Abstract: In this work, we study the PopulationProtocol model of Angluin et al. from the perspective of protocol verification. In particular, we are interested in algorithmically solving the problem of determining whether a given populationprotocol conforms to its specifications. Since this is the first work on verification of populationprotocols, we redefine most notions of populationprotocols in order to make them suitable for algorithmic verification. Moreover, we formally define the general verification problem and some interesting special cases. All these problems are shown to be NP-hard. We next propose some first algorithmic solutions for a natural special case. Finally, we conduct experiments and algorithmic engineering in order to improve our verifiers' running times.

Abstract: This work focuses on the computational power of the Mediated PopulationProtocol model on complete communication graphs and initially identical edges (SMPP). In particular, we investigate the class MPS of all predicates that are stably computable by the SMPP model. It is already known that MPS is in the symmetric subclass of NSPACE(n^2). Here we prove that this inclusion holds with equality, thus, providing the following exact characterization for MPS: A predicate is in MPS iff it is symmetric and is in NSPACE(n^2).

Abstract: We work on an extension of the PopulationProtocol model
of Angluin et al. [1] that allows edges of the communication graph, G, to
have states that belong to a constant size set. In this extension, the so
called Mediated PopulationProtocol model (MPP) [2,3], both uniformity
and anonymity are preserved.We here study a simplified version of MPP,
the Graph Decision Mediated PopulationProtocol model (GDM), in
order to capture MPP's ability to decide graph languages. We also prove
some first impossibility results both for weakly connected and possibly
disconnected communication graphs.

Abstract: We survey here some recent computational models for networks of tiny artifacts. In particular, we focus on networks consisting of artifacts with sensing capabilities. We first imagine the artifacts moving passively, that is, being mobile but unable to control their own movement. This leads us to the populationprotocol model of Angluin et al. (2004) [16]. We survey this model and some of its recent enhancements. In particular, we also present the mediated populationprotocol model in which the interaction links are capable of storing states and the passively mobile machines model in which the finite state nature of the agents is relaxed and the agents become multitape Turing machines that use a restricted space. We next survey the sensor field model, a general model capturing some identifying characteristics of many sensor network˘s settings. A sensor field is composed of kinds of devices that can communicate one to the other and also to the environment through input/output data streams. We, finally, present simulation results between sensor fields and populationprotocols and analyze the capability of their variants to decide graph properties

Abstract: Here we survey various computational models for Wireless Sensor Networks (WSNs). The populationprotocol model (PP) considers networks of tiny mobile finite-state artifacts that can sense the environment and communicate in pairs to perform a computation. The mediated populationprotocol model (MPP) enhances the previous model by allowing the communication links to have a constant size buffer, providing more computational power. The graph decision MPP model (GDM) is a special case of MPP that focuses on the MPP's ability to decide graph properties of the network. Another direction towards enhancing the PP is followed by the PALOMA model in which the artifacts are no longer finite-state automata but Turing Machines of logarithmic memory in the population size. A different approach to modeling WSNs is the static synchronous sensor field model (SSSF) which describes devices communicating through a fixed communication graph and interacting with their environment via input and output data streams. In this survey, we present the computational capabilities of each model and provide directions for further research.

Abstract: We work on an extension of the PopulationProtocol model of Angluin et al. that allows edges of the communication graph, G, to have states that belong to a constant size set. In this extension, the so called Mediated PopulationProtocol model (MPP), both uniformity and anonymity are preserved. We here study a simplified version of MPP, the Graph Decision Mediated PopulationProtocol model (GDM), in order to capture MPP's ability to decide (stably compute) graph languages (sets of communication graphs). To understand properties of the communication graph is an important step in almost any distributed system. We prove that any graph language is undecidable if we allow disconnected communication graphs. As a result, we focus on studying the computational limits of the GDM model in (at least) weakly connected communication graphs only and give several examples of decidable graph languages in this case. To do so, we also prove that the class of decidable graph languages is closed under complement, union and intersection operations. Node and edge parity, bounded out-degree by a constant, existence of a node with more incoming than outgoing neighbors and existence of some directed path of length at least k=O(1) are some examples of properties whose decidability is proven. To prove the decidability of graph languages we provide protocols (GDMs) for them and exploit the closure results. Finally, we prove the existence of symmetry in two specific communication (sub)graphs which we believe is the first step towards the proof of impossibility results in the GDM model. In particular, we prove that there exists no GDM, whose states eventually stabilize, to decide whether G contains some directed cycle of length 2 (2-cycle).

