is suitably generalized, one has a [[topological automaton]].

			An example of such a generalization is the [[quantum finite automaton]]; here, the automaton state is represented by a point in [[complex projective space]], while the transition matrices are a fixed set chosen from the [[unitary group]]. The cut-point is understood as a limit on the maximum value of the [[quantum angle]].

			==References==
			<references/>
			*Arto Salomaa, ''Theory of Automata'' (1969) Pergamon Press, Oxford ''(See chapter 2)''.

			[[Category:Automata theory]]
			[[Category:Probabilistic models]]

141.223.38.41: /* Flag h-vector and cd-index */

2012-02-28T01:37:07Z

Flag h-vector and cd-index

New page

Wilber Berryhill is what his wife loves to contact him and he completely enjoys this name. She functions as a travel agent but soon she'll be on her personal. As a lady what she [http://medialab.zendesk.com/entries/54181460-Will-You-Often-End-Up-Bored-Try-One-Of-These-Hobby-Ideas- certified psychics] really likes is fashion and she's been performing it for fairly a while. Mississippi is where her house is but her spouse wants [http://c045.danah.co.kr/home/index.php?document_srl=1356970&mid=qna love psychic] them to transfer.<br><br>Here is my page :: tarot card readings - [http://jplusfn.gaplus.kr/xe/qna/78647 http://jplusfn.gaplus.kr/xe/qna/78647] -

H-vector - Revision history

en>Paxinum: /* Definition */ added ref to ehrhart poly

en>David Eppstein: link Peter McMullen

141.223.38.41: /* Flag h-vector and cd-index */