Towards Efficient Verification of Population Protocols

TL;DR

This paper introduces the WS3 class of protocols, enabling efficient automatic verification of well-specified population protocols by reducing the problem to solving boolean combinations of linear constraints.

Contribution

It defines the WS3 class, proves its computational equivalence to general protocols, and develops the first software for automatic verification of well-specification.

Findings

WS3 captures standard protocols from literature

Membership problem reduces to solving linear constraints

Software can automatically verify well-specification

Abstract

Population protocols are a well established model of computation by anonymous, identical finite state agents. A protocol is well-specified if from every initial configuration, all fair executions reach a common consensus. The central verification question for population protocols is the well-specification problem: deciding if a given protocol is well-specified. Esparza et al. have recently shown that this problem is decidable, but with very high complexity: it is at least as hard as the Petri net reachability problem, which is EXPSPACE-hard, and for which only algorithms of non-primitive recursive complexity are currently known. In this paper we introduce the class WS3 of well-specified strongly-silent protocols and we prove that it is suitable for automatic verification. More precisely, we show that WS3 has the same computational power as general well-specified protocols, and captures standard protocols from the literature. Moreover, we show that the membership problem for WS3 reduces to solving boolean combinations of linear constraints over N. This allowed us to develop the first software able to automatically prove well-specification for all of the infinitely many possible inputs.

Equations

$$I(X)(q)$$

$$t_{AB}$$

$$t_{Ab}$$

$$t_{Ba}$$

$$t_{ba}$$

$$(T_1,T_2,\dots ,T_n)$$

$$T_1$$

$$T_2$$

$$C^{\prime }(q)=C(q)+\sum _{t\in T}\mathbf{x}(t)\cdot (\mathrm{post}(t)(q)-\mathrm{pre}(t)(q)).$$

$$\mathrm{pre}(u^{\prime })\nleqq \mathrm{pre}(t)+(\mathrm{pre}(u)\text{varominus}\mathrm{post}(t)).$$

$$P(x_1,\dots ,x_k)=\left(\sum _{i=1}^k{a}_i{x}_i<c\right),$$

$$P(x_1,\dots ,x_k)=\left(\sum _{i=1}^k{a}_i{x}_i\equiv c(\mathrm{mod}m)\right),$$

$$v_{\text{max}}\stackrel{\text{def}}{=}\max (|a_1|,|a_2|,\dots ,|a_k|,|c|+1)$$

$$f(m,n)$$

$$g(m,n)$$

$$b(m,n)$$

$$Q$$

$$\Sigma$$

$$I(x_i)$$

$$O(\ell ,n,o)$$

$$(1,n,o),(l,n^{\prime },o^{\prime })$$

$$T_1$$

$$T_2$$

$$S_1$$

$$S_2$$

$$\sum _{i=1}^k{a}_i{x}_i\equiv c(\text{mod }m).$$

$$Q$$

$$\Sigma$$

$$I(x_i)$$

$$O(q)$$

$$(n,n^{\prime })$$

$$(n,b)$$

$$S_1$$

$$S_2$$

$$\mathrm{pre}(u^{\prime })\nleqq \mathrm{pre}(s)+(\mathrm{pre}(u)\text{varominus}\mathrm{post}(s))\text{ for every }{u}^{\prime }\in U\text{ such that }\mathrm{post}(u^{\prime })\ne \mathrm{pre}(u^{\prime }).$$

$$\mathrm{pre}(u^{\prime })$$

$$\le \mathrm{pre}(s_n)+(C_n\text{varominus}\mathrm{post}(s_n))$$

$$\le C_n\text{varominus}\mathrm{post}(s_n)+\mathrm{pre}(s_n)$$

$$=C_{n-1}$$

$$\mathrm{val}(C^{\prime })$$

Full text

In this paper we introduce the class ${\it WS}^{3}$ of well-specified strongly-silent protocols and we prove that it is suitable for automatic verification. More precisely, we show that ${\it WS}^{3}$ has the same computational power as general well-specified protocols, and captures standard protocols from the literature. Moreover, we show that the membership problem for ${\it WS}^{3}$ reduces to solving boolean combinations of linear constraints over $\mathbb{N}$. This allowed us to develop the first software able to automatically prove well-specification for all of the infinitely many possible inputs.

1 Introduction

Population protocols [1, 2] are a model of distributed computation by many anonymous finite-state agents. They were initially introduced to model networks of passively mobile sensors [1, 2], but are now also used to describe chemical reaction networks (see e.g. [20, 7]).

In each computation step of a population protocol, a fixed number of agents are chosen nondeterministically, and their states are updated according to a joint transition function. Since agents are anonymous and identical, the global state of a protocol is completely determined by the number of agents at each local state, called a configuration. A protocol computes a boolean value $b$ for a given initial configuration $C_{0}$ if in all fair executions starting at $C_{0}$, all agents eventually agree to $b$ — so, intuitively, population protocols compute by reaching consensus under a certain fairness condition. A protocol is well-specified if it computes a value for each of its infinitely many initial configurations (also called inputs). The predicate computed by a protocol is the function that assigns to each input the corresponding consensus value. In a famous series of papers, Angluin et al. [1, 2] have shown that well-specified protocols compute exactly the predicates definable in Presburger arithmetic [1, 2, 3, 4].

In this paper we search for efficient algorithms for the well-specification problem: Given a population protocol, is it well-specified? This is a question about an infinite family of finite-state systems. Indeed, for every input the semantics of a protocol is a finite graph with the reachable configurations as nodes. Deciding if the protocol reaches consensus for a fixed input only requires to inspect one of these graphs, and can be done automatically using a model checker. This approach has been followed in a number of papers [21, 25, 6, 8], but it only shows well-specification for some inputs. There has also been work in formalizing well-specification proofs in interactive theorem provers [10], but this approach is not automatic: a human prover must first come up with a proof for each particular protocol.

Recently, the second author, together with other co-authors, has shown that the well-specification problem is decidable [13]. That is, there is an algorithm that decides if for all inputs the protocol stabilizes to a boolean value. The proof uses deep results of the theory of Petri nets, a model very close to population protocols. However, the same paper shows that the well-specification problem is at least as hard as the reachability problem for Petri nets, a famously difficult problem. More precisely, the problem is known to be \EXPSPACE-hard, and all known algorithms for it have non-primitive recursive complexity [23]. In particular, there are no stable implementations of any of these algorithms, and they are considered impractical for nearly all applications.

For this reason, in this paper we search for a class of well-specified protocols satisfying three properties:

• (a)

No loss of expressive power: the class should compute all Presburger-definable predicates.

• (b)

Natural: the class should contain most protocols discussed in the literature.

• (c)

Feasible membership problem: membership for the class should have reasonable complexity.

The class ${\it WS}$ of all well-specified protocols obviously satisfies  (a) and  (b), but not (c). So we introduce a new class ${\it WS}^{3}$, standing for Well-Specified Strongly Silent protocols. We show that ${\it WS}^{3}$ still satisfies (a) and (b), and then prove that the membership problem for ${\it WS}^{3}$ is in the complexity class \DP; the class of languages $L$ such that $L=L_{1}\cap L_{2}$ for some languages $L_{1}\in\NP$ and $L_{2}\in\coNP$. This is a dramatic improvement with respect to the \EXPSPACE-hardness of the membership problem for ${\it WS}$.

Our proof that the problem is in \DP reduces membership for ${\it WS}^{3}$ to checking (un)satisfiability of two systems of boolean combinations of linear constraints over the natural numbers. This allowed us to implement our decision procedure on top of the constraint solver Z3 [9], yielding the first software able to automatically prove well-specification for all inputs. We tested our implementation on the families of protocols studied in [21, 25, 6, 8]. These papers prove well-specification for some inputs of protocols with up to 9 states and 28 transitions. Our approach proves well-specification for all inputs of protocols with up to 20 states in less than one second, and protocols with 70 states and 2500 transitions in less than one hour. In particular, we can automatically prove well-specification for all inputs in less time than previous tools needed to check one single large input.

The verification problem for population protocols naturally divides into two parts: checking that a given protocol is well specified, and checking that a given well-specified protocol computes the desired predicate. While in this paper we are concerned with well-specification, our implementation is already able to solve the second problem for all the families of protocols described above. This is achieved by adding to the second system of constraints used to check well-specification further linear constraints describing the sets of input configurations for which the protocol should return true or false. An extension of the software that, given a protocol and an arbitrary Presburger predicate, checks whether the protocol computes the predicate, requires to solve implementation problems related to Presburger arithmetic, and is left for future research.

The paper is organized as follows. Section 2 contains basic definitions. Section 3 introduces an intermediate class ${\it WS}^{2}$ of silent well-specified protocols, and shows that its membership problem is still as hard as for ${\it WS}$. In Section 4, we characterize ${\it WS}^{2}$ in terms of two properties which are then strengthened to define our new class ${\it WS}^{3}$. We then show that the properties defining ${\it WS}^{3}$ can be tested in \NP and \coNP, and so that membership for ${\it WS}^{3}$ is in \DP. Section 5 proves that ${\it WS}^{3}$-protocols compute all Presburger predicates. Section 6 reports on our experimental results, and Section 7 presents conclusions.

2 Preliminaries

Multisets.

A multiset over a finite set $E$ is a mapping $M:E\to\mathbb{N}$. The set of all multisets over $E$ is denoted $\mathbb{N}^{E}$. For every $e\in E$, $M(e)$ denotes the number of occurrences of $e$ in $M$. We sometimes denote multisets using a set-like notation, e.g. $\Lbag f,g,g\Rbag$ is the multiset $M$ such that $M(f)=1$, $M(g)=2$ and $M(e)=0$ for every $e\in E\setminus\{f,g\}$. The support of $M\in\mathbb{N}^{E}$ is $\llbracket M\rrbracket\stackrel{{\scriptstyle\scriptscriptstyle\text{def}}}{{=}}\{e\in E:M(e)>0\}$. The size of $M\in\mathbb{N}^{E}$ is $|M|\stackrel{{\scriptstyle\scriptscriptstyle\text{def}}}{{=}}\sum_{e\in E}M(e)$. Addition and comparison are extended to multisets componentwise, i.e. $(M\mathbin{+}M^{\prime})(e)\stackrel{{\scriptstyle\scriptscriptstyle\text{def}}}{{=}}M(e)+M^{\prime}(e)$ for every $e\in E$, and $M\leq M^{\prime}\stackrel{{\scriptstyle\scriptscriptstyle\text{def}}}{{\iff}}M(e)\leq M(e)$ for every $e\in E$. We define multiset difference as $(M\mathbin{\varominus}M^{\prime})(e)\stackrel{{\scriptstyle\scriptscriptstyle\text{def}}}{{=}}\max(M(e)-M^{\prime}(e),0)$ for every $e\in E$. The empty multiset is denoted $\bm{0}$, and for every $e\in E$ we write $\bm{e}\stackrel{{\scriptstyle\scriptscriptstyle\text{def}}}{{=}}\Lbag e\Rbag$.

Population protocols.

A population $P$ over a finite set $E$ is a multiset $P\in\mathbb{N}^{E}$ such that $|P|\geq 2$. The set of all populations over $E$ is denoted by $\mathrm{Pop}(E)$. A population protocol is a tuple $\mathcal{P}=(Q,T,\Sigma,I,O)$ where

• •

$Q$ is a non-empty finite set of states,

• •

$T\subseteq Q^{2}\times Q^{2}$ is a set of transitions such that for every $(p,q)\in Q^{2}$ there exists at least a pair $(p^{\prime},q^{\prime})\in Q^{2}$ such that $(p,q,p^{\prime},q^{\prime})\in T$,

• •

$\Sigma$ is a non-empty finite input alphabet,

• •

$I:\Sigma\to Q$ is the input function mapping input symbols to states,

• •

$O:Q\to\{0,1\}$ is the output function mapping states to boolean values.

Following the convention of previous papers, we call the populations of $\mathrm{Pop}(Q)$ configurations. Intuitively, a configuration $C$ describes a collection of identical finite-state agents with $Q$ as set of states, containing $C(q)$ agents in state $q$ for every $q\in Q$, and at least two agents in total.

Pairs of agents111While protocols only model interactions between two agents, $k$-way interactions for a fixed $k>2$ can be simulated by adding additional states. interact using transitions. For every $t=(p,q,\linebreak p^{\prime},q^{\prime})\in T$, we write $(p,q)\mapsto(p^{\prime},q^{\prime})$ to denote $t$, and we define $\mathrm{pre}(t)\stackrel{{\scriptstyle\scriptscriptstyle\text{def}}}{{=}}\Lbag p,q\Rbag$ and $\mathrm{post}(t)\stackrel{{\scriptstyle\scriptscriptstyle\text{def}}}{{=}}\Lbag p^{\prime},q^{\prime}\Rbag$. For every configuration $C$ and transition $t\in T$, we say that $t$ is enabled at $C$ if $C\geq\mathrm{pre}(t)$. Note that by definition of $T$, every configuration enables at least one transition. A transition $t\in T$ enabled at $C$ can occur, leading to the configuration $C\mathbin{\varominus}\mathrm{pre}(t)+\mathrm{post}(t)$. Intuitively, a pair of agents in states $\mathrm{pre}(t)$ move to states $\mathrm{post}(t)$. We write $C\xrightarrow{t}C^{\prime}$ to denote that $t$ is enabled at $C$ and that its occurrence leads to $C^{\prime}$. A transition $t\in T$ is silent if $\mathrm{pre}(t)=\mathrm{post}(t)$, i.e., if it cannot change the current configuration.

For every sequence of transitions $w=t_{1}t_{2}\cdots t_{k}$, we write $C\xrightarrow{w}C^{\prime}$ if there exists a sequence of configurations $C_{0},C_{1},\ldots,C_{k}$ such that $C=C_{0}\xrightarrow{t_{1}}C_{1}\cdots\xrightarrow{t_{k}}C_{k}=C^{\prime}$. We also write $C\xrightarrow{}C^{\prime}$ if $C\xrightarrow{t}C^{\prime}$ for some transition $t\in T$, and call $C\xrightarrow{}C^{\prime}$ a step. We write $C\xrightarrow{*}C^{\prime}$ if $C\xrightarrow{w}C^{\prime}$ for some $w\in T^{*}$. We say that $C^{\prime}$ is reachable from $C$ if $C\xrightarrow{*}C^{\prime}$. An execution is an infinite sequence of configurations $C_{0}C_{1}\cdots$ such that $C_{i}\xrightarrow{}C_{i+1}$ for every $i\in\mathbb{N}$. An execution $C_{0}C_{1}\cdots$ is fair if for every step $C\xrightarrow{}C^{\prime}$, if $C_{i}=C$ for infinitely many indices $i\in\mathbb{N}$, then $C_{j}=C$ and $C_{j+1}=C^{\prime}$ for infinitely many indices $j\in\mathbb{N}$. We say that a configuration $C$ is

• •

terminal if $C\xrightarrow{*}C^{\prime}$ implies $C=C^{\prime}$, i.e., if every transition enabled at $C$ is silent;

• •

a consensus configuration if $O(p)=O(q)$ for every $p,q\in\llbracket C\rrbracket$.

For every consensus configuration $C$, let $O(C)$ denote the unique output of the states in $\llbracket C\rrbracket$. An execution $C_{0}C_{1}\cdots$ stabilizes to $b\in\{0,1\}$ if there exists $n\in\mathbb{N}$ such that $C_{i}$ is a consensus configuration and $O(C_{i})=b$ for every $i\geq n$.

Predicates computable by population protocols.

Every input $X\in\mathrm{Pop}(\Sigma)$ is mapped to the configuration $I(X)\in\mathrm{Pop}(Q)$ defined by

[TABLE]

A configuration $C$ is said to be initial if $C=I(X)$ for some input $X$. A population protocol is well-specified if for every input $X$, there exists $b\in\{0,1\}$ such that every fair execution of $\mathcal{P}$ starting at $I(X)$ stabilizes to $b$. We say that $\mathcal{P}$ computes a predicate $\varphi$ if for every input $X$, every fair execution of $\mathcal{P}$ starting at $I(X)$ stabilizes to $\varphi(X)$. It is readily seen that $\mathcal{P}$ computes a predicate if and only if it is well-specified.

Example 1**.**

We consider the majority protocol of [3] as a running example. Initially, agents of the protocol can be in either state $A$ or $B$. The protocol computes whether there are at least as many agents in state $B$ as there are in state $A$. The states and the input alphabet are $Q=\left\{A,B,a,b\right\}$ and $\Sigma=\{A,B\}$ respectively. The input function is the identity function, and the output function is given by $O(B)=O(b)=1$ and $O(A)=O(a)=0$. The set of transitions $T$ consists of:

[TABLE]

and of silent transitions for the remaining pairs of states. Transition $t_{AB}$ ensures that every fair execution eventually reaches a configuration $C$ such that $C(A)=0$ or $C(B)=0$. If $C(A)=0=C(B)$, then there were initially equally many agents in $A$ and $B$. Transition $t_{ba}$ then acts as tie breaker, resulting in a terminal configuration populated only by $b$. If, say, $C(A)>0$ and $C(B)=0$, then there were initially more $A$s than $B$s, and $t_{Ab}$ ensures that every fair execution eventually reaches a terminal configuration populated only by $A$ and $a$.