Abstract: We study here the problem of determining the majority type in an arbitrary connected network, each vertex of which has initially two possible types. The vertices may have a few additional possible states and can interact in pairs only if they share an edge. Any (population) protocol is required to stabilize in the initial majority. We first present and analyze a protocol with 4 states per vertex that always computes the initial majority value, under any fair scheduler. As we prove, this protocol is optimal, in the sense that there is no populationprotocol that always computes majority with fewer than 4 states per vertex. However this does not rule out the existence of a protocol with 3 states per vertex that is correct with high probability. To this end, we examine a very natural majority protocol with 3 states per vertex, introduced in [Angluin et al. 2008] where its performance has been analyzed for the clique graph. We study the performance of this protocol in arbitrary networks. We prove that, when the two initial states are put uniformly at random on the vertices, this protocol converges to the initial majority with probability higher than the probability of converging to the initial minority. In contrast, we present an infinite family of graphs, on which the protocol can fail whp, even when the difference between the initial majority and the initial minority is n−Θ(lnn). We also present another infinite family of graphs in which the protocol of Angluin et al. takes an expected exponential time to converge. These two negative results build upon a very positive result concerning the robustness of the protocol on the clique. Surprisingly, the resistance of the clique to failure causes the failure in general graphs. Our techniques use new domination and coupling arguments for suitably defined processes whose dynamics capture the antagonism between the states involved.

Abstract: In the near future, it is reasonable to expect that new types of systems will appear, of massive scale, expansive and permeating their environment, of very heterogeneous nature, and operating in a constantly changing networked environment. We expect that most such systems will have the form of a very large society of unimpressive networked artefacts. Yet by cooperation, they will be organized in large societies to accomplish tasks that are difficult or beyond the capabilities of todays conventional centralized systems.
The PopulationProtocol model of Angluin et. al. introduced a novel approach towards the study of such systems by assuming that each artefact is an agent, so limited, that can be represented as a finite-state sensor of constant (O(1)) total storage capacity. Such agents are passively mobile and communicate in pairs using a low-power wireless signal. It has been proven that, although such systems consist of extremely limited, cheap and bulk-produced hardware devices, they are still capable of carrying out very useful nontrivial computations. Based on this approach we investigate many new intriguing directions.

Abstract: We extend here the PopulationProtocol model of Angluin et al. [2004,2006] in order to model more powerful networks of resource-limited agents that are possibly mobile. The main feature of our extended model, called the Mediated PopulationProtocol (MPP) model, is to allow edges of the communication graph to have states that belong to a constant size set. We then allow the protocol rules for pairwise interactions to modify the corresponding edge state. Protocol specifications preserve both uniformity and anonymity. We first focus on the computational power of the MPP model on complete communication graphs and initially identical edges. We provide the following exact characterization for the class MPS of stably computable predicates: A predicate is in MPS iff it is symmetric and is in NSPACE(n^2)$. We finally ignore the input to the agents and study MPP's ability to compute graph properties.

Abstract: We extend here the PopulationProtocol model of Angluin et al. [2004] in order to model more powerful networks of very small resource-limited artefacts (agents) that are possibly mobile. Communication can happen only between pairs of artefacts. A communication graph (or digraph) denotes the permissible pairwise interactions. The main feature of our extended model is to allow edges of the communication graph, G, to have states that belong to a constant size set. We also allow edges to have readable only costs, whose values also belong to a constant size set. We then allow the protocol rules for pairwise interactions to modify the corresponding edge state. Thus, our protocol specifications are still independent of the population size and do not use agent ids, i.e. they preserve scalability, uniformity and anonymity. Our Mediated PopulationProtocols (MPP) can stably compute graph properties of the communication graph. We show this for the properties of maximal matchings (in undirected communication graphs), also for finding the transitive closure of directed graphs and for finding all edges of small cost. We demonstrate that our mediated protocols are stronger than the classical populationprotocols, by presenting a mediated protocol that stably computes the product of two positive integers, when G is the complete graph. This is not a semilinear predicate. To show this fact, we state and prove a general Theorem about the Composition of two stably computing mediated populationprotocols. We also show that all predicates stably computable in our model are (non-uniformly) in the class NSPACE(m), where m is the number of edges of the communication graph. We also define Randomized MPP and show that, any Peano predicate accepted by a MPP, can be verified in deterministic Polynomial Time.

