Generalized nets - Revision history

Nikolay Ikonomov at 10:45, 2 April 2022

2022-04-02T10:45:24Z

← Older revision		Revision as of 13:45, 2 April 2022
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	* a dynamic structure,		* a dynamic structure,
	* temporal components.		* temporal components.

	~~[[File:Priroda 4 2021 cover 1.pdf\|thumb]]~~

	The '''''static structure''''' consists of objects called ''[[transition]]s'', which have input and output ''[[place]]s''. Two transitions can share a place, but every place can be an input of at most one transition and can be an output of at most one transition.		The '''''static structure''''' consists of objects called ''[[transition]]s'', which have input and output ''[[place]]s''. Two transitions can share a place, but every place can be an input of at most one transition and can be an output of at most one transition.

Nikolay Ikonomov at 10:44, 2 April 2022

2022-04-02T10:44:08Z

← Older revision		Revision as of 13:44, 2 April 2022
Line 8:		Line 8:
	* a dynamic structure,		* a dynamic structure,
	* temporal components.		* temporal components.

			[[File:Priroda 4 2021 cover 1.pdf\|thumb]]

	The '''''static structure''''' consists of objects called ''[[transition]]s'', which have input and output ''[[place]]s''. Two transitions can share a place, but every place can be an input of at most one transition and can be an output of at most one transition.		The '''''static structure''''' consists of objects called ''[[transition]]s'', which have input and output ''[[place]]s''. Two transitions can share a place, but every place can be an input of at most one transition and can be an output of at most one transition.

Vassia Atanassova: /* Formal description */

2010-01-08T07:12:03Z

Formal description

← Older revision		Revision as of 10:12, 8 January 2010
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	* <math>\pi_{L}</math> is a function giving the ''priorities of the places'', i.e. <math>\pi_{L} : L \rightarrow N</math>		* <math>\pi_{L}</math> is a function giving the ''priorities of the places'', i.e. <math>\pi_{L} : L \rightarrow N</math>
	* <math>c</math> is a function giving the ''place capacities'', i.e. <math>c : L \rightarrow N</math>		* <math>c</math> is a function giving the ''place capacities'', i.e. <math>c : L \rightarrow N</math>
	* <math>c</math> is a function giving the ''truth value of the predicates''. In the basic case, it may obtain values "true" (1) and "false" (0). In [[fuzzy generalized net]]s its domain is the [0;1] interval (see [[fuzzy set]]) and in the [[intuitionistic fuzzy generalized net]]s its domain is the set [0;1]×[0;1] (see [[intuitionistic fuzzy set]]).		* <math>f</math> is a function giving the ''truth value of the predicates''. In the basic case, it may obtain values "true" (1) and "false" (0). In [[fuzzy generalized net]]s its domain is the [0;1] interval (see [[fuzzy set]]) and in the [[intuitionistic fuzzy generalized net]]s its domain is the set [0;1]×[0;1] (see [[intuitionistic fuzzy set]]).
	* <math>\theta_{1}</math> is a function giving the ''next time moment'' when a given transition will be fired (will become active). Hence, <math>\theta_{1}(t) = t'</math> where <math>t, t' \in [T, T+t^*]; t \le t'</math>. The value of this function is being recalculated in the moment when the transition's active state ceases.		* <math>\theta_{1}</math> is a function giving the ''next time moment'' when a given transition will be fired (will become active). Hence, <math>\theta_{1}(t) = t'</math> where <math>t, t' \in [T, T+t^*]; t \le t'</math>. The value of this function is being recalculated in the moment when the transition's active state ceases.
	* <math>\theta_{2}</math> is a function giving the ''duration of the transition's active state''. Hence, <math>\theta_{2}(t) = t'</math> where <math>t, t' \in [T, T+t^*]; t' \ge 0</math>. The value of this function is calculated in the moment when the transition's active state begins.		* <math>\theta_{2}</math> is a function giving the ''duration of the transition's active state''. Hence, <math>\theta_{2}(t) = t'</math> where <math>t, t' \in [T, T+t^*]; t' \ge 0</math>. The value of this function is calculated in the moment when the transition's active state begins.