3 Well-specified silent protocols

Silent protocols222Silent protocols are also referred to as protocols with stabilizing states and silent transitions are called ineffective in [19, 18]. were introduced in [12]. Loosely speaking, a protocol is silent if communication between agents eventually ceases, i.e. if every fair execution eventually stays in the same configuration forever. Observe that a well-specified protocol need not be silent: fair executions may keep alternating from a configuration to another as long as they are consensus configurations with the same output.

More formally, we say that an execution $C_{0}C_{1}\cdots$ is silent if there exists $n\in\mathbb{N}$ and a configuration $C$ such that $C_{i}=C$ for every $i\geq n$. A population protocol $\mathcal{P}$ is silent if every fair execution of $\mathcal{P}$ is silent, regardless of the starting configuration. We call a protocol that is well-specified and silent a ${\it WS}^{2}$-protocol, and denote by ${\it WS}^{2}$ the set of all ${\it WS}^{2}$-protocols.

Example 2**.**

As explained in Example 1, every fair execution of the majority protocol is silent. This implies that the protocol is silent. If, for example, we add a new state $b^{\prime}$ where $O(b^{\prime})=1$, and transitions $(b,b)\mapsto(b^{\prime},b^{\prime}),(b^{\prime},b^{\prime})\mapsto(b,b)$, then the protocol is no longer silent since the execution where two agents alternate between states $b$ and $b^{\prime}$ is fair but not silent.

Being silent is a desirable property. While in arbitrary protocols it is difficult to determine if an execution has already stabilized, in silent protocols it is simple: one just checks if the current configuration only enables silent transitions. Even though it is was not observed explicitely, the protocols introduced in  [1] to characterize the expressive power of population protocols belong to ${\it WS}^{2}$. Therefore, ${\it WS}^{2}$-protocols can compute the same predicates as general ones.

Unfortunately, a slight adaptation of [14, Theorem 10] shows that the complexity of the membership problem for ${\it WS}^{2}$-protocols is still as high as for the general case:

{restatable}

propositionpropSilentHardness The reachability problem for Petri nets is reducible in polynomial time to the membership problem for ${\it WS}^{2}$. In particular, membership for ${\it WS}^{2}$ is \EXPSPACE-complete.

To circumvent this high complexity, we will show in the next section how ${\it WS}^{2}$ can be refined into a smaller class of well-specified protocols with the same expressive power, and a membership problem of much lower complexity.

4 A finer class of silent well-specified protocols: ${\it WS}^{3}$

It can be shown that ${\it WS}^{2}$-protocols are exactly the protocols satisfying the two following properties:

• •

Termination: for every configuration $C$, there exists a terminal configuration $C^{\prime}$ such that $C\xrightarrow{*}C^{\prime}$.

• •

Consensus: for every initial configuration $C$, there exists $b\in\{0,1\}$ such that every terminal configuration $C^{\prime}$ reachable from $C$ is a consensus configuration with output $b$, i.e. $C\xrightarrow{*}C^{\prime}$ implies $O(C^{\prime})=b$.

We will introduce the new class ${\it WS}^{3}$ as a refinement of ${\it WS}^{2}$ obtained by strengthening Termination and Consensus into two new properties called LayeredTermination and StrongConsensus. We introduce these properties in Section 4.1 and Section 4.2, and show that their decision problems belong to \NP and \coNP respectively.

Before doing so, let us introduce some useful notions. Let $\mathcal{P}=(Q,T,\Sigma,I,O)$ be a population protocol. For every $S\subseteq T$, $\mathcal{P}[S]$ denotes the protocol induced by $S$, i.e. $\mathcal{P}[S]\stackrel{{\scriptstyle\scriptscriptstyle\text{def}}}{{=}}(Q,S\cup T^{\prime},\Sigma,I,O)$ where $T^{\prime}\stackrel{{\scriptstyle\scriptscriptstyle\text{def}}}{{=}}\left\{(p,q,p,q):p,q\in Q\right\}$ is added to ensure that any two states can interact. Let $\xrightarrow{}_{S}$ denote the transition relation of $\mathcal{P}[S]$. An ordered partition of $T$ is a tuple $(T_{1},T_{2},\ldots,T_{n})$ of nonempty subsets of $T$ such that $T=\bigcup_{i=1}^{n}T_{i}$ and $T_{i}\cap T_{j}=\emptyset$ for every $1\leq i<j\leq n$.

4.1 Layered termination

We replace Termination by a stronger property called LayeredTermination, and show that deciding LayeredTermination belongs to \NP. The definition of LayeredTermination is inspired by the typical structure of protocols found in the literature. Such protocols are organized in layers such that transitions of higher layers cannot be enabled by executing transitions of lower layers. In particular, if the protocol reaches a configuration of the highest layer that does not enable any transition, then this configuration is terminal. For such protocols, Termination can be proven by showing that every (fair or unfair) execution of a layer is silent.

Definition 3**.**

A population protocol $\mathcal{P}=(Q,T,\Sigma,I,O)$ satisfies LayeredTermination if there is an ordered partition

[TABLE]

of $T$ such that the following properties hold for every $i\in[n]$:

• (a)

For every configuration $C$, every (fair or unfair) execution of $\mathcal{P}[T_{i}]$ starting at $C$ is silent.

• (b)

For every configurations $C$ and $C^{\prime}$, if $C\xrightarrow{*}_{T_{i}}C^{\prime}$ and $C$ is terminal in $\mathcal{P}[T_{1}\cup T_{2}\cup\cdots\cup T_{i-1}]$, then $C^{\prime}$ is also terminal in $\mathcal{P}[T_{1}\cup T_{2}\cup\cdots\cup T_{i-1}]$.

Example 4**.**

The majority protocol satisfies LayeredTermination. Indeed, consider the ordered partition $(T_{1},T_{2})$, where

[TABLE]

All executions of $\mathcal{P}[T_{1}]$ and $\mathcal{P}[T_{2}]$ are silent. For every terminal configuration $C$ of $\mathcal{P}[T_{1}]$, we have $\llbracket C\rrbracket\subseteq\{A,a\}$ or $\llbracket C\rrbracket\subseteq\{B,a,b\}$. In the former case, no transition of $T_{2}$ is enabled; in the latter case, taking transitons of $T_{2}$ cannot enable $T_{1}$.

As briefly sketched above, LayeredTermination implies Termination. In the rest of this section, we prove that checking LayeredTermination is in \NP. We do this by showing that conditions (a) and (b) of Definition 3 can be tested in polynomial time.

We recall a basic notion of Petri net theory recast in the terminology of population protocols. For every step $C\xrightarrow{t}C^{\prime}$ and every state $q$ of a population protocol, we have $C^{\prime}(q)=C(q)+\mathrm{post}(t)(q)-\mathrm{pre}(t)(q)$. This observation can be extended to sequences of transitions. Let $|w|_{t}$ denote the number of occurrences of transition $t$ in a sequence $w$. We have $C^{\prime}(q)=C(q)+\sum_{t\in T}|w|_{t}\cdot(\mathrm{post}(t)(q)-\mathrm{pre}(t)(q))$. Thus, a necessary condition for $C\xrightarrow{w}C^{\prime}$ is the existence of some $\bm{x}:T\to\mathbb{N}$ such that

[TABLE]

We call (1) the flow equation for state $q$.

Proposition 5**.**

Let $\mathcal{P}=(Q,T,\Sigma,I,O)$ be a population protocol. Deciding whether an ordered partition $(T_{1},T_{2},\ldots,T_{n})$ of $T$ satisfies condition (a) of Definition 3 can be done in polynomial time.

Proof.

Let $i\in[n]$ and let $U_{i}$ be the set of non silent transitions of $T_{i}$. It can be shown that $\mathcal{P}[T_{i}]$ is non silent if and only if there exists $\bm{x}:U_{i}\to\mathbb{Q}$ such that $\sum_{t\in U_{i}}\bm{x}(t)\cdot(\mathrm{post}(t)(q)-\mathrm{pre}(t)(q))=0$ and $\bm{x}(q)\geq 0$ for every $q\in Q$, and $\bm{x}(q)>0$ for some $q\in Q$. Therefore, since linear programming is in ¶, we can check for the (non) existence of an appropriate rational solution $\bm{x}_{i}$ for every $i\in[n]$. ∎

We show how to check condition (b) of Definition 3 in polynomial time. Let $U\subseteq T$ be a set of transitions. A configuration $C\in\mathrm{Pop}(Q)$ is $U$-dead if for every $t\in U$, $C\xrightarrow{t}C^{\prime}$ implies $C^{\prime}=C$. We say that $\mathcal{P}$ is $U$-dead from $C_{0}\in\mathrm{Pop}(Q)$ if every configuration reachable from $C_{0}$ is $U$-dead, i.e. $C_{0}\xrightarrow{*}C$ implies that $C$ is $U$-dead. Finally, we say that $\mathcal{P}$ is $U$-dead if it is $U$-dead from every $U$-dead configuration $C_{0}\in\mathrm{Pop}(Q)$.

Proposition 6**.**

Let $\mathcal{P}=(Q,T,\Sigma,I,O)$ be a population protocol. Deciding whether an ordered partition $(T_{1},\ldots,T_{n})$ of $T$ satisfies condition (b) of Definition 3 can be done in polynomial time.

Proof.

Let $i\in[n]$ and let $U\stackrel{{\scriptstyle\scriptscriptstyle\text{def}}}{{=}}T_{1}\cup T_{2}\cup\cdots\cup T_{i-1}$. $\mathcal{P}[T_{i}]$ satisfies condition (b) if and only if $\mathcal{P}[T_{i}]$ is $U$-dead. The latter can be tested in polynomial time through the following characterization: $\mathcal{P}[T_{i}]$ is not $U$-dead if and only if there exist $t\in T_{i}$ and non silent $u\in U$ such that for every non silent $u^{\prime}\in U$:

[TABLE]

Propositions 5 and 6 yield an \NP procedure to decide LayeredTermination. Indeed, it suffices to guess an ordered partition and to check whether it satisfies conditions (a) and (b) of Definition 3 in polynomial time.

Corollary 7**.**

Deciding if a protocol satisfies LayeredTermination is in \NP.

4.2 Strong consensus

To overcome the high complexity of reachability in population protocols, we strengthen Consensus by replacing the reachability relation in its definition by an over-approximation, i.e., a relation $\mathrel{\mathchoice{\mathrel{}\mathrel{\mathop{\dabar@\dabar@}\limits}\mathrel{\mathchar 0\relax\symAMSa 04B}{\!\!}}{\mathrel{}\mathrel{\mathop{\dabar@\dabar@}\limits}\mathrel{\mathchar 0\relax\symAMSa 04B}{\!\!}}{\mathrel{}\mathrel{\mathop{\dabar@\dabar@}\limits}\mathrel{\mathchar 0\relax\symAMSa 04B}{\!\!}}{\mathrel{}\mathrel{\mathop{\dabar@\dabar@}\limits}\mathrel{\mathchar 0\relax\symAMSa 04B}{\!\!}}}$ over configurations such that $C\xrightarrow{*}C^{\prime}$ implies $C\mathrel{\mathchoice{\mathrel{}\mathrel{\mathop{\dabar@\dabar@}\limits}\mathrel{\mathchar 0\relax\symAMSa 04B}{\!\!}}{\mathrel{}\mathrel{\mathop{\dabar@\dabar@}\limits}\mathrel{\mathchar 0\relax\symAMSa 04B}{\!\!}}{\mathrel{}\mathrel{\mathop{\dabar@\dabar@}\limits}\mathrel{\mathchar 0\relax\symAMSa 04B}{\!\!}}{\mathrel{}\mathrel{\mathop{\dabar@\dabar@}\limits}\mathrel{\mathchar 0\relax\symAMSa 04B}{\!\!}}}C^{\prime}$. Observe that the flow equations provide an over-approximation of the reachability relation. Indeed, as mentioned earlier, if $C\xrightarrow{*}C^{\prime}$, then there exists $\bm{x}:T\to\mathbb{N}$ such that $(C,C^{\prime},\bm{x})$ satisfies all of the flow equations. However, this over-approximation alone is too crude for the verification of protocols.

Example 8**.**

For example, let us consider the configurations $C=\Lbag A,B\Rbag$ and $C^{\prime}=\Lbag a,a\Rbag$ of the majority protocol. The flow equations are satisfied by the mapping $\bm{x}$ such that $\bm{x}(t_{AB})=\bm{x}(t_{Ab})=1$ and $\bm{x}(t_{Ba})=\bm{x}(t_{ba})=0$. Yet, $C\xrightarrow{*}C^{\prime}$ does not hold.

To obtain a finer reachability over-approximation, we introduce so-called traps and siphons constraints borrowed from the theory of Petri nets [11, 16, 15] and successfully applied to a number of analysis problems (see e.g. [16, 15, 5]). Intuitively, for some subset of transitions $U\subseteq T$, a $U$-trap is a set of states $P\subseteq Q$ such that every transition of $U$ that removes an agent from $P$ also moves an agent into $P$. Conversely, a $U$-siphon is a set $P\subseteq Q$ such that every transition of $U$ that moves an agent into $P$ also removes an agent from $P$. More formally, let ${{}^{\bullet}R}\stackrel{{\scriptstyle\scriptscriptstyle\text{def}}}{{=}}\{t\in T:\llbracket\mathrm{post}(t)\rrbracket\cap R\not=\emptyset\}$ and ${R^{\bullet}}\stackrel{{\scriptstyle\scriptscriptstyle\text{def}}}{{=}}\{t\in T:\llbracket\mathrm{pre}(t)\rrbracket\cap R\not=\emptyset\}$. $U$-siphons and $U$-traps are defined as follows:

Definition 9**.**

A subset of states $P\subseteq Q$ is a $U$-trap if ${P^{\bullet}}\cap U\subseteq{{}^{\bullet}P}$, and a $U$-siphon if ${{}^{\bullet}P}\cap U\subseteq{P^{\bullet}}$.

For every configuration $C\in\mathrm{Pop}(Q)$ and $P\subseteq Q$, let $C(P)\stackrel{{\scriptstyle\scriptscriptstyle\text{def}}}{{=}}\sum_{q\in P}C(q)$. Consider a sequence of steps $C_{0}\xrightarrow{t_{1}}C_{1}\xrightarrow{t_{2}}\cdots\xrightarrow{t_{n}}C_{n}$ where $t_{1},\ldots,t_{n}\in U$. It follows from Definition 9 that if some transition $t_{i}$ moves an agent to a $U$-trap $P$, then $C_{j}(P)>0$ for every $j\geq i$. Similarly, if some transition $t_{i}$ removes an agent from a $U$-siphon, then $C_{j}(P)>0$ for every $j<i$. In particular:

Observation 10**.**

Let $U\subseteq T$, and let $C$ and $C^{\prime}$ be configurations such that $C\xrightarrow{*}_{U}C^{\prime}$. For every $U$-trap $P$, if $C^{\prime}(P)=0$, then ${{}^{\bullet}P}\cap U=\emptyset$. For every $U$-siphon $P$, if $C(\mathit{S})=0$, then ${P^{\bullet}}\cap U=\emptyset$.

We obtain a necessary condition for $C\xrightarrow{*}_{U}C^{\prime}$ to hold, which we call potential reachability:

Definition 11**.**

Let $C,C^{\prime}$ be two configurations, let $\bm{x}:T\to\mathbb{N}$, and let $U=\llbracket\bm{x}\rrbracket$. We say that $C^{\prime}$ is potentially reachable from $C$ through $\bm{x}$, denoted C\mathrel{\mathchoice{\mathrel{}\mathrel{\mathop{\dabar@\dabar@}\limits^{\hbox{\set@color\raisebox{-1.1pt}{$\scriptstyle\bm{x}\!\!$}}}}\mathrel{\mathchar 0\relax\symAMSa 04B}{\!\!}}{\mathrel{}\mathrel{\mathop{\dabar@\dabar@}\limits^{\hbox{\set@color\raisebox{-1.1pt}{$\scriptstyle\bm{x}\!\!$}}}}\mathrel{\mathchar 0\relax\symAMSa 04B}{\!\!}}{\mathrel{}\mathrel{\mathop{\dabar@\dabar@}\limits^{\hbox{\set@color\raisebox{-1.1pt}{$\scriptscriptstyle\bm{x}\!\!$}}}}\mathrel{\mathchar 0\relax\symAMSa 04B}{\!\!}}{\mathrel{}\mathrel{\mathop{\dabar@\dabar@}\limits^{\hbox{\set@color\raisebox{-1.1pt}{$\scriptstyle\bm{x}\!\!$}}}}\mathrel{\mathchar 0\relax\symAMSa 04B}{\!\!}}}C^{\prime}, if

• (a)

the flow equation (1) holds for every $q\in Q$,

• (b)

$C^{\prime}(P)=0$ implies ${{}^{\bullet}P}\cap U=\emptyset$ for every $U$-trap $P$, and

• (c)

$C(P)=0$ implies ${P^{\bullet}}\cap U=\emptyset$ for every $U$-siphon $P$.

