S. Dali (1952) Galatea of the Spheres

Amorphous Computing Systems Jiří Wiedermann & Lukáš Petrů Academy of Sciences Charles University Prague, CZ Institute of Computer Science Academy of Sciences of the Czech Republic Partially supported by grant 1ET

Scientific fiction: The Black Cloud by Fred Hoyle, 1957 The Black Cloud by Fred Hoyle, 1957 A Deepness in the Sky by Vernor Vinge, 1999 A Deepness in the Sky by Vernor Vinge, 1999 Device Miniaturization Technology allows producing very small energy efficient processors (MEMS, NEMS) equipped by sensors, transmitters and a power source They can be built in large numbers → Wireless (sensor) networks → Amorphous computers (Smart Dust) Cellular engineering Synthetic biology (Biots) Motivation

A collection (of order 10^6 up to 10^12) of computational particles dispersed irregularly on a surface or throughout a volume creating a self-organizing computing network Particles: have no a priori knowledge about their positions, no identifiers are possibly faulty, contain sensors and effectors, might be mobile are not synchronized but operate at the same speed are identically programmed Each particle: has energy source, modest computing power and modest amount of memory, random number generator, plus sender/receiver apparatus Can communicate only with nearby neighbors (via radio, chemically, optically, acoustically,…) communication is unreliable, no assurance of message delivery

m 1/1000 mm 1/1000  m 1/1000 nm Å.. The Scale

Possible applications: Traffic control, people and item tracking in buildings, monitoring plant yield in agriculture, environment monitoring, weather forecast, ocean, atmosphere and underwater flow, intruder detection, battle field surveillance, artificial immune systems in the blood stream, space prospecting, etc.

Amorphous Computer A bag containing a large number of computing elements

Amorphous Computer

A simple model: synchronous static amorphous computer all processors are the same, have no identifiers, and can be modeled by finite automata the nodes communicate via a single channel radio with each other only within a bounded distance the nodes are synchronized Assumptions: a uniformly distributed set of processors in a bounded area no “big holes” in the actual distribution of nodes the number of neighbors of a node is bounded by a fixed constant Experiment: wave propagation

Simulating a two-dimensional cellular automaton: establish a regular 2D grid consisting of so-called virtual cells over the nodes of the amorphous computer in one round simulate one (parallel) step of a cellular automaton by letting the nodes in each virtual cell communicate with the nodes from the neighboring virtual cells Rows and columns formation in an amorphous computer:

Grid formation with node density 20:

Theorem 1: Let C be an amorphous computer whose target square area consists of M squares of size r  r. The probability that some square is not covered by any node or that nodes of one square cannot communicate with any node in some adjacent square is not greater than 5M(1-1/5M)^N. Theorem 2: Any n  n cellular automaton can be simulated by a well-formed synchronous amorphous computer with O(n  n) nodes with a constant slow-down. Well-formed: there is a constant c  1 such the square area of the amorphous computer can be covered by a regular grid in such a way that each virtual cell contains at least one and at most c nodes of the amorphous computer.

Registers Size O(log N) bits Control unit … Random number Radio Transceiver Local clock, not synchronized with other nodes No identifiers Radio transceiver: single channel limited range r No collision detection (a node cannot distinguish the case of no broadcasting from the case of two or more nodes broadcasting simultaneously) A node of an amorphous computer A more realistic model: asynchronous AC

Random Topology When the node topology is random, what are the necessary conditions for the communication among nodes to work? For low densities, majority of nodes will be isolated

Random Topology For medium densities, there will be many fixed- sizes components Node density: average number of nodes in the communication area of one node: d=N  r²/a where a is the size of A

Random Topology High density For high densities, above so-called percolation threshold, one large component containing almost all nodes will emerge Experiments: d=6 gives components of size ~0.5N in 98% of cases

Important properties of random communication graphs Connectivity: one large connected component (“giant component”) containing majority of nodes (a density above percolation threshold is needed) Remark: Determining percolation threshold is an important open problem, with many applications Small diameter; asymptotically, D=  N (verified experimentally) Bounded node degree; asymptotically, deg=~logN/loglog N with probability ~1/N (proved analytically)

Jungle telegraph No synchronization no addresses no sense of direction no collision detection

Protocol Send transmitting a message to node’s neighbours; the key ideas: – send sporadically (in order to avoid broadcast conflicts), – Repeat sending a few times (increases the probability of error- free message receiving) procedure Send (m:message; p:probability; k:integer) if the processor is not locked w.r.t m then for i := 1 to k do { Wait for time 2T; With probability p do Send message m}; lock the processor w.r.t m; Theorem: If all nodes of the network use algorithm Send, then the probability that a message fails to be delivered from a sender to any of its neighbours is at most ε. The protocol works in time O (Q log (1/ε) ).

