Population protocols with unreliable communication

TL;DR

This paper investigates how unreliable communication affects the computational power of population protocols, showing that many models become less expressive under message loss, with immediate observation protocols remaining robust.

Contribution

It demonstrates that unreliable communication limits the expressive power of population protocols, establishing that many models are equivalent to immediate observation protocols in such settings.

Findings

Unreliable protocols with constant storage can only compute predicates of immediate observation protocols.

Immediate observation protocols are inherently tolerant of unreliable communication.

Many more expressive message-based models become less expressive under message loss.

Abstract

Population protocols are a model of distributed computation intended for the study of networks of independent computing agents with dynamic communication structure. Each agent has a finite number of states, and communication opportunities occur nondeterministically, allowing the agents involved to change their states based on each other's states. In the present paper we study unreliable models based on population protocols and their variations from the point of view of expressive power. We model the effects of message loss. We show that for a general definition of unreliable protocols with constant-storage agents such protocols can only compute predicates computable by immediate observation population protocols (sometimes also called one-way protocols). Immediate observation population protocols are inherently tolerant of unreliable communication and keep their expressive power under a wide range of fairness conditions. We also prove that a large class of message-based models that are generally more expressive than immediate observation becomes strictly less expressive than immediate observation in the unreliable case.

Full text

In the present paper we study unreliable models based on population protocols and their variations from the point of view of expressive power. We model the effects of message loss. We show that for a general definition of protocols with unreliable communication with constant-storage agents such protocols can only compute predicates computable by immediate observation (IO) population protocols (sometimes also called one-way protocols). Immediate observation population protocols are inherently tolerant to unreliable communication and keep their expressive power under a wide range of fairness conditions. We also prove that a large class of message-based models that are generally more expressive than IO becomes strictly less expressive than IO in the unreliable case.

1 Introduction

Population protocols have been introduced in [1, 2] as a restricted yet useful subclass of general distributed protocols. In population protocols each agent has a constant amount of local storage, and during the protocol execution pairs of agents are selected and permitted to interact. The selection of pairs is assumed to be done by an adversary bound by a fairness condition. The fairness condition ensures that the adversary cannot trivially stall the protocol. A typical fairness condition requires that every configuration that stays reachable during an infinite execution is reached infinitely many times.

Population protocols have been studied from various points of view, such as expressive power [5], verification complexity [19], time to convergence [3, 17], privacy [13], impact of different interaction scheduling [10] etc. Multiple related models have been introduced. Some of them change or restrict the communication structure: this is the case for immediate, delayed, and queued transmission and observation [5], as well as for broadcast protocols [18]. Some explore the implications of adding limited amounts of storage (below the usual linear or polynomial storage permitted in traditional distributed protocols): this is the case for community protocols [23] (which allow an agent to recognise a constant number of other agents), PALOMA [11] (permitting logarithmic amount of local storage), mediated population protocols [26] (giving some constant amount of common storage to every pair of agents), and others.

The original target application of population protocols and related models is modelling networks of restricted sensors, starting from the original paper [1] on population protocols. On the other hand, verifying distributed algorithms benefits from translating the algorithms in question or their parts into a restricted setting, as most problems are undecidable in the unrestricted case. Both applications motivate study of fault tolerance. Some papers on population protocols and related models [12, 23, 4, 24] consider questions of fault tolerance, but in the context of expressive power the fault is typically expected to be either a total agent failure or a Byzantine failure. There are some exceptions such as a study of fine-grained notions of unreliability [15, 14] in the context of step-by-step simulation of population protocols by distributed systems with binary interactions. However, these studies answer a completely different set of questions, as they are concerned with simulating a protocol as a process as opposed to designing a protocol to achieve a given result no matter in what way.

In a practical context, many distributed algorithms pay attention to a specific kind of failure: message loss. While the eventual convergence approach typical in study of population protocols escapes the question of availability during a temporary network partition (the problem studied, for example, in [22]), the onset of a network partition may include message loss in the middle of an interaction. In such a situation the participants do not always agree whether the interaction has succeeded or failed. In terms of population protocols, one of the agents assumes that an interaction has happened and updates the local state, while a counterparty thinks the interaction has failed and keeps the old state.

In the present paper we study the expressive power of a very wide class of models with interacting constant-storage agents when unreliability of communication is introduced. This unreliability corresponds to the loss of atomicity of interactions due to message loss. Indeed, in the distributed systems ensuring that both sides agree on whether the interaction has taken place is often the costliest part; a special case of it is “exactly-once” message arrival, known to be much more complex to ensure than “at most-once”. We model such loss of atomicity by allowing some agents to update their state based on an interaction, while other agents keep their original state because they assume the interaction has failed. For a bit more generality, corresponding, for example, to request-response interactions with the response being impossible if the request is lost, we allow to require that some agents can only update their state if the others do.

We consider the expressive power in the context of computing predicates by protocols with eventual convergence of individual opinions. We show that under very general conditions the expressive power of protocols with unreliable communication coincides with the expressive power of immediate observation population protocols. Immediate observation population protocols, modelling interactions where an agent can observe the state of another one without the observee noticing, provide a model that inherently tolerates unreliability and is considered a relatively weak model in the fully reliable case. This model also has other nice properties, such as relatively low complexity ($\mathbf{PSPACE}$-complete) of verification tasks [21]. Our results hold under any definition of fairness satisfying two general assumptions (see Definition 10), including all the usually used versions of fairness.

We prove it by observing a general structural property shared by all protocols with unreliable communication. Informally speaking, protocols with unreliable communication have some special fair executions which can be extended by adding an additional agent with the same initial and final state as a chosen existing one. This property is similar to the copycat arguments used, for example, for proving the exact expressive power of immediate observation protocols. The usual structure of the copycat arguments includes a proof that we can pick an agent in an execution and add another agent (copycat) which will repeat all the state transitions of the chosen one. In the immediate observation case the corresponding property is almost self-evident once defined. A slightly stronger but still straightforward argument is needed in the case of reconfigurable broadcast networks [8]. The latter model is equivalent to unreliable broadcast networks; a sender broadcasts a message and changes the local state, and an arbitrary set of receivers react to the message (immediately). However, unlike all the previous uses of the copycat-like arguments in the context of population protocols and similar models, proving the necessary copycat-like property for a general notion of protocols with unreliable communication (sufficient to handle assymetry of message loss where loss for sender requires loss for receiver) requires careful analysis using different techniques.

Note that although the natural way to design population protocols for our setting involves the use of immediate observation population protocols, we still need to rule out additional opportunities arising from the fact that eventually a two-agent interaction with both agents correctly updated will happen. However, in contrast to self-stabilising protocols [16, 6], the protocols cannot rely on the message loss being absent for an arbitrarily long time.

Surprisingly, asynchronous transmission and receipt of messages, which provides more expressive power than immediate observation population protocols in the reliable setting, turns out to have strictly less expressive power in the unreliable setting. Note that message reordering is allowed already in the reliable setting, while unreliability is essentially a generalisation of message loss. One could say that an unbounded delay in message delivery becomes a liability instead of an asset once there is message loss.