Example 12**.**

Let us reconsider Example 8. Let $U=\llbracket\bm{x}\rrbracket=\{t_{AB},t_{Ab}\}$ and $P=\{A,b\}$. Recall that $t_{AB}=(A,B)\mapsto(a,b)$ and $t_{Ab}=(A,b)\mapsto(A,a)$. We have ${P^{\bullet}}\cap U=U$ which implies that $P$ is a $U$-trap. This means that Definition 11(b) is violated as $C^{\prime}(P)=0$ and ${{}^{\bullet}P}\cap U=U\not=\emptyset$. Therefore, \Lbag A,B\Rbag\mathrel{\mathchoice{\mathrel{}\mathrel{\mathop{\dabar@\dabar@}\limits^{\hbox{\set@color\raisebox{-1.1pt}{$\scriptstyle\bm{x}\!\!$}}}}\mathrel{\mathchar 0\relax\symAMSa 04B}{\!\!}}{\mathrel{}\mathrel{\mathop{\dabar@\dabar@}\limits^{\hbox{\set@color\raisebox{-1.1pt}{$\scriptstyle\bm{x}\!\!$}}}}\mathrel{\mathchar 0\relax\symAMSa 04B}{\!\!}}{\mathrel{}\mathrel{\mathop{\dabar@\dabar@}\limits^{\hbox{\set@color\raisebox{-1.1pt}{$\scriptscriptstyle\bm{x}\!\!$}}}}\mathrel{\mathchar 0\relax\symAMSa 04B}{\!\!}}{\mathrel{}\mathrel{\mathop{\dabar@\dabar@}\limits^{\hbox{\set@color\raisebox{-1.1pt}{$\scriptstyle\bm{x}\!\!$}}}}\mathrel{\mathchar 0\relax\symAMSa 04B}{\!\!}}}\Lbag a,a\Rbag does not hold.

We write $C\mathrel{\mathchoice{\mathrel{}\mathrel{\mathop{\dabar@\dabar@}\limits}\mathrel{\mathchar 0\relax\symAMSa 04B}{\!\!}}{\mathrel{}\mathrel{\mathop{\dabar@\dabar@}\limits}\mathrel{\mathchar 0\relax\symAMSa 04B}{\!\!}}{\mathrel{}\mathrel{\mathop{\dabar@\dabar@}\limits}\mathrel{\mathchar 0\relax\symAMSa 04B}{\!\!}}{\mathrel{}\mathrel{\mathop{\dabar@\dabar@}\limits}\mathrel{\mathchar 0\relax\symAMSa 04B}{\!\!}}}C^{\prime}$ if C\mathrel{\mathchoice{\mathrel{}\mathrel{\mathop{\dabar@\dabar@}\limits^{\hbox{\set@color\raisebox{-1.1pt}{$\scriptstyle\bm{x}\!\!$}}}}\mathrel{\mathchar 0\relax\symAMSa 04B}{\!\!}}{\mathrel{}\mathrel{\mathop{\dabar@\dabar@}\limits^{\hbox{\set@color\raisebox{-1.1pt}{$\scriptstyle\bm{x}\!\!$}}}}\mathrel{\mathchar 0\relax\symAMSa 04B}{\!\!}}{\mathrel{}\mathrel{\mathop{\dabar@\dabar@}\limits^{\hbox{\set@color\raisebox{-1.1pt}{$\scriptscriptstyle\bm{x}\!\!$}}}}\mathrel{\mathchar 0\relax\symAMSa 04B}{\!\!}}{\mathrel{}\mathrel{\mathop{\dabar@\dabar@}\limits^{\hbox{\set@color\raisebox{-1.1pt}{$\scriptstyle\bm{x}\!\!$}}}}\mathrel{\mathchar 0\relax\symAMSa 04B}{\!\!}}}C^{\prime} for some $\bm{x}:T\to\mathbb{N}$. As an immediate consequence of Observation 10, for every configurations $C$ and $C^{\prime}$, if $C\xrightarrow{*}C^{\prime}$, then $C\mathrel{\mathchoice{\mathrel{}\mathrel{\mathop{\dabar@\dabar@}\limits}\mathrel{\mathchar 0\relax\symAMSa 04B}{\!\!}}{\mathrel{}\mathrel{\mathop{\dabar@\dabar@}\limits}\mathrel{\mathchar 0\relax\symAMSa 04B}{\!\!}}{\mathrel{}\mathrel{\mathop{\dabar@\dabar@}\limits}\mathrel{\mathchar 0\relax\symAMSa 04B}{\!\!}}{\mathrel{}\mathrel{\mathop{\dabar@\dabar@}\limits}\mathrel{\mathchar 0\relax\symAMSa 04B}{\!\!}}}C^{\prime}$. This allows us to strengthen Consensus by redefining it in terms of potential reachability instead of reachability:

Definition 13**.**

A protocol satisfies StrongConsensus if for every initial configuration $C$, there exists $b\in\{0,1\}$ such that every terminal configuration $C^{\prime}$ potentially reachable from $C$ is a consensus configuration with output $b$, i.e. $C\mathrel{\mathchoice{\mathrel{}\mathrel{\mathop{\dabar@\dabar@}\limits}\mathrel{\mathchar 0\relax\symAMSa 04B}{\!\!}}{\mathrel{}\mathrel{\mathop{\dabar@\dabar@}\limits}\mathrel{\mathchar 0\relax\symAMSa 04B}{\!\!}}{\mathrel{}\mathrel{\mathop{\dabar@\dabar@}\limits}\mathrel{\mathchar 0\relax\symAMSa 04B}{\!\!}}{\mathrel{}\mathrel{\mathop{\dabar@\dabar@}\limits}\mathrel{\mathchar 0\relax\symAMSa 04B}{\!\!}}}C^{\prime}$ implies $O(C^{\prime})=b$.

Since the number of $U$-traps and $U$-siphons of a protocol can be exponential in the number of states, checking trap and siphon constraints by enumerating them may take exponential time. Fortunately, this can be avoided. By definition, it follows that the union of two $U$-traps is again a $U$-trap, and similarly for siphons. Therefore, given a configuration $C$, there exists a unique maximal $U$-siphon $P_{\text{max}}$ such that $C(P_{\text{max}})=0$, and a unique maximal $U$-trap $P^{\prime}_{\text{max}}$ such that $C(P^{\prime}_{\text{max}})=0$. Moreover, $P_{\text{max}}$ and $P^{\prime}_{\text{max}}$ can be computed in linear time by means of a simple greedy algorithm (see e.g. [11, Ex. 4.5]). This simplifies the task of checking traps and siphons constraints, and yields a \coNP procedure for testing StrongConsensus:

{restatable}

propositionstrconscoNP Deciding if a protocol satisfies StrongConsensus is in \coNP.

Proof.

Testing whether a protocol does not satisfy StrongConsensus can be done by guessing $C_{0},C,C^{\prime}\in\mathrm{Pop}(Q)$, $b\in\{0,1\}$, $q,q^{\prime}\in Q$ and $\bm{x},\bm{x}^{\prime}:T\to\mathbb{N}$, and testing whether

• (a)

$C_{0}$ is initial, $C$ is terminal, $C^{\prime}$ is terminal, $q\in\llbracket C\rrbracket$, $q^{\prime}\in\llbracket C^{\prime}\rrbracket$, $O(q)\not=O(q^{\prime})$, and

• (b)

C_{0}\mathrel{\mathchoice{\mathrel{}\mathrel{\mathop{\dabar@\dabar@}\limits^{\hbox{\set@color\raisebox{-1.1pt}{$\scriptstyle\bm{x}\!\!$}}}}\mathrel{\mathchar 0\relax\symAMSa 04B}{\!\!}}{\mathrel{}\mathrel{\mathop{\dabar@\dabar@}\limits^{\hbox{\set@color\raisebox{-1.1pt}{$\scriptstyle\bm{x}\!\!$}}}}\mathrel{\mathchar 0\relax\symAMSa 04B}{\!\!}}{\mathrel{}\mathrel{\mathop{\dabar@\dabar@}\limits^{\hbox{\set@color\raisebox{-1.1pt}{$\scriptscriptstyle\bm{x}\!\!$}}}}\mathrel{\mathchar 0\relax\symAMSa 04B}{\!\!}}{\mathrel{}\mathrel{\mathop{\dabar@\dabar@}\limits^{\hbox{\set@color\raisebox{-1.1pt}{$\scriptstyle\bm{x}\!\!$}}}}\mathrel{\mathchar 0\relax\symAMSa 04B}{\!\!}}}C and C_{0}\mathrel{\mathchoice{\mathrel{}\mathrel{\mathop{\dabar@\dabar@}\limits^{\hbox{\set@color\raisebox{-1.1pt}{$\scriptstyle\bm{x}^{\prime}\!\!$}}}}\mathrel{\mathchar 0\relax\symAMSa 04B}{\!\!}}{\mathrel{}\mathrel{\mathop{\dabar@\dabar@}\limits^{\hbox{\set@color\raisebox{-1.1pt}{$\scriptstyle\bm{x}^{\prime}\!\!$}}}}\mathrel{\mathchar 0\relax\symAMSa 04B}{\!\!}}{\mathrel{}\mathrel{\mathop{\dabar@\dabar@}\limits^{\hbox{\set@color\raisebox{-1.1pt}{$\scriptscriptstyle\bm{x}^{\prime}\!\!$}}}}\mathrel{\mathchar 0\relax\symAMSa 04B}{\!\!}}{\mathrel{}\mathrel{\mathop{\dabar@\dabar@}\limits^{\hbox{\set@color\raisebox{-1.1pt}{$\scriptstyle\bm{x}^{\prime}\!\!$}}}}\mathrel{\mathchar 0\relax\symAMSa 04B}{\!\!}}}C^{\prime}.

Since there is no a priori bound on the size of $C_{0},C,C^{\prime}$ and $\bm{x},\bm{x}^{\prime}$, we guess them carefully. First, we guess whether $D(p)=0$, $D(p)=1$ or $D(p)\geq 2$ for every $D\in\{C_{0},C,C^{\prime}\}$ and $p\in Q$. This gives enough information to test (a). Then, we guess $\llbracket\bm{x}\rrbracket$ and $\llbracket\bm{x}^{\prime}\rrbracket$. This allows to test traps/siphons constraints as follows. Let $U\stackrel{{\scriptstyle\scriptscriptstyle\text{def}}}{{=}}\llbracket\bm{x}\rrbracket$, let $P_{\text{max}}$ be the maximal $U$-trap such that $C(P_{\text{max}})=0$, and let $P^{\prime}_{\text{max}}$ be the maximal $U$-siphon such that $C_{0}(P^{\prime}_{\text{max}})=0$. Conditions (b) and (c) of Definition 11 hold if and only if ${{}^{\bullet}(P_{\text{max}})}\cap U=\emptyset$ and ${(P^{\prime}_{\text{max}})^{\bullet}}\cap U=\emptyset$, which can be tested in polynomial time. The same is done for $\bm{x}^{\prime}$. If (a) and siphons/traps constraints hold, we build the system $\mathcal{S}$ of linear equations/inequalities obtained from the conjunction of the flow equations together with the constraints already guessed. By standard results on integer linear programming (see e.g. [24, Sect. 17]), if $\mathcal{S}$ has a solution, then it has one of polynomial size, and hence we may guess it. ∎

4.3 ${\it WS}^{3}$-protocols

We say that a protocol belongs to ${\it WS}^{3}$ if it satisfies LayeredTermination and StrongConsensus. Since ${\it WS}^{3}$ \subseteq$${\it WS}^{2} \subseteq$${\it WS} holds, every ${\it WS}^{3}$-protocol is well-specified. Recall that a language $L$ belongs to the class \DP [22] if there exist languages $L_{1}\in\NP$ and $L_{2}\in\coNP$ such that $L=L_{1}\cap L_{2}$. By taking $L_{1}$ and $L_{2}$ respectively as the languages of population protocols satisfying LayeredTermination and StrongConsensus, Corollary 7 and Proposition 13 yield:

Theorem 14**.**

The membership problem for ${\it WS}^{3}$-protocols is in \DP.

5 ${\it WS}^{3}$ is as expressive as ${\it WS}$

In a famous result, Angluin et al. [3] have shown that a predicate is computable by a population protocol if and only if it is definable in Presburger arithmetic, the first-order theory of addition [1, 3]. In particular, [1] constructs protocols for Presburger-definable predicates by means of a well-known result: Presburger-definable predicates are the smallest set of predicates containing all threshold and remainder predicates, and closed under boolean operations. A threshold predicate is a predicate of the form

[TABLE]

where $k\geq 1$ and $a_{1},\ldots,a_{k},c\in\mathbb{Z}$. A remainder predicate is a predicate of the form

[TABLE]

where $k\geq 1$, $m\geq 2$ and $a_{1},\ldots,a_{k},c,m\in\mathbb{Z}$. Here, we show that these predicates can be computed by ${\it WS}^{3}$-protocols, and that ${\it WS}^{3}$ is closed under negation and conjunction. As a consequence, we obtain that ${\it WS}^{3}$-protocols are as expressive as ${\it WS}$, the class of all well-specified protocols.

Threshold.

We describe the protocol given in [1] to compute the threshold predicate $\sum_{i=1}^{k}a_{i}x_{i}<c$. Let

[TABLE]

and define

[TABLE]

The protocol is $\mathcal{P}_{\text{thr}}\stackrel{{\scriptstyle\scriptscriptstyle\text{def}}}{{=}}(Q,T,\Sigma,I,O)$, where

[TABLE]

and $T$ contains

[TABLE]

for every $n,n^{\prime}\in[-v_{\text{max}},v_{\text{max}}]$, $\ell,o,o^{\prime}\in\{0,1\}$. Intuitively, a state $(\ell,n,o)$ indicates that the agent has value $n$, opinion $o$, and that it is a leader if and only if $\ell=1$. When a leader $q$ and a state $r$ interact, $r$ becomes a non leader, and $q$ increases its value as much as possible by substracting from the value of $r$. Moreover, a leader can change the opinion of any non leader.

{restatable}

propositionpropThresholdConsensus $\mathcal{P}_{\text{thr}}$ satisfies StrongConsensus.

Proof.

Let $\mathrm{val}(q)\stackrel{{\scriptstyle\scriptscriptstyle\text{def}}}{{=}}n$ for every state $q=(\ell,n,o)\in Q$, and let $\mathrm{val}(C)\stackrel{{\scriptstyle\scriptscriptstyle\text{def}}}{{=}}\sum_{q\in Q}C(q)\cdot\mathrm{val}(q)$ for every configuration $C\in\mathrm{Pop}(Q)$. The following holds for every $C,C^{\prime}\in\mathrm{Pop}(Q)$:

• (a)

If $(C,C^{\prime},\bm{x})$ is a solution to the flow equations for some $\bm{x}:T\to\mathbb{N}$, then $\mathrm{val}(C)=\mathrm{val}(C^{\prime})$.

• (b)

If $C,C^{\prime}$ are terminal, $C$ and $C^{\prime}$ contain a leader, and $\mathrm{val}(C)=\mathrm{val}(C^{\prime})$, then $O(C)=O(C^{\prime})$.

Suppose for the sake of contradiction that $\mathcal{P}$ does not satisfy StrongConsensus. There exist $C_{0},C,C^{\prime}\in\mathrm{Pop}(Q)$, $q,q^{\prime}\in Q$ and $\bm{x},\bm{x}^{\prime}:T\to\mathbb{N}$ such that C_{0}\mathrel{\mathchoice{\mathrel{}\mathrel{\mathop{\dabar@\dabar@}\limits^{\hbox{\set@color\raisebox{-1.1pt}{$\scriptstyle\bm{x}\!\!$}}}}\mathrel{\mathchar 0\relax\symAMSa 04B}{\!\!}}{\mathrel{}\mathrel{\mathop{\dabar@\dabar@}\limits^{\hbox{\set@color\raisebox{-1.1pt}{$\scriptstyle\bm{x}\!\!$}}}}\mathrel{\mathchar 0\relax\symAMSa 04B}{\!\!}}{\mathrel{}\mathrel{\mathop{\dabar@\dabar@}\limits^{\hbox{\set@color\raisebox{-1.1pt}{$\scriptscriptstyle\bm{x}\!\!$}}}}\mathrel{\mathchar 0\relax\symAMSa 04B}{\!\!}}{\mathrel{}\mathrel{\mathop{\dabar@\dabar@}\limits^{\hbox{\set@color\raisebox{-1.1pt}{$\scriptstyle\bm{x}\!\!$}}}}\mathrel{\mathchar 0\relax\symAMSa 04B}{\!\!}}}C, C_{0}\mathrel{\mathchoice{\mathrel{}\mathrel{\mathop{\dabar@\dabar@}\limits^{\hbox{\set@color\raisebox{-1.1pt}{$\scriptstyle\bm{x}^{\prime}\!\!$}}}}\mathrel{\mathchar 0\relax\symAMSa 04B}{\!\!}}{\mathrel{}\mathrel{\mathop{\dabar@\dabar@}\limits^{\hbox{\set@color\raisebox{-1.1pt}{$\scriptstyle\bm{x}^{\prime}\!\!$}}}}\mathrel{\mathchar 0\relax\symAMSa 04B}{\!\!}}{\mathrel{}\mathrel{\mathop{\dabar@\dabar@}\limits^{\hbox{\set@color\raisebox{-1.1pt}{$\scriptscriptstyle\bm{x}^{\prime}\!\!$}}}}\mathrel{\mathchar 0\relax\symAMSa 04B}{\!\!}}{\mathrel{}\mathrel{\mathop{\dabar@\dabar@}\limits^{\hbox{\set@color\raisebox{-1.1pt}{$\scriptstyle\bm{x}^{\prime}\!\!$}}}}\mathrel{\mathchar 0\relax\symAMSa 04B}{\!\!}}}C^{\prime}, $C_{0}$ is initial, $C$ and $C^{\prime}$ are terminal consensus configurations, $q\in\llbracket C\rrbracket$, $q^{\prime}\in\llbracket C^{\prime}\rrbracket$ and $O(q)\neq O(q^{\prime})$. Note that $(C_{0},C,\bm{x})$ and $(C_{0},C^{\prime},\bm{x}^{\prime})$ both satisfy the flow equations. Thus, by (a), $\mathrm{val}(C)=\mathrm{val}(C_{0})=\mathrm{val}(C^{\prime})$. Since $C_{0}$ is initial, it contains a leader. Since the set of leaders forms a $U$-trap for every $U\subseteq T$, and $(C_{0},C,\bm{x})$ and $(C_{0},C^{\prime},\bm{x})$ satisfy trap constraints, $C$ and $C^{\prime}$ contain a leader. By (b), $C$ and $C^{\prime}$ are consensus configurations with $O(C)=O(C^{\prime})$, which is a contradiction. ∎

{restatable}

propositionpropThresholdLayered $\mathcal{P}_{\text{thr}}$ satisfies LayeredTermination.