Abstract: Wireless sensor networks are about to be part of everyday life. Homes and workplaces capable of self-controlling and adapting air-conditioning for different temperature and humidity levels, sleepless forests ready to detect and react in case of a fire, vehicles able to avoid sudden obstacles or possibly able to self-organize routes to avoid congestion, and so on, will probably be commonplace in the very near future. Mobility plays a central role in such systems and so does passive mobility, that is, mobility of the network stemming from the environment itself. The populationprotocol model was an intellectual invention aiming to describe such systems in a minimalistic and analysis-friendly way. Having as a starting-point the inherent limitations but also the fundamental establishments of the populationprotocol model, we try in this monograph to present some realistic and practical enhancements that give birth to some new and surprisingly powerful (for this kind of systems) computational models.

Abstract: Wireless Sensor Networks (WSNs) constitute a recent and promising new
technology that is widely applicable. Due to the applicability of this
technology and its obvious importance for the modern distributed
computational world, the formal scientific foundation of its inherent laws
becomes essential. As a result, many new computational models for WSNs
have been proposed. PopulationProtocols (PPs) are a special category of
such systems. These are mainly identified by three distinctive
characteristics: the sensor nodes (agents) move passively, that is, they
cannot control the underlying mobility pattern, the available memory to
each agent is restricted, and the agents interact in pairs. It has been
proven that a predicate is computable by the PP model iff it is
semilinear. The class of semilinear predicates is a fairly small class. In
this work, our basic goal is to enhance the PP model in order to improve
the computational power. We first make the assumption that not only the
nodes but also the edges of the communication graph can store restricted
states. In a complete graph of n nodes it is like having added O(n2)
additional memory cells which are only read and written by the endpoints
of the corresponding edge. We prove that the new model, called Mediated
PopulationProtocol model, can operate as a distributed nondeterministic
Turing machine (TM) that uses all the available memory. The only
difference from a usual TM is that this one computes only symmetric
languages. More formally, we establish that a predicate is computable by
the new model iff it is symmetric and belongs to NSPACE(n2). Moreover, we
study the ability of the new model to decide graph languages (for general
graphs). The next step is to ignore the states of the edges and provide
another enhancement straight away from the PP model. The assumption now is
that the agents are multitape TMs equipped with infinite memory, that can
perform internal computation and interact with other agents, and we define
space-bounded computations. We call this the Passively mobile Machines
model. We prove that if each agent uses at most f(n) memory for f(n)={\`U}(log
n) then a predicate is computable iff it is symmetric and belongs to
NSPACE(nf(n)). We also show that this is not the case for f(n)=o(log n).
Based on these, we show that for f(n)={\`U}(log n) there exists a space
hierarchy like the one for classical symmetric TMs. We also show that the
latter is not the case for f(n)=o(loglog n), since here the corresponding
class collapses in the class of semilinear predicates and finally that for
f(n)={\`U}(loglog n) the class becomes a proper superset of semilinear
predicates. We leave open the problem of characterizing the classes for
f(n)={\`U}(loglog n) and f(n)=o(log n).

Abstract: We propose a novel, generic definition of probabilistic schedulers for populationprotocols. We then identify the consistent probabilistic schedulers, and prove that any consistent scheduler that assigns a non-zero probability to any transition i->j, where i and j are configurations satisfying that i is not equal to j, is fair with probability 1. This is a new theoretical framework that aims to simplify proving specific probabilistic schedulers fair. In this paper we propose two new schedulers, the State Scheduler and the Transition Function Scheduler. Both possess the significant capability of being protocol-aware, i.e. they can assign transition probabilities based on information concerning the underlying protocol. By using our framework we prove that the proposed schedulers, and also the Random Scheduler that was defined by Angluin et al., are all fair with probability 1. We also define and study equivalence between schedulers w.r.t. performance (time equivalent schedulers) and correctness (computationally equivalent schedulers). Surprisingly, we prove the following.
1. The protocol-oblivious (or agnostic) Random Scheduler is not time equivalent to the State and Transition Function Schedulers, although all three are fair probabilistic schedulers (with probability 1). To prove the statement we study the performance of the One-Way Epidemic Protocol (OR Protocol) under these schedulers. To illustrate the unexpected performance variations of protocols under different fair probabilistic schedulers, we additionally modify the State Scheduler to obtain a fair probabilistic scheduler, called the Modified Scheduler, that may be adjusted to lead the One-Way Epidemic Protocol to arbitrarily bad performance.
2. The Random Scheduler is not computationally equivalent to the Transition Function Scheduler. To prove the statement we study the Majority Protocol w.r.t. correctness under the Transition Function Scheduler. It turns out that the minority may win with constant probability under the same initial margin for which the majority w.h.p. wins under the Random Scheduler (as proven by Angluin et al.).