Vassia Atanassova: /* Extended generalized nets */

2009-11-29T12:49:24Z

Extended generalized nets

@@ Line 70: / Line 70: @@
 == Reduced generalized nets ==
-== Extended generalized nets ==
+== Extensions of generalized nets ==
 == Theoretical aspects of generalized nets ==

Vassia Atanassova: /* Algorithms related to generalized nets */

2009-11-22T15:48:05Z

Algorithms related to generalized nets

← Older revision		Revision as of 18:48, 22 November 2009
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	* '''[[Algorithm for generalized net functioning]]'''.		* '''[[Algorithm for generalized net functioning]]'''.

	The algorithm for transition functioning was [[Algorithm for transition functioning#~~Modification~~\|later modified]] in 2007 <ref>Atanassov K., Tasseva V., Trifonov T., [[Issue:Modification of the algorithm for token transfer in generalized nets\|]], Cybernetics and Information Technologies, Vol. 7, 2007, Np. 1, 62-66</ref>.		The algorithm for transition functioning was [[Algorithm for transition functioning#Modified algorithm, 2007\|later modified]] in 2007 <ref>Atanassov K., Tasseva V., Trifonov T., [[Issue:Modification of the algorithm for token transfer in generalized nets\|]], Cybernetics and Information Technologies, Vol. 7, 2007, Np. 1, 62-66</ref>.

	''Theorem 20'' in the book "[[On Generalized Nets Theory]]" from 2007 year <ref>Atanassov K., [[On Generalized Nets Theory]], "Professor Marin Drinov" Academic Publishing House, 2007, ISBN 978-954-322-237-7 </ref> states that the functioning and the results of the work of a given GN transition are equal for both algorithms, yet the difference between both algorithms is that in almost any case the modified algorithm yields results more quickly.		''Theorem 20'' in the book "[[On Generalized Nets Theory]]" from 2007 year <ref>Atanassov K., [[On Generalized Nets Theory]], "Professor Marin Drinov" Academic Publishing House, 2007, ISBN 978-954-322-237-7 </ref> states that the functioning and the results of the work of a given GN transition are equal for both algorithms, yet the difference between both algorithms is that in almost any case the modified algorithm yields results more quickly.

Vassia Atanassova: /* Algorithms related to generalized nets */

2009-11-22T14:47:57Z

Algorithms related to generalized nets

← Older revision		Revision as of 17:47, 22 November 2009
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	In the the book "[[Generalized Nets (World Scientific)\|Generalized Nets]]" from 1991 year <ref>Atanassov K., "Generalized Nets", World Scientific, Singapore, 1991, ISBN 978-981-02-0598-0</ref>, there were formulated two major algorithms, related to generalized nets:		In the the book "[[Generalized Nets (World Scientific)\|Generalized Nets]]" from 1991 year <ref>Atanassov K., "Generalized Nets", World Scientific, Singapore, 1991, ISBN 978-981-02-0598-0</ref>, there were formulated two major algorithms, related to generalized nets:
	* '''[[Algorithm for transition functioning]]''', which is to some extent equivalent to the tokens transfer algorithm in Petri nets, and		* '''[[Algorithm for transition functioning]]''', which is to some extent equivalent to the tokens transfer algorithm in Petri nets, and
	* '''[[Algorithm for GN functioning]]'''.		* '''[[Algorithm for generalized net functioning]]'''.

	The algorithm for transition functioning was [[Algorithm for transition functioning#Modification\|later modified]] in 2007 <ref>Atanassov K., Tasseva V., Trifonov T., [[Issue:Modification of the algorithm for token transfer in generalized nets\|]], Cybernetics and Information Technologies, Vol. 7, 2007, Np. 1, 62-66</ref>.		The algorithm for transition functioning was [[Algorithm for transition functioning#Modification\|later modified]] in 2007 <ref>Atanassov K., Tasseva V., Trifonov T., [[Issue:Modification of the algorithm for token transfer in generalized nets\|]], Cybernetics and Information Technologies, Vol. 7, 2007, Np. 1, 62-66</ref>.