Proof.

Let $L_{0}\stackrel{{\scriptstyle\scriptscriptstyle\text{def}}}{{=}}\{(1,x,0):c\leq x\leq v_{\text{max}}\}$, $L_{1}\stackrel{{\scriptstyle\scriptscriptstyle\text{def}}}{{=}}\{(1,x,1):-v_{\text{max}}\leq x<c\}$, $N_{0}\stackrel{{\scriptstyle\scriptscriptstyle\text{def}}}{{=}}\{(0,0,0)\}$ and $N_{1}\stackrel{{\scriptstyle\scriptscriptstyle\text{def}}}{{=}}\{(0,0,1)\}$. It can be shown that the following ordered partitions satisfy layered termination for $c>0$ and $c\leq 0$ respectively:

[TABLE]

Remainder.

We give a protocol for the remainder predicate

[TABLE]

The protocol is $\mathcal{P}_{\text{rmd}}=(Q,T,\Sigma,I,O)$, where

[TABLE]

and where $T$ contains the following transitions for every $n,n^{\prime}\in[0,m)$ and $b\in\{\mathrm{false},\mathrm{true}\}$:

[TABLE]

In the appendix we show that $\mathcal{P}_{\text{rmd}}$ belongs to ${\it WS}^{3}$ by adapting the proof for $\mathcal{P}_{\text{thr}}$.

Negation and conjunction.

Let $\mathcal{P}_{1}=(Q_{1},T_{1},\Sigma,I_{1},O_{1})$ and $\mathcal{P}_{2}=(Q_{2},T_{2},\Sigma,I_{2},O_{2})$ be ${\it WS}^{3}$-protocols computing predicates $\varphi_{1}$ and $\varphi_{2}$ respectively. We may assume that $\mathcal{P}_{1}$ and $\mathcal{P}_{2}$ are defined over identical $\Sigma$, for we can always extend the input domain of threshold/remainder predicates by variables with coefficients of value zero. The predicate $\neg\varphi_{i}$ can be computed by replacing $O_{i}$ by the new output function $O_{i}^{\prime}$ such that $O_{i}^{\prime}(q)\stackrel{{\scriptstyle\scriptscriptstyle\text{def}}}{{=}}\neg O_{i}(q)$ for every $q\in Q_{i}$. To compute $\varphi_{1}\land\varphi_{2}$, we build an asynchronous product where steps of $\mathcal{P}_{1}$ and $\mathcal{P}_{2}$ can be executed independently.

More formally, the conjunction of $\mathcal{P}_{1}$ and $\mathcal{P}_{2}$ is defined as the population protocol $\mathcal{P}\stackrel{{\scriptstyle\scriptscriptstyle\text{def}}}{{=}}(Q,S,I,\Sigma,O)$ such that $Q\stackrel{{\scriptstyle\scriptscriptstyle\text{def}}}{{=}}Q_{1}\times Q_{2}$, $S\stackrel{{\scriptstyle\scriptscriptstyle\text{def}}}{{=}}S_{1}\cup S_{2}$, $I(\sigma)\stackrel{{\scriptstyle\scriptscriptstyle\text{def}}}{{=}}(I_{1}(\sigma),I_{2}(\sigma))$ and $O(p,q)\stackrel{{\scriptstyle\scriptscriptstyle\text{def}}}{{=}}O_{1}(p)\land O_{2}(q)$ where

[TABLE]

In the appendix we show that $\mathcal{P}$ is in ${\it WS}^{3}$ since terminal/consensus configurations, flow equations, and traps and siphons constraints are preserved by projections from $\mathcal{P}$ onto $\mathcal{P}_{1}$ and $\mathcal{P}_{2}$.

6 Experimental results

We have developed a tool called Peregrine 333Peregrine and benchmarks are available from https://gitlab.lrz.de/i7/peregrine/. to check membership in ${\it WS}^{3}$. Peregrine is implemented on top of the SMT solver Z3 [9].

Peregrine reads in a population protocol $\mathcal{P}=(Q,T,\Sigma,I,O)$ and constructs two sets of constraints. The first set is satisfiable if and only if LayeredTermination holds, and the second is unsatisfiable if and only if StrongConsensus holds.

For LayeredTermination, our tool Peregrine iteratively constructs constraints checking the existence of an ordered partition of size 1,$$2, $\ldots,|T|$ and decides if they are satisfiable. To check that the execution of a layer is silent, the constraints mentioned in the proof of Proposition 5 are transformed using Farkas’ lemma (see e.g. [24]) into a version that is satisfiable if and only if all the executions of the layer are silent. Also, the constraints for condition (b) of Definition 3 are added.

For StrongConsensus, Peregrine initially constructs the constraints for the flow equation for three configurations $C_{0},C_{1},C_{2}$ and vectors $\bm{x}_{1}$ and $\bm{x}_{2}$, with additional constraints to guarantee that $C_{0}$ is initial, $C_{1}$ and $C_{2}$ are terminal, and $C_{1}$ and $C_{2}$ are consensus of different values. If these constraints are unsatisfiable, the protocol satisfies StrongConsensus. Otherwise, Peregrine searches for a $U$-trap or $U$-siphon to show that either C_{0}\mathrel{\mathchoice{\mathrel{}\mathrel{\mathop{\dabar@\dabar@}\limits^{\hbox{\set@color\raisebox{-1.1pt}{$\scriptstyle\bm{x}_{1}\!\!$}}}}\mathrel{\mathchar 0\relax\symAMSa 04B}{\!\!}}{\mathrel{}\mathrel{\mathop{\dabar@\dabar@}\limits^{\hbox{\set@color\raisebox{-1.1pt}{$\scriptstyle\bm{x}_{1}\!\!$}}}}\mathrel{\mathchar 0\relax\symAMSa 04B}{\!\!}}{\mathrel{}\mathrel{\mathop{\dabar@\dabar@}\limits^{\hbox{\set@color\raisebox{-1.1pt}{$\scriptscriptstyle\bm{x}_{1}\!\!$}}}}\mathrel{\mathchar 0\relax\symAMSa 04B}{\!\!}}{\mathrel{}\mathrel{\mathop{\dabar@\dabar@}\limits^{\hbox{\set@color\raisebox{-1.1pt}{$\scriptstyle\bm{x}_{1}\!\!$}}}}\mathrel{\mathchar 0\relax\symAMSa 04B}{\!\!}}}C_{1} or C_{0}\mathrel{\mathchoice{\mathrel{}\mathrel{\mathop{\dabar@\dabar@}\limits^{\hbox{\set@color\raisebox{-1.1pt}{$\scriptstyle\bm{x}_{2}\!\!$}}}}\mathrel{\mathchar 0\relax\symAMSa 04B}{\!\!}}{\mathrel{}\mathrel{\mathop{\dabar@\dabar@}\limits^{\hbox{\set@color\raisebox{-1.1pt}{$\scriptstyle\bm{x}_{2}\!\!$}}}}\mathrel{\mathchar 0\relax\symAMSa 04B}{\!\!}}{\mathrel{}\mathrel{\mathop{\dabar@\dabar@}\limits^{\hbox{\set@color\raisebox{-1.1pt}{$\scriptscriptstyle\bm{x}_{2}\!\!$}}}}\mathrel{\mathchar 0\relax\symAMSa 04B}{\!\!}}{\mathrel{}\mathrel{\mathop{\dabar@\dabar@}\limits^{\hbox{\set@color\raisebox{-1.1pt}{$\scriptstyle\bm{x}_{2}\!\!$}}}}\mathrel{\mathchar 0\relax\symAMSa 04B}{\!\!}}}C_{2} does not hold. If, say, a $U$-siphon $\mathit{S}$ is found, then Peregrine adds the constraint $C_{0}(\mathit{S})>0$ to the set of initial constraints. This process is repeated until either the constraints are unsatisfiable and StrongConsensus is shown, or all possible $U$-traps and $U$-siphons are added, in which case StrongConsensus does not hold. We use this refinement-based approach instead of the \coNP approach described in Proposition 13, as that could require a quadratic number of variables and constraints, and we generally expect to need a small number of refinement steps.

We evaluated Peregrine on a set of benchmarks: the threshold and remainder protocols of [2], the majority protocol of [3], the broadcast protocol of [8] and two versions of the flock of birds444The variant from [8] is referred to as threshold-n by its authors. protocol from [8, 6]. We checked the parametrized protocols for increasing values of their primary parameter until we reached a timeout. For the threshold and remainder protocols, we set the secondary parameter $c$ to $1$ since it has no incidence on the size of the protocol, and since the variation in execution time for different values of $c$ was negligible. Moreover, we assumed that all possible values for $a_{i}$ were present in the inputs, which represents the worst case.

All experiments were performed on the same machine equipped with an Intel Core i7-4810MQ CPU and 16 GB of RAM. The time limit was set to 1 hour. The results are shown in Table 1. In all cases where we terminated within the time limit, we were able to show membership for ${\it WS}^{3}$. Generally, showing StrongConsensus took much less time than showing LayeredTermination, except for the flock of birds protocols, where we needed linearly many $U$-traps.

As an extension, we also tried proving correctness after proving membership in ${\it WS}^{3}$. For this, we constructed constraints for the existence of an input $X$ and configuration $C$ with I(X)\mathrel{\mathchoice{\mathrel{}\mathrel{\mathop{\dabar@\dabar@}\limits^{\hbox{\set@color\raisebox{-1.1pt}{$\scriptstyle\bm{x}\!\!$}}}}\mathrel{\mathchar 0\relax\symAMSa 04B}{\!\!}}{\mathrel{}\mathrel{\mathop{\dabar@\dabar@}\limits^{\hbox{\set@color\raisebox{-1.1pt}{$\scriptstyle\bm{x}\!\!$}}}}\mathrel{\mathchar 0\relax\symAMSa 04B}{\!\!}}{\mathrel{}\mathrel{\mathop{\dabar@\dabar@}\limits^{\hbox{\set@color\raisebox{-1.1pt}{$\scriptscriptstyle\bm{x}\!\!$}}}}\mathrel{\mathchar 0\relax\symAMSa 04B}{\!\!}}{\mathrel{}\mathrel{\mathop{\dabar@\dabar@}\limits^{\hbox{\set@color\raisebox{-1.1pt}{$\scriptstyle\bm{x}\!\!$}}}}\mathrel{\mathchar 0\relax\symAMSa 04B}{\!\!}}}C and $\varphi(X)\neq O(C)$. We were able to prove correctness for all the protocols in our set of benchmarks. The correctness check was faster than the well-specification check for broadcast, majority, threshold and both flock of birds protocols, and slower for the remainder protocol, where we reached a timeout for $m=70$.

7 Conclusion and further work

We have presented ${\it WS}^{3}$, the first class of well-specified population protocols with a membership problem of reasonable complexity (i.e. in \DP) and with the full expressiveness of well-specified protocols. Previous work had shown that the membership problem for the general class of well-specified protocols is decidable, but at least \EXPSPACE-hard with algorithms of non primitive recursive complexity.

We have shown that ${\it WS}^{3}$ is a natural class that contains many standard protocols from the literature, like flock-of-birds, majority, threshold and remainder protocols. We implemented the membership procedure for ${\it WS}^{3}$ on top of the SMT solver Z3, yielding the first software able to automatically prove well-specification of population protocols for all (of the infinitely many) inputs. Previous work could only prove partial correctness of protocols with at most 9 states and 28 transitions, by trying exhaustively a finite number of inputs [21, 25, 6, 8]. Our algorithm deals with all inputs and can handle larger protocols with up to 70 states and over 2500 transitions.

Future work will concentrate on three problems: improving the performance of our tool; automatically deciding if a ${\it WS}^{3}$-protocol computes the predicate described by a given Presburger formula; and the diagnosis problem: when a protocol does not belong to ${\it WS}^{3}$, delivering an explanation, e.g. a non-terminating fair execution. We think that our constraint-based approach provides an excellent basis for attacking these questions.

Appendix A Missing proofs of Section 3

For the proof of Proposition 2 we need to introduce Petri nets. Intuitively, Petri nets are similar to population protocols, but their transitions can also create and destroy agents.

A Petri net $N=(P,T,F)$ consists of a finite set $P$ of places, a finite set $T$ of transitions, and a flow function $F\colon(P\times T)\cup(T\times P)\rightarrow\mathbb{N}$. Given a transition $t\in T$, the multiset $\mathrm{pre}(t)$ of input places of $t$ is defined by $\mathrm{pre}(t)(p)=F(p,t)$, and the multiset $\mathrm{post}(t)$ of output places by $\mathrm{post}(t)(p)=F(t,p)$. A marking $M$ of a net $N$ is a multiset of places. Given a place $p$, we say that $M$ puts $M(p)$ tokens in $p$. A transition $t\in T$ is enabled at a marking $M$ if $\mathrm{pre}(t)\leq M$. A transition $t$ enabled at $M$ can fire, yielding the marking $M^{\prime}=M-\mathrm{pre}(t)+\mathrm{post}(t)$. We write this fact as $M\xrightarrow{t}M^{\prime}$. We extend enabledness and firing to sequences of transitions as follows. Let $\sigma=t_{1}\ldots t_{k}$ be a finite sequence of transitions $t_{j}\in T$. We write $M\xrightarrow{\sigma}M^{\prime}$ and call it a firing sequence if there exists a sequence $M_{0},\ldots,M_{k}$ of markings such that $M=M_{0}\xrightarrow{t_{1}}M_{1}\cdots\xrightarrow{t_{k}}M_{k}=M^{\prime}$. In that case, we say that $M^{\prime}$ is reachable from $M$ and denote by ${\it Reach}(N,M)$ the set of markings reachable from $M$.

\propSilentHardness

Proof.

The proof is very similar to the one of [14, Theorem 10]. However, since the proof requires small modifications at different places, we give it in full for completeness. The proof constructs a sequence of reductions from the Petri net reachability problem. Each step in the sequence transforms a problem on Petri nets into an equivalent problem closer to the model of population protocols. The first step uses a well-known result of Hack [17]. The reachability problem for Petri nets can be reduced in polynomial time to the single-place-zero-reachability problem:

Given a Petri net $N_{0}$, a marking $M_{0}$, and a place $\hat{p}$: decide whether some marking $M\in{\it Reach}(N_{0},M_{0})$ satisfies $M(\hat{p})=0$.

We introduce a normal form for Petri nets. A Petri net $N=(P,T,F)$ is said to be in normal form if $F(x,y)\in\{0,1\}$ for every $x,y\in(P\times T)\cup(T\times P)$, and every transition $t$ satisfies $1\leq|\mathrm{pre}(t)|\leq 2$ and $1\leq|\mathrm{post}(t)|\leq 2$. For every Petri net $N=(P,T,F)$ and markings $M_{1},M_{2}$, one can construct a normal form Petri net $N^{\prime}=(P^{\prime},T^{\prime},F^{\prime})$ with $P\subseteq P^{\prime}$ such that $M_{2}$ is reachable from $M_{1}$ in $N$ if and only if $M_{2}^{\prime}$ is reachable from $M_{1}^{\prime}$ in $N^{\prime}$, and $M_{i}^{\prime}=M_{i}+\bm{\ell}$, where $\ell$ is a special lock place. Intuitively, each transition $t$ of $N$ with more than two input and/or output places is simulated in $N^{\prime}$ by a widget. The widget starts and finishes its execution by acquiring the lock and releasing it, respectively. This guarantees no two widgets are executing concurrently. When simulating transition $t$, its widget first consumes, one by one, the tokens consumed by $t$ (as given by $\mathrm{pre}(t)$), and then produces, one by one, the tokens produced by $t$ (as given by $\mathrm{post}(t)$). Figure 1 shows a transition and its widget. Observe that all transitions of the widget are in normal form.

