Abstract We study distributed agreement in microbial distributed systems under stochastic population dynamics and competitive interactions. Motivated by recent applications in synthetic biology, we examine how the presence and absence of direct competition among microbial species influences their ability to reach majority consensus . In this problem, two species are designated as input species, and the goal is to guarantee that eventually only the input species which had the highest initial count prevails. We show that direct competition dynamics reach majority consensus with high probability even when the initial gap between the species is small, i.e., $$\Omega (\sqrt{n\log n})$$ , where n is the initial population size. In contrast, we show that absence of direct competition is not robust: solving majority consensus with constant probability requires a large initial gap of $$\Omega (n)$$ . To corroborate our analytical results, we use simulations to show that these consensus dynamics occur within practical biological time scales.

Biochemical Reaction Networks (BCRNs) model species and their interactions via reactions. They have been extensively used in chemistry and extended to biological settings by generalizing the reactions' kinetics. However, detailed models of biochemical processes tend to result in complex BCRN models. We present the Meta-species Oriented Biosystem Syntax (MobsPy), a language designed to simplify the modeling process using the concept of meta-species. Meta-species are constructed using a bottom-up approach from base species, which represent elementary, simple characteristics. These characteristics are then combined to create meta-species with all their complex behavior. The combined species have characteristics that are the Cartesian product of the base species' characteristics and feature inheritance of reactions involving the base species. New reactions can involve all the states of a meta-species or only a subset that is selected via a query. In particular, reactions of meta-species can express a state change of one of the reactants. MobsPy is deployed as a Python package. We showcase its modeling capabilities by building concise models for biochemical systems from the literature.

Permutation equivariance (PE) is a property widely present in mathematics and machine learning. Classic deep reinforcement learning (DRL) algorithms, such as Deep Q-Network (DQN), require thoroughly exploring the state space to achieve optimal performance. For a PE problem such as the Multi-Armed Bandit (MAB) problem, the PE property helps reduce the space that needs to be explored. This paper proposes PEDQN, a PE DRL framework based on DQN by applying a PE neural network structure. Our MAB experiments show that PEDQN has clear advantages compared to DQN with a fully connected network and achieves the same or better performance than UCB1 when tested in the same environment as the training.

One of the key challenges in synthetic biology is devising robust signaling primitives for engineered microbial consortia. In such systems, a fundamental signal amplification problem is the majority consensus problem: given a system with two input species with initial difference of Δ in population sizes, what is the probability that the system reaches a state in which only the initial majority species is present?

We provide a complete characterization of both uniform and non-uniform deterministic consensus solvability in distributed systems with benign process and communication faults using point-set topology. More specifically, we non-trivially extend the approach introduced by Alpern and Schneider in 1985, by introducing novel fault-aware topologies on the space of infinite executions: the process-view topology, induced by a distance function that relies on the local view of a given process in an execution, and the minimum topology, which is induced by a distance function that focuses on the local view of the process that is the last to distinguish two executions. Consensus is solvable in a given model if and only if the sets of admissible executions leading to different decision values is disconnected in these topologies. By applying our approach to a wide range of different applications, we provide a topological explanation of a number of existing algorithms and impossibility results and develop several new ones, including a general equivalence of the strong and weak validity conditions.

Thresholded mode-switched ODEs are restricted dynamical systems that switch ODEs depending on digital input signals only, and produce a digital output signal by thresholding some internal signal. Such systems arise in recent digital circuit delay models, where the analog signals within a gate are governed by ODEs that change depending on the digital inputs.

The famous asynchronous computability theorem (ACT) relates the existence of an asynchronous wait-free shared memory protocol for solving a task with the existence of a simplicial map from a subdivision of the simplicial complex representing the inputs to the simplicial complex representing the allowable outputs. The original theorem relies on a correspondence between protocols and simplicial maps in round-structured models of computation that induce a compact topology. This correspondence, however, is far from obvious for computation models that induce a non-compact topology, and indeed previous attempts to extend the ACT have failed. This paper shows that in every non-compact model, protocols solving tasks correspond to simplicial maps that need to be continuous. It first proves a generalized ACT for sub-IIS models, some of which are non-compact, and applies it to the set agreement task. Then it proves that in general models too, protocols are simplicial maps that need to be continuous, hence showing that the topological approach is universal. Finally, it shows that the approach used in ACT that equates protocols and simplicial complexes actually works for every compact model. Our study combines, for the first time, combinatorial and point-set topological aspects of the executions admitted by the computation model.