The rest of the present paper is organised as follows. First, in Section 2 we define a general protocol framework generalising many previously studied approaches. Then in Section 3 we summarise the results from the literature on the expressive power of various models covered by this framework. Afterwards in Section 4 we formally define our general notion of a protocol with unreliable communication. Then in Section 5 we formalise the common limitation of all the protocols with unreliable communication, and provide the proof sketches of this restriction and the main result. Afterwards in Section 6 we show that fully asynchronous (message-based) models become strictly less powerful than immediate observation in the unreliable setting. The paper ends with a brief conclusion and some possible future directions.

1.1 Main results (preview)

The precise statements of our results require the detailed definitions introduced later. However, we can roughly summarise them as follows.

First, we characterise the expressive power of all fixed-memory protocols given unreliable comunication.

Proposition 1**.**

Adding unreliability of communication to population protocols restricts the predicates they can express to boolean combinations of comparisons of arguments with constants.

This is the same expressive power as the immediate observation protocols.

Next we show that unreliability changes the expressive power non-monotonically for some natural classes.

Proposition 2**.**

Queued transmission protocols with unreliable communication are strictly less expressive than immediate observation population protocols (with or without unreliable communication).

Note that without unreliability queued transmission protocols are strictly more expressive than immediate observation population protocols.

2 Basic definitions

2.1 Protocols

We consider various models of distributed computation where the number of agents is constant during protocol execution, each agent has a constant amount of local storage, and agents cannot distinguish each other except via the states. We provide a general framework for describing such protocols. Note that we omit some very natural restrictions (such as decidability of correctness of a finite execution) because they are irrelevant for the problems we study. We allow agents to be distinguished and tracked individually for the purposes of analysis, even though they cannot identify each other during the execution of the protocol.

We will use the following problem to illustrate our definitions: the agents have states $q_{0}$ and $q_{1}$ corresponding to input symbols [math] and $1$ and aim to find out if all the agents have the same input. They have an additional state $q_{\bot}$ to represent the observation that both input symbols were present. We will define four protocols for this problem using different communication primitives.

• •

Two agents interact and both switch to $q_{\bot}$ unless they are in the same state (population protocol interaction).

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An agent observes another agent and switches to $q_{\bot}$ if they are in different states (immediate observation).

• •

An agent can send a message with its state, $q_{0}$, $q_{1}$ or $q_{\bot}$. An agent in a state $q_{0}$ or $q_{1}$ can receive a message (any of the pending messages, regardless of order); the agent switches to $q_{\bot}$ if the message contains a state different from its own (queued transmission).

• •

An agent broadcasts its state without changing it; each other agent receives the broadcast simultaneouly and switches to $q_{\bot}$ if its state is different from the broadcast state (broadcast protocol interaction).

Definition 1**.**

A protocol is specified by a tuple $(Q,M,\Sigma,I,o,\mathrm{Tr},\Phi)$, with components being a finite nonempty set $Q$ of (individual agent) states, a finite (possibly empty) set $M$ of messages, a finite nonempty input alphabet $\Sigma$, an input mapping function $I:\Sigma\to Q$, an individual output function $o:Q\to\{true,false\}$, a transition relation $\mathrm{Tr}$ (which is described in more details below), and a fairness condition $\Phi$ on executions.

The protocol defines evolution of populations of agents (possibly with some message packets being present).

Definition 2**.**

A population is a pair of sets: $A$ of agents and $P$ of packets. A configuration $C$ is a population together with two functions, $C_{A}:A\to Q$ provides agent states, and $C_{P}:P\to M$ provides packet contents.

Note that if $M$ is empty, then $P$ must also be empty. As the set of agents is the domain of the function $C_{A}$, we use the notation $\mathrm{Dom}(C_{A})$ for it. The same goes for the set of packets $\mathrm{Dom}(C_{P})$. Without loss of generality $\mathrm{Dom}(C_{P})$ is a subset of a fixed countable set of possible packets.

The message packets are only used for asynchronous communication; instant interaction between agents (such as in the classical rendezvous-based population protocols or in broadcast protocols) does not require describing the details of communication in the configurations.

Example 1**.**

The four example protocols have the same set of states $Q=\{q_{0},q_{1},q_{\bot}\}$. The first two protocols have the empty set of messages, and the last two have the set of messages $M=\{m_{0},m_{1},m_{\bot}\}$. The example protocols all have the same input alphabet $\Sigma=\{0,1\}$, input mapping $I:i\mapsto q_{i}$, and output mapping $o:q_{0}\mapsto\mathrm{true},q_{1}\mapsto\mathrm{true},q_{\bot}\mapsto\mathrm{false}$.

The definition of the transition relation uses the following notation.

Definition 3**.**

For a function $f$ and $x\notin\mathrm{Dom}(f)$ let $f\cup\{x\mapsto y\}$ denote the function $g$ defined on $\mathrm{Dom}(f)\cup\{x\}$ such that $g\mid_{\mathrm{Dom}(f)}=f$ and $g(x)=y$. For $u\in\mathrm{Dom}(f)$ let $f[u\mapsto v]$ denote the function $h$ defined on $\mathrm{Dom}(f)$ such that $h\mid_{\mathrm{Dom}(f)\setminus\{u\}}=f\mid_{Dom(f)\setminus\{u\}}$ and $h(u)=v$. For symmetry, if $w=f(u)$ let $f\setminus\{u\mapsto w\}$ denote restriction $f\mid_{\mathrm{Dom}(f)\setminus\{u\}}$.

Use of this notation implies an assertion of correctness, i.e. $x\notin\mathrm{Dom}(f)$, $u\in\mathrm{Dom}(f)$, and $w=f(u)$. We use the same notation with a configuration $C$ instead of a function if it is clear from context whether $C_{A}$ or $C_{P}$ is modified.

Now we can describe the transition relation that tells us which configurations can be obtained from a given one via a single interaction. In order to cover broadcast protocols we define the transition relation as a relation on configurations. The restrictions on the transition relation ensure that the protocol behaves like a distributed system with arbitrarily large number of anonymous agents.

Definition 4**.**

The transition relation of a protocol is a set of triples $(C,A^{\odot},C^{\prime})$, called transitions, where $C$ and $C^{\prime}$ are configurations and $A^{\odot}\subset\mathrm{Dom}(C_{A})$ is the set of active agents (of the transition); agents in $\mathrm{Dom}(C_{A})\setminus A^{\odot}$, are called passive. We write $C\xrightarrow{A^{\odot}}C^{\prime}$ for $(C,A^{\odot},C^{\prime})\in\mathrm{Tr}$, and let $C\to C^{\prime}$ denote the projection of $\mathrm{Tr}$: $C\to{}C^{\prime}\Leftrightarrow{}\exists{A^{\odot}}:C\xrightarrow{A^{\odot}}C^{\prime}$. The transition relation must satisfy the following conditions for every transition $C\xrightarrow{A^{\odot}}C^{\prime}$:

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Agent conservation. $\mathrm{Dom}(C_{A})=\mathrm{Dom}(C^{\prime}_{A})$.

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Agent and packet anonymity. If $h_{A}$ and $h_{P}$ are bijections such that $D_{A}=C_{A}\circ h_{A}$, $D^{\prime}_{A}=C^{\prime}_{A}\circ h_{A}$, $D_{P}=C_{P}\circ h_{P}$, and $D^{\prime}_{P}=C^{\prime}_{P}\circ h_{P}$, then $D\xrightarrow{h^{-1}(A^{\odot})}D^{\prime}$.

