Binary Cellular Automata Approach to Anonymous, Self-Stabilizing Leader Election on Rings

1. Binary Cellular Automata Approach toAnonymous, Self-Stabilizing Leader Election on Rings Peter Banda banda@fmph.uniba.sk

2. Agenda • Introduction • Leader Election Problem • Cellular Automaton • Computational Mechanics • Cellular Automata Evolution • Leader Election By Cellular Automata • Upper Bound Performance

3. Introduction • Reductionism • Analytical decomposition – system reduced to its parts • Superstars of 20th century science – particle physics and molecular biology • Not sufficient • Complexity Approach • Holism – organization principles • Emergence - one H2O is not water, one neuron is not consciousness → system is > Σ constituting components • Circular causality, feedback loops • Unpredictability – small change in the cause implies dramatic effects • Structural simplicity and uniformity vs. complex global dynamics- Cellular automaton (CA) • Leader election problem – fundamental distributed problem

4. Goals • Find solution for CA instance of leader election problem by employing genetic algorithms. • Analyze CA dynamics to understand basis of global, collective computation – Computational Mechanics. • Find barriers (upper bound performance) of binary CA approach. • Track complex phenomena and uncover information processing in nature.

5. Leader Election – Distributed Algorithms • Definition of Leader Election • Distributed system is required to reach configuration in which the state of exactly one processor is, and then remains, within defined subset. The states of all other processors remain outside the subset. • Important prerequisite of other distributed algorithms • UID-Based Protocols • Processors are required to be distinguishable by (comparable) unique identifiers (UIDs); Le Lann, Chang & Roberts, Hirshberg & Sinclair etc. • Anonymous Protocols • (Angluin) No anonymous, deterministic algorithm (full symmetry of system). • Randomized model - random assignment of pseudo-IDs. • Self-stabilization • System eventually reaches legitimate state, no matter what initial configuration is.

6. Leader Election – Our CA Model • Our distributed model of binary cellular automaton • Anonymous, deterministic, uniform, synchronous, self-stabilizing and shared-memory based operating on bidirectional rings. • Main aspects: • Deals with one of fully symmetric instances of problem that are principally insolvable. • Besides the inherent limitations on initial configurations, it is a one of the most fundamental system capable of leader election reaching very satisfactory performance of 94 - 99% on arbitrary initial configuration. • Minimal possible memory - binary state; O(N) time complexity • Compared to other distributed models does not require any additional prerequisites as centralized demon, oracle, randomization etc.

7. Leader Election in Nature • Important role for global coordination, decision making and spatial orientation of variety of social and biological systems (sociobiology, development biology) • Shared consensus vs. despotic decision

8. Cellular Automaton - Definition ECA 110 • Structurally simplest, discrete, dynamical system • Consists of the lattice of cells (size N) • Cell state sti (i  N), conf. st = (st0,…,stN-1) • Neighborhood: N → n • Transition rule : n → st+1i = (ti) • Global transition rule : N→ Nst+1 = (st) • Ensemble operator : 2→ 2  = N • CA – regular language processor Neighborhood r = 3 Local State sti Configuration st t Global transition rule  Transition table  Configuration st+1 t+1

9. Cellular Automaton - Dynamics • Phenomenology • Wolfram’s classes, Edge of chaos concept • Global evolution • Ensemble operator update: L(Mt+1) = t+1 = (t) • FME algorithm: Mt+1 = [To Mt]OUT • Limit sets -  = ∩t=0t • Algebraic methods • Qualitative theory of dynamical systems • Computational Mechanics

10. Computational Mechanics of CA • Analysis using concepts from the computation and dynamical system theories • Introduced by Crutchfield, Young and Hanson at SFI (1993) • The global, collective dynamics of CA can be understood and described in terms of space-time structures: • Regular domains – regular background of computation • Particles – carrier of information • Particle interactions – information processing

11. Computational Mechanics – Regular Domains • Homogenous space-time region containing the same set of configurations appearing invariantly over and over again • Regular domain jis a process language consisting of a set of spatial configurations that fulfils the following two properties: • 1. Temporal invariance - CA dynamics represented by  maps jto itself, i.e. p(j) = j, (time periodicity) • 2. Spatial homogeneity - jis spatial translation invariant, the process language of a jis strongly connected, (spatial periodicity) • Can be identified either by eye inspection or by employing ε-machine reconstruction • Domains  = {0, 1,…} with associated DFAs can be filtered out from space-time diagrams by so called domain filter