Abstract Computing via synthetically engineered bacteria is a vibrant and active field with numerous applications in bio-production, bio-sensing, and medicine. Motivated by the lack of robustness and by resource limitation inside single cells, distributed approaches with communication among bacteria have recently gained in interest. In this paper, we focus on the problem of population growth happening concurrently, and possibly interfering, with the desired bio-computation. Specifically, we present a fast protocol in systems with continuous population growth for the majority consensus problem and prove that it correctly identifies the initial majority among two inputs with high probability if the initial difference is $$\varOmega (\sqrt{n\log n})$$ Ω ( n log n ) where n is the total initial population. We also present a fast protocol that correctly computes the Nand of two inputs with high probability. By combining Nand gates with the majority consensus protocol as an amplifier, it is possible to compute arbitrary Boolean functions. Finally, we extend the protocols to several biologically relevant settings. We simulate a plausible implementation of a noisy Nand gate with engineered bacteria. In the context of continuous cultures with a constant outflow and a constant inflow of fresh media, we demonstrate that majority consensus is achieved only if the flow is slower than the maximum growth rate. Simulations suggest that flow increases consensus time over a wide parameter range. The proposed protocols help set the stage for bio-engineered distributed computation that directly addresses continuous stochastic population growth.

We introduce the Composable Involution Delay Model (CIDM) for fast and accurate digital simulation. It is based on the Involution Delay Model (IDM) [Függer et al., IEEE TCAD 2020], which has been shown to be the only existing candidate model for faithful glitch propagation. The IDM, however, has shortcomings that limit its applicability. Our CIDM thus reduces the characterization effort by allowing independent discretization thresholds, improves composability and increases the modeling power by exposing canceled pulse trains at the gate interconnect. We formally show that, despite these improvements, the CIDM still retains the IDM's faithfulness.

We consider the standard population protocol model, where (a priori) indistinguishable and anonymous agents interact in pairs according to uniformly random scheduling. The self-stabilizing leader election problem requires the protocol to converge on a single leader agent from any possible initial configuration. We initiate the study of time complexity of population protocols solving this problem in its original setting: with probability 1, in a complete communication graph. The only previously known protocol by Cai, Izumi, and Wada [Theor. Comput. Syst. 50] runs in expected parallel time Θ(n2) and has the optimal number of n states in a population of n agents. The existing protocol has the additional property that it becomes silent, i.e., the agents' states eventually stop changing.

Agreeing on a common value among a set of agents is a fundamental problem in distributed computing, which occurs in several variants: In contrast to exact consensus, approximate variants are studied in systems where exact agreement is not possible or required, e.g., in human-made distributed control systems and in the analysis of natural distributed systems, such as bird flocking and opinion dynamics. We study the time complexity of two classical agreement problems: non-terminating asymptotic consensus and terminating approximate consensus. Asymptotic consensus, requires agents to repeatedly set their outputs such that the outputs converge to a common value within the convex hull of initial values; approximate consensus requires agents to eventually stop setting their outputs, which must then lie within a predefined distance of each other. We prove tight lower bounds on the contraction ratios of asymptotic consensus algorithms subject to oblivious message adversaries, from which we deduce bounds on the time complexity of approximate consensus algorithms. In particular, the obtained bounds show optimality of asymptotic and approximate consensus algorithms presented by Charron-Bost et al. (ICALP’16) for certain systems, including the strongest oblivious message adversary in which asymptotic and approximate consensus are solvable. As a corollary we also obtain asymptotically tight bounds for asymptotic consensus in the classical asynchronous model with crashes. Central to the lower-bound proofs is an extended notion of valency, the set of reachable limits of an asymptotic consensus algorithm starting from a given configuration. We further relate topological properties of valencies to the solvability of exact consensus, shedding some light on the relation of these three fundamental problems in dynamic networks.