Vassia Atanassova: /* Algorithms related to generalized nets */

2009-11-22T14:45:04Z

Algorithms related to generalized nets

← Older revision		Revision as of 17:45, 22 November 2009
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	* <math>b</math> is a function giving the ''maximal number of characteristics'', which a given token may obtain during its movement throughout the net, i.e. <math>b : K \rightarrow N</math>. In general, <math>b</math> may possess four different values: 0, 1, <math>s</math> or <math>\infty</math> meaning that the token keeps, respectively: none of its characteristics, its last characteristic, its last <math>s</math> characteristics, or all of its characteristics obtained during its movement in the net.		* <math>b</math> is a function giving the ''maximal number of characteristics'', which a given token may obtain during its movement throughout the net, i.e. <math>b : K \rightarrow N</math>. In general, <math>b</math> may possess four different values: 0, 1, <math>s</math> or <math>\infty</math> meaning that the token keeps, respectively: none of its characteristics, its last characteristic, its last <math>s</math> characteristics, or all of its characteristics obtained during its movement in the net.

	=== Algorithms for tokens transfer ~~===~~		=== Algorithms related to generalized nets ===
			In a Petri net implementation, parallelism is reduced to a sequential firing of the net transitions and in general the order of their activation is probabilistic or dependent on the transitions' priorities, if any. The respective algorithms for generalized nets enable a more detailed modelling of the described process.

			In the the book "[[Generalized Nets (World Scientific)\|Generalized Nets]]" from 1991 year <ref>Atanassov K., "Generalized Nets", World Scientific, Singapore, 1991, ISBN 978-981-02-0598-0</ref>, there were formulated two major algorithms, related to generalized nets:
			* '''[[Algorithm for transition functioning]]''', which is to some extent equivalent to the tokens transfer algorithm in Petri nets, and
			* '''[[Algorithm for GN functioning]]'''.

			The algorithm for transition functioning was [[Algorithm for transition functioning#Modification\|later modified]] in 2007 <ref>Atanassov K., Tasseva V., Trifonov T., [[Issue:Modification of the algorithm for token transfer in generalized nets\|]], Cybernetics and Information Technologies, Vol. 7, 2007, Np. 1, 62-66</ref>.

			''Theorem 20'' in the book "[[On Generalized Nets Theory]]" from 2007 year <ref>Atanassov K., [[On Generalized Nets Theory]], "Professor Marin Drinov" Academic Publishing House, 2007, ISBN 978-954-322-237-7 </ref> states that the functioning and the results of the work of a given GN transition are equal for both algorithms, yet the difference between both algorithms is that in almost any case the modified algorithm yields results more quickly.

	== Reduced generalized nets ==		== Reduced generalized nets ==

Vassia Atanassova: /* Formal description */

2009-11-08T23:11:34Z

Formal description

← Older revision		Revision as of 02:11, 9 November 2009
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	; ''2. Dynamic structure''		; ''2. Dynamic structure''
	* <math>K</math> is the set of tokens of the generalized net. In certain cases it is more convenient to denote this set as <math>K = \bigcup_{l \in Q^I} K_{l} </math> where <math>K_{l}</math> is the set of all GN tokens which are waiting to enter place <math>l</math> and <math>Q^I</math> is the set of all input places in the net.		* <math>K</math> is the ''set of tokens'' of the generalized net. In certain cases it is more convenient to denote this set as <math>K = \bigcup_{l \in Q^I} K_{l} </math> where <math>K_{l}</math> is the set of all GN tokens which are waiting to enter place <math>l</math> and <math>Q^I</math> is the set of all input places in the net.
	* <math>\pi_K</math> is a function giving the priorities of the tokens, i.e. <math>\pi_{K} : K \rightarrow N</math>		* <math>\pi_K</math> is a function giving the ''priorities of the tokens'', i.e. <math>\pi_{K} : K \rightarrow N</math>
	* <math>\theta_K</math> is a function giving the moment of time when a given token may enter the GN, i.e. <math>\theta_K (\alpha)=t</math> where <math>\alpha \in K, t \in [T; T+t^*]</math>		* <math>\theta_K</math> is a function giving the ''moment of time when a given token may enter'' the GN, i.e. <math>\theta_K (\alpha)=t</math> where <math>\alpha \in K, t \in [T; T+t^*]</math>


