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'''Generalized nets''' (''GNs'') constitute a discrete tool for universal description of adaptable, flexible, structured and reusable models of complex systems with many different and interacting components, not necessarily of the homogeneous structure and origin, usually involved in parallel, simultaneous activities. Generalized nets represent a significant extension and generalization of the concept of [[Petri nets]], as well as of other [[Extensions of Petri nets|Petri nets extensions and modifications]].

== Components of a generalized net ==
[[Image:GN-transition-mxn.png|right|thumb|25oxp|A GN transition with ''m'' inputs and ''n'' outputs]]

A generalized net consists of:
* a static structure,
* a dynamic structure,
* temporal components.

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 '''''dynamic structure''''' consists of ''[[token]]s'', which act as information carriers and can occupy a single place at every moment of the GN execution. The tokens pass through the transition from one input to another output place; such an ordered pair of places is called ''[[transition arc]]''. The tokens' movement is governed by conditions (''predicates''), contained in the [[predicate matrix]] of the transition. 

The information carried by a token is contained in its [[token characteristics|characteristics]], which can be viewed as an associative array of characteristic names and values. The values of the token characteristics change in time according to specific rules, called ''[[characteristic function]]s''. Every place possesses at most one characteristic function, which assigns new characteristics to the incoming tokens. Apart from movement in the net and change of the characteristics, tokens can also split and merge in the places.

The '''''temporal components''''' describe the time scale of GN execution. ''Temporal conditions'' control the transitions' moments of activation and duration of active state. Various other tools for fine tuning of the GN functioning are provided in the form of ''priorities'' of separate transitions, places and tokens, as well as ''capacities'' of places and transitions arcs.

=== Graphic representation ===

== Formal description ==

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

<center><math>
< \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{ <T, t^{0}, t^{*} >}_{3. \ Time},
  \underbrace{ <X, \Phi, b >}_{4. \ Memory} >
</math></center>

where:
; ''1. Static structure''
* <math>A</math> is a ''set of transitions'' (see the [[Transition#Formal description|formal definition of a transition]])
* <math>\pi_{A}</math> is a function giving the ''priorities of the transitions'', i.e. <math>\pi_{A} : A \rightarrow N</math> where N = {0, 1, 2, ...} ∪ {∞}
* <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>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_{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.


; ''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>\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>


; ''3. Time''
* <math>T</math> is the moment of time when the generalized net starts functioning. This moment is determined according to a fixed global timescale.
* <math>t^0</math> is the elementary time step of the fixed global timescale (the interval with which time increments in the timescale).
* <math>t^*</math> is the total duration of functioning of the net.


; ''4. Memory''
* <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>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.