Let $N_{1}$ be the result of normalizing $N_{0}$. Let $P_{aux}$ be the set of places of $N_{1}$ that are not places of $N_{0}$, and are different from the lock place $\ell$. The single-place-zero-reachability problem reduces to

• (P1)

Does some marking $M\in{\it Reach}(N_{1},M_{0}+\bm{\ell})$ satisfy $M(\hat{p})=0$ and $M(P_{aux})=0$?

(Observe that $M(P_{aux})=0$ guarantees that no widget is in the middle of its execution.)

Now we add to $N_{1}$ a new place $p_{0}$ and a new widget simulating a transition $t_{0}$ with $\mathrm{pre}(t_{0})=\bm{p_{0}}$ and $\mathrm{post}(t_{0})=M_{0}+\bm{\ell}$. Let the resulting net be $N_{2}$. Then (P1) reduces to:

• (P2)

Does some marking $M\in{\it Reach}(N_{2},\bm{p_{0}})$ satisfy $M(\hat{p})=M(p_{0})=M(P_{aux})=0$?

For our next step, we “reverse” $N_{2}$: define $N_{3}$ as the result of reversing all arcs of ${N}_{2}$, i.e., $P_{3}=P_{2}$, $T_{3}=T_{2}$ but $F_{3}(x,y)=F_{2}(y,x)$ for every two nodes $x,y$. Clearly, $N_{3}$ is in normal form when $N_{2}$ is. The problem (P2) reduces to:

• (P3)

Is $\bm{p_{0}}\in{\it Reach}(N_{3},M)$ for some marking $M$ of $N_{3}$ satisfying $M(\hat{p})=M(p_{0})=M(P_{aux})=0$?

In the last step we reduce (P3) to the membership problem for ${\it WS}^{2}$. Let $N_{3}=(P_{3},T_{3},F_{3})$. We construct a population protocol $\mathcal{P}=(Q,T,\Sigma,I,O)$, defined as follows:

• •

$Q=P_{3}\cup\{{\it Fresh},{\it Used},{\it Collect}\}$. That is, $\mathcal{P}$ contains a state for each place of $N_{3}$, plus three auxiliary places.

• •

$\Sigma=Q\setminus(\{\hat{p},p_{0}\}\cup P_{aux})$, and $I$ is the identity mapping.

• •

$O(p_{0})=1$ and $O(q)=0$ for every $q\neq p_{0}$.

• •

$T=T_{3}^{\prime}\cup T_{P}\cup T_{s}$.

These sets are formally described below. Intuitively, the transitions of $T_{3}^{\prime}$ simulate the Petri net transitions of $T_{3}$, the transitions of $T_{P}$ guarantee that a terminal consensus is reachable from every configuration that does not represent $\bm{p_{0}}$, and the $T_{s}$ are additional silent actions to make the protocol well-formed.

The transitions of $T_{3}^{\prime}$ simulate the behaviour of $N_{3}$. For this, $T_{3}^{\prime}$ contains a transition $t^{\prime}$ for every net transition $t\in T_{3}$. If $t\in T_{3}$ has two input places $p_{1},p_{2}$ and two output places $p_{1}^{\prime},p_{2}^{\prime}$, then $t^{\prime}=(p_{1},p_{2})\xrightarrow{}(p_{1}^{\prime},p_{2}^{\prime})$, The other cases are: if $t$ has one input place $p_{1}$ and two output places $p_{1}^{\prime},p_{2}^{\prime}$, then $t^{\prime}=(p_{1},{\it Fresh})\xrightarrow{}(p_{1}^{\prime},p_{2}^{\prime})$; if $t$ has two input places $p_{1},p_{2}$ and one output place $p_{1}^{\prime}$, then $t^{\prime}=(p_{1},p_{2})\xrightarrow{}(p_{1}^{\prime},{\it Used})$; if $t$ has one input place $p_{1}$ and one output place $p_{1}^{\prime}$, then $t^{\prime}=(p_{1},{\it Fresh})\xrightarrow{}(p_{1}^{\prime},{\it Used})$.

The transitions of $T_{P}$ test the presence of tokens anywhere, apart from one single token in $p_{0}$. For every pair $(q,q^{\prime})\in\left(\left(P_{3}\setminus\left\{p_{0}\right\}\right)\times Q\right)\cup\left\{(p_{0},p_{0})\right\}$, the set $T_{P}$ contains a transition $(q,q^{\prime})\mapsto({\it Collect},{\it Collect})$. Further, for every place $q\in Q$, the set $T_{P}$ contains a transition $(q,{\it Collect})\mapsto({\it Collect},{\it Collect})$. Intuitively, these transitions guarantee that as long as the current marking of $N_{3}$ is different from $\mathbf{p}_{0}$, the protocol $\mathcal{P}$ can reach a terminal configuration with all agents in state ${\it Collect}$.

The set $T_{s}$ contains a silent transition $(q,q^{\prime})\mapsto(q,q^{\prime})$ for every pair $(q,q^{\prime})$ of states.

Assume that $\bm{p_{0}}\in{\it Reach}(N_{3},M)$ for some marking $M$ such that $M(\hat{p})=M(p_{0})=M(P_{aux})=0$. Let $\sigma$ be a firing sequence such that $M\xrightarrow{\sigma}\bm{p_{0}}$. Observe that $\sigma$ is nonempty, and must end with a firing of transition $t_{0}$. Let $K$ be the number of times that transitions with only one input place occur in $\sigma$. We claim that the initial configuration $C$ given by $C({\it Fresh})=K$, and $C(p)=M(p)$ for every $p\in P_{3}$ has a fair execution that does not reach a consensus. Indeed, the finite execution of $\mathcal{P}$ that simulates $\sigma$ by executing the corresponding transitions of $T_{3}^{\prime}$ (and which, abusing language, we also denote $\sigma$), reaches a configuration $C^{\prime}$ with $C^{\prime}(p_{0})=1$, $C^{\prime}({\it Fresh})=0$, $C^{\prime}({\it Used})>0$ (because every transition that moves an agent to $p_{0}$ also moves an agent to ${\it Used}$), and $C^{\prime}(p)=0$ for any other place $p$. Since $O(p_{0})=1$ and $O({\it Used})=0$, the configuration $C^{\prime}$ is not a consensus configuration. Since no transition of $T_{3}^{\prime}\cup T_{P}$ is enabled at $C^{\prime}$, all transitions enabled at $C^{\prime}$ are silent, and therefore from $C^{\prime}$ it is not possible to reach a consensus.

Assume now that $\bm{p_{0}}\notin{\it Reach}(N_{3},M)$ for any marking $M$ such that $M(\hat{p})=M(p_{0})=M(P_{aux})=0$. Then every configuration reachable from any initial configuration enables some transition of $T_{P}$. By fairness, every fair execution from any initial configuration contains at least one transition of $T_{P}$, and so some configuration reached along the execution populates state Collect. But then, again by fairness, the execution gets eventually trapped in a terminal configuration $C$ of the form $C({\it Collect})>0$ and $C(q)=0$ for every $q\notin{\it Collect}$. So every fair execution is silent and stabilizes to [math], and therefore the protocol belongs to ${\it WS}^{2}$.

∎

Appendix B Missing proofs of Section 4

{restatable}

propositionpropWSS A protocol belongs to ${\it WS}^{2}$ if and only if it satisfies Termination and Consensus.

Proof.

We prove a stronger result:

• (a)

A protocol is silent if and only if it satisfies Termination.

• (b)

A silent protocol is well-specified if and only if it satisfies Consensus.

((a) $\Rightarrow$): Follows immediately from the definitions.

((a) $\Leftarrow$): Let $C_{0}$ be an arbitrary configuration, and let $\gamma=C_{0}C_{1}C_{2}\cdots$ be a fair execution of the protocol. Let $\mathcal{C}_{\bot}$ be the set of terminal configurations reachable from $C_{0}$. Since Termination holds for every configuration, and so in particular for all of $C_{1},C_{2},\ldots$, all configurations of $\gamma$ can reach some cofiguration of $\mathcal{C}_{\bot}$.

For every $C_{i}$, let $d(C_{i})$ be the length of a shortest path from $C_{i}$ to some configuration of $\mathcal{C}_{\bot}$. We claim that for every $n\geq 0$, there are infinitely many indices $i$ such that $d(C_{i})\leq n$. Since there are only finitely many configurations reachable from $C_{0}$, say $K$, we have $d(C_{i})\leq K$ for every index $i\geq 0$. So it suffices to show that if there are infinitely many indices $i$ such that $d(C_{i})\leq n$, then there are infinitely many indices $j$ such that $d(C_{j})\leq n-1$.

Let $i_{1}\leq i_{2}\leq i_{3}\cdots$ be an infinite collection of indices such that $d(C_{i_{j}})\leq n$ for every $j\geq 1$. By definition of $d$, for every configuration $C_{i_{j}}$ there is a step $C_{i_{j}}\xrightarrow{}C_{i_{j}}^{\prime}$ such that $d(C_{i_{j}}^{\prime})=n-1$. By fairness, we have $C_{i_{j}}^{\prime}=C_{i_{j}+1}$ for infinitely many $j\geq 1$, and the claim is proved. By this claim, there are infinitely many indices $i$ such that $d(C_{i})\leq 0$, i.e., $C_{i}\in\mathcal{C}_{\bot}$. Let $i_{0}$ be one of them. Since $\mathcal{C}_{\bot}$ only contains terminal configurations, we have $C_{i}=C_{i_{0}}$ for every $i\geq i_{0}$, and so $\gamma$ converges to $C_{\bot}$.

((b) $\Rightarrow$) Let $\mathcal{P}$ be a silent and well-specified protocol. Let $C_{0}$ be an initial configuration of $\mathcal{P}$, and let $C_{0}C_{1}\cdots C_{n}$ be a finite prefix of an execution such that $C_{n}$ is terminal. The execution $C_{0}C_{1}\cdots(C_{n})^{\omega}$ is fair. Since the protocol is well-specified, $C_{n}$ is a consensus configuration.

((b) $\Leftarrow$) Let $\mathcal{P}$ be a silent protocol satisfying Consensus. By convergence, every fair execution starting at an initial configuration $C$ eventually reaches a terminal configuration. Since $\mathcal{P}$ satisfies Consensus, all these configurations are consensus configurations, and moreover they all agree to the same boolean value. ∎

{restatable}

propositionlayterimpter LayeredTermination implies Termination.

Proof.

Let $\mathcal{P}=(Q,T,\Sigma,I,O)$ be a population protocol satisfying LayeredTermination, and let $C$ be an arbitrary configuration of $\mathcal{P}$. Let $(T_{1},T_{2},\ldots,T_{n})$ be the ordered partition of $T$ for LayeredTermination. There exists a sequence $w_{1}\in T_{1}^{*}$ such that $C\xrightarrow{w_{1}}C_{1}$, and $C_{1}$ is a terminal configuration of $\mathcal{P}[T_{1}]$. By the same reasoning, there exists a sequence $w_{2}\in T_{2}^{*}$ such that $C_{1}\xrightarrow{w_{2}}C_{2}$, and $C_{2}$ is a terminal configuration of $\mathcal{P}[T_{2}]$; further, by condition (b) of Definition 3, $C_{2}$ is also a terminal configuration of $T_{1}$. Iterating this process we find $C_{1}\xrightarrow{w_{1}\ldots w_{n}}C_{n}$ such that $C_{n}$ is a terminal configuration of all of $\mathcal{P}[T_{1}],\mathcal{P}[T_{2}],\ldots,\mathcal{P}[T_{n}]$. Therefore, $C_{n}$ is a terminal configuration of $\mathcal{P}$. ∎

We prove the claim made in the proof of Proposition 5:

{restatable}

lemmatinv Let $\mathcal{P}=(Q,T,\Sigma,I,O)$ be a population protocol and let $U$ be the set of non silent transitions of $T$. $\mathcal{P}$ has a non silent execution if and only if there exists $\bm{x}:U\to\mathbb{Q}$ such that $\sum_{t\in U}\bm{x}(t)\cdot(\mathrm{post}(t)(q)-\mathrm{pre}(t)(q))=0$ and $\bm{x}(q)\geq 0$ for every $q\in Q$, and $\bm{x}(q)>0$ for some $q\in Q$.

Proof.

$\Rightarrow$) Let $C_{0}C_{1}\cdots$ be a non silent execution. Since taking a silent transition does not change the current configuration, we can assume that the transitions occurring in the execution are non silent. Since the total number of agents of a configuration is left unchanged by transitions, there exist indices $i<j$ such that $C_{i}=C_{j}$. Let $w$ be the non empty sequence of transitions leading from $C_{i}$ to itself. By the flow equation with $C\stackrel{{\scriptstyle\scriptscriptstyle\text{def}}}{{=}}C_{i}$ and $C^{\prime}\stackrel{{\scriptstyle\scriptscriptstyle\text{def}}}{{=}}C_{i}$ we have $\sum_{t\in T}|w|_{t}\cdot(\mathrm{post}(t)(q)-\mathrm{pre}(t)(q))=0$ for every state $q$. Define $\bm{x}(t)\stackrel{{\scriptstyle\scriptscriptstyle\text{def}}}{{=}}|w|_{t}$.

$\Leftarrow$) Without loss of generality, we can assume $\bm{x}(q)\in\mathbb{N}$. If this is not the case, we multiply $\bm{x}$ by a suitable coefficient which yields another vector satisfying the conditions. Let $w\in U^{*}$ be any sequence of transitions such that $|w|_{t}=\bm{x}(t)$ for every $t\in U$, and $|w|_{t}=0$ otherwise. Choose a configuration $C$ such that $C\xrightarrow{w}C^{\prime}$ for some configuration $C^{\prime}$. Observe that $C$ exists, for example it suffices to take $C(q)>2\cdot|w|$ for every state $q$. By the flow equation, we have $C^{\prime}=C$. So $w^{\omega}$ is a non silent execution of $\mathcal{P}$ from $C$. ∎

We prove the claim made in the proof of Proposition 6:

{restatable}

lemmaudead Let $\mathcal{P}=(Q,T,\Sigma,I,O)$ be a protocol, and let $U\subseteq T$. It can be decided in polynomial time if $\mathcal{P}$ is $U$-dead.

Proof.

Let $S=T\setminus U$. We claim that $\mathcal{P}$ is not $U$-dead if and only if there exist $s\in S$ and non silent $u\in U$ such that

[TABLE]

The polynomial time upper bound follows by simply testing this condition on every pair $(s,u)\in S\times U$. Let us now prove the claim.

$\Leftarrow$) Suppose there exist $s\in S$ and non silent $u\in U$ such that (2) holds. Let $C_{0}\in\mathrm{Pop}(Q)$ be the configuration such that $C_{0}\stackrel{{\scriptstyle\scriptscriptstyle\text{def}}}{{=}}\mathrm{pre}(s)+(\mathrm{pre}(u)\mathbin{\varominus}\mathrm{post}(s))$. By (2), $C_{0}$ is $U$-dead. Moreover, $C_{0}\xrightarrow{s}C$ where $C=(\mathrm{pre}(u)\mathbin{\varominus}\mathrm{post}(s))+\mathrm{post}(s)$. Therefore, $u$ can be taken from $C$ since $\mathrm{pre}(u)\leq C$. We conclude that $\mathcal{P}$ is not $U$-dead from $C_{0}$.

$\Rightarrow$) If $\mathcal{P}$ is not $U$-dead, then there exists $C_{0}\xrightarrow{s_{1}}C_{1}\xrightarrow{s_{2}}\cdots\xrightarrow{s_{n}}C_{n}$ such that $C_{0},\ldots,C_{n-1}$ are $U$-dead, $s_{1},s_{2},\dots,s_{n}\in S$, and $C_{n}$ is not $U$-dead. Let $u\in U$ be some non silent transition that can be taken from $C_{n}$, i.e. such that $\mathrm{pre}(u)\leq C_{n}$. Suppose there exists some $u^{\prime}\in U$ such that $\mathrm{post}(u^{\prime})\neq\mathrm{pre}(u^{\prime})$ and $\mathrm{pre}(u^{\prime})\leq\mathrm{pre}(s_{n})+(\mathrm{pre}(u)\mathbin{\varominus}\mathrm{post}(s_{n}))$. We obtain

[TABLE]

Therefore, $C_{n-1}\xrightarrow{u^{\prime}}C$ for some $C\in\mathrm{Pop}(Q)$. Moreover, $C\neq C_{n-1}$ since $\mathrm{post}(u^{\prime})\neq\mathrm{pre}(u^{\prime})$. This contradicts the fact that $C_{n-1}$ is $U$-dead, hence (2) holds. ∎

Appendix C Missing proofs of Section 5

C.1 Threshold protocol

We first prove the two claims made in the proof of Prop. 5 and then give a full proof of Prop. 5.

Proposition 15**.**

For every $C,C^{\prime}\in\mathrm{Pop}(Q)$ and $\bm{x}:T\to\mathbb{N}$, if $(C,C^{\prime},\bm{x})$ is a solution to the flow equations, then $\mathrm{val}(C)=\mathrm{val}(C^{\prime})$.

Proof.

