Abstract Broadcast consensus protocols (BCPs) are a model of computation, in which anonymous, identical, finite-state agents compute by sending/receiving global broadcasts. BCPs are known to compute all number predicates in $$\mathsf {NL}=\mathsf {NSPACE}(\log n)$$ NL = NSPACE ( log n ) where n is the number of agents. They can be considered an extension of the well-established model of population protocols. This paper investigates execution time characteristics of BCPs. We show that every predicate computable by population protocols is computable by a BCP with expected $$\mathcal {O}(n \log n)$$ O ( n log n ) interactions, which is asymptotically optimal. We further show that every log-space, randomized Turing machine can be simulated by a BCP with $$\mathcal {O}(n \log n \cdot T)$$ O ( n log n · T ) interactions in expectation, where T is the expected runtime of the Turing machine. This allows us to characterise polynomial-time BCPs as computing exactly the number predicates in $$\mathsf {ZPL}$$ ZPL , i.e. predicates decidable by log-space, randomised Turing machine with zero-error in expected polynomial time where the input is encoded as unary.

Abstract Population protocols (Angluin et al. in PODC, 2004) are a model of distributed computation in which indistinguishable, finite-state agents interact in pairs to decide if their initial configuration, i.e., the initial number of agents in each state, satisfies a given property. In a seminal paper Angluin et al. classified population protocols according to their communication mechanism, and conducted an exhaustive study of the expressive power of each class, that is, of the properties they can decide (Angluin et al. in Distrib Comput 20(4):279–304, 2007). In this paper we study the correctness problem for population protocols, i.e., whether a given protocol decides a given property. A previous paper (Esparza et al. in Acta Inform 54(2):191–215, 2017) has shown that the problem is decidable for the main population protocol model, but at least as hard as the reachability problem for Petri nets, which has recently been proved to have non-elementary complexity. Motivated by this result, we study the computational complexity of the correctness problem for all other classes introduced by Angluin et al., some of which are less powerful than the main model. Our main results show that for the class of observation models the complexity of the problem is much lower, ranging from $$\varPi _2^p$$ Π 2 p to .

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 of the protocol 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 -hard, and for which only algorithms of non-primitive recursive complexity are currently known. In this paper we introduce the class $${ WS}^3$$ WS 3 of well-specified strongly-silent protocols and we prove that it is suitable for automatic verification. More precisely, we show that $${ WS}^3$$ 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 and correctness problems for $${ WS}^3$$ WS 3 reduce to solving boolean combinations of linear constraints over $${\mathbb {N}}$$ N . This allowed us to develop the first software able to automatically prove correctness for all of the infinitely many possible inputs.

We analyze affine reachability problems in dimensions 1 and 2. We show that the reachability problem for 1-register machines over the integers with affine updates is PSPACE-hard, hence PSPACE-complete, strengthening a result by Finkel et al. that required polynomial updates. Building on recent results on two-dimensional integer matrices, we prove NP-completeness of the mortality problem for 2-dimensional integer matrices with determinants +1 and 0. Motivated by tight connections with 1-dimensional affine reachability problems without control states, we also study the complexity of a number of reachability problems in finitely generated semigroups of 2-dimensional upper-triangular integer matrices.

Angluin et al. proved that population protocols compute exactly the predicates definable in Presburger arithmetic (PA), the first-order theory of addition. As part of this result, they presented a procedure that translates any formula $φ$ of quantifier-free PA with remainder predicates (which has the same expressive power as full PA) into a population protocol with $2^{O(\text{poly}(|φ|))}$ states that computes $φ$. More precisely, the number of states of the protocol is exponential in both the bit length of the largest coefficient in the formula, and the number of nodes of its syntax tree. In this paper, we prove that every formula $φ$ of quantifier-free PA with remainder predicates is computable by a leaderless population protocol with $O(\text{poly}(|φ|))$ states. Our proof is based on several new constructions, which may be of independent interest. Given a formula $φ$ of quantifier-free PA with remainder predicates, a first construction produces a succinct protocol (with $O(|φ|^3)$ leaders) that computes $φ$; this completes the work initiated in [STACS'18], where we constructed such protocols for a fragment of PA. For large enough inputs, we can get rid of these leaders. If the input is not large enough, then it is small, and we design another construction producing a succinct protocol with one leader that computes $φ$. Our last construction gets rid of this leader for small inputs.

Population protocols are a formal model of computation by identical, anonymous mobile agents interacting in pairs. Their computational power is rather limited: Angluin et al. have shown that they can only compute the predicates over $\mathbb{N}^k$ expressible in Presburger arithmetic. For this reason, several extensions of the model have been proposed, including the addition of devices called cover-time services, absence detectors, and clocks. All these extensions increase the expressive power to the class of predicates over $\mathbb{N}^k$ lying in the complexity class NL when the input is given in unary. However, these devices are difficult to implement, since they require that an agent atomically receives messages from all other agents in a population of unknown size; moreover, the agent must know that they have all been received. Inspired by the work of the verification community on Emerson and Namjoshi's broadcast protocols, we show that NL-power is also achieved by extending population protocols with reliable broadcasts, a simpler, standard communication primitive.

We introduce P eregrine , the first tool for the analysis and parameterized verification of population protocols. Population protocols are a model of computation very much studied by the distributed computing community, in which mobile anonymous agents interact stochastically to achieve a common task. P eregrine allows users to design protocols, to simulate them both manually and automatically, to gather statistics of properties such as convergence speed, and to verify correctness automatically. This paper describes the features of P eregrine and their implementation.

In this interactive paper, which you should preferably read connected to the Internet, the Black Ninjas introduce you to population protocols, a fundamental model of distributed computation, and to recent work by the authors and their colleagues on their automatic verification.

Population protocols are a well established model of distributed computation by mobile finite-state agents with very limited storage. A classical result establishes that population protocols compute exactly predicates definable in Presburger arithmetic. We initiate the study of the minimal amount of memory required to compute a given predicate as a function of its size. We present results on the predicates $x \geq n$ for $n \in \mathbb{N}$, and more generally on the predicates corresponding to systems of linear inequalities. We show that they can be computed by protocols with $O(\log n)$ states (or, more generally, logarithmic in the coefficients of the predicate), and that, surprisingly, some families of predicates can be computed by protocols with $O(\log\log n)$ states. We give essentially matching lower bounds for the class of 1-aware protocols.

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 of the protocol 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.