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Possibility to ignore extra packets. For every $p\notin\mathrm{Dom}(C_{P})\cup\mathrm{Dom}(C^{\prime}_{P})$ and $m\in M$: $C\cup\{p\mapsto m\}\xrightarrow{A^{\odot}}C^{\prime}\cup\{p\mapsto m\}$.

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Possibility to add passive agents. For every agent $a\notin\mathrm{Dom}(C_{A})$ and $q\in Q$ there exists $q^{\prime}\in Q$ such that: $C\cup\{a\mapsto q\}\xrightarrow{A^{\odot}}C^{\prime}\cup\{a\mapsto q^{\prime}\}$.

Informally speaking, the active agents are the agents that transmit something during the interaction. The passive agents can still observe other agents and change their state. The choice of active agents is used for the definition of protocols with unreliable communication, as a failure to transmit precludes success of reception. The formal interpretation will be provided in Definition 13.

Many models studied in the literature have the transition relation defined using pairwise interaction. In these models the transitions are always changing the states of two agents based on their previous states. When discussing such protocols, we will use the notation $(p,q)\to(p^{\prime},q^{\prime})$ for a transition where agents in the states $p$ and $q$ switch to states $p^{\prime}$ and $q^{\prime}$, correspondingly.

Example 2**.**

The four example protocols have the following transition relations.

• •

In the first protocol for a configuration $C$ and two agents $a,a^{\prime}\in\mathrm{Dom}(C_{A})$ such that $C_{A}(a)\neq C_{A}(a^{\prime})$ we have $C\xrightarrow{\{a,a^{\prime}\}}C[a\mapsto q_{\bot}][a^{\prime}\mapsto q_{\bot}]$ (in other notation, $(C,\{a,a^{\prime}\},C[a\mapsto q_{\bot}][a^{\prime}\mapsto q_{\bot}])\in\mathrm{Tr}$).

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In the second protocol for a configuration $C$ and two agents $a,a^{\prime}\in\mathrm{Dom}(C_{A})$ such that $C_{A}(a)\neq C_{A}(a^{\prime})$ we have $C\xrightarrow{\{a\}}C[a\mapsto q_{\bot}]$. We can say that $a$ observes $a^{\prime}$ in a different state and switches to $q_{\bot}$.

• •

In the third protocol there are two types of transitions. Let a configuration $C$ be fixed. For an agent $a\in\mathrm{Dom}(C_{A})$, $i\in\{0,1,\bot\}$ such that $C_{A}(a)=q_{i}$, and a new message identity $p\notin\mathrm{Dom}(C_{P})$ we have $C\xrightarrow{\{a\}}C\cup\{p\mapsto m_{i}\}$ (sending a message). If $C_{A}(a)=q_{i}$ for some $i\in\{0,1\}$, for each message $p\in\mathrm{Dom}(C_{P})$, we also have $C\xrightarrow{\{a\}}C[a\mapsto q^{\prime}]\setminus\{p\mapsto C_{P}(p)\}$ where $q^{\prime}$ is equal to $q_{i}$ if $C_{P}(p)=m_{i}$ and $q_{\bot}$ otherwise (receiving a message).

• •

In the fourth protocol, for a configuration $C$ and an agent $a\in\mathrm{Dom}(C_{A})$ we can construct $C^{\prime}$ by replacing $C_{A}$ with $C^{\prime}_{A}$ that maps each $a^{\prime}\in\mathrm{Dom}(C_{A})$ to $C_{A}(a^{\prime})$ if $C_{A}(a)=C_{A}(a^{\prime})$ and $q_{\bot}$ otherwise. Then we have $C\xrightarrow{\{a\}}C^{\prime}$. We can say that $a$ broadcasts its state and all the agents in the other states switch to $q_{\bot}$.

2.2 Definitions of the protocol classes studied in the literature

Many previously studied models can be defined inside out framework. Among such models are population protocols, immediate transmission population protocols, immediate observation population protocols, queued transmission protocols, broadcast protocols. These general definitions are similar to the definitions for specific protocols provided as exampled, and our results do not depend on these definitions. We provide them as a corroboration of sufficient generality of our framework.

First we translate the initial definition of the population protocols [1].

Definition 5**.**

A population protocol is described by an interaction relation $\delta\subseteq Q^{2}\times Q^{2}$. The set of messages is empty. A configuration $C^{\prime}$ can be obtained from $C$, if there are agents $a_{1},a_{2}\in\mathrm{Dom}(C_{A})$ and states $q_{1},q_{2},q_{3},q_{4}\in Q$ such that $C_{A}(a_{1})=q_{1}$, $C_{A}(a_{2})=q_{2}$, $C^{\prime}=C[a_{1}\mapsto q_{3}][a_{2}\mapsto q_{4}]$, and $((q_{1},q_{2}),(q_{3},q_{4}))\in\delta$. The set of active agents $A^{\odot}$ is $\{a_{1},a_{2}\}$.

Now we proceed to the variants of the population protocols appearing in the paper on expressive power of population protocols and their variants [5].

Definition 6**.**

An immediate transmission population protocol is a population protocol such that $q_{3}$ depends only on $q_{1}$, i.e. the following two conditions hold. If $((q_{1},q_{2}),(q_{3},q_{4}))\in\delta$ and $((q_{1},q^{\prime}_{2}),(q^{\prime}_{3},q^{\prime}_{4}))\in\delta$ then $q_{3}=q^{\prime}_{3}$. If $((q_{1},q_{2}),(q_{3},q_{4}))\in\delta$ then for every $q^{\prime}_{2}$ there exists $q^{\prime}_{4}$ such that $((q_{1},q^{\prime}_{2}),(q_{3},q^{\prime}_{4}))\in\delta$.

Definition 7**.**

An immediate observation population protocol is an immediate transmission population protocol such that every possible interaction $((q_{1},q_{2}),(q_{3},q_{4}))\in\delta$ has $q_{1}=q_{3}$.

We can consider only the first agent to be active.

Definition 8**.**

Queued transmission protocol has a nonempty set $M$ of messages. It has two transition relations: $\delta_{s}\subseteq Q\times(Q\times M)$ describing sending the messages, and $\delta_{r}\subseteq(Q\times M)\times Q$ describing receiving the messages. If agent $a$ has state $q=C_{A}(a)$ and $(q,(q^{\prime},m))\in\delta_{s}$, it can send a message $m$ as a fresh packet $p$ and switch to state $q^{\prime}$: $C\xrightarrow{\{a\}}C[a\mapsto q^{\prime}]\cup\{p\mapsto m\}$. If agent $a$ has state $q=C_{A}(a)$, packet $p$ contains message $m=C_{P}(p)$ and $((q,m),q^{\prime}))\in\delta_{r}$, agent $a$ can receive the message: $C\xrightarrow{\{a\}}C[a\mapsto q^{\prime}]\setminus\{p\mapsto m\}$.

Delayed transmission protocol is a queued transmission protocol where every message can always be received by every agent, i.e. the projection of $\delta_{r}$ to $Q\times M$ is the entire $Q\times M$.

Delayed observation protocol is a delayed transmission protocol where sending a message doesn’t change state, i.e. $(q,(q^{\prime},m))\in\delta_{s}$ implies $q=q^{\prime}$.

As the last example, we consider broadcast protocols [18].