Abstract: We propose a new theoretical model for passively mobile Wireless Sensor Networks. We
call it the PALOMA model, standing for PAssively mobile LOgarithmic space MAchines. The main
modification w.r.t. the PopulationProtocol model [2] is that agents now, instead of being automata, are
Turing Machines whose memory is logarithmic in the population size n. Note that the new model is still
easily implementable with current technology. We focus on complete communication graphs. We define
the complexity class PLM, consisting of all symmetric predicates on input assignments that are stably
computable by the PALOMA model. We assume that the agents are initially identical. Surprisingly, it
turns out that the PALOMA model can assign unique consecutive ids to the agents and inform them
of the population size! This allows us to give a direct simulation of a Deterministic Turing Machine
of O(n log n) space, thus, establishing that any symmetric predicate in SPACE(n log n) also belongs
to PLM. We next prove that the PALOMA model can simulate the Community Protocol model [15],
thus, improving the previous lower bound to all symmetric predicates in NSPACE(n log n). Going
one step further, we generalize the simulation of the deterministic TM to prove that the PALOMA
model can simulate a Nondeterministic TM of O(n log n) space. Although providing the same lower
bound, the important remark here is that the bound is now obtained in a direct manner, in the sense
that it does not depend on the simulation of a TM by a Pointer Machine. Finally, by showing that a
Nondeterministic TM of O(n log n) space decides any language stably computable by the PALOMA
model, we end up with an exact characterization for PLM: it is precisely the class of all symmetric
predicates in NSPACE(n log n).

Abstract: We propose a new theoretical model for passively mobile Wireless Sensor Networks, called PM, standing for Passively mobile Machines. The main modification w.r.t. the PopulationProtocol model [Angluin et al. 2006] is that agents now, instead of being automata, are Turing Machines. We provide general definitions for unbounded memories, but we are mainly interested in computations upper-bounded by plausible space limitations. However, we prove that our results hold for more general cases. We focus on \emph{complete interaction graphs} and define the complexity classes PMSPACE(f(n)) parametrically, consisting of all predicates that are stably computable by some PM protocol that uses O(f(n)) memory in each agent. We provide a protocol that generates unique identifiers from scratch only by using O(log n) memory, and use it to provide an exact characterization of the classes PMSPACE(f(n)) when f(n) = Ω(log n): they are precisely the classes of all symmetric predicates in NSPACE(nf(n)). As a consequence, we obtain a space hierarchy of the PM model when the memory bounds are Ω(log n). We next explore the computability of the PM model when the protocols use o(loglog n) space per machine and prove that SEM = PMSPACE(f(n)) when f(n) = o(loglog n), where SEM denotes the class of the semilinear predicates. Finally, we establish that the minimal space requirement for the computation of non-semilinear predicates is O(log log n).

Abstract: We propose a new theoretical model for passively mobile Wireless Sensor Networks, called PM, standing for Passively mobile Machines. The main modification w.r.t. the PopulationProtocol model [Angluin et al. 2006] is that the agents now, instead of being automata, are Turing Machines. We provide general definitions for unbounded memories, but we are mainly interested in computations upper-bounded by plausible space limitations. However, we prove that our results hold for more general cases. We focus on complete interaction graphs and define the complexity classes PMSPACE(f(n)) parametrically, consisting of all predicates that are stably computable by some PM protocol that uses O(f(n)) memory in each agent. We provide a protocol that generates unique identifiers from scratch only by using O(log n) memory, and use it to provide an exact characterization of the classes PMSPACE(f(n)) when f(n)=Omega(log n): they are precisely the classes of all symmetric predicates in NSPACE(nf(n)). As a consequence, we obtain a space hierarchy of the PM model when the memory bounds are Omega(log n). Finally, we establish that the minimal space requirement for the computation of non-semilinear predicates is O(log log n).