Protocol Flood Purpose: message m is to be delivered from node X to all other network nodes Assumption: all nodes, except of X, are not locked w.r.t m; Algorithm: – X sends m with probability  /N using SEND – Upon receiving m, the respective node also starts sending m – After re-sending m once, any node locks itself w.r.t m; – This continues until the base station after time O(DQ log(N/  ) floods the network by another signal

Protocol Flood Theorem: On a network of diameter D, in which all nodes, are not locked w.r.t. m, algorithm Flood started at X sending m will run in time O(D · Q · log (N/ε)), and with probability 1-ε, message m from X will be delivered to each node at the end, all nodes will be locked w.r.t m

Simulating a RAM Assumptions: The simulating AC has one large connected component of size N of diameter D and maximum neighborhood size Q containing the base station The unit time complexity of our RAM is T(n); each RAM register is of size O(log T(n))=O(log N) The nodes are used as RAM registers while RAM accumulators are in the base station The simulation is controlled by the base station where the RAM program also resides Simulation consists of three phases: 1. Address assignment using the random no. generators in nodes 2. Input reading and register initialization – via the base node 3. Simulation – using protocol FLOOD from communication between the nodes with the respective addresses

The complexity of simulation: Theorem: a unit-cost RAM of time complexity T(n) and space complexity S(n) can be simulated by an amorphous computer of size N>2S(n) in time O(T(n) D Q log(2NT(n)/  )), with probability of failure at most  D – network diameter Q – neighbourhood size Remark: our AC is a non-uniform model

Communicating mobile nano-machines Nano-machines molecular cell-sized devices or engineered organisms produced by self-assembly or self-replication capable of performing simple tasks such as actuation and sensing CommunicationMolecularConventional CarrierMoleculeElectromagnetic waves Signal typeChemicalDigital PropagationSlow (by diffusion)Fast (light speed) EnvironmentAqueousAirborne or cable Energy costsLowHigh ReliabilityLow (stochastic)High accuracy

A finite number of nanomachines freely moving (actively or passively) in a closed liquid environment no external control interaction via molecular communication Computational/Communication Scenario: Recei- ver Sender

Nano-machine Computational Model Timed probabilistic finite-state automaton – each transition has a probability value and a timer assigned determining the likelihood that the transition occurs and the time at which it does so; a timer can be reset by a transition and since that time the timer starts running A biomolecular realization of probabilistic automata has been described, e.g., in Adar R., Benenson Y., Linshiz G., Rozner A, Tishby N., Shapiro E.: Stochastic computing with biomolecular automata. Proc. Natl. Acad. Sci. USA, 101, pp , 2004 Open: biomolecular timer realization

Distributed computing through nano-machines Idea: represent the input in unary, i.e., each machines reads a symbol from {0,1} via a special sensor and stores it in its state simulate a so-called counter machine: Elementary operations: add and subtract 1 test for zero

Counter – a set of nanomachines in the same state Both adding and subtracting 1 requires determining of a single nanomachine – so-called leader Determining a leader by randomized halving of a set: Theorem: a nanonet can correctly simulate a counter machine with a high probability Throughout, protocols SEND and FLOOD are used

Flying amorphous computer Model: the same as in the case of a static asynchronous amorphous computer (“a RAM with distributed registers”), but this time the nodes are mobile. The nodes move in a square area in random directions, with a constant speed, bouncing when reaching a boundary: Obviously, protocol SEND will also work in this case, but FLOOD need not work at all, because there is a new kind of problems: A node’s temporal inaccessibility…

We say that a flying amorphous computer is a lively flying computer if any of its nodes cannot stay isolated (unreachable) from other nodes for an infinite time. That is, for any node the following property holds: if a message is repeatedly broadcast from that node then after a finite time the message will reach all other nodes. No upper bound on the time complexity for actions of lively flying amorphous computers can be given

Broadcasting in a flying amorphous computer

Simulating a RAM 1.Address assignment: repeatedly use the leader selection strategy for assigning addresses 1,2,…N 0. 2.Reading the input into the first registers (see the next item) and initializing the rest of the registers 3.The base node simulates the individual RAM instructions: It alternates between sending a request to perform an instruction in a specific register and expecting the acknowledgement from it until such an acknowledgement is received (this must happen in a finite time)

Theorem: Let M1 be a RAM with N 0 registers of length s1 bits. Let M2 be a lively flying amorphous computer with N  N 0 nodes. Let the nodes of M2 have assigned unique addresses and contain the input values. Then there exists a simulation algorithm such that M2 correctly simulates the computation of M1 on a given input. The simulation ends in a finite time.

Experimental validation of the communication protocol Average time needed to send a message and to receive a reply. Parameter l (decreased after each broadcast) ranging from 50 to 180 denotes the number of the broadcast steps of the Broadcast protocol; parameter p (probability of sending) ranges from 0.3 to 0.9. Darker color represents better values. N=400 d=6 Average time needed to send a message and to receive a reply. Speed v goes from to 1.

Can a Turing machine simulate a flying amorphous computer? Scenario: the AC processors fly about, read the (changeable) input data from the environment and compute some function of the data The problem of the “mobile parallel input” ?

Conclusion We have … Introduced a new class of computing devices: amorphous computers inspired both by nature and scientific imagination Shown the universal computing power of families of such computers Members of the respective families are among the simplest computational devices possessing universal computing power Further research is needed to better understand the paradigm of amorphous computing