Definition 9**.**

Broadcast protocol is defined by two relations: $\delta_{s}\subseteq Q\times Q$ describing a sender transition, and $\delta_{r}\subseteq(Q\times Q)\times Q$. To perform a transition from a configuration $C$, we pick an agent $a\in{}\mathrm{Dom}(C_{A})$ with state $q$ and change its state to $q^{\prime}$ such that $(q,q^{\prime})\in\delta_{s}$. At the same time, we simultaneously update the state of all other agents, in such a way that an agent in state $q_{j}$ can switch to any state $q^{\prime}_{j}$ such that $((q_{j},q),q^{\prime}_{j})\in\delta_{r}$.

We consider the transmitting agent to be the only active one.

Remark 1*.*

In the literature, the relations $\delta$, $\delta_{s}$, $\delta_{r}$ and $\delta_{s}$ are sometimes required to be partial functions. As we use relations in the general case, we use relations here for consistency.

2.3 Fair executions

In this section we define the notion of fairness. This notion is traditionally used to exclude the most pathological cases without a complete probabilistic analysis of the model. For the population protocols fairness has been a part of the definition since the introduction [1, 2]. However, in the general study of distributed computation there has long been some interest in comparing effects of different approaches to fairness in execution scheduling [7]. For example, the distinction between weak fairness and strong fairness and the conditions where one can be made to model the other has been studied in [25]. The difference between weak and strong scheduling is that strong fairness executes infinitely often every interaction that is enabled infinitely often, while weak fairness only guarantees anything for continuously enabled interactions. As there are multiple notion of fairness in use, we define their basic common traits. Our results hold for all notions of fairness satisfying these basic requirements, including all the notions of fairness used in the literature, as well as much stronger and much weaker fairness conditions.

Definition 10**.**

An execution is a sequence (finite or infinite) $C_{n}$ of configurations such that at each moment $i$ either nothing changes, i.e. $C_{i}=C_{i+1}$ or a single interaction occurs, i.e. $C_{i}\to C_{i+1}$. A configuration $C^{\prime}$ is reachable from configuration $C$ if there exists an execution $C_{0},\ldots,C_{n}$ with $C_{0}=C$ and $C_{n}=C^{\prime}$ (and unreachable otherwise).

A protocol defines a fairness condition $\Phi$ which is a predicate on executions. It should satisfy the following properties.

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A fairness condition is eventual, i.e. every finite execution can be continued to an infinite fair execution.

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A fairness condition ensures activity, i.e. if an execution contains only configuration $C$ after some moment, only $C$ itself is reachable from $C$.

Definition 11**.**

The default fairness condition accepts an execution if every configuration either becomes unreachable after some moment, or occurs infinitely many times.

Example 3**.**

The example protocols use the default fairness condition.

It is clear that the default fairness condition ensures activity.

Lemma 1** **(adapted from

[5]).

Default fairness condition is eventual.

Proof.

Consider a configuration after a finite execution. Then there is a countable set of possible configurations (note that the set of potential packets is at most countable). Consider an arbitrary enumeration of configurations that mentions each configuration infinitely many times.

We repeat the following procedure: skip unreachable configurations in the enumeration, then perform the transitions necessary to reach the next reachable one. If we skip a configuration, it can never become reachable again. Therefore all the configurations that stay reachable infinitely long are never skipped and therefore they are reached infinitely many times. ∎

The fairness condition is sometimes said to be an approximation of probabilistic behaviour. In our general model the default fairness condition provides executions similar to random ones for protocols without messages but not always for protocols with messages. The arguments from [20] with minimal modification prove this. The core idea in the case without messages is observing we have a finite state space reachable from any given configuration; a random walk eventually gets trapped in some strongly connected component, visiting all of its states infinitely many time. If we do have messages, the message count might behave like a biased random walk; while consuming all the messages stays possible in principle, with probability one it only happens a finite number of times.

2.4 Functions implemented by protocols

In this section we recall the standard notion of a function evaluated by a protocol. Here the standard definition generalises trivially.

Definition 12**.**

An input configuration is a configuration where there are no packets and all agents are in input states, i.e. $P=\varnothing$ and $\mathrm{Im}(C_{A})\subseteq\mathrm{Im}(I)$ where $\mathrm{Im}$ denotes the image of a function. We extend $I$ to be applicable to multisets of input symbols. For every $\overline{x}\in\mathbb{N}^{\Sigma}$, we define $I(\overline{x})$ to be a configuration of $|\overline{x}|$ agents with $\sum_{I(\sigma)=q_{i}}\overline{x}(\sigma)$ agents in input state $q_{i}$ (and no packets).

A configuration $C$ is a consensus if the individual output function yields the same value for the states of all agents, i.e. $\forall a,a^{\prime}\in\mathrm{Dom}(C_{A}):o(C_{A}(a))=o(C_{A}(a^{\prime}))$. This value is the output value for the configuration. $C$ is a stable consensus if all configurations reachable from $C$ are consensus configurations with the same value.

A protocol implements a predicate $\varphi:\mathbb{N}^{\Sigma}\to\{true,false\}$ if for every $\overline{x}\in\mathbb{N}^{\Sigma}$ every fair execution starting from $I(\overline{x})$ reaches a stable consensus with the output value $\varphi(\overline{x})$. A protocol is well-specified if it implements some predicate.

Example 4**.**

It is easy to see that each of the four example protocols implements the predicate $\varphi(\overline{x})\Leftrightarrow(x(0)=0)\vee(x(1)=0)$ on $\mathbb{N}^{2}$. In other words, the protocol accepts the input configurations where one of the two input states has zero agents and rejects the configurations where both input states occur.

This framework is general enough to define the models studied in the literature, such as population protocols, immediate transmission protocols, immediate observation population protocols, delayed transmission protocols, delayed observation protocols, queued transmission protocols, and broadcast protocols.

3 Expressive power of population protocols and related models

In this section we give an overview of previously known results on expressive power of various models related to population protocols. We only consider predicates, i.e. functions with the output values being $\mathrm{true}$ and $\mathrm{false}$ because the statements of the theorems become more straightforward in that case.

The expressive power of models related to population protocols is expressed in terms of semilinear, $\mathbf{coreMOD}$, and counting predicates. Semilinear predicates on tuples of natural numbers can be expressed using the addition function, remainders modulo constants, and the order relation, such as $x+x\geq{}y+3$ or $x\mod 7=3$. Roughly speaking, $\mathbf{coreMOD}$ is the class of predicates that become equivalent to modular equality for inputs with only large and zero components. An example could be $(z=1\wedge x\geq{}y)\vee(x+y\mod 2=0)$, a semilinear predicate which becomes a modular equality whenever $z=0$ or $z$ is large (i.e. $z\geq{}2$). Counting predicates are logical combinations of inequalities including one coordinate and one constant each, for example, $x\geq{3}$.

Theorem 1** (see [5] for details).**

Population protocols and queued transmission protocols can implement precisely semilinear predicates.

Immediate transmission population protocols and delayed transmission protocols can implement precisely all the semilinear predicates that are also in $\mathbf{coreMOD}$.

Immediate observation population protocols implement counting predicates.

Delayed observation protocols implement the counting predicates where every constant is equal to $1$.

Theorem 2** (see [9] for details).**

Broadcast protocols implement precisely the predicates computable in nondeterministic linear space.