Abstract: In this work we extend the populationprotocol model of Angluin et al., in
order to model more powerful networks of very small resource limited
artefacts (agents) that is possible to follow some unpredictable passive
movement. These agents communicate in pairs according to the commands of
an adversary scheduler. A directed (or undirected) communication graph
encodes the following information: each edge (u,\~{o}) denotes that during the
computation it is possible for an interaction between u and \~{o} to happen in
which u is the initiator and \~{o} the responder. The new characteristic of
the proposed mediated populationprotocol model is the existance of a
passive communication provider that we call mediator. The mediator is a
simple database with communication capabilities. Its main purpose is to
maintain the permissible interactions in communication classes, whose
number is constant and independent of the population size. For this reason
we assume that each agent has a unique identifier for whose existence the
agent itself is not informed and thus cannot store it in its working
memory. When two agents are about to interact they send their ids to the
mediator. The mediator searches for that ordered pair in its database and
if it exists in some communication class it sends back to the agents the
state corresponding to that class. If this interaction is not permitted to
the agents, or, in other words, if this specific pair does not exist in
the database, the agents are informed to abord the interaction. Note that
in this manner for the first time we obtain some control on the safety of
the network and moreover the mediator provides us at any time with the
network topology. Equivalently, we can model the mediator by communication
links that are capable of keeping states from a edge state set of constant
cardinality. This alternative way of thinking of the new model has many
advantages concerning the formal modeling and the design of protocols,
since it enables us to abstract away the implementation details of the
mediator. Moreover, we extend further the new model by allowing the edges
to keep readable only costs, whose values also belong to a constant size
set. We then allow the protocol rules for pairwise interactions to modify
the corresponding edge state by also taking into account the costs. Thus,
our protocol descriptions are still independent of the population size and
do not use agent ids, i.e. they preserve scalability, uniformity and
anonymity. The proposed Mediated PopulationProtocols (MPP) can stably
compute graph properties of the communication graph. We show this for the
properties of maximal matchings (in undirected communication graphs), also
for finding the transitive closure of directed graphs and for finding all
edges of small cost. We demonstrate that our mediated protocols are
stronger than the classical populationprotocols. First of all we notice
an obvious fact: the classical model is a special case of the new model,
that is, the new model can compute at least the same things with the
classical one. We then present a mediated protocol that stably computes
the product of two nonnegative integers in the case where G is complete
directed and connected. Such kind of predicates are not semilinear and it
has been proven that classical populationprotocols in complete graphs can
compute precisely the semilinear predicates, thus in this manner we show
that there is at least one predicate that our model computes and which the
classical model cannot compute. To show this fact, we state and prove a
general Theorem about the composition of two mediated populationprotocols, where the first one has stabilizing inputs. We also show that
all predicates stably computable in our model are (non-uniformly) in the
class NSPACE(m), where m is the number of edges of the communication
graph. Finally, we define Randomized MPP and show that, any Peano
predicate accepted by a Randomized MPP, can be verified in deterministic
polynomial time.

Abstract: This is a joint work with Ioannis Chatzigiannakis and Othon Michail.
We discuss here the populationprotocol model and most of its well-known extensions. The populationprotocol model aims to represent sensor networks consisting of tiny computational devices with sensing capabilities that follow some unpredictable and uncontrollable mobility pattern. It adopts a minimalistic approach and, thus, naturally computes a quite restricted class of predicates and exhibits almost no fault-tolerance. Most recent approaches make extra realistic and implementable assumptions, in order to gain more computational power and/or speed-up the time to convergence and/or improve fault-tolerance. In particular, the mediated populationprotocol model, the community protocol model, and the PALOMA model, which are all extensions of the populationprotocol model, are thoroughly discussed. Finally, the inherent difficulty of verifying the correctness of populationprotocols that run on complete communication graphs is revealed, but a promising algorithmic solution is presented.