Computing with synthetically engineered bacteria is a vibrant and active field with numerous applications in bio-production, bio-sensing, and medicine. Motivated by the lack of robustness and by resource limitation inside single cells, distributed approaches with communication among bacteria have recently gained in interest. In this paper, we focus on the problem of population growth happening concurrently, and possibly interfering, with the desired bio-computation. Specifically, we present a fast protocol in systems with continuous population growth for the majority consensus problem and prove that it correctly identifies the initial majority among two inputs with high probability if the initial difference is $Ω(\sqrt{n\log n})$ where $n$ is the total initial population. We also present a fast protocol that correctly computes the NAND of two inputs with high probability. We demonstrate that combining the NAND gate protocol with the continuous-growth majority consensus protocol, using the latter as an amplifier, it is possible to implement circuits computing arbitrary Boolean functions.

A nonsplit graph is a directed graph where each pair of nodes has a common incoming neighbor. We show that the radius of such graphs is in O(loglogn), where n is the number of nodes. This is an exponential improvement on the previously best known upper bound of O(logn). We then generalize the result to products of nonsplit graphs. The analysis of nonsplit graph products has direct implications in the context of distributed systems, where processes operate in rounds and communicate via message passing in each round: communication graphs in several distributed systems naturally relate to nonsplit graphs and the graph product concisely represents relaying messages in such networks. Applying our results, we obtain improved bounds on the dynamic radius of such networks, i.e., the maximum number of rounds until all processes have received a message from a common process, if all processes relay messages in each round. We finally connect the dynamic radius to lower bounds for achieving consensus in dynamic networks.

Függer et al. (2016) proved that no existing digital circuit model, including those based on pure and inertial delay channels, faithfully captures glitch propagation: for the short-pulse filtration (SPF) problem similar to that of building a one-shot inertial delay, they showed that every member of the broad class of bounded single-history channels either contradicts the unsolvability of SPF in bounded time or the solvability of SPF in unbounded time in physical circuits. In this article, we propose binary circuit models based on novel involution channels that do not suffer from this deficiency. Namely, in sharp contrast to bounded single-history channels, SPF cannot be solved in bounded time with involution channels, whereas it is easy to provide an unbounded SPF implementation. Hence, binary-valued circuit models based on involution channels allow to solve SPF precisely when this is possible in physical circuits. Additionally, using both SPICE simulations and physical measurements of an inverter chain instrumented by high-speed analog amplifiers, we demonstrate that our model provides good modeling accuracy with respect to real circuits as well. Consequently, our involution channel model is not only a promising basis for sound formal verification but also allows to seamlessly improve existing dynamic timing analysis.

In this paper, we provide a rigorous characterization of consensus solvability in synchronous directed dynamic networks controlled by an arbitrary message adversary using point-set topology: We extend the approach introduced by Alpern and Schneider in 1985 by introducing two novel topologies on the space of infinite executions: the process-view topology, induced by a distance function that relies on the local view of a given process in an execution, and the minimum topology, which is induced by a distance function that focuses on the local view of the process that is the last to distinguish two executions. We establish some simple but powerful topological results, which not only lead to a topological explanation of bivalence arguments, but also provide necessary and sufficient topological conditions on the admissible graph sequences of a message adversary for solving consensus. In particular, we characterize consensus solvability in terms of connectivity of the set of admissible graph sequences. For non-compact message adversaries, which are not limit-closed in the sense that there is a convergent sequence of graph sequences whose limit is not permitted, this requires the exclusion of all "fair'' and "unfair'' limit sequences that coincide with the forever bivalent runs constructed in bivalence proofs. For both compact and non-compact message adversaries, we also provide tailored characterizations of consensus solvability, i.e., tight conditions for impossibility and existence of algorithms, based on the broadcastability of the connected components of the set of admissible graph sequences.

