Transition - Revision history

Vassia Atanassova at 17:21, 22 November 2009

2009-11-22T17:21:46Z

← Older revision		Revision as of 20:21, 22 November 2009
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	When tokens enter the input place of a transition, it becomes ''potentially fireable'' and at the moment of their transfer towards the transition's output places, the transition is being ''fired''.		When tokens enter the input place of a transition, it becomes ''potentially fireable'' and at the moment of their transfer towards the transition's output places, the transition is being ''fired''.

			The tokens' transfer through a transition is described by a [[Algorithm for transition functioning\|special formal algorithm]].

	== Formal description ==		== Formal description ==

Vassia Atanassova: /* Formal description */

2009-04-18T15:09:41Z

Formal description

← Older revision		Revision as of 18:09, 18 April 2009
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	* <math>t_2</math> is the current value of the duration of its active state.		* <math>t_2</math> is the current value of the duration of its active state.
	* <math>r</math> is the transition's ''condition'', determining which tokens will transfer from the transition's inputs to its outputs. The parameter has the form of an [[index matrix]]:		* <math>r</math> is the transition's ''condition'', determining which tokens will transfer from the transition's inputs to its outputs. The parameter has the form of an [[index matrix]]:
			*: <math> r =
	<math> r =
	\begin{array}{c\|c c c c c} & l''_1 & ... & l''_j & ... & l''_n \\		\begin{array}{c\|c c c c c} & l''_1 & ... & l''_j & ... & l''_n \\
	\hline		\hline
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	\end{array}		\end{array}
	</math>		</math>
			*: where <math>r_{i,j}</math> are predicates, <math>1 \le i \le m, 1 \le j \le n</math>
	: where <math>r_{i,j}</math> are predicates, <math>1 \le i \le m, 1 \le j \le n</math>

	* <math>M</math> is the index matrix of the capacities of the transition's arcs:		* <math>M</math> is the index matrix of the capacities of the transition's arcs:
			*: <math> M =
	<math> M =
	\begin{array}{c\|c c c c c} & l''_1 & ... & l''_j & ... & l''_n \\		\begin{array}{c\|c c c c c} & l''_1 & ... & l''_j & ... & l''_n \\
	\hline		\hline
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	\end{array}		\end{array}
	</math>		</math>
			*: where <math>M_{i,j} \ge 0</math> are natural numbers or <math>\infty</math>, <math>1 \le i \le m, 1 \le j \le n</math>
	: where <math>M_{i,j} \ge 0</math> are natural numbers or <math>\infty</math>, <math>1 \le i \le m, 1 \le j \le n</math>

	* <math>\Box</math> is called transition's type, an object having a form similar to a Boolean expression. It may contain as variables the symbols that serve as labels for transition's input places, and it is an expression constructed of variables and the Boolean connectives <math>\land</math> and <math>\lor</math> determining the following conditions:		* <math>\Box</math> is called transition's type, an object having a form similar to a Boolean expression. It may contain as variables the symbols that serve as labels for transition's input places, and it is an expression constructed of variables and the Boolean connectives <math>\land</math> and <math>\lor</math> determining the following conditions:
	*: <math>\land(l_{i_1}, l_{i_2},...,l_{i_u})</math> - each of the places <math>l_{i_1}, l_{i_2},...,l_{i_u}</math> must contain at least one token,		*: <math>\land(l_{i_1}, l_{i_2},...,l_{i_u})</math> - each of the places <math>l_{i_1}, l_{i_2},...,l_{i_u}</math> must contain at least one token,

Vassia Atanassova: New page: A GN transition with ''m'' inputs and ''n'' outputs '''Transition''' in the context of generalized nets is an object from the static s...

2009-04-18T15:01:13Z

New page: right|thumb|200px|A GN transition with ''m'' inputs and ''n'' outputs '''Transition''' in the context of generalized nets is an object from the static s...

New page

[[Image:GN-transition-mxn.png|right|thumb|200px|A GN transition with ''m'' inputs and ''n'' outputs]]
'''Transition''' in the context of [[generalized nets]] is an object from the static structure of the net, which comprises the conditions of [[token]]s' transfer from the transition's input [[place]]s to its output places.

When tokens enter the input place of a transition, it becomes ''potentially fireable'' and at the moment of their transfer towards the transition's output places, the transition is being ''fired''.

== Formal description ==
Formally, every transition is described by a 7-tuple:

<div align="center"><math>Z = \langle L', L'', t_1, t_2, r, M, \Box \rangle</math></div>

where:
* <math>L', L''</math> are finite, non-empty sets of places: the transition's input and output places, respectively.
* <math>t_1</math> is the current time-moment of the transition's firing.
* <math>t_2</math> is the current value of the duration of its active state.
* <math>r</math> is the transition's ''condition'', determining which tokens will transfer from the transition's inputs to its outputs. The parameter has the form of an [[index matrix]]:

<math> r =
\begin{array}{c|c c c c c} & l''_1 & ... & l''_j & ... & l''_n \\
\hline
l'_1 & & & & & \\
... & & & & & \\
l'_i & & & r_{i,j} & & \\
... & & & & & \\
l'_m & & & & & \\
\end{array}
</math>

: where <math>r_{i,j}</math> are predicates, <math>1 \le i \le m, 1 \le j \le n</math>

* <math>M</math> is the index matrix of the capacities of the transition's arcs:

<math> M =
\begin{array}{c|c c c c c} & l''_1 & ... & l''_j & ... & l''_n \\
\hline
l'_1 & & & & & \\
... & & & & & \\
l'_i & & & M_{i,j} & & \\
... & & & & & \\
l'_m & & & & & \\
\end{array}
</math>

: where <math>M_{i,j} \ge 0</math> are natural numbers or <math>\infty</math>, <math>1 \le i \le m, 1 \le j \le n</math>

* <math>\Box</math> is called transition's type, an object having a form similar to a Boolean expression. It may contain as variables the symbols that serve as labels for transition's input places, and it is an expression constructed of variables and the Boolean connectives <math>\land</math> and <math>\lor</math> determining the following conditions:
*: <math>\land(l_{i_1}, l_{i_2},...,l_{i_u})</math> - each of the places <math>l_{i_1}, l_{i_2},...,l_{i_u}</math> must contain at least one token,
*: <math>\lor(l_{i_1}, l_{i_2},...,l_{i_u})</math> - there must be at least one token in the set of places <math>l_{i_1}, l_{i_2},...,l_{i_u}</math> where <math>\lbrace l_{i_1}, l_{i_2},...,l_{i_u} \rbrace \subset L'</math>
*: When the value of a type (calculated as a Boolean expression) is ''"true"'', the transition can become active, otherwise it cannot.

== References ==
* [[On Generalized Nets Theory]], [[Krassimir Atanassov]], Prof. Marin Drinov Academic Publishing House, Sofia, 2007

[[Category:Generalized nets]]