Abstract: With a rapidly aging population, the health-care community will
soon face a severe medical personnel shortage. It is imperative that automated
health monitoring technologies be developed to help meet this
shortage. In this direction, we are developing Ayushman, a health monitoring
infrastructure and testbed. The vision behind its development
is two-fold: first, to develop a wireless sensor-based automated health
monitoring system that can be used in diverse situations, from homebased
care, to disaster situations, without much customization; second,
to provide a testbed for implementing and testing communication protocols
and systems. Ayushman provides a collection of services which
enables it to perform this dual role. It possess a hierarchical cluster
topology which provides a fault-tolerant and reliable system by ensuring
that each tier in the hierarchy is self-contained and can survive on its
own in case of network partition. Ayushman is being implemented using
off-the-shelf and diverse hardware and software components, which
presents many challenges in system integration and operational reliability.
This is an ongoing project at the IMPACT lab at Arizona State
University1, and in this paper, we present our system˘s architecture and
some of our experiences in the development of its initial prototype.

Abstract: The populationprotocol model (PP) proposed by Angluin et al. [2] describes sensor networks consisting of passively mobile finite-state agents. The agents sense their environment and communicate in pairs to carry out some computation on the sensed values. The mediated populationprotocol model (MPP) [13] extended the PP model by communication links equipped with a constant size buffer. The MPP model was proved in [13] to be stronger than the PP model. However, its most important contribution is that it provides us with the ability to devise optimizing protocols, approximation protocols and protocols that decide properties of the communication graph on which they run. The latter case, suggests a simplified model, the GDM model, that was formally defined and studied in [11]. GDM is a special case of MPP that captures MPP's ability to decide properties of the communication graph. Here we survey recent advances in the area initiated by the proposal of the PP model and at the same time we provide new protocols, novel ideas and results.

Abstract: In this work, we study protocols so that populations of distributed processes can construct networks. In order to highlight the basic principles of distributed network construction, we keep the model minimal in all respects. In particular, we assume finite-state processes that all begin from the same initial state and all execute the same protocol. Moreover, we assume pairwise interactions between the processes that are scheduled by a fair adversary. In order to allow processes to construct networks, we let them activate and deactivate their pairwise connections. When two processes interact, the protocol takes as input the states of the processes and the state of their connection and updates all of them. Initially all connections are inactive and the goal is for the processes, after interacting and activating/deactivating connections for a while, to end up with a desired stable network. We give protocols (optimal in some cases) and lower bounds for several basic network construction problems such as spanning line, spanning ring, spanning star, and regular network. The expected time to convergence of our protocols is analyzed under a uniform random scheduler. Finally, we prove several universality results by presenting generic protocols that are capable of simulating a Turing Machine (TM) and exploiting it in order to construct a large class of networks. We additionally show how to partition the population into k supernodes, each being a line of log k nodes, for the largest such k. This amount of local memory is sufficient for the supernodes to obtain unique names and exploit their names and their memory to realize nontrivial constructions.

Abstract: In this work, we study protocols (i.e. distributed algorithms) so that populations of distributed processes can construct networks. In order to highlight the basic principles of distributed network construction we keep the model minimal in all respects. In particular, we assume finite-state processes that all begin from the same initial state and all execute the same protocol (i.e. the system is homogeneous). Moreover, we assume pairwise interactions between the processes that are scheduled by an adversary. The only constraint on the adversary scheduler is that it must be fair, intuitively meaning that it must assign to every reachable configuration of the system a non-zero probability to occur. In order to allow processes to construct networks, we let them activate and deactivate their pairwise connections. When two processes interact, the protocol takes as input the states of the processes and the state of their connection and updates all of them. In particular, in every interaction, the protocol may activate an inactive connection, deactivate an active one, or leave the state of a connection unchanged. Initially all connections are inactive and the goal is for the processes, after interacting and activating/deactivating connections for a while, to end up with a desired stable network (i.e. one that does not change any more). We give protocols (optimal in some cases) and lower bounds for several basic network construction problems such as spanning line, spanning ring, spanning star, and regular network. We provide proofs of correctness for all of our protocols and analyze the expected time to convergence of most of them under a uniform random scheduler that selects the next pair of interacting processes uniformly at random from all such pairs. Finally, we prove several universality results by presenting generic protocols that are capable of simulating a Turing Machine (TM) and exploiting it in order to construct a large class of networks. Our universality protocols use a subset of the population (waste) in order to distributedly construct there a TM able to decide a graph class in some given space. Then, the protocols repeatedly construct in the rest of the population (useful space) a graph equiprobably drawn from all possible graphs. The TM works on this and accepts if the presented graph is in the class. We additionally show how to partition the population into k supernodes, each being a line of log k nodes, for the largest such k. This amount of local memory is sufficient for the supernodes to obtain unique names and exploit their names and their memory to realize nontrivial constructions. Delicate composition and reinitialization issues have to be solved for these general constructions to work.

