# Parameterized Analysis of Immediate Observation Petri Nets

## TL;DR

This paper introduces immediate observation Petri nets, analyzes their parameterized reachability, coverability, and liveness problems, and proves the PSPACE-completeness of the correctness problem for related population protocols.

## Contribution

It establishes the complexity of parameterized problems for immediate observation Petri nets and solves an open problem on population protocol correctness.

## Key findings

- All three problems are in PSPACE for infinite initial markings.
- Problems are PSPACE-hard even for singleton initial markings.
- Correctness of immediate observation population protocols is PSPACE-complete.

## Abstract

We introduce immediate observation Petri nets, a class of interest in the study of population protocols (a model of distributed computation), and enzymatic chemical networks. In these areas, relevant analysis questions translate into parameterized Petri net problems: whether an infinite set of Petri nets with the same underlying net, but different initial markings, satisfy a given property. We study the parameterized reachability, coverability, and liveness problems for immediate observation Petri nets. We show that all three problems are in PSPACE for infinite sets of initial markings defined by counting constraints, a class sufficiently rich for the intended application. This is remarkable, since the problems are already PSPACE-hard when the set of markings is a singleton, i.e., in the non-parameterized case. We use these results to prove that the correctness problem for immediate observation population protocols is PSPACE-complete, answering a question left open in a previous paper.