Consider a distributed system with n processors out of which f can be Byzantine faulty. In the approximate agreement task, each processor i receives an input value x_i and has to decide on an output value y_i such that 1) the output values are in the convex hull of the non-faulty processors' input values, 2) the output values are within distance d of each other. Classically, the values are assumed to be from an m-dimensional Euclidean space, where m >= 1. In this work, we study the task in a discrete setting, where input values with some structure expressible as a graph. Namely, the input values are vertices of a finite graph G and the goal is to output vertices that are within distance d of each other in G, but still remain in the graph-induced convex hull of the input values. For d=0, the task reduces to consensus and cannot be solved with a deterministic algorithm in an asynchronous system even with a single crash fault. For any d >= 1, we show that the task is solvable in asynchronous systems when G is chordal and n > (omega+1)f, where omega is the clique number of G. In addition, we give the first Byzantine-tolerant algorithm for a variant of lattice agreement. For synchronous systems, we show tight resilience bounds for the exact variants of these and related tasks over a large class of combinatorial structures.

Accurate delay models are important for static and dynamic timing analysis of digital circuits, and mandatory for formal verification. However, Függer et al. [IEEE TC 2016] proved that pure and inertial delays, which are employed for dynamic timing analysis in state-of-the-art tools like ModelSim, NC-Sim and VCS, do not yield faithful digital circuit models. Involution delays, which are based on delay functions that are mathematical involutions depending on the previous-output-to-input time offset, were introduced by Függer et al. [DATE'15] as a faithful alternative (that can easily be used with existing tools). Although involution delays were shown to predict real signal traces reasonably accurately, any model with a deterministic delay function is naturally limited in its modeling power. In this paper, we thus extend the involution model, by adding non-deterministic delay variations (random or even adversarial), and prove analytically that faithfulness is not impaired by this generalization. Albeit the amount of non-determinism must be considerably restricted to ensure this property, the result is surprising: the involution model differs from non-faithful models mainly in handling fast glitch trains, where small delay shifts have large effects. This originally suggested that adding even small variations should break the faithfulness of the model, which turned out not to be the case. Moreover, the results of our simulations also confirm that this generalized involution model has larger modeling power and, hence, applicability.

This paper improves pulse-coupled synchronization for highly dynamic wireless networks, where vehicles may move unpredictably, causing topological changes of the network. For pulse-coupled methods, vehicles broadcast zero-bit pulses to estimate the clock differences to their neighbors. Vehicles then update local clocks according to the average of difference of pulses from its neighbors. The proposed algorithm further introduces (1) a time wheel to improve the robustness of the synchronization protocol and (2) an additional drift compensation mechanism to reduce clock skew. We compare our new algorithm to previous works via simulation. Both static and dynamic networks are simulated and compared. Different frequencies and clock drifts are analyzed. The proposed algorithm adapts to highly dynamic vehicle networks more quickly and more robustly than previous algorithms.

We study the problem of asymptotic consensus as it occurs in a wide range of applications in both man-made and natural systems. In particular, we study systems with directed communication graphs that may change over time. We recently proposed a new family of convex combination algorithms in dimension one whose weights depend on the received values and not only on the communication topology. Here, we extend this approach to arbitrarily high dimensions by introducing two new algorithms: the ExtremePoint and the Centroid algorithm. Contrary to classical convex combination algorithms, both have component-wise contraction rates that are constant in the number of agents. Paired with a speed-up technique for convex combination algorithms, we get a convergence time linear in the number of agents, which is optimal. Besides their respective contraction rates, the two algorithms differ in the fact that the Centroid algorithm's update rule is independent of any coordinate system while the ExtremePoint algorithm implicitly assumes a common agreed-upon coordinate system among agents. The latter assumption may be realistic in some man-made multi-agent systems but is highly questionable in systems designed for the modelization of natural phenomena. Finally we prove that our new algorithms also achieve asymptotic consensus under very weak connectivity assumptions, provided that agent interactions are bidirectional.