Assume $(C,C^{\prime},\bm{x})$ is a solution to the flow equations. For every $m,n\in[-v_{\text{max}},v_{\text{max}}]$, we have $g(m,n)+f(m,n)=m+n$. Therefore, $\mathrm{val}(\mathrm{pre}(t))=\mathrm{val}(\mathrm{post}(t))$ for every $t\in T$. This implies:

[TABLE]

Proposition 16**.**

Let $C,C^{\prime}\in\mathrm{Pop}(Q)$ be terminal configuration that contain a leader. Both $C$ and $C^{\prime}$ are consensus configurations. Moreover, if $\mathrm{val}(C)=\mathrm{val}(C^{\prime})$, then $O(C)=O(C^{\prime})$.

Proof.

We prove the first claim for $C$. The argument is identical for $C^{\prime}$. Suppose that $C$ is not a consensus configuration. Let $(1,m,o)\in\llbracket C\rrbracket$ be a leader of $C$. Since $C$ is not a consensus configuration, there exists $(\ell,n,\neg o)\in\llbracket C\rrbracket$. Therefore, the following transition $t$ is enabled at $C$:

[TABLE]

Moreover, $t$ is non silent which contradicts the fact that $C$ is terminal. Thus, $C$ is a consensus configuration.

Assume that $\mathrm{val}(C)=\mathrm{val}(C^{\prime})$. Suppose that $O(C)\neq O(C^{\prime})$ for the sake of contradiction. Without loss of generality, we may assume that $O(C)=1$ and $O(C^{\prime})=0$. Let $p_{C},p_{C^{\prime}}\in Q$ be respectively leaders of $C$ and $C^{\prime}$. We have $\mathrm{val}(p_{C})<c<v_{\text{max}}$ and $\mathrm{val}(p_{C^{\prime}})\geq c>-v_{\text{max}}$. We claim that

[TABLE]

To see that the claim holds, suppose that $\mathrm{val}(p_{C})<\mathrm{val}(C)$. There exists some $q_{C}\in\llbracket C\rrbracket$ such that $\mathrm{val}(q_{C})>0$. Since $\mathrm{val}(p_{C})<v_{\text{max}}$, some part of the value of $q_{C}$ can be transferred to $p_{C}$, i.e. there exists a non silent transition $t\in T$ with $\mathrm{pre}(t)=\Lbag p_{C},q_{C}\Rbag$, which contradicts that $C$ is terminal. Thus, $\mathrm{val}(p_{C})\geq\mathrm{val}(C)$ holds. The case $\mathrm{val}(p_{C^{\prime}})\leq\mathrm{val}(C^{\prime})$ follows by a similar argument.

Now, by (3) we have $\mathrm{val}(C)\leq\mathrm{val}(p_{C})<c$ and $\mathrm{val}(C^{\prime})\geq\mathrm{val}(p_{C^{\prime}})\geq c$ which is a contradiction since $\mathrm{val}(C)=\mathrm{val}(C^{\prime})$. Therefore, $O(C)=O(C^{\prime})$.∎

\propThresholdConsensus

Proof.

Let $\mathrm{val}(q)\stackrel{{\scriptstyle\scriptscriptstyle\text{def}}}{{=}}n$ for every state $q=(\ell,n,o)\in Q$, and let $\mathrm{val}(C)\stackrel{{\scriptstyle\scriptscriptstyle\text{def}}}{{=}}\sum_{q\in Q}C(q)\cdot\mathrm{val}(q)$ for every configuration $C\in\mathrm{Pop}(Q)$. Suppose for the sake of contradiction that $\mathcal{P}_{\text{thr}}$ does not satisfy StrongConsensus. There are two cases to consider.

• •

There exist $C,C^{\prime}\in\mathrm{Pop}(Q)$ such that $C\mathrel{\mathchoice{\mathrel{}\mathrel{\mathop{\dabar@\dabar@}\limits}\mathrel{\mathchar 0\relax\symAMSa 04B}{\!\!}}{\mathrel{}\mathrel{\mathop{\dabar@\dabar@}\limits}\mathrel{\mathchar 0\relax\symAMSa 04B}{\!\!}}{\mathrel{}\mathrel{\mathop{\dabar@\dabar@}\limits}\mathrel{\mathchar 0\relax\symAMSa 04B}{\!\!}}{\mathrel{}\mathrel{\mathop{\dabar@\dabar@}\limits}\mathrel{\mathchar 0\relax\symAMSa 04B}{\!\!}}}C^{\prime}$, $C$ is initial, $C^{\prime}$ is terminal and $C^{\prime}$ is not a consensus configuration. Since $C$ is initial, it contains a leader. It is readily seen that the set of leaders forms a $U$-trap for every $U\subseteq T$, which implies that $C^{\prime}$ contains a leader as $(C,C^{\prime},\bm{x})$ satisfying $U$-trap constraints for all $U$. By Prop. 16, $C^{\prime}$ is a consensus configuration, which is a contradiction.

• •

There exist $C_{0},C,C^{\prime}\in\mathrm{Pop}(Q)$ and $\bm{x},\bm{x}^{\prime}:T\to\mathbb{N}$ such that C_{0}\mathrel{\mathchoice{\mathrel{}\mathrel{\mathop{\dabar@\dabar@}\limits^{\hbox{\set@color\raisebox{-1.1pt}{$\scriptstyle\bm{x}\!\!$}}}}\mathrel{\mathchar 0\relax\symAMSa 04B}{\!\!}}{\mathrel{}\mathrel{\mathop{\dabar@\dabar@}\limits^{\hbox{\set@color\raisebox{-1.1pt}{$\scriptstyle\bm{x}\!\!$}}}}\mathrel{\mathchar 0\relax\symAMSa 04B}{\!\!}}{\mathrel{}\mathrel{\mathop{\dabar@\dabar@}\limits^{\hbox{\set@color\raisebox{-1.1pt}{$\scriptscriptstyle\bm{x}\!\!$}}}}\mathrel{\mathchar 0\relax\symAMSa 04B}{\!\!}}{\mathrel{}\mathrel{\mathop{\dabar@\dabar@}\limits^{\hbox{\set@color\raisebox{-1.1pt}{$\scriptstyle\bm{x}\!\!$}}}}\mathrel{\mathchar 0\relax\symAMSa 04B}{\!\!}}}C, C_{0}\mathrel{\mathchoice{\mathrel{}\mathrel{\mathop{\dabar@\dabar@}\limits^{\hbox{\set@color\raisebox{-1.1pt}{$\scriptstyle\bm{x}^{\prime}\!\!$}}}}\mathrel{\mathchar 0\relax\symAMSa 04B}{\!\!}}{\mathrel{}\mathrel{\mathop{\dabar@\dabar@}\limits^{\hbox{\set@color\raisebox{-1.1pt}{$\scriptstyle\bm{x}^{\prime}\!\!$}}}}\mathrel{\mathchar 0\relax\symAMSa 04B}{\!\!}}{\mathrel{}\mathrel{\mathop{\dabar@\dabar@}\limits^{\hbox{\set@color\raisebox{-1.1pt}{$\scriptscriptstyle\bm{x}^{\prime}\!\!$}}}}\mathrel{\mathchar 0\relax\symAMSa 04B}{\!\!}}{\mathrel{}\mathrel{\mathop{\dabar@\dabar@}\limits^{\hbox{\set@color\raisebox{-1.1pt}{$\scriptstyle\bm{x}^{\prime}\!\!$}}}}\mathrel{\mathchar 0\relax\symAMSa 04B}{\!\!}}}C^{\prime}, $C_{0}$ is initial, $C$ and $C^{\prime}$ are terminal consensus configurations, and $O(C)\neq O(C^{\prime})$. Note that $(C_{0},C,\bm{x})$ and $(C_{0},C^{\prime},\bm{x}^{\prime})$ both satisfy the flow equations. Therefore, by Prop. 15, $\mathrm{val}(C)=\mathrm{val}(C_{0})=\mathrm{val}(C^{\prime})$. Again, since $C_{0}$ is initial, it contains a leader, which implies that both $C$ and $C^{\prime}$ contain a leader. Since $\mathrm{val}(C)=\mathrm{val}(C^{\prime})$, Prop. 16 yields $O(C)=O(C^{\prime})$ which is a contradiction.∎

We now give a full proof of Proposition 5.

\propThresholdLayered

Proof.

Assume $c>0$. The case where $c\leq 0$ follows by a symmetric argument. Let $L_{0}\stackrel{{\scriptstyle\scriptscriptstyle\text{def}}}{{=}}\{(1,x,0):c\leq x\leq v_{\text{max}}\}$ and $N_{1}\stackrel{{\scriptstyle\scriptscriptstyle\text{def}}}{{=}}\{(0,0,1)\}$. We claim that the following ordered partition satisfies layered termination:

[TABLE]

We first show that every execution of $\mathcal{P}_{\text{thr}}[T_{1}]$ is fair. For the sake of contradiction, assume this is not the case. There exists a non silent execution $C_{0}\xrightarrow{t_{1}}C_{1}\xrightarrow{t_{2}}\cdots$ where $C_{0},C_{1},\ldots\in\mathrm{Pop}(Q)$ and $t_{1},t_{2},\ldots\in T_{1}$. For every $i\in\mathbb{N}$, let

[TABLE]

As observed in [1], $a_{0}\geq a_{1}\geq\cdots$. Therefore, there exists $n_{1}\in\mathbb{N}$ such that $a_{i}=a_{i-1}$ for every $i>n_{1}$. Similarly, no transition of $\mathcal{P}_{\text{thr}}$ increases the number of leaders. Thus, there exists $n_{2}\in\mathbb{N}$ such that $\mathrm{num}\text{-}\mathrm{leaders}(C_{i})=\mathrm{num}\text{-}\mathrm{leaders}(C_{i-1})$ for every $i>n_{2}$. Let $L_{\text{err}}\stackrel{{\scriptstyle\scriptscriptstyle\text{def}}}{{=}}\{(1,x,b):-v_{\text{max}}\leq x\leq v_{\text{max}},b\not=(x<c)\}$ be the set of leaders whose opinion is inconsistent with their value. Since no transition of $\mathcal{P}_{\text{thr}}$ produces states from $L_{\text{err}}$, transitions involving a state from $L_{\text{err}}$ can only be taken in finitely many steps. More formally, there exists $n_{3}\in\mathbb{N}$ such that $\llbracket\mathrm{pre}(t_{i})\rrbracket\cap L_{\text{err}}=\emptyset$ for every $i>n_{3}$. Let $n\stackrel{{\scriptstyle\scriptscriptstyle\text{def}}}{{=}}\max(n_{1},n_{2},n_{3})$. Any non silent transition $t_{i}$ such that $i>n$ must be of the form:

[TABLE]

for some $x<c$, as otherwise one of the above observations would be violated. But such transitions set the opinion of non leaders to [math], which can only occur for finitely many steps. Therefore, there exists $n^{\prime}\geq n$ such that every transition enabled in $C_{n^{\prime}}$ is silent. This is a contradiction.

It is readily seen that any execution of $\mathcal{P}_{\text{thr}}[T_{2}]$ is silent since each transition of $T_{2}$ is of the form:

[TABLE]

for some $c\leq x\leq v_{\text{max}}$. Therefore, it remains to prove that $\mathcal{P}_{\text{thr}}[T_{2}]$ is $T_{1}$-dead. Let $C\in\mathrm{Pop}(Q)$ be a $T_{1}$-dead configuration. For the sake of contradiction, suppose there exists $w\in T_{2}^{+}$ and $C^{\prime}\in\mathrm{Pop}(Q)$ such that $C\xrightarrow{w}C^{\prime}$ and $C^{\prime}$ enables some non silent transition $t\in T_{1}$. Since $C$ is $T_{1}$-dead, transition $t$ must be of the form

[TABLE]

for some $y<c$. Moreover, $(1,y,1)$ already appeared in $C$. This means that $C$ contains one leader of opinion [math], and one leader of opinion $1$. Therefore, $C$ is not $T_{1}$-dead, which is a contradiction. ∎

C.2 Remainder protocol

Let $\mathrm{val}(C)\stackrel{{\scriptstyle\scriptscriptstyle\text{def}}}{{=}}\left(\sum_{n\in[0,m)}C(n)\cdot n\right)\text{ mod }m$ for every $C\in\mathrm{Pop}(Q)$.

Proposition 17**.**

$\mathcal{P}_{\text{rmd}}$* satisfies StrongConsensus.*

Proof.

For every $C^{\prime},C^{\prime}\in\mathrm{Pop}(Q)$,

• (a)

if $(C,C^{\prime},\bm{x})$ is a solution to the flow equations for some $\bm{x}:T\to\mathbb{N}$, then $\mathrm{val}(C)=\mathrm{val}(C^{\prime})$.

• (b)

if $C,C^{\prime}\in\mathrm{Pop}(Q)$ are terminal configuration that contain a numerical value, then both $C$ and $C^{\prime}$ are consensus configurations, and if $\mathrm{val}(C)=\mathrm{val}(C^{\prime})$, then $O(C)=O(C^{\prime})$.

The proof of these two claims follows from the definition of $\mathcal{P}_{\text{rmd}}$ as in the case of the threshold protocol.

Suppose for the sake of contradiction that $\mathcal{P}_{\text{rmd}}$ does not satisfy StrongConsensus. There are two cases to consider.

• •

There exist $C,C^{\prime}\in\mathrm{Pop}(Q)$ such that $C\mathrel{\mathchoice{\mathrel{}\mathrel{\mathop{\dabar@\dabar@}\limits}\mathrel{\mathchar 0\relax\symAMSa 04B}{\!\!}}{\mathrel{}\mathrel{\mathop{\dabar@\dabar@}\limits}\mathrel{\mathchar 0\relax\symAMSa 04B}{\!\!}}{\mathrel{}\mathrel{\mathop{\dabar@\dabar@}\limits}\mathrel{\mathchar 0\relax\symAMSa 04B}{\!\!}}{\mathrel{}\mathrel{\mathop{\dabar@\dabar@}\limits}\mathrel{\mathchar 0\relax\symAMSa 04B}{\!\!}}}C^{\prime}$, $C$ is initial, $C^{\prime}$ is terminal and $C^{\prime}$ is not a consensus configuration. Since $C_{0}$ is initial, it only contains numerical values. Since numerical values form a $U$-trap for every $U\subseteq T$, $C$ contains a numerical value. By (b), $C$ is a consensus configuration, which is a contradiction.

• •

There exist $C_{0},C,C^{\prime}\in\mathrm{Pop}(Q)$ and $\bm{x},\bm{x}^{\prime}:T\to\mathbb{N}$ such that C_{0}\mathrel{\mathchoice{\mathrel{}\mathrel{\mathop{\dabar@\dabar@}\limits^{\hbox{\set@color\raisebox{-1.1pt}{$\scriptstyle\bm{x}\!\!$}}}}\mathrel{\mathchar 0\relax\symAMSa 04B}{\!\!}}{\mathrel{}\mathrel{\mathop{\dabar@\dabar@}\limits^{\hbox{\set@color\raisebox{-1.1pt}{$\scriptstyle\bm{x}\!\!$}}}}\mathrel{\mathchar 0\relax\symAMSa 04B}{\!\!}}{\mathrel{}\mathrel{\mathop{\dabar@\dabar@}\limits^{\hbox{\set@color\raisebox{-1.1pt}{$\scriptscriptstyle\bm{x}\!\!$}}}}\mathrel{\mathchar 0\relax\symAMSa 04B}{\!\!}}{\mathrel{}\mathrel{\mathop{\dabar@\dabar@}\limits^{\hbox{\set@color\raisebox{-1.1pt}{$\scriptstyle\bm{x}\!\!$}}}}\mathrel{\mathchar 0\relax\symAMSa 04B}{\!\!}}}C, C_{0}\mathrel{\mathchoice{\mathrel{}\mathrel{\mathop{\dabar@\dabar@}\limits^{\hbox{\set@color\raisebox{-1.1pt}{$\scriptstyle\bm{x}^{\prime}\!\!$}}}}\mathrel{\mathchar 0\relax\symAMSa 04B}{\!\!}}{\mathrel{}\mathrel{\mathop{\dabar@\dabar@}\limits^{\hbox{\set@color\raisebox{-1.1pt}{$\scriptstyle\bm{x}^{\prime}\!\!$}}}}\mathrel{\mathchar 0\relax\symAMSa 04B}{\!\!}}{\mathrel{}\mathrel{\mathop{\dabar@\dabar@}\limits^{\hbox{\set@color\raisebox{-1.1pt}{$\scriptscriptstyle\bm{x}^{\prime}\!\!$}}}}\mathrel{\mathchar 0\relax\symAMSa 04B}{\!\!}}{\mathrel{}\mathrel{\mathop{\dabar@\dabar@}\limits^{\hbox{\set@color\raisebox{-1.1pt}{$\scriptstyle\bm{x}^{\prime}\!\!$}}}}\mathrel{\mathchar 0\relax\symAMSa 04B}{\!\!}}}C^{\prime}, $C_{0}$ is initial, $C$ and $C^{\prime}$ are terminal consensus configurations, and $O(C)\neq O(C^{\prime})$. Note that $(C_{0},C,\bm{x})$ and $(C_{0},C^{\prime},\bm{x}^{\prime})$ both satisfy the flow equations. Therefore, by (a), $\mathrm{val}(C)=\mathrm{val}(C_{0})=\mathrm{val}(C^{\prime})$. Again, since $C_{0}$ is initial, it contains a numerical value, which implies that both $C$ and $C^{\prime}$ contain a numerical value. Since $\mathrm{val}(C)=\mathrm{val}(C^{\prime})$, (b) yields $O(C)=O(C^{\prime})$ which is a contradiction.∎

Proposition 18**.**

$\mathcal{P}_{\text{rmd}}$* satisfies LayeredTermination.*

Proof.

