Homogeneous Equations of Algebraic Petri Nets

TL;DR

This paper characterizes stability of homogeneous linear equations in algebraic Petri nets, showing that stability is decidable when the coefficients form a cyclic group, thus advancing the understanding of correctness conditions.

Contribution

It provides a complete characterization of stability for homogeneous linear equations in algebraic Petri nets with cyclic group coefficients.

Findings

Stability of homogeneous linear equations is fully characterized.

Decidability of stability is established for cyclic group coefficients.

The work extends the understanding of correctness conditions in algebraic Petri nets.

Abstract

Algebraic Petri nets are a formalism for modeling distributed systems and algorithms, describing control and data flow by combining Petri nets and algebraic specification. One way to specify correctness of an algebraic Petri net model $N$ is to specify a linear equation $E$ over the places of $N$ based on term substitution, and coefficients from an abelian group $G$. Then, $E$ is valid in $N$ iff $E$ is valid in each reachable marking of $N$ . Due to the expressive power of Algebraic Petri nets, validity is generally undecidable. Stable linear equations form a class of linear equations for which validity is decidable. Place invariants yield a well-understood but incomplete characterization of all stable linear equations. In this paper, we provide a complete characterization of stability for the subclass of homogeneous linear equations, by restricting ourselves to the interpretation of terms over the Herbrand structure without considering further equality axioms. Based thereon, we show that stability is decidable for homogeneous linear equations if $G$ is a cyclic group.