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. t − → M . Reachability and coverability Given σ = t 1 . . . t n we write M σ − → M n when M t1 − → M 1 t2 − → M 2 . . . tn − → M n , and call σ a firing sequence. We write M * − → M if M σ − → M for some σ ∈ T * , and say that M is reachable from M . A marking M covers another marking M , written M ≥ M if M (p) ≥ M (p) for all places p. A marking M is coverable from M if there exists a marking M such that M * − → M ≥ M .
We show the equivalence of two distributed computing models, namely reconfigurable broadcast networks (RBN) and asynchronous shared-memory systems (ASMS), that were introduced independently. Both RBN and ASMS are systems in which a collection of anonymous, finite-state processes run the same protocol. In RBN, the processes communicate by selective broadcast: a process can broadcast a message which is received by all of its neighbors, and the set of neighbors of a process can change arbitrarily over time. In ASMS, the processes communicate by shared memory: a process can either write to or read from a shared register. Our main result is that RBN and ASMS can simulate each other, i.e. they are equivalent with respect to parameterized reachability, where we are given two (possibly infinite) sets of configurations C and C ′ defined by upper and lower bounds on the number of processes in each state and we would like to decide if some configuration in C can reach some configuration in C ′ . Using this simulation equivalence, we transfer results of RBN to ASMS and vice versa. Finally, we show that RBN and ASMS can simulate a third distributed model called immediate observation (IO) nets. Moreover, for a slightly stronger notion of simulation (which is satisfied by all the simulations given in this paper), we show that IO nets cannot simulate RBN.
Population protocols (Angluin et al. in PODC, 2004) are a model of distributed computation in which indistinguishable, finite-state agents interact in pairs to decide if their initial configuration, i.e., the initial number of agents in each state, satisfies a given property. In a seminal paper Angluin et al. classified population protocols according to their communication mechanism, and conducted an exhaustive study of the expressive power of each class, that is, of the properties they can decide (Angluin et al. in Distrib Comput 20(4):279–304, 2007). In this paper we study the correctness problem for population protocols, i.e., whether a given protocol decides a given property. A previous paper (Esparza et al. in Acta Inform 54(2):191–215, 2017) has shown that the problem is decidable for the main population protocol model, but at least as hard as the reachability problem for Petri nets, which has recently been proved to have non-elementary complexity. Motivated by this result, we study the computational complexity of the correctness problem for all other classes introduced by Angluin et al., some of which are less powerful than the main model. Our main results show that for the class of observation models the complexity of the problem is much lower, ranging from $$\varPi _2^p$$ Π 2 p to .
Reconfigurable broadcast networks (RBN) are a model of distributed computation in which agents can broadcast messages to other agents using some underlying communication topology which can change arbitrarily over the course of executions. In this paper, we conduct parameterized analysis of RBN. We consider cubes, (infinite) sets of configurations in the form of lower and upper bounds on the number of agents in each state, and we show that we can evaluate boolean combinations over cubes and reachability sets of cubes in . In particular, reachability from a cube to another cube is a -complete problem.To prove the upper bound for this parameterized analysis, we prove some structural properties about the reachability sets and the symbolic graph abstraction of RBN, which might be of independent interest. We justify this claim by providing two applications of these results. First, we show that the almost-sure coverability problem is -complete for RBN, thereby closing a complexity gap from a previous paper [3]. Second, we define a computation model using RBN, à la population protocols, called RBN protocols. We characterize precisely the set of predicates that can be computed by such protocols.
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