Abstract: We work on an extension of the PopulationProtocol model of Angluin et al. that allows edges of the communication graph, G, to have states that belong to a constant size set. In this extension, the so called Mediated PopulationProtocol model (MPP), both uniformity and anonymity are preserved. We study here a simplified version of MPP in order to capture MPP's ability to stably compute graph properties. To understand properties of the communication graph is an important step in almost any distributed system. We prove that any graph property is not computable if we allow disconnected communication graphs. As a result, we focus on studying (at least) weakly connected communication graphs only and give several examples of computable properties in this case. To do so, we also prove that the class of computable properties is closed under complement, union and intersection operations. Node and edge parity, bounded out-degree by a constant, existence of a node with more incoming than outgoing neighbors, and existence of some directed path of length at least k=O(1) are some examples of properties whose computability is proven. Finally, we prove the existence of symmetry in two specific communication graphs and, by exploiting this, we prove that there exists no protocol, whose states eventually stabilize, to determine whether G contains some directed cycle of length 2.

Abstract: In this work, we consider a \emph{solution of automata} similar to \emph{PopulationProtocols} and \emph{Network Constructors}. The automata (also called \emph{nodes}) move passively in a well-mixed solution without being capable of controlling their movement. However, the nodes can \emph{cooperate} by interacting in pairs. Every such interaction may result in an update of the local states of the nodes. Additionally, the nodes may also choose to connect to each other in order to start forming some required structure. We may think of such nodes as the \emph{smallest possible programmable pieces of matter}, like tiny nanorobots or programmable molecules. The model that we introduce here is a more applied version of Network Constructors, imposing \emph{physical} (or \emph{geometrical}) \emph{constraints} on the connections that the nodes are allowed to form. Each node can connect to other nodes only via a very limited number of \emph{local ports}, which implies that at any given time it has only a \emph{bounded number of neighbors}. Connections are always made at \emph{unit distance} and are \emph{perpendicular to connections of neighboring ports}. Though such a model cannot form abstract networks like Network Constructors, it is still capable of forming very practical \emph{2D or 3D shapes}. We provide direct constructors for some basic shape construction problems, like \emph{spanning line}, \emph{spanning square}, and \emph{self-replication}. We then develop \emph{new techniques} for determining the computational and constructive capabilities of our model. One of the main novelties of our approach, concerns our attempt to overcome the inability of such systems to detect termination. In particular, we exploit the assumptions that the system is well-mixed and has a unique leader, in order to \emph{give terminating protocols that are correct with high probability}. This allows us to develop terminating subroutines that can be \emph{sequentially composed} to form larger \emph{modular protocols} (which has not been the case in the relevant literature). One of our main results is a \emph{terminating protocol counting the size $n$ of the system} with high probability. We then use this protocol as a subroutine in order to develop our \emph{universal constructors}, establishing that \emph{it is possible for the nodes to become self-organized with high probability into arbitrarily complex shapes while still detecting termination of the construction}.

Abstract: We extend the populationprotocol model with a cover-time service that informs a walking state every time it covers the whole network. This represents a known upper bound on the cover time of a random walk. The cover-time service allows us to introduce termination into populationprotocols, a capability that is crucial for any distributed system. By reduction to an oracle-model we arrive at a very satisfactory lower bound on the computational power of the model: we prove that it is at least as strong as a Turing Machine of space log n with input commutativity, where n is the number of nodes in the network. We also give a log n-space, but nondeterministic this time, upper bound. Finally, we prove interesting similarities of this model to linear bounded automata.