We claim that the following ordered partition satisfies layered termination:

[TABLE]

We first show that every execution of $\mathcal{P}_{\text{rmd}}[T_{1}]$ is silent. For the sake of contradiction, assume it is not the case. There exists a non silent execution $C_{0}\xrightarrow{t_{1}}C_{1}\xrightarrow{t_{2}}\cdots$ where $C_{0},C_{1},\ldots\in\mathrm{Pop}(Q)$ and $t_{1},t_{2},\ldots\in T_{1}$. For every $i\in\mathbb{N}$, let $\mathrm{numerical}(C_{i})\stackrel{{\scriptstyle\scriptscriptstyle\text{def}}}{{=}}\sum_{n\in[0,m)}C_{i}(n)$. It is readily seen that $\mathrm{numerical}(C_{0})\geq\mathrm{numerical}(C_{1})\geq\cdots$. Therefore, there exists $\ell\in\mathbb{N}$ such that $\mathrm{numerical}(C_{i})=\mathrm{numerical}(C_{i-1})$ for every $i>\ell$. This implies that, for every $i>\ell$, if $t_{i}$ is non silent, then it is of the form $(n,\mathrm{false})\mapsto(n,\mathrm{true})$ for some $n\in[0,m)$. But, these non silent transitions can only occur for a finite amount of steps, which is a contradiction.

It is readily seen that every execution of $\mathcal{P}_{\text{rmd}}[T_{2}]$ is silent since non silent transitions of $T_{2}$ are all of the form $(n,\mathrm{true})\mapsto(n,\mathrm{false})$ for some $n\in[0,m)$. Therefore, it remains to prove that $\mathcal{P}_{\text{rmd}}[T_{2}]$ is $T_{1}$-dead. Let $C\in\mathrm{Pop}(Q)$ be a $T_{1}$-dead configuration. For the sake of contradiction, suppose there exists $w\in T_{2}^{+}$ and $C^{\prime}\in\mathrm{Pop}(Q)$ such that $C\xrightarrow{w}C^{\prime}$ and $C^{\prime}$ enables some non silent transition $t\in T_{1}$. We have $C(\mathrm{true})>0$ and $C(n)>0$ for some $n\in[0,m)$ such that $O(n)=\mathrm{false}$. Moreover, since $C$ is $T_{1}$-dead, $\mathrm{numerical}(C)=1$. Therefore $t$ must be of the form $(n,\mathrm{false})\mapsto(n,\mathrm{false})$. We obtain a contradiction since $t$ is non silent. ∎

C.3 Conjunction protocol

For the rest of this subsection, let us fix some population protocols $\mathcal{P}_{1}=(Q_{1},T_{1},\Sigma,I_{1},O_{1})$ and $\mathcal{P}_{2}=(Q_{2},T_{2},\Sigma,I_{2},O_{2})$. For every transition transition $t=(q,r)\mapsto(q^{\prime},r^{\prime})$, and every pair of states $(p,s)$, let $t\mathbin{\varotimes}(p,s)$ denote the transition lifted to $(p,s)$:

[TABLE]

Similarly let $(p,s)\mathbin{\varotimes}t$ denote the lifted transition

[TABLE]

Definition 19**.**

The conjunction of $\mathcal{P}_{1}$ and $\mathcal{P}_{2}$ is defined as the population protocol $\mathcal{P}\stackrel{{\scriptstyle\scriptscriptstyle\text{def}}}{{=}}(Q,S,I,\Sigma,O)$ such that $Q\stackrel{{\scriptstyle\scriptscriptstyle\text{def}}}{{=}}Q_{1}\times Q_{2}$, $S\stackrel{{\scriptstyle\scriptscriptstyle\text{def}}}{{=}}S_{1}\cup S_{2}$, $I(\sigma)\stackrel{{\scriptstyle\scriptscriptstyle\text{def}}}{{=}}(I_{1}(\sigma),I_{2}(\sigma))$ and $O(p,q)\stackrel{{\scriptstyle\scriptscriptstyle\text{def}}}{{=}}O_{1}(p)\land O_{2}(q)$ where

[TABLE]

The projection of $q\in Q$ on $Q_{i}$ is the state $\pi_{i}(q)\stackrel{{\scriptstyle\scriptscriptstyle\text{def}}}{{=}}q_{i}$ where $q=(q_{1},q_{2})$. The projection of $t\in S_{i}$ on $T_{i}$ is the transition $\pi_{i}(t)\stackrel{{\scriptstyle\scriptscriptstyle\text{def}}}{{=}}(\pi_{i}(p),\pi_{i}(q),\pi_{i}(p^{\prime}),\pi_{i}(q^{\prime}))$ where $t=(p,q,p^{\prime},q^{\prime})$. We lift projections to $\mathrm{Pop}(Q)$ and $S\to\mathbb{N}$ as follows. For every $C\in\mathrm{Pop}(Q)$ and $\bm{x}:S\to\mathbb{N}$, the projections $\pi_{i}(C)\in\mathrm{Pop}(Q_{i})$ and $\pi_{i}(\bm{x}):T_{i}\to\mathbb{N}$ are respectively the configuration and mapping such that

[TABLE]

Let $\mathcal{I}_{\mathcal{P}}\in\mathbb{N}^{Q\times T}$ be the matrix such that $\mathcal{I}_{\mathcal{P}}(q,t)\stackrel{{\scriptstyle\scriptscriptstyle\text{def}}}{{=}}\mathrm{post}(t)(q)-\mathrm{pre}(t)(q)$ for every $q\in Q$ and $t\in T$. It is readily seen that $(C,C^{\prime},\bm{x})$ satisfies the flow equations if and only if $C^{\prime}=C+\mathcal{I}_{\mathcal{P}}\cdot\bm{x}$. The same holds for the matrices $\mathcal{I}_{\mathcal{P}_{1}}$ and $\mathcal{I}_{\mathcal{P}_{1}}$ defined similarly for $\mathcal{P}_{1}$ and $\mathcal{P}_{2}$. The following holds:

Proposition 20**.**

For every $i\in\{1,2\}$, $C,C^{\prime}\in\mathrm{Pop}(Q)$ and $\bm{x}\in S\to\mathbb{N}$ we have:

• (a)

$\pi_{i}(C+C^{\prime})=\pi_{i}(C)+\pi_{i}(C^{\prime})$, and

• (b)

$\pi_{i}\left(\mathcal{I}_{\mathcal{P}}\cdot\bm{x}\right)=\mathcal{I}_{\mathcal{P}_{i}}\cdot\pi_{i}(\bm{x})$.

Proof.

For every $q\in Q$, we have

[TABLE]

This proves (a). Let us now prove (b). Let $i\in\{1,2\}$ and $q\in Q_{i}$. By definition of $S$, we have

[TABLE]

Therefore,

[TABLE]

{restatable}

propositionpropConjPtrans For every $C,C^{\prime}\in\mathrm{Pop}(Q)$, $\bm{x}:S\to\mathbb{N}$ and $i\in\{1,2\}$, if C\mathrel{\mathchoice{\mathrel{}\mathrel{\mathop{\dabar@\dabar@}\limits^{\hbox{\set@color\raisebox{-1.1pt}{$\scriptstyle\bm{x}\!\!$}}}}\mathrel{\mathchar 0\relax\symAMSa 04B}{\!\!}}{\mathrel{}\mathrel{\mathop{\dabar@\dabar@}\limits^{\hbox{\set@color\raisebox{-1.1pt}{$\scriptstyle\bm{x}\!\!$}}}}\mathrel{\mathchar 0\relax\symAMSa 04B}{\!\!}}{\mathrel{}\mathrel{\mathop{\dabar@\dabar@}\limits^{\hbox{\set@color\raisebox{-1.1pt}{$\scriptscriptstyle\bm{x}\!\!$}}}}\mathrel{\mathchar 0\relax\symAMSa 04B}{\!\!}}{\mathrel{}\mathrel{\mathop{\dabar@\dabar@}\limits^{\hbox{\set@color\raisebox{-1.1pt}{$\scriptstyle\bm{x}\!\!$}}}}\mathrel{\mathchar 0\relax\symAMSa 04B}{\!\!}}}C^{\prime}, then \pi_{i}(C)\mathrel{\mathchoice{\mathrel{}\mathrel{\mathop{\dabar@\dabar@}\limits^{\hbox{\set@color\raisebox{-1.1pt}{$\scriptstyle\pi_{i}(\bm{x})\!\!$}}}}\mathrel{\mathchar 0\relax\symAMSa 04B}{\!\!}}{\mathrel{}\mathrel{\mathop{\dabar@\dabar@}\limits^{\hbox{\set@color\raisebox{-1.1pt}{$\scriptstyle\pi_{i}(\bm{x})\!\!$}}}}\mathrel{\mathchar 0\relax\symAMSa 04B}{\!\!}}{\mathrel{}\mathrel{\mathop{\dabar@\dabar@}\limits^{\hbox{\set@color\raisebox{-1.1pt}{$\scriptscriptstyle\pi_{i}(\bm{x})\!\!$}}}}\mathrel{\mathchar 0\relax\symAMSa 04B}{\!\!}}{\mathrel{}\mathrel{\mathop{\dabar@\dabar@\dabar@}\limits^{\hbox{\set@color\raisebox{-1.1pt}{$\scriptstyle\pi_{i}(\bm{x})\!\!$}}}}\mathrel{\mathchar 0\relax\symAMSa 04B}{\!\!}}}\pi_{i}(C^{\prime}).

Proof.

Flow equations: We have $C^{\prime}=C+\mathcal{I}_{\mathcal{P}}\cdot\bm{x}_{j}$. Therefore, for every $i\in\{1,2\}$,

[TABLE]

Trap constraints: For the sake of contradiction, suppose there exists $i\in\{1,2\}$ such that a $U$-trap constraint is violated by $(\pi_{i}(C),\pi_{i}(C^{\prime}),\pi_{i}(\bm{x}))$ for some $P\subseteq Q_{i}$. As both cases are symmetric, we may assume without loss of generality that $i=1$. We have

[TABLE]

Let $R\stackrel{{\scriptstyle\scriptscriptstyle\text{def}}}{{=}}P\times Q_{2}$. By definition of projections, we have

[TABLE]

We claim that

[TABLE]

If (8) and (9) hold, then we are done since the $\llbracket\bm{x}\rrbracket$-trap constraint for $R$ is violated. So, it remains to prove these claims. To prove (8), let $t\in{{}^{\bullet}P}\cap\llbracket\pi_{1}(\bm{x})\rrbracket$. By assumption, such a $t$ must exist. Since $t\in{{}^{\bullet}P}$, we have that $t\colon(p,p^{\prime})\mapsto(r,r^{\prime})$ with $r\in P$ or $r^{\prime}\in P$. Moreover, since $t\in\llbracket\pi_{1}(\bm{x})\rrbracket$, by definition of projections there must exist some $t^{\prime}\in\llbracket\bm{x}\rrbracket$ given by

[TABLE]

for some $q,q^{\prime}\in Q_{2}$. It remains to show that $t^{\prime}\in{{}^{\bullet}R}$ in order to prove that  (8) holds. Indeed, since $r\in P$ or $r^{\prime}\in P$, we have that $(r,q)\in R$ or $(r^{\prime},q^{\prime})\in R$, and thus $t^{\prime}\in{{}^{\bullet}R}$.

Now we show that  (9) holds. To this end, let $t\in{R^{\bullet}}\cap\llbracket\bm{x}\rrbracket\subseteq{{}^{\bullet}P\times Q_{2}}$. There exist $p\in P$ and $q\in Q_{2}$ such that $(p,q)\in{{}^{\bullet}t}$. Moreover, $\bm{x}(t)>0$. We must prove $t\in{{}^{\bullet}R}$. We consider two cases

• •

Assume $t\in S_{2}$. By definition of $S_{2}$, $t$ is of the form

[TABLE]

for some $p^{\prime}\in Q_{1}$ and $q^{\prime},r,r^{\prime}\in Q_{2}$. In particular, we have $(p,r)\in{t^{\bullet}}$ which implies $t\in{{}^{\bullet}(P\times Q_{2})}={{}^{\bullet}R}$.

• •

Assume $t\in S_{1}$. Let $s\stackrel{{\scriptstyle\scriptscriptstyle\text{def}}}{{=}}\pi_{1}(t)$. By definition of $S_{2}$, $t$ is of the form

[TABLE]

where $\mathrm{pre}(s)=\Lbag p,p^{\prime}\Rbag$, $\mathrm{post}(s)=\Lbag r,r^{\prime}\Rbag$ and $q^{\prime}\in Q_{2}$. This implies that $s\in{p^{\bullet}}\subseteq{P^{\bullet}}$. Moreover, since $t\in\llbracket\bm{x}\rrbracket$, we have $s\in\llbracket\pi_{1}(\bm{x})\rrbracket$. Therefore, by (6), we have $s\in{{}^{\bullet}P}$. This implies that either $r\in P$ or $r^{\prime}\in P$, which in turn implies that $t\in{{}^{\bullet}R}$.

$U$-Siphon constraints: Symmetric to $U$-trap constraints. ∎

Proposition 21**.**

For every $i\in\{1,2\}$, $C\in\mathrm{Pop}(Q)$ and $t\in T_{i}$, $t$ is enabled in $\pi_{i}(C)$ if and only if there exists $s\in S_{i}$ such that $\pi_{i}(s)=t$ and $s$ is enabled in $C$.

Proof.

We only prove the claim for $i=1$, as the case $i=2$ is symmetric. Let $p,q\in Q_{1}$ be such that $\mathrm{pre}(t)=\Lbag p_{1},q_{1}\Rbag$. By definition of $\pi_{1}$, we have

[TABLE]

This implies that

[TABLE]

$\Rightarrow$) Assume $t$ is enabled in $\pi_{1}(C)$. By (10), $C\geq\Lbag(p_{1},p_{2}),(q_{1},q_{2})\Rbag$ for some $p_{2},q_{2}\in Q_{2}$. Let

[TABLE]

We have $s\in S_{1}$. Moreover, $s$ is enabled at $C$.

$\Leftarrow$) Assume there exists $s\in S_{1}$ such that $\pi_{1}(s)=t$ and $s$ is enabled at $C$. By definition of $S_{1}$,

[TABLE]

for some $p_{2},q_{2}\in Q_{2}$. Since $s$ is enabled at $C$, we have $C\geq\Lbag(p_{1},p_{2}),(q_{1},q_{2})\Rbag$. By (10), this implies $\pi_{1}(C)\geq\Lbag p_{1},q_{1}\Rbag$, which in turn implies that $t$ is enabled at $\pi_{1}(C)$. ∎

Corollary 22**.**

For every $C\in\mathrm{Pop}(Q)$, if $C$ is terminal in $\mathcal{P}$, then $\pi_{1}(C)$ and $\pi_{2}(C)$ are respectively terminal in $\mathcal{P}_{1}$ and $\mathcal{P}_{2}$.

Proof.

Let $C\in\mathrm{Pop}(Q)$ be such that $C$ is terminal in $\mathcal{P}$. For the sake of contradiction, suppose there exists $i\in\{1,2\}$ such that $\pi_{i}(C)$ is not terminal in $\mathcal{P}_{i}$. There exists $t\in T_{i}$ such that $t$ is non silent and enabled in $\pi_{i}(C)$. By Proposition 21, there exists $s\in S_{i}$ such that $\pi_{i}(s)=t$ and $s$ is enabled at $C$. We have $s=t\mathbin{\varotimes}q$ for some $q\in Q_{2}\times Q_{2}$. This implies that $s$ is non silent, since $t$ is non silent. We conclude that $C$ is non terminal which is contradiction. ∎

{restatable}

lemmalemmaConjSplit If $\mathcal{P}_{1}$ and $\mathcal{P}_{2}$ satisfy StrongConsensus, then $\mathcal{P}$ satisfies StrongConsensus.

Proof.

We prove the contrapositive: if $\mathcal{P}$ does not satisfy StrongConsensus, then at least one of $\mathcal{P}_{1}$ and $\mathcal{P}_{2}$ does not satisfy StrongConsensus. Assume $\mathcal{P}$ does not satisfy StrongConsensus. There are two cases to consider.