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	; ''4. Memory''		; ''4. Memory''
	* <math>X</math> is the set of initial characteristics, which tokens may exhibit when they enter the net for first.		* <math>X</math> is the ''set of initial characteristics'', which tokens may exhibit when they enter the net for first.
	* <math>\Phi</math> is a characteristic function, which assigns a new characteristic to each token when it makes the transfer from an input to an output place of a given transition.		* <math>\Phi</math> is a ''characteristic function'', which assigns a new characteristic to each token when it makes the transfer from an input to an output place of a given transition.
	* <math>b</math> is a function giving the maximal number of characteristics, which a given token may obtain during its movement throughout the net, i.e. <math>b : K \rightarrow N</math>. In general, <math>b</math> may possess four different values: 0, 1, <math>s</math> or <math>\infty</math> meaning that the token keeps, respectively: none of its characteristics, its last characteristic, its last <math>s</math> characteristics, or all of its characteristics obtained during its movement in the net.		* <math>b</math> is a function giving the ''maximal number of characteristics'', which a given token may obtain during its movement throughout the net, i.e. <math>b : K \rightarrow N</math>. In general, <math>b</math> may possess four different values: 0, 1, <math>s</math> or <math>\infty</math> meaning that the token keeps, respectively: none of its characteristics, its last characteristic, its last <math>s</math> characteristics, or all of its characteristics obtained during its movement in the net.



	=== Algorithms for tokens transfer ===		=== Algorithms for tokens transfer ===

Vassia Atanassova: /* Formal description */

2009-11-08T23:09:07Z

Formal description

← Older revision		Revision as of 02:09, 9 November 2009
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	=== Graphic representation ===		=== Graphic representation ===

	== Formal ~~definition~~ ==		== Formal description ==

	Formally described, the generalized net is represented by the following four-tuple:		Formally described, the generalized net is represented by the following four-tuple:

	<math>		<center><math>
	< \underbrace{ <A, \pi_{A}, \pi_{L}, c, f, \theta_{1}, \theta_{2} >}_{1.\ Static \ structure},		< \underbrace{ <A, \pi_{A}, \pi_{L}, c, f, \theta_{1}, \theta_{2} >}_{1.\ Static \ structure},
	\underbrace{ <K, \pi_{K}, \theta_{K} >}_{2. \ Dynamic \ structure},		\underbrace{ <K, \pi_{K}, \theta_{K} >}_{2. \ Dynamic \ structure},
	\underbrace{ <T, t^{0}, t^{*} >}_{3. \ Time},		\underbrace{ <T, t^{0}, t^{*} >}_{3. \ Time},
	\underbrace{ <X, \Phi, b >}_{4. \ ~~Characteristics~~}		\underbrace{ <X, \Phi, b >}_{4. \ Memory} >
	>		</math></center>
	</math>

	where:		where:
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	; ''4. ~~Characteristics~~''		; ''4. Memory''
	* <math>X</math> is the set of initial characteristics, which tokens may exhibit when they enter the net for first.		* <math>X</math> is the set of initial characteristics, which tokens may exhibit when they enter the net for first.
	* <math>\Phi</math> is a characteristic function, which assigns a new characteristic to each token when it makes the transfer from an input to an output place of a given transition.		* <math>\Phi</math> is a characteristic function, which assigns a new characteristic to each token when it makes the transfer from an input to an output place of a given transition.

Vassia Atanassova: /* Formal definition */

2009-11-08T23:04:11Z

Formal definition

@@ Line 19: / Line 19: @@
 === Graphic representation ===
-== Formal description ==
+== Formal definition ==
 === Algorithms for tokens transfer ===