[Paper Review] Synthesis of Reduced Asymmetric Choice Petri Nets

Harro Wimmel|arXiv (Cornell University)|Nov 20, 2019

Petri Nets in System Modeling22 references4 citations

TL;DR

This paper presents a synthesis method for reduced asymmetric choice (RAC) Petri nets, a subclass of asymmetric choice nets where postset containment is restricted to singleton and doubleton sets. By adapting region-based synthesis and solving separation problems via integer linear programs, the approach enables polynomial-time synthesis under specific constraints, marking a key step toward efficient synthesis for asymmetric choice Petri nets.

ABSTRACT

A Petri net is choice-free if any place has at most one transition in its postset (consuming its tokens) and it is (extended) free-choice (EFC) if the postsets of any two places are either equal or disjoint. Asymmetric choice (AC) extends EFC such that two places may also have postsets where one is contained in the other. In reduced AC nets this containment is limited: If the postsets are neither disjoint nor equal, one is a singleton and the other has exactly two transitions. The aim of Petri net synthesis is to find an unlabelled Petri net in some target class with a reachability graph isomorphic to a given finite labelled transition system (lts). Choice-free nets have strong properties, allowing to often easily detect when synthesis will fail or at least to quicken the synthesis. With EFC as the target class, only few properties can be checked ahead and there seem to be no short cuts lowering the complexity of the synthesis (compared to arbitrary Petri nets). For AC nets no synthesis procedure is known at all. We show here how synthesis to a superclass of reduced AC nets (not containing the full AC net class) can be done.

Motivation & Objective

• To develop a synthesis procedure for a subclass of asymmetric choice Petri nets, specifically reduced asymmetric choice (RAC) nets, where postset containment is limited to singleton and doubleton sets.
• To address the lack of known synthesis algorithms for asymmetric choice (AC) nets by targeting a tractable superclass, BRAC (block-reduced asymmetric choice nets).
• To enable efficient synthesis by leveraging structural analysis of the labelled transition system (LTS) and reducing the complexity of separation problems.
• To identify conditions under which synthesis can be performed in polynomial time, particularly when self-loops are limited or arc weights are allowed to exceed one.
• To extend the applicability of region-based synthesis to asymmetric choice structures, which previously resisted formulation in separation problem frameworks.

Proposed method

• Adapts region-based synthesis to the RAC net class by formulating state separation problems (SSPs) and event-state separation problems (ESSPs) as integer linear inequality systems.
• Uses spanning trees of the LTS to compute Parikh vectors for chords, enabling the detection of structural constraints in the net.
• Solves ESSPs by testing disjointness and containment of postsets, with at most |Σ| inequality systems per ESSP (s,a), where Σ is the label set.
• Applies SMT solvers or Karmarkar's algorithm for rational solutions, which are scaled to integers to construct valid Petri net places.
• Introduces a block-reduction mechanism to simplify the structure of asymmetric choice nets, reducing combinatorial explosion in state separation.
• Handles self-loops in the LTS by allowing arc weights >1, which enables polynomial-time handling of self-loop edges in BRAC nets.

Experimental results

Research questions

• RQ1 Can synthesis be achieved for a meaningful subclass of asymmetric choice Petri nets, given that no synthesis procedure exists for the full AC class?
• RQ2 Under what conditions can region-based synthesis be adapted to handle asymmetric choice structures, particularly those with nested postset containment?
• RQ3 Is it possible to achieve polynomial-time synthesis for RAC nets when the LTS has a bounded number of self-loop labels?
• RQ4 How do arc weights and structural constraints affect the complexity of solving separation problems in asymmetric choice net synthesis?
• RQ5 Can the use of rational solutions and scaling techniques mitigate the NP-completeness of integer linear programming in this context?

Key findings

• Synthesis to the BRAC (block-reduced asymmetric choice) net class is feasible, providing a constructive approach where none existed before.
• For LTSs with only a fixed number of self-loop labels, synthesis to WCP (weighted choice Petri nets) can be performed in polynomial time.
• When arc weights are allowed to exceed one, self-loops can be handled in polynomial time within the BRAC framework.
• The method identifies two sources of potential non-polynomial runtime: the number of inequality systems and the NP-completeness of integer linear programming.
• The use of rational solutions followed by scaling to integers enables efficient construction of Petri net places, especially when constant terms in inequalities are zero.
• Structural analysis of the LTS via spanning trees and Parikh vectors allows for effective detection of chordal transitions and postset containment patterns essential for asymmetric choice nets.