• (a)

There exist $C,C^{\prime}\in\mathrm{Pop}(Q)$ such that $C\mathrel{\mathchoice{\mathrel{}\mathrel{\mathop{\dabar@\dabar@}\limits}\mathrel{\mathchar 0\relax\symAMSa 04B}{\!\!}}{\mathrel{}\mathrel{\mathop{\dabar@\dabar@}\limits}\mathrel{\mathchar 0\relax\symAMSa 04B}{\!\!}}{\mathrel{}\mathrel{\mathop{\dabar@\dabar@}\limits}\mathrel{\mathchar 0\relax\symAMSa 04B}{\!\!}}{\mathrel{}\mathrel{\mathop{\dabar@\dabar@}\limits}\mathrel{\mathchar 0\relax\symAMSa 04B}{\!\!}}}C^{\prime}$, $C$ is initial, $C^{\prime}$ is a terminal non consensus configuration. Since $C^{\prime}$ is a non consensus configuration, there exist $(p,q),(p^{\prime},q^{\prime})\in\llbracket C^{\prime}\rrbracket$ such that $O_{1}(p)\land O_{1}(q)=O(p,q)\not=O(p^{\prime},q^{\prime})=O_{2}(p^{\prime})\land O_{2}(q^{\prime})$. Without loss of generality, we can assume that $O_{1}(p)\not=O_{1}(p^{\prime})$. By Corollary 22, $\pi_{1}(C^{\prime})$ is terminal in $\mathcal{P}_{1}$. Moreover, since $p,p^{\prime}\in\pi_{1}(C^{\prime})$, $\pi_{1}(C^{\prime})$ is a non consensus configuration. Therefore, $\pi_{1}(C^{\prime})$ is a terminal non consensus configuration of $\mathcal{P}_{1}$. Moreover, by Prop. 20 \pi_{1}(C)\mathrel{\mathchoice{\mathrel{}\mathrel{\mathop{\dabar@\dabar@}\limits^{\hbox{\set@color\raisebox{-1.1pt}{$\scriptstyle\pi_{1}(\bm{x})\!\!$}}}}\mathrel{\mathchar 0\relax\symAMSa 04B}{\!\!}}{\mathrel{}\mathrel{\mathop{\dabar@\dabar@}\limits^{\hbox{\set@color\raisebox{-1.1pt}{$\scriptstyle\pi_{1}(\bm{x})\!\!$}}}}\mathrel{\mathchar 0\relax\symAMSa 04B}{\!\!}}{\mathrel{}\mathrel{\mathop{\dabar@\dabar@}\limits^{\hbox{\set@color\raisebox{-1.1pt}{$\scriptscriptstyle\pi_{1}(\bm{x})\!\!$}}}}\mathrel{\mathchar 0\relax\symAMSa 04B}{\!\!}}{\mathrel{}\mathrel{\mathop{\dabar@\dabar@\dabar@}\limits^{\hbox{\set@color\raisebox{-1.1pt}{$\scriptstyle\pi_{1}(\bm{x})\!\!$}}}}\mathrel{\mathchar 0\relax\symAMSa 04B}{\!\!}}}\pi_{1}(C^{\prime}) which implies that $\mathcal{P}_{1}$ does not satisfy StrongConsensus.

• (b)

There exist $C_{0},C,C^{\prime}\in\mathrm{Pop}(Q)$ and $\bm{x},\bm{x}^{\prime}:T\to\mathbb{N}$ such that C_{0}\mathrel{\mathchoice{\mathrel{}\mathrel{\mathop{\dabar@\dabar@}\limits^{\hbox{\set@color\raisebox{-1.1pt}{$\scriptstyle\bm{x}\!\!$}}}}\mathrel{\mathchar 0\relax\symAMSa 04B}{\!\!}}{\mathrel{}\mathrel{\mathop{\dabar@\dabar@}\limits^{\hbox{\set@color\raisebox{-1.1pt}{$\scriptstyle\bm{x}\!\!$}}}}\mathrel{\mathchar 0\relax\symAMSa 04B}{\!\!}}{\mathrel{}\mathrel{\mathop{\dabar@\dabar@}\limits^{\hbox{\set@color\raisebox{-1.1pt}{$\scriptscriptstyle\bm{x}\!\!$}}}}\mathrel{\mathchar 0\relax\symAMSa 04B}{\!\!}}{\mathrel{}\mathrel{\mathop{\dabar@\dabar@}\limits^{\hbox{\set@color\raisebox{-1.1pt}{$\scriptstyle\bm{x}\!\!$}}}}\mathrel{\mathchar 0\relax\symAMSa 04B}{\!\!}}}C, C_{0}\mathrel{\mathchoice{\mathrel{}\mathrel{\mathop{\dabar@\dabar@}\limits^{\hbox{\set@color\raisebox{-1.1pt}{$\scriptstyle\bm{x}^{\prime}\!\!$}}}}\mathrel{\mathchar 0\relax\symAMSa 04B}{\!\!}}{\mathrel{}\mathrel{\mathop{\dabar@\dabar@}\limits^{\hbox{\set@color\raisebox{-1.1pt}{$\scriptstyle\bm{x}^{\prime}\!\!$}}}}\mathrel{\mathchar 0\relax\symAMSa 04B}{\!\!}}{\mathrel{}\mathrel{\mathop{\dabar@\dabar@}\limits^{\hbox{\set@color\raisebox{-1.1pt}{$\scriptscriptstyle\bm{x}^{\prime}\!\!$}}}}\mathrel{\mathchar 0\relax\symAMSa 04B}{\!\!}}{\mathrel{}\mathrel{\mathop{\dabar@\dabar@}\limits^{\hbox{\set@color\raisebox{-1.1pt}{$\scriptstyle\bm{x}^{\prime}\!\!$}}}}\mathrel{\mathchar 0\relax\symAMSa 04B}{\!\!}}}C^{\prime}, $C_{0}$ is initial, $C$ and $C^{\prime}$ are terminal consensus configurations, and $O(C)\neq O(C^{\prime})$. Since $C$ and $C^{\prime}$ have different opinions, there exist $(p,q)\in\llbracket C\rrbracket$ and $(p^{\prime},q^{\prime})\in\llbracket C^{\prime}\rrbracket$ such that $O(p,q)\not=O(p^{\prime},q^{\prime})$. Without loss of generality, we can assume that $O_{1}(p)\not=O_{1}(p^{\prime})$. By Corollary (22), $\pi_{1}(C)$ and $\pi_{1}(C^{\prime})$ are terminal in $\mathcal{P}_{1}$. Moreover, since $p\in\pi_{1}(C)$ and $p^{\prime}\in\pi_{1}(C^{\prime})$, $\pi_{1}(C)$ and $\pi_{1}(C^{\prime})$ are terminal configuration with different consensus. Moreover, by Prop. 20, \pi_{1}(C)\mathrel{\mathchoice{\mathrel{}\mathrel{\mathop{\dabar@\dabar@}\limits^{\hbox{\set@color\raisebox{-1.1pt}{$\scriptstyle\pi_{1}(\bm{x})\!\!$}}}}\mathrel{\mathchar 0\relax\symAMSa 04B}{\!\!}}{\mathrel{}\mathrel{\mathop{\dabar@\dabar@}\limits^{\hbox{\set@color\raisebox{-1.1pt}{$\scriptstyle\pi_{1}(\bm{x})\!\!$}}}}\mathrel{\mathchar 0\relax\symAMSa 04B}{\!\!}}{\mathrel{}\mathrel{\mathop{\dabar@\dabar@}\limits^{\hbox{\set@color\raisebox{-1.1pt}{$\scriptscriptstyle\pi_{1}(\bm{x})\!\!$}}}}\mathrel{\mathchar 0\relax\symAMSa 04B}{\!\!}}{\mathrel{}\mathrel{\mathop{\dabar@\dabar@\dabar@}\limits^{\hbox{\set@color\raisebox{-1.1pt}{$\scriptstyle\pi_{1}(\bm{x})\!\!$}}}}\mathrel{\mathchar 0\relax\symAMSa 04B}{\!\!}}}\pi_{1}(C^{\prime}) which implies that $\mathcal{P}_{1}$ does not satisfy StrongConsensus. ∎

{restatable}

propositionpropConjLayered If $\mathcal{P}_{1}$ and $\mathcal{P}_{2}$ satisfy LayeredTermination, then $\mathcal{P}$ satisfies LayeredTermination.

Proof.

Let $X_{1},X_{2},\ldots,X_{m}$ and $Y_{1},Y_{2},\ldots,Y_{n}$ be ordered partitions respectively for LayeredTermination in $\mathcal{P}_{1}$ and $\mathcal{P}_{2}$. We may assume without loss of generality that $m\geq n$. For every $n<i\leq m$, we define $Y_{i}\stackrel{{\scriptstyle\scriptscriptstyle\text{def}}}{{=}}\emptyset$.

For every $i\in[m]$, we let

[TABLE]

We claim that $Z_{1},Z_{2},\ldots,Z_{m}$ is an ordered partition for LayeredTermination in $\mathcal{P}$. Let $i\in[m]$. Let us show that every execution of $\mathcal{P}[Z_{i}]$ is silent. Suppose for the sake of contradiction that there exist $C_{0},C_{1},\ldots\in\mathrm{Pop}(Q)$ and $t_{0},t_{1},\ldots\in Z_{i}$ such that $C_{0}\xrightarrow{t_{0}}C_{1}\xrightarrow{t_{1}}\cdots$ is non silent. There exists $j\in\{1,2\}$ such that infinitely many non silent transitions $t_{i}$ belong to $S_{j}$. Let $i_{0}<i_{1}<\cdots$ be all indices such that $t_{i_{k}}\in S_{j}$. We have

[TABLE]

which is an infinite non silent execution of $\mathcal{P}_{1}[X_{i}]$ or $\mathcal{P}_{2}[Y_{i}]$ depending on $j$. This is a contradiction.

Let $W\stackrel{{\scriptstyle\scriptscriptstyle\text{def}}}{{=}}(Z_{1}\cup\cdots\cup Z_{i-1})$. Let us now prove that $\mathcal{P}[Z_{i}]$ is $W$-dead. For the sake of contradiction, assume it is not. There exist $C,C^{\prime}\in\mathrm{Pop}(Q)$, $w\in Z_{i}^{+}$ and $t\in W$ such that $C$ is $W$-dead, $C\xrightarrow{w}C^{\prime}$ and $t$ is enabled at $C^{\prime}$. We have $t\in S_{j}$ for some $j\in\{1,2\}$. We may assume without loss of generality that $j=1$. Since $C$ is $W$-dead, $\pi_{j}(C)$ is $(X_{1}\cup\cdots\cup X_{i-1})$-dead. But then, $\pi_{1}(C)\xrightarrow{*}\pi_{1}(C^{\prime})$ and $t\in X_{1}\cup\cdots\cup X_{i-1}$ is enabled at $C^{\prime}$ which is a contradiction. ∎

{restatable}

corollarycorConjWSSS If $\mathcal{P}_{1}$ and $\mathcal{P}_{2}$ belong to ${\it WS}^{3}$, then $\mathcal{P}$ belongs to ${\it WS}^{3}$ and is correct.

Proof.

By Lemma 22 and Prop. 22, $\mathcal{P}$ belongs to ${\it WS}^{3}$. Let $w\in\mathrm{Pop}(\Sigma)$, $C\stackrel{{\scriptstyle\scriptscriptstyle\text{def}}}{{=}}I(w)$, $C_{1}\stackrel{{\scriptstyle\scriptscriptstyle\text{def}}}{{=}}I_{1}(w)$ and $C_{2}\stackrel{{\scriptstyle\scriptscriptstyle\text{def}}}{{=}}I_{2}(w)$. Note that all three protocols are well-specified since they belong to ${\it WS}^{3}$. Therefore, there exist terminal consensus configurations $C^{\prime}\in\mathrm{Pop}(Q)$, $C_{1}^{\prime}\in\mathrm{Pop}(Q_{1})$ and $C_{2}^{\prime}\in\mathrm{Pop}(Q_{2})$ such that $C\xrightarrow{*}C^{\prime}$, $C_{1}\xrightarrow{*}C_{1}^{\prime}$ and $C_{2}\xrightarrow{*}C_{2}^{\prime}$.

We must prove that $O(C^{\prime})=O_{1}(C_{1}^{\prime})\land O_{2}(C_{2}^{\prime})$. Let $j\in\{1,2\}$. Since $C\xrightarrow{*}C^{\prime}$, we have $C\mathrel{\mathchoice{\mathrel{}\mathrel{\mathop{\dabar@\dabar@}\limits}\mathrel{\mathchar 0\relax\symAMSa 04B}{\!\!}}{\mathrel{}\mathrel{\mathop{\dabar@\dabar@}\limits}\mathrel{\mathchar 0\relax\symAMSa 04B}{\!\!}}{\mathrel{}\mathrel{\mathop{\dabar@\dabar@}\limits}\mathrel{\mathchar 0\relax\symAMSa 04B}{\!\!}}{\mathrel{}\mathrel{\mathop{\dabar@\dabar@}\limits}\mathrel{\mathchar 0\relax\symAMSa 04B}{\!\!}}}C^{\prime}$. By Prop. 20, $\pi_{j}(C)\mathrel{\mathchoice{\mathrel{}\mathrel{\mathop{\dabar@\dabar@}\limits}\mathrel{\mathchar 0\relax\symAMSa 04B}{\!\!}}{\mathrel{}\mathrel{\mathop{\dabar@\dabar@}\limits}\mathrel{\mathchar 0\relax\symAMSa 04B}{\!\!}}{\mathrel{}\mathrel{\mathop{\dabar@\dabar@}\limits}\mathrel{\mathchar 0\relax\symAMSa 04B}{\!\!}}{\mathrel{}\mathrel{\mathop{\dabar@\dabar@}\limits}\mathrel{\mathchar 0\relax\symAMSa 04B}{\!\!}}}\pi_{j}(C^{\prime})$. By definition of $I$, we have $C_{j}=\pi_{j}(C)$. Therefore, $C_{j}\mathrel{\mathchoice{\mathrel{}\mathrel{\mathop{\dabar@\dabar@}\limits}\mathrel{\mathchar 0\relax\symAMSa 04B}{\!\!}}{\mathrel{}\mathrel{\mathop{\dabar@\dabar@}\limits}\mathrel{\mathchar 0\relax\symAMSa 04B}{\!\!}}{\mathrel{}\mathrel{\mathop{\dabar@\dabar@}\limits}\mathrel{\mathchar 0\relax\symAMSa 04B}{\!\!}}{\mathrel{}\mathrel{\mathop{\dabar@\dabar@}\limits}\mathrel{\mathchar 0\relax\symAMSa 04B}{\!\!}}}\pi_{j}(C^{\prime})$. Moreover, by Corollary 22, $\pi_{j}(C^{\prime})$ is terminal in $\mathcal{P}_{j}$. Since $\mathcal{P}_{j}$ belongs to ${\it WS}^{3}$, $\pi_{j}(C^{\prime})$ is a consensus configuration such that $O_{j}(\pi_{j}(C^{\prime}))=O_{j}(C_{j}^{\prime})$. Altogether, we obtain

[TABLE]

Appendix D Full set of constraints for Section 6

We detail the constraints tested with the SMT solver in our implementation. Given a population protocol $\mathcal{P}=(Q,T,\Sigma,I,O)$, let $U\stackrel{{\scriptstyle\scriptscriptstyle\text{def}}}{{=}}\{t\in T\colon\mathrm{post}(t)\neq\mathrm{pre}(t)\}$ be the set of non silent transitions.

D.1 Constraints for LayeredTermination

For a given number of layers $k\in[1,|T|]$, the constraints for LayeredTermination use variables $\bm{y}_{i}\colon Q\rightarrow\mathbb{N}$ for each $i\in[1,k]$ and $\bm{b}\colon T\rightarrow\mathbb{N}$. Vector $\bm{b}$ assigns transitions to layers.

For a pair of transitions $t,u$, define $U^{\prime}(t,u)\stackrel{{\scriptstyle\scriptscriptstyle\text{def}}}{{=}}\{u^{\prime}\in U\colon\mathrm{pre}(u^{\prime})\leq\mathrm{pre}(t)+(\mathrm{pre}(u)\mathbin{\varominus}\mathrm{post}(t))\}$. These sets are precomputed. The full set of constraints is:

[TABLE]

If there is a solution, then each layer $i\in[1,k]$ is given by $T_{i}\stackrel{{\scriptstyle\scriptscriptstyle\text{def}}}{{=}}\{t\in T\mid\bm{b}(t)=i\}$, and each vector $\bm{y}_{i}$ assigns a valuation $\bm{y}_{i}(C)\stackrel{{\scriptstyle\scriptscriptstyle\text{def}}}{{=}}\sum_{q\in Q}\bm{y}_{i}(q)\cdot C(q)$ to a given configuration $C$. For any configuration $C$, we have $\bm{y}_{i}(C)\geq 0$ and for any $C\xrightarrow{t}C^{\prime}$ with $t\in T_{i}$, we have $\bm{y}_{i}(C)\geq\bm{y}_{i}(C^{\prime})$ and additionally $\bm{y}_{i}(C)>\bm{y}_{i}(C^{\prime})$ if $t$ is non silent. This is a proof that every execution of $\mathcal{P}[T_{i}]$ is silent.

D.2 Constraints for StrongConsensus

For a vector $\bm{c}:Q\rightarrow\mathbb{N}$, we define the following constraints:

[TABLE]

For sets of states $R,S\subseteq Q$ and vectors $\bm{c},\bm{c^{\prime}}:Q\rightarrow\mathbb{N}$, and $\bm{x}:T\rightarrow\mathbb{N}$, we define the following constraints:

[TABLE]

The constraints for StrongConsensus use the variables $\bm{c}_{0},\bm{c}_{1},\bm{c}_{2}\colon Q\rightarrow\mathbb{N}$ and $\bm{x}_{1},\bm{x}_{2}\colon T\rightarrow\mathbb{N}$. With given sets $\mathcal{R}$ of $U$-traps and $\mathcal{S}$ of $U$-siphons, the constraints are as follows:

[TABLE]