Abstract: We extend the populationprotocol model with a cover-time service that informs a walking state every time it covers the whole network. This is simply a known upper bound on the cover time of a random walk. This allows us to introduce termination into populationprotocols, a capability that is crucial for any distributed system. By reduction to an oracle-model we arrive at a very satisfactory lower bound on the computational power of the model: we prove that it is at least as strong as a Turing Machine of space logn with input commutativity, where n is the number of nodes in the network. We also give a logn-space, but nondeterministic this time, upper bound. Finally, we prove interesting similarities of this model to linear bounded automata.

Abstract: We explore the capability of a network of extremely limited
computational entities to decide properties about any of its subnetworks.
We consider that the underlying network of the interacting
entities (devices, agents, processes etc.) is modeled by a complete in-
teraction graph and we devise simple graph protocols that can decide
properties of some input subgraph provided by some preprocessing on
the network. The agents are modeled as nite-state automata and run
the same global graph protocol. Each protocol is a xed size grammar,
that is, its description is independent of the size (number of agents) of
the network. This size is not known by the agents. We propose a simple
model, the Mediated Graph Protocol (MGP) model, similar to the PopulationProtocol model of Angluin et al., in which each network link is
characterized by a state taken from a nite set. This state can be used
and updated during each interaction between the corresponding agents.
We provide some interesting properties of the MGP model among which
is the ability to decide properties on stabilizing (initially changing for a
nite number of steps) input graphs and we show that the MGP model
has the ability to decide properties of disconnected input graphs. We
show that the computational power within the connected components is
fairly restricted. Finally, we give an exact characterization of the class
GMGP, of graph languages decidable by the MGP model: it is equal
to the class of graph languages decidable by a nondeterministic Turing
Machine of linear space that receives its input graph by its adjacency
matrix representation.

Abstract: We explore the capability of a network of extremely limited computational entities to decide properties about itself or any of its subnetworks. We consider that the underlying network of the interacting entities (devices, agents, processes etc.) is modeled by an interaction graph that reflects the network’s connectivity. We examine the following two cases: First, we consider the case where the input graph is the whole interaction graph and second where it is some subgraph of the interaction graph given by some preprocessing on the network. In each case, we devise simple graph protocols that can decide properties of the input graph. The computational entities, that are called agents, are modeled as finite-state automata and run the same global graph protocol. Each protocol is a fixed size grammar, that is, its description is independent of the size (number of agents) of the network. This size is not known by the agents. We present two simple models (one for each case), the Graph Decision Mediated PopulationProtocol (GDMPP) and the Mediated Graph Protocol (MGP) models, similar to the PopulationProtocol model of Angluin et al., where each network link (edge of the interaction graph) is characterized by a state taken from a finite set. This state can be used and updated during each interaction between the corresponding agents. We provide some example protocols and some interesting properties for the two models concerning the computability of graph languages in various settings (disconnected input graphs, stabilizing input graphs). We show that the computational power within the family of all (at least) weakly-connected input graphs is fairly restricted. Finally, we give an exact characterization of the class of graph languages decidable by the MGP model in the case of complete interaction graphs: it is equal to the class of graph languages decidable by a nondeterministic Turing Machine of linear space that receives its input graph by its adjacency matrix representation.

Abstract: In the near future, it is reasonable to expect that new types of systems will appear, of massive scale that will operating in a constantly changing networked environment. We expect that most such systems will have the form of a large society of tiny networked artefacts. Angluin et al. introduced the notion of "Probabilistic PopulationProtocols'' (PPP) in order to model the behavior of such systems where extremely limited agents are represented as finite state machines that interact in pairs under the control of an adversary scheduler. We propose to study the dynamics of Probabilistic PopulationProtocols, via the differential equations approach. We provide a very general model that allows to examine the continuous dynamics of populationprotocols and we show that it includes the model of Angluin et. al., under certain conditions, with respect to the continuous dynamics of the two models. Our main proposal here is to exploit the powerful tools of continuous nonlinear dynamics in order to examine the behavior of such systems. We also provide a sufficient condition for stability.

Abstract: Angluin et al. [1] introduced the notion of ``Probabilistic PopulationProtocols'' where extremely limited agents are represented as finite state machines that interact in pairs under the control of an adversary scheduler. We provide a very general model that allows to examine the continuous dynamics of populationprotocols and we show that it includes the model of [1], under certain conditions, with respect to the continuous dynamics of the two models.