In this work we study the performance of asymptotic and approximate consensus algorithms in dynamic networks. The asymptotic consensus problem requires a set of agents to repeatedly set their outputs such that the outputs converge to a common value within the convex hull of initial values. This problem, and the related approximate consensus problem, are fundamental building blocks in distributed systems where exact consensus among agents is not required, e.g., man-made distributed control systems, and have applications in the analysis of natural distributed systems, such as flocking and opinion dynamics. We prove new nontrivial lower bounds on the contraction rates of asymptotic consensus algorithms, from which we deduce lower bounds on the time complexity of approximate consensus algorithms. In particular, the obtained bounds show optimality of asymptotic and approximate consensus algorithms presented in [Charron-Bost et al., ICALP'16] for certain classes of networks that include classical failure assumptions, and confine the search for optimal bounds in the general case. Central to our lower bound proofs is an extended notion of valency, the set of reachable limits of an asymptotic consensus algorithm starting from a given configuration. We further relate topological properties of valencies to the solvability of exact consensus, shedding some light on the relation of these three fundamental problems in dynamic networks.

We introduce a new class of distributed algorithms for the approximate consensus problem in dynamic rooted networks, which we call amortized averaging algorithms. They are deduced from ordinary averaging algorithms by adding a value-gathering phase before each value update. This results in a drastic drop in decision times, from being exponential in the number n of processes to being polynomial under the assumption that each process knows n. In particular, the amortized midpoint algorithm is the first algorithm that achieves a linear decision time in dynamic rooted networks with an optimal contraction rate of 1/2 at each update step. We then show robustness of the amortized midpoint algorithm under violation of network assumptions: it gracefully degrades if communication graphs from time to time are non rooted, or under a wrong estimate of the number of processes. Finally, we prove that the amortized midpoint algorithm behaves well if processes can store and send only quantized values, rendering it well-suited for the design of dynamic networked systems. As a corollary we obtain that the 2-set consensus problem is solvable in linear time in any dynamic rooted network model.

We show that no existing continuous-time, binary value-domain model for digital circuits is able to correctly capture glitch propagation. Prominent examples of such models are based on pure delay channels (P), inertial delay channels (I), or the elaborate Delay Degradation Model (DDM) channels proposed by Bellido-Diaz et al. We accomplish our goal by considering the border between solvability and non-solvability of a simple problem called short-pulse filtration (SPF), which is closely related to arbitration and synchronization. On one hand, we prove that SPF is solvable in bounded time in any such model that provides channels with non constant delay, like I and DDM. This is in opposition to the impossibility of solving bounded SPF in real (physical) circuit models. On the other hand, for binary circuit models with constant-delay channels, we prove that SPF cannot be solved even in unbounded time; again in opposition to physical circuit models. Consequently, indeed none of the binary value-domain models proposed so far (and that we are aware of) faithfully captures glitch propagation of real circuits. We finally show that these modeling mismatches do not hold for the weaker eventual SPF problem.

Fast digital timing simulations based on continuous-time, digital-value circuit models are an attractive and heavily used alternative to analog simulations. Models based on analytic delay formulas are particularly interesting here, as they also facilitate formal verification and delay bound synthesis of complex circuits. Recently, Függer et al. (arXiv:1406.2544 [cs.OH]) proposed a circuit model based on so-called involution channels. It is the first binary circuit model that realistically captures solvability of short-pulse filtration, a non-trivial glitch propagation problem related to building one-shot inertial delays.

International audience

We study sequences of optimal walks of a growing length in weighted digraphs, or equivalently, sequences of entries of max-algebraic matrix powers with growing exponents. It is known that these sequences are eventually periodic when the digraphs are strongly connected. The transient of such periodicity depends, in general, both on the size of digraph and on the magnitude of the weights. In this paper, we show that some bounds on the indices of periodicity of (unweighted) digraphs, such as the bounds of Wielandt, Dulmage–Mendelsohn, Schwarz, Kim and Gregory–Kirkland–Pullman, apply to the weights of optimal walks when one of their ends is a critical node.

Designing algorithms for distributed systems that provide a round abstraction is often simpler than designing for those that do not provide such an abstraction. Further, distributed systems need to tolerate various kinds of failures. The concept of a synchronizer deals with both: It constructs rounds and allows masking of transmission failures. One simple way of dealing with transmission failures is to retransmit a message until it is known that the message was successfully received. We calculate the exact value of the average rate of a retransmission-based synchronizer in environments with probabilistic message loss, within which the synchronizer shows nontrivial timing behavior. We show how to make this calculation efficient, and present analytical results on the convergence speed. The theoretic results, based on Markov theory, are backed up with Monte Carlo simulations.