4 Our models

4.1 Proposed models

We propose a general notion of an unreliable communication version of a protocol. Our notion models transient failures, so the set of agents is preserved. The intuition we formalise is the idea that for every possible transition some agents may fail to update their states (and keep their corresponding old states). We also require that for some passive agent to receive a transmission, the transmission has to occur (and active agents who transmit do not update their state if they fail to transmit, although a successful transmission can still fail to be received).

Definition 13**.**

A protocol with unreliable communication, corresponding to a protocol $\mathcal{P}$, is a protocol that differs from $\mathcal{P}$ only in the transition relation. For every allowed transition $C\xrightarrow{A^{\odot}}C^{\prime}$ we also allow all the transitions $C\xrightarrow{A^{\odot}}C^{\prime\prime}$ where $C^{\prime\prime}$ satisfies the following conditions.

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Population preservation. $\mathrm{Dom}(C^{\prime\prime}_{A})=\mathrm{Dom}(C^{\prime}_{A})$, $\mathrm{Dom}(C^{\prime\prime}_{P})=\mathrm{Dom}(C^{\prime}_{P})$.

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State preservation. For every agent $a\in\mathrm{Dom}(C^{\prime\prime}_{A})$: $C^{\prime\prime}_{A}(a)\in\{C_{A}(a),C^{\prime}_{A}(a)\}$.

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Message preservation. For every packet $p\in\mathrm{Dom}(C^{\prime\prime}_{P})$: $C^{\prime\prime}_{P}(p)=C^{\prime}_{P}(p)$.

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Reliance on active agents. Either for every agent $a\notin A^{\odot}$ we have $C^{\prime\prime}_{A}(a)=C_{A}(a)$, or for every agent $a\in A^{\odot}$ we have $C^{\prime\prime}_{A}(a)=C^{\prime}_{A}(a)$.

Example 5**.**

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Population protocols with unreliable communication allow an interaction to update the state of only one of the two agents.

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Immediate transmission population protocols with unreliable communication allow the sender to update the state with no receiving agents.

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Immediate observation population protocols with unreliable communication do not differ from ordinary immediate observation population protocols, because each transition changes the state of only one agent. Failing to change the state means a no-change transition which is already allowed anyway.

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Queued transmission protocols with unreliable communication allow messages to be discarded with no effect. Note that for delayed observation protocols unreliable communication doesn’t change much, as sending the messages also has no effect.

• •

Broadcast protocols with unreliable communication allow a broadcast to be received by an arbitrary subset of agents.

4.2 The main result

Our main result is that no class of protocols with unreliable communication can be more expressive than immediate observation protocols.

Definition 14**.**

A cube is a subset of $\mathbb{N}^{k}$ defined by a lower and upper (possibly infinite) bound for each coordinate. A counting set is a finite union of cubes.

A counting predicate is a membership predicate for some counting set. Alternatively, we can say it is a predicate that can be computed using comparisons of input values with constants and logical operations.

Theorem 3**.**

The set of predicates that can be implemented by protocols with unreliable communication is the set of counting predicates. All counting predicates can be implemented by (unreliable) immediate observation protocols.

5 Proof of the main result

Our main lemma is generalises of the copycat lemma normally applied to specific models such as immediate observation protocols. The idea is that for every initial configuration there is a fair execution that can be extended to a possibly unfair execution by adding a copy of a chosen agent. In some special cases, for example, broadcast protocols with unreliable communication, a simple proof can be given by saying that if the original agent participates in an interaction, the copy should do the same just before the original without anyone ever receiving the broadcasts from the copy. The copycat arguments are usually applied to models where a similar proof suffices. The situation is more complex for models like immediate transmission protocols with unreliable communication. As a message cannot be received without being sent, the receiver cannot update its state if the sender doesn’t. We present an argument applicable in the general case.

Definition 15**.**

Let $E$ be an arbitrary execution of protocol $P$ with initial configuration $C$. Let $a\in{}\mathrm{Dom}(C_{A})$ be an agent in this execution. Let $a^{\prime}\notin\mathrm{Dom}(C_{A})$ be an agent, and $C^{\prime}=C\cup\{a^{\prime}\mapsto C_{A}(a)\}$. A set $\mathfrak{E}_{a}$ of executions starting in configuration $C^{\prime}$ is a shadow extension of the execution $E$ around the agent $a$ if the following conditions hold:

• •

removing $a^{\prime}$ from each configuration in any execution in $\mathfrak{E}_{a}$ yields $E$;

• •

for each moment during the execution, there is a corresponding execution in $\mathfrak{E}_{a}$ such that $a$ and $a^{\prime}$ have the same state at that moment.

The added agent $a^{\prime}$ is a shadow agent, and elements of $\mathfrak{E}_{a}$ are shadow executions. A protocol $P$ is shadow-permitting if for every configuration $C$ there is a fair execution starting from $C$ that has a shadow extension around each agent $a\in{}\mathrm{Dom}(C_{A})$.

Note that the executions in $\mathfrak{E}_{a}$ might not be fair even if $E$ is fair.

Not all population protocols are shadow-permitting. For example, consider a protocol with one input state $q_{0}$, additional states $q_{+}$ and $q_{-}$, and one transition $(q_{0},q_{0})\to(q_{+},q_{-})$. As the number of agents in the states $q_{+}$ and $q_{-}$ is always the same, one can’t add a single extra agent going from state $q_{0}$ to state $q_{+}$.

Lemma 2**.**

All protocols with unreliable communication are shadow-permitting.

The intuition behind the proof is the following. We construct a fair execution together with the shadow executions and keep track what states can be reached by the shadow agents. The set of reachable states will not shrink, as the shadow agent can always just fail to update. If an agent $a$ tries to move from a state $q$ to a state $q^{\prime}$ not reachable by the corresponding shadow agent in any of the shadow executions, we “split” the shadow execution reaching $q$: one copy just stays in place, and in the other the shadow agent $a^{\prime}$ takes the place of $a$ in the transition while $a$ keeps the old state. In the main execution there is no $a^{\prime}$ so $a$ participates in the interaction but fails to update. Afterwards we restart the process of building a fair execution.

Proof.

We construct an execution and the families $\mathfrak{E}_{a}$ in parallel, then show that the resulting execution $E$ is fair. We say that a state $q$ is $a$-reachable after $k$ transitions, if there is an execution in $\mathfrak{E}_{a}$ such that $a^{\prime}$ has state $q$ after $k$ transitions. The goal of the construction is to ensure that the set of $a$-reachable states grows as $k$ increases and contains the state of $a$ after $k$ transitions.

Consider an initial configuration $C$. We build the execution $E$ and its shadow extensions $\mathfrak{E}_{a}$ for each $a\in\mathrm{Dom}(C_{A})$ step by step. Initially, $E=(C)$ and $\mathfrak{E}_{a}$ has exactly one execution, namely $(C\cup\{a^{\prime}\mapsto{}C_{A}(a)\})$. We pick an arbitrary fair continuation $E^{\infty}$ starting with $E$.

At each step we extend $E=(E_{0}=C,E_{1},\ldots,E_{k})$ by one configuration and update $\mathfrak{E}_{a}$ for each $a\in\mathrm{Dom}(C_{A})$. Consider the next configuration in $E^{\infty}$, which we can denote $E^{\infty}_{k+1}$. By definition there exists a set of agents $A^{\odot}$ such that $E^{\infty}_{k}\xrightarrow{A^{\odot}}E^{\infty}_{k+1}$. We consider the following cases.

