In this paper, we propose a modification of Generalised Stochastic Petri Nets that includes only immediate transitions, referred to as Probabilistic Selection Petri Nets ( s). This formalism enables the modelling of realworld systems in such a way that the probability of executing a given action within the model aligns with empirical observations. The paper presents an analysis of the key properties of s and demonstrates their application in simulating chess game openings.
{} = ∑︀
(:∈ℰ()) = for ∈ , ℰ ( )- set of enabled transitions at . (1)
Given reachable marking , the priority vector = (1, . . . , ) and = |ℰ ( )|, we consider probability mass function ( ) = (1, 2, . . . , ). Note that is a vector which entries fulfill the condition = { = } for ∈ {1, . . . , }, corresponds to the -th enabled transition, according to the order in . A transition is executed when a sample drawn from the corresponds to its assigned execution probability.
Remark 1. (a) s are conceptually similar to Stochastic Petri Nets ( s) [
A standard reachability graph can be constructed for s in much the same way as for classical Petri Nets. However, without incorporating information about the execution probabilities of transitions, such a graph would be indistinguishable from the classical version. To address this limitation, we introduce the concept of the Probabilistic Selection Reachability Graph.
Definition 2. For a given = (, , , 0, ), its Probabilistic Selection Reachability Graph is a tuple = (, ), where: (1) = ( ) is a classical Reachability Graph of , (2) is a probability function, assigning the probability of executing an action in a given marking to the edges of the graph, such that (, , ′) = is the probability of the execution of (enabled) transition at marking , leading to marking ′, defined in Formula 1.
The probability function can be extended to entire execution paths in the , starting from the initial marking. Let = (, , , 0, ) be a , and let = (1, 2, . . . , ) denote a path in its Reachability Graph, then the probability of following the path is defined as ( ) = ∏︀ =1 () for ∈ {0, . . . , }, 0[ ⟩+1.
Definition 3. Let = (, , , 0, ) be with = (, ). Let , ′ ∈ [0⟩ be two reachable markings of . The probability Π(, ′) of reaching state ′ from state in is the sum of the labelling of all short paths from state to state ′ – Π(, ′) = ∑︀ ( ) for all ’s being short paths between and ′. Moreover, Π(, ) = 1. A marking probability function Π is a function assigning the probability of reaching state in the net (i.e. from the initial marking) defined as follows Π( ) = Π(0, ). Of course, Π(0) = 1.
Since can be considered as without timed transitions and priorities, they are indeed Petri nets. Moreover Probabilistic Selection Petri Nets can be “simulated” using classical Petri Nets.
Definition 4. We call two s 1 = (1, 1, 1, 01 , 1) and 2 = (2, 2, 2, 02 , 2) probabilistically equivalent when their sets of places are the same (1 = 2), every marking reachable in 1 is also reachable in 2 (i.e. reachability sets in 1 and 2 are the same, denoted by [01 ⟩1 =[02 ⟩2 and it results in 01 = 02 ), and the probability of reaching in 1 equals the probability of reaching in 1, i.e. Π( )1 = Π( )2 .
Proposition 1. Proposition 1. For every , there exists a ′ that is probabilistically equivalent to , in which the priority function assigns the value 1 to all transitions.
To demonstrate the properties of s, we developed an application called PSPN Chess [
In this paper, we have presented s — a specific modification of s. As they omit timed transitions and inhibitor arcs, s may be viewed as a relatively direct extension of classical Petri Nets, bearing similarities to s. In s, the probability of firing an enabled transition is determined by the value of its priority function. Although the underlying idea of s is conceptually simple, we believe they provide a valuable and flexible modelling framework. They are particularly well suited to representing scenarios in which multiple outcomes are possible, but not equally likely. We demonstrated the applicability of s using chess openings as an example, where transitions model moves with difering frequencies. Our future work will focus on modelling real-world systems — especially those found in biology — where probabilistic behaviour with varying likelihoods is commonly observed.
Declaration on Generative AI The author(s) have not employed any Generative AI tools.