12. Computational Mechanics – Domain Filter Domains identification ε-machine reconstruction Domain filter construction Filtering space-time diagram

13. Computational Mechanics - Particles • Particle (marked by letter of Greek alphabet) is a spatially localized and temporally periodic structure at the boundary of two domains • Displacement d- the number of cells, particle is shifted during one period • Velocityvis calculated as v= d / p • Particles code information about domains they separate • Geometric (“billiard”) model of computation

14. Computational Mechanics - Interactions • Embedded Information processing in the computational mechanics • Interaction of particle α with particle β resulting to the production of particle  is denoted as α + β→ • Interaction types: • Decay - α→β +  • React - α + β→ + δ • Annihilate - α + β→ (i) • Different results according particle phases

15. Computational Mechanics – Particle Catalog

16. Evolutionary Cellular Automata • First performed by Packard in ‘88 • Main research formed by EvCA research group at SFI • Chromosome • binary vector of length k2r+1 coding a transition table outputs (i) • Fitness defined by computational task T: • Performance • fraction of correctly computed ICs (in terms of computational task) to the total number of test ICs I (TMAX = 2N) • One-point cross-over and elitist selection • Example: Density classification, synchronization, prison dilemma etc.

17. Evolutionary Cellular Automata (cont.) Population Individual Genetic Algorithm Γ Cellular Automaton 

18. Evolution of Leader Election • Computational task T: • Binary CA with radius r = 3 • Jump from the fitness of 0.4 to 0.8 – complexity transition • Various CA strategies found by evolutions were identified and examined • Localistic strategies • Strategy of mandatory function • Density reduction • Divide and eliminate strategy • Particle-based strategies • First particle-based strategy • Strategy of mirror particles

19. Localistic Strategies • Strategy of mandatory function • includes 2r + 2 mandatory transitions securing the final leader configuration to be a stable point • Density reduction • reduces the number of active cells (density) to the minimum • contains zeros at the position of almost all bits (except (0001000) and (1111111)) • Divide and eliminate strategy • division splits continuous regions of active or inactive cells into (10)+1, resp. (100)+1 • elimination operates on these sequences and reduces them from one or both sides • Performance: , • Performance decreases rapidly with regard to N

20. First Particle-Based Strategy • Leader itself is considered as particle -  • The crucial particle  provides • Leader election function:  +  • Cleaning up functions:  +   ,  +   ,  +    • Main issues • All particles except  have positive velocities • Particle  itself is very slow (just -1/3) • Particles  and  do not check the whole configuration before they create leader → more leaders possible • Performance decreases very slowly with regard to N

21. First Particle-Based Strategy (Cont.)

22. Strategy of Mirror Particles • Occurrence of "mirror" particles with opposite velocities • Leader election mechanism • Interaction  +  +  and  +  • Collision of  +  indicates that automaton is ready to enter the final phase • Particles  and  shift around the whole configuration and check remaining particles • Particles generations: • Interaction  +  and  +  +  • Particle phases (phase shift of ) • High velocities of colliding particles, fosters overall performance.

23. Strategy of Mirror Particles (Cont.)

24. Strategy of Mirror Particles - N  5 mod 6 limitation • The product of crucial leader electing interactions  +  and  +  depends on the phases of colliding particles

25. Upper Bound Performance – Homogenous ICs • Homogenous IC (0N, 1N) • Symmetry breaking issue (Angluin) • insolvable by any 1D, deterministic, uniform CA • All cells are always in the same state → no leader can be elected

26. Upper Bound Performance – Stable Point Limit. • Leader Election definition → Mandatory transition function • ICs with loose-coupled active cells that keep a distance  2r + 1 must be stable points, hence they are insolvable

27. Upper Bound Performance • Upper bound performance of binary CA with ICs generated as • Density-uniform • Uniform where

28. Upper Bound Performance • Comparison with performance of Improved strategy of mirror particles:

29. Thank you for your attention:)

30. Questions ???