Case 1: For each agent $a$ the state $E^{\infty}_{k+1}(a)$ is $a$-reachable (after $k$ transitions).

We set $E_{k+1}=E^{\infty}_{k+1}(a)$ and keep the same $E^{\infty}$. In other words, we just copy the next transition from $E^{\infty}$. Then for each agent $a\in\mathrm{Dom}(C_{A})$ and for each $E^{\prime}_{a}\in\mathfrak{E}_{a}$ we set $(E^{\prime}_{a})_{k+1}=E_{k+1}\cup\{a^{\prime}\mapsto(E^{\prime}_{a})_{k}(a^{\prime})\}$, i.e. say that $a^{\prime}$ fails to update its state.

Case 2: For each active agent $a^{\odot}\in{}A^{\odot}$ the state $E^{\infty}_{k+1}(a^{\odot})$ is $a^{\odot}$-reachable, but there is a passive agent $a\notin{}A^{\odot}$ such that the state $E^{\infty}_{k+1}(a)$ is not $a$-reachable (after $k$ transitions).

We construct $E_{k+1}$ such that $E_{k+1}(a^{\odot})=E^{\infty}_{k+1}(a^{\odot})$ for each active $a^{\odot}\in{}A^{\odot}$, and $E_{k+1}(a)=E_{k}(a)$ for each passive agent $a\in\mathrm{Dom}(C_{A})\setminus{}A^{\odot}$. In other words, all the active agents perform the update, but all the passive agents fail to update. The message packets are still consumed or created as if we performed the transition $E_{k}=E^{\infty}_{k}\xrightarrow{A^{\odot}}E^{\infty}_{k+1}$, i.e. $(E_{k+1})_{P}=(E^{\infty}_{k+1})_{P}$. As $E^{\infty}$ is now not a continuation of $E$, we replace $E^{\infty}$ with an arbitrary fair continuation of our new $E$. Then for each $E^{\prime}_{a}\in\mathfrak{E}_{a}$ we set $(E^{\prime}_{a})_{k+1}=E_{k+1}\cup\{a^{\prime}\mapsto(E^{\prime}_{a})_{k}(a^{\prime})\}$ like in the previous case. Also, for each passive agent $a\in\mathrm{Dom}(C_{A})\setminus{}A^{\odot}$ we add a trajectory $E^{\prime\prime}_{a}$ to $\mathfrak{E}_{a}$ obtained by modifying an existing trajectory $E^{\prime}_{a}\in\mathfrak{E}_{a}$ such that $(E^{\prime}_{a})_{k}(a^{\prime})=(E^{\prime}_{a})_{k}(a)$. We set $(E^{\prime\prime}_{a})_{k+1}(a^{\prime})=E^{\infty}_{k+1}(a)$, and keep everything else the same as in $E^{\prime}_{a}$. In other words, we make $a^{\prime}$ perform the update that $a$ would perform in $E^{\infty}$.

Case 3: There is an active agent $a\in{}A^{\odot}$ such that the state $E^{\infty}_{k+1}(a)$ is not $a$-reachable (after $k$ transitions).

We set $(E_{k+1})_{A}=(E_{k})_{A}$, i.e. we say that all the agents fail to update. The message packets are still consumed or created as if we performed the transition $E_{k}=E^{\infty}_{k}\xrightarrow{A^{\odot}}E^{\infty}_{k+1}$, i.e. $(E_{k+1})_{P}=(E^{\infty}_{k+1})_{P}$. As $E^{\infty}$ is now not a continuation of $E$, we replace $E^{\infty}$ with an arbitrary fair continuation of our new $E$. Then for each $E^{\prime}_{a}\in\mathfrak{E}_{a}$ we set $(E^{\prime}_{a})_{k+1}=E_{k+1}\cup\{a^{\prime}\mapsto(E^{\prime}_{a})_{k}(a^{\prime})\}$ (like in the previous two cases). Also, for each active agent $a\in{}A^{\odot}$ we add a trajectory $E^{\prime\prime}_{a}$ to $\mathfrak{E}_{a}$ obtained by modifying an existing trajectory $E^{\prime}_{a}\in\mathfrak{E}_{a}$ such that $(E^{\prime}_{a})_{k}(a^{\prime})=(E^{\prime}_{a})_{k}(a)$. We set $(E^{\prime\prime}_{a})_{k+1}(a^{\prime})=E^{\infty}_{k+1}(a)$, and keep everything else the same as in $E^{\prime}_{a}$. In other words, we allow $a^{\prime}$ to update its state in the way $a$ would do in $E^{\infty}$.

We now prove that the above construction is always correctly defined and yields a fair execution $E$ together with shadow extensions around each agent.

First we show that we always continue $E$ in a valid way, i.e. $E_{k}\xrightarrow{A^{\odot}}E_{k+1}$. In the first case it is true by construction as $E_{k}=E^{\infty}_{k}$ and $E_{k+1}=E^{\infty}_{k+1}$. In the second and the third case, we modify the states of some agents in the second configuration of a valid transition $E^{\infty}_{k}\xrightarrow{A^{\odot}}E^{\infty}_{k+1}$ by assigning them the states from the first configuration. Such changes clearly cannot violate population preservation and message preservation. State preservation is satisfied because we replace the agent’s state in the second configuration with the state from the first configuration. The case split between the cases 2 and 3 ensures reliance on active agents; we either make sure that all the active agents update their state, or none of them. Therefore, all the conditions of the Definition 13 are satisfied and the changed transition is also present in the protocol with unreliable communication.

As the updated execution $E$ is a valid finite execution, we can find a fair continuation $E^{\infty}$ as the fairness condition is eventual.

When we extend the executions in the shadow extensions by repeating the same state, we just use possibility to add passive agents to add $a^{\prime}$ to the valid transition from $E$, then observe that making a passive agent fail to update is always allowed in an protocol with unreliable communication.

When we add new trajectories in cases 2 and 3, we use possibility to add passive agents to add $a^{\prime}$ to the valid transition from $E$, then we use agent anonymity to swap the state changes of $a$ and $a^{\prime}$, then we use unreliability to make the (passive) agent $a$ fail to update the state, as well as either all the passive or all the agents from $\mathrm{Dom}(C_{A})$.

So far we know that the construction can be performed and yields a valid execution $E$ and some valid executions in each $\mathfrak{E}_{a}$. Now we check that each $\mathfrak{E}_{a}$ is a shadow extension around $a$, and $E$ is fair. We observe that our construction indeed only increases the set of $a$-reachable states as the number of transitions grows. Furthermore, at each step either agent $a$ moves to an $a$-reachable state, or $a$ stays in an $a$-reachable state, thus $\mathfrak{E}_{a}$ is indeed a shadow extension around the agent $a$. Whenever the fair continuation $E^{\infty}$ is changed, for at least one agent $a$ the set of $a$-reachable states strictly increases. As the set of agents is finite and cannot change by agent conservation, and the set of states is finite, all but a finite number of steps correspond to the case 1. Therefore from some point on $E^{\infty}$ does not change and $E$ coincides with it, and therefore $E$ is fair.

This concludes the proof of the lemma. ∎

We also use a straightforward generalisation of the truncation lemma from [5]. The lemma says that all large amounts of agents are equivalent for the notion of stable consensus.

Definition 16**.**

A protocol is truncatable if there exists a number $K$ such that for every stable consensus adding an extra agent with a state $q$ that is already represented by at least $K$ other agents yields a stable consensus.

Lemma 3** (adapted from [5]).**

All protocols (not necessarily with unreliable communication) are truncatable.

Proof.

Every configuration can be summarised by an element of $\mathbb{N}^{Q\cup{}M}$ (each state is mapped to the number of agents in this state, each message is mapped to the number of packets with this message). In other words, we can forget the identities and consider the multiset of states and messages. If a configuration is a consensus (correspondingly, stable consensus), all the configurations with the same multiset of states and messages are also consensus configurations (correspondingly, stable consensus configurations). The set $\overline{ST}$ of elements of $\mathbb{N}^{Q\cup{}M}$ not representing stable consensus configurations is upwards closed, because reaching a state with a different local output value cannot be impeded by adding agents or packets. Indeed, if we can reach a configuration $C_{\overline{ST}}$ with some state $q$ present, we can always use addition of passive agents to each transition of the path and still have a path of valid transitions from a larger configuration to some configuration $C^{*}_{\overline{ST}}$ with state $q$ still present. By possibility to ignore extra packets, we can also allow additional packets in the initial configuration. By Dickson’s lemma, the set $\overline{ST}$ of non-stable-consensus state multisets has a finite set of minimal elements $\overline{ST}_{min}$. We can take $K$ larger than all coordinates of all minimal elements. Then adding more agents with the state that already has at least $K$ agents leads to increasing a component larger than $K$ in the multiset of states. This cannot change any component-wise comparisons with multisets from $\overline{ST}_{min}$, and therefore belonging to $\overline{ST}$ and being or not a stable consensus. ∎

Remark 2*.*

A specific bound on the truncation threshold $K$ can be obtained using the Rackoff’s bound for the size of configuration necessary for covering in general Vector Addition Systems [27].

Lemma 4**.**

If a predicate $\varphi$ can be implemented by a shadow-permitting truncatable protocol, then $\varphi$ is a counting predicate.

Proof.

Let $K$ be the truncation constant. We claim that $\varphi$ can be expressed as a combination of threshold predicates with thresholds no larger than $|Q|\times K$.

More specifically, we prove an equivalent statement: adding $1$ to an argument already larger than $|Q|\times K$ doesn’t change the output value of $\varphi$. Let us call the state corresponding to this argument $q$. Indeed, consider any corresponding input configuration. We can build a fair execution starting in it with shadow extensions around each agent. As the predicate is correctly implemented, this fair execution has to reach a stable consensus. By assumption (and pigeonhole principle), more than $K$ agents from the state $q$ end up in the same state. By definition of shadow extension, there is an execution starting with one more agent in the state $q$, and reaching the same stable consensus but with one more agent in a state with more than $K$ other ones (which doesn’t break the stable consensus). Continuing this finite execution to a fair execution we see that the value of $\varphi$ must be the same. This concludes the proof. ∎

For the lower bound, we adapt the following lemma from [5].

Lemma 5**.**

All counting predicates can be implemented by immediate observation protocols (possibly with unreliable communication), even if the fairness condition is replaced with an arbitrary different (activity-ensuring) one.

Proof.

We have already observed that immediate observation population protocols do not change if we add unreliability. It was shown in [5] that immediate observation population protocols implement all counting predicates. Moreover, the protocol $(k,k)\mapsto(k+1,k);(k,n)\mapsto(n,n)$ provided there for threshold predicates has the state of each agent increase monotonically. It is easy to see that ensuring activity is enough for this protocol to converge to a state where no more configuration-changing transitions can be taken. Also, the construction for boolean combination of predicates via direct product of protocols used in [5] converges as long as the protocols for the two arguments converge. Therefore it doesn’t need any extra restrictions on the fairness condition. ∎

Theorem 3 now follows from the fact that all the protocols with unreliable communication are shadow-permitting (by Lemma 2) and truncatable (by Lemma 3), therefore they only implement counting predicates. By Lemma 5 all counting predicates can be implemented.

6 Non-monotonic impact of unreliability

In this section we observe that, surprisingly, while delayed transmission protocols and queued transmission protocols are more powerful than immediate observation population protocols, their unreliable versions are strictly less expressive than immediate observation population protocols (possibly with unreliable communication).

Definition 17**.**

A protocol is fully asynchronous if for each allowed transition $(C,A^{\odot},C^{\prime})$ the following conditions hold.

• •

There is exactly one active agent, i.e. $|A^{\odot}|=1$.

• •

No passive agents change their states.

• •

Either the packets are only sent or the packets are only consumed, i.e. $\mathrm{Dom}(C_{P})\subseteq\mathrm{Dom}(C^{\prime}_{P})$ or $\mathrm{Dom}(C_{P})\supseteq\mathrm{Dom}(C^{\prime}_{P})$. Packet contents do not change, i.e. $C_{P}\mid_{\mathrm{Dom}(C_{P})\cap\mathrm{Dom}(C^{\prime}_{P})}=C^{\prime}_{P}\mid_{\mathrm{Dom}(C_{P})\cap\mathrm{Dom}(C^{\prime}_{P})}$.

It turns out that given unreliable communication such protocols can check presence of states but cannot count. As our old notion of ensuring activity doesn’t force any messages to be ever received, we need a slightly stronger fairness condition for any positive claims.

Definition 18**.**

A fairness condition ensures communication if the following two conditions hold in every fair run.

If the agent states $C_{A}$ do not change after some moment, from each configuration occurring after some later moment there is no possible transition changing $C_{A}$. 2. 2.

If the set of messages present in $C_{P}$ (ignoring multiplicities) does not change after some moment, then for each configuration after some later moment there is no possible transition that creates a packet with a new message.

Theorem 4**.**

Fully asynchronous protocols with unreliable communication compute exactly the predicates that are boolean combinations of positivity of single coordinates.

The upper bound holds under any eventual fairness condition, while the lower bound requires a fairness conditions that ensures communication.

The core idea of the proof is to ensure that in a reachable situation rare messages do not exist and cannot be created. In other words, if there is a packet with some message, or if such a packet can be created, then there are many packets with the same message. This makes irrelevant both the production of new messages by agents, and the exact number of agents needing to follow a particular sequence of transitions. This idea has some similarity with the message saturation construction from [20], but here the production of new messages might require consuming some of the old ones. We choose the threshold for “many” packets depending on the number of messages that do not yet have “many” packets. The threshold ensures that a new message will become abundant before we exhaust the packets for any previously numerous message.

Definition 19**.**

The in-degree of a fully asynchronous protocol is the maximum number of messages consumed in a single transition.

The supply of a message $m\in M$ in configuration $C$ is the number of packets in $C$ with the message $m$, i.e. $|C_{P}^{-1}(m)|$.

Let $F(x,y,z,n,k)=(32(xyzn+1))^{32(xyzn+1)-2k}$. An abundance set is the largest set $M^{\infty}\subseteq M$ such that the supply of each message in $M^{\infty}$ is at least $F(|Q|,|M|,d,|C_{A}|,|M^{\infty}|)$ where $d$ is the in-degree. As $F$ decreases in the last argument, the abundance set $M^{\infty}(C)$ is well-defined. A message $m$ is abundant in configuration $C$ if it is in the abundance set, i.e. $m\in M^{\infty}(C)$. A message $m$ is expendable at some moment in execution $E$ if it is abundant in some configuration that has occurred in $E$ before that moment. A packet is expendable if it bears an expendable message.

An execution $E$ is careful if no transition that decreases the supply of non-expendable messages changes agent states.

Remark 3*.*

The function $F$ is chosen to make its rate of growth obviously sufficient in the following calculations. A much smaller function would suffice for a more tedious analysis.

Lemma 6**.**

Every fully asynchronous protocol with unreliable communication has a careful fair execution starting from any configuration without message packets.

Moreover, if the protocol is well-specified, there is a careful fair execution that runs each packet-consuming transition twice in a row, failing to update the state the first time, until stable consensus is reached.

Proof.

We start with an execution with only the initial configuration.

In the first phase, as long as it is possible to create a packet with a non-expendable message (without making the execution careless), we do it while consuming the minimal possible number of packets with expendable messages. After creating each packet we increase the abundance set if possible.

In the second phase, a long as it is possible to consume a packet with a non-expendable message, we do it (but fail to update the agent states).

In the third phase we reach a stable consensus by consuming the minimal number of packets. We call the end of the third phase the target moment. Afterwards we pick an arbitrary fair continuation.

We now prove that each abundance set with a new message obtained during the first phase includes all the previous abundance sets. We only use the ways to create a new non-expendable packet that do not require consuming any non-expendable packets. Indeed, consuming a non-expendable packet is not allowed to change the internal state by definition of carefulness, and cannot create any new messages by definition of a fully asynchronous protocol. Note that reaching the internal state that can create a new non-expendable packet can take most $|Q|\times n$ transitions as all the expendable packets are already available for consumption and thus there is no reason to repeat the same internal state of the same agent twice. Therefore creating an additional non-expendable packet can consume at most $|Q|\times n\times d$ packets. To make the supply of some message reach $F(|Q|,|M|,d,n,k+1)$, we need to repeat this at most $F(|Q|,|M|,d,n,k+1)\times|M|$ times consuming at most $F(|Q|,|M|,d,n,k+1)\times M|\times|Q|\times n\times d$ expendable packets. We might consume twice as many expendable packets if we want to fail every other packet consumption transition. As $3\times F(|Q|,|M|,n,d,k+1)\times|M|\times|Q|\times n\times d<F(|Q|,|M|,d,n,k)$, all the expendable messages together with this message form an abundance set.

In the second phase, we run consumption in at most $|Q|$ states; reaching each of them requires at most $|Q|$ transitions. Thus the state changes consume at most $|Q|^{2}\times d$ expendable packets. Note that consuming a non-expendable packet requires consuming at most $d$ expendable packets. As the supply of each non-expendable message is less than $F(|Q|,|M|,d,n,|M^{\infty}|+1)$, we consume at most $d\times(|Q|^{2}+|M|\times F(|Q|,|M|,d,n,|M^{\infty}|+1))$. We also could have spent twice as many expendable messages if the non-expendable messages were not the limiting factor. Therefore we still have more than $F(|Q|,|M|,d,n,|M^{\infty}|+1)>4\times|Q|\times n\times d$ packets with each expendable message left by the time there are no non-expendable packets that can be received in a reachable state and no possibility to create a non-expendable packet.

A reachable stable consensus exists if the protocol computes some predicate. As it is impossible to produce or consume new non-expendable messages, we cannot violate the carefulness property. Moreover, we can reach it while spending at most $|Q|\times n\times d$ expendable packets (or twice as many if we fail to update the state every second time). That many packets are available, so producing new expendable packets is not required.

We see that the construction indeed provides a careful fair execution. Thus the lemma is proven. ∎

As the execution obtained via the previous lemma wastes a lot of messages, we can add one more agent to make use of those messages.

Lemma 7**.**

Consider a fully asynchronous protocol with unreliable communication that computes some predicate.

Then for any input configuration, adding one more agent in an already present input state cannot change the value of the predicate.

Proof.

Consider a careful execution constructed by Lemma 6. Consider an extra agent that we want to use as a copycat of an existing one, which we call target.

If a transition performed by the target agent sends messages, so does the copycat agent. If a transition requires receiving messages and the target agent updates the state, we cancel the previous transition where the target agent failed to update the state after consuming the same messages, and let the copycat agent receive those messages and update the state. Thus the copycat agent always mimics the state of the target agent.

Additionally, we extend phase two of the execution to consume the non-expendable messages sent by the copycat agent. They are the same as the target agent has sent, and there is a reserve of expendable messages for consuming these non-expendable messages (those that can be consumed in some reachable state).

As consumption of expendable messages did not allow to emit any non-expendable messages after reaching the stable consensus, the same must be true when we add the copycat agent as the set of reachable agent states without producing or consuming non-expendable messages is the same. But then the set of all the reachable states is the same, and we get a stable consensus with the same answer.

As the protocol is well-specified, this concludes the proof. ∎

Corollary 1**.**

A predicate computed by a fully asynchronous protocol with unreliable communication only depends on which coordinates are positive.

Proof.

Consider two configurations with the same set of represented input states. By repeated addition of copycat agents we can prove that the predicate value for either of configurations is the same as the predicate value for their union. ∎

It is clear that the predicates that only depend on the set of positive coordinates can be computed.

Lemma 8**.**

For any fairness condition ensuring communication, and for any predicate only depending on positivity of arguments, there is a fully asynchronous protocol computing that predicate.

Proof.

We just describe the protocol informally. The messages correspond to the input states. The states correspond to nonempty states of the input state (which are known to the agent to be initially present). An agent can send a message corresponding to an initial state in the agent’s set. An agent can receive a message and add the corresponding initial state to the set. An agent has output value equal to the value of the predicate on the input where all the input states from the agent’s set get the value $1$, while the others get [math].

Ensuring communication implies that the only stable situation is when all the initially present input states are reflected in message packets, and are also reflected in the sets of all the agents. ∎

The theorem now follows from Corollary 1 and Lemma 8.

Remark 4*.*

This result doesn’t mean that fundamentally asynchronous nature of communication prevents us from using any expressive models for verification of unreliable systems. It is usually possible to keep enough state to implement, for example, immediate observation via request and response.

7 Conclusion and future directions

We have studied unreliability based on message loss, a practically motivated approach to fault tolerance in population protocols. We have shown that inside a general framework of defining protocols with unreliable communication we can prove a specific structural property that bounds the expressive power of protocols with unreliable communication by the expressive power of immediate observation population protocols. Immediate observation population protocols permit verification of many useful properties, up to well-specification, correctness and reachability between counting sets, in polynomial space. We think that relatively low complexity of verification together with inherent unreliability tolerance and locally optimal expressive power under atomicity violations motivate further study and use of such protocols.

It is also interesting to explore if for any class of protocols adding unreliability makes some of the verification tasks easier. Both complexity and expressive power implications of unreliability can be studied for models with larger per-agent memory, such as community protocols, PALOMA and mediated population protocols. We also believe that some models even more restricted than community protocols but still permitting a multi-interaction conversation are an interesting object of study both in the reliable and unreliable settings.

Acknowledgements

I thank Javier Esparza for useful discussions and the feedback on the drafts of the present article. I thank Chana Weil-Kennedy for useful discussions.

This work is an extended version of [28], differing in the inclusion of full proofs as well as more precise characterisation of expressive power of fully asynchronous protocols. I thank the anonymous reviewers both of the previous and of the current version for